http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Perry&feedformat=atomSoft-Matter - User contributions [en]2020-07-06T10:32:41ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13557Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-02T03:45:39Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: <math>I</math> is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescence, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile, Poiseuille or plug, in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13556Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-02T03:45:20Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: <math>I</math> is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescence, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiseuille or plug- in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13555Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-02T03:44:01Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: <math>I</math> is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescence, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13554Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-02T03:43:01Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: <math>I</math> is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13553Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-02T03:42:35Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: <math>I</math> is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13552Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-02T03:41:40Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiseuille flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13494Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:46:14Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers use a creative method to observe the propagation of the added fluid through the dry foam. The added fluid includes fluorescein salt which is imaged using UV light and a CCD camera. The experimenters find that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so are able to see and record the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the column of foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates in the time of the experiment and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors say that the rigid walls assumption is invalid because the slipping condition fits the data better. But, couldn't it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used [http://www.geom.uiuc.edu/software/evolver/ Surface Evolver] from the University of Minnesota Geometry Center.<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13493Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:41:04Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes. The resulting power law is:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. As seen in figure 3, front velocity increases with bubble size. This is not intuitive to me. It seems to me that larger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors seem to say that the rigid walls assumption must be wrong, because the slipping condition fits the data better. But, could it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13492Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:38:44Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip boundary conditions which result in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors seem to say that the rigid walls assumption must be wrong, because the slipping condition fits the data better. But, could it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13491Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:37:56Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors seem to say that the rigid walls assumption must be wrong, because the slipping condition fits the data better. But, could it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13490Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:37:27Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors seem to say that the rigid walls assumption must be wrong, because the slipping condition fits the data better. But, could it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13489Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:36:59Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math> . A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors seem to say that the rigid walls assumption must be wrong, because the slipping condition fits the data better. But, could it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13488Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:32:00Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
It would be great to verify the flow profile- Poiselle or plug- in a channel. The authors seem to say that the rigid walls assumption must be wrong, because the slipping condition fits the data better. But, could it be a different aspect of the initial analysis that is wrong? To confirm the new theory, I think we need to see which kind of flow happens in the channels.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13487Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:29:59Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other of two damping terms dominates (equation 4 in [1]). Such approximations are often made when analyzing complex systems such as those in soft matter. Clearly, careful consideration must be given to which terms are important as the results can be quite different!<br />
<br />
The authors use dimensional analysis to in developing their new theory focused on damping in the nodes.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13486Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:26:06Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
<br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13485Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:25:39Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory in which slip is allowed. Channel-slip theory leads to plug flow and viscous damping in the nodes.<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math><br />
<br />
with <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
Note: I is dimensionless and representative of viscous forces in the nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory comes close to the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area, and thus a slower flow. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13484Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:22:14Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<br />
with <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13483Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:21:56Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=\frac{1}{2}</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math><br />
<math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13482Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:20:50Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedron (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=1/2</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13481Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:20:29Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers seem interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from gravity-driven bulk flow through the foam: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The researchers consider two models of flow through a monodisperse foam with [[Tetrakaidecahedral (Kelvin Cell)|tetrakaidecahedral]] cells. The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=1/2</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13480Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T03:17:19Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|The structure of a network element in a model foam. Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=1/2</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13478Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:38:05Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=1/2</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3. The front velocity increases with bubble size. This is not intuitive to me. It seems to me that bigger bubbles would have fewer channels per tube cross-sectional area. However, for the same volume fraction, the larger bubbles must have bigger channels and nodes.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13477Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:35:38Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=1/2</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
The two different theories come from assuming that one or the other term in a sum dominates. Such approximations are often made when analyzing complex systems in soft matter. Clearly, careful consideration must be given as the results can be quite different!<br />
<br />
The authors use dimensional analysis to develop their equation 7.</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13476Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:31:07Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and a new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) is made up of the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the velocity profiles in the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls and no-slip which results in Poiselle flow through the channels. In this theory, the nodes are not important. This theory predicts <math>\alpha=1/2</math>:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
'''Next Steps?'''</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13475Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:17:08Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
'''Next Steps?'''</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13474Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:16:45Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration]]), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
'''Next Steps?'''</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13473Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:16:29Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the [[Critical Micelle Concentration | critical micelle concentration), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
'''Next Steps?'''</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13472Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:15:02Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
The researchers used a creative method to observe the propagation of the wetter flow through the dry foam. The added fluid had fluorescein salt which was imaged using UV light and a CCD camera. The experimenters found that the liquid fraction of the foam is proportional to the intensity of the fluorescing, so were able to "see" the front of wet foam progress.<br />
<br />
To ensure that the experimental work is robust, the authors look at distilled water, tap water, various soap concentrations (all above the critical micelle concentration), and determine that these changes had no effect.<br />
<br />
To keep the foam from coarsening (which it naturally would over time), the researchers constantly generate foam at the bottom of the tube so that the volume fraction at any height is constant. I'm not clear on the relative rates and how foam moving up the tube would affect the volume of fluid propagating down the tube.<br />
<br />
'''Theoretical Methods:'''<br />
To create Figure 1, Koehler ''et. al.'' used Surface Evolver from the University of Minnesota Geometry Center: [http://www.geom.uiuc.edu/software/evolver/].<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
'''Next Steps?'''</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13471Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:04:17Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|right|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
surface evolver <br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13470Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:03:50Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|left|Figure 3 from [1]]]<br />
<br />
The new theory is termed channel-slip theory, leads to plug flow, and viscous damping in the nodes.<br />
<br />
Equation 9 from [1]:<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
The 1/3 power which comes out of the channel-slip theory matches the exponent of approximately .36 determined experimentally and presented in Figure 3.<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
surface evolver <br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13469Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T02:01:18Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. A quantity <math>V_s</math> is defined as the volumetric flow into the top of the tube divided by the cross-sectional area of the tube. The experimenters looked at three sizes of bubbles.<br />
<br />
The penetration speed <math>\nu_f</math> increases with <math>V_s</math> as <math>\nu_f=\nu_0V_s^\alpha</math>, but with what power <math>\alpha</math>?<br />
<br />
The older theory is based on assuming rigid channel walls which results in Poiselle flow through the channels. In this theory, the nodes are not important.<br />
<br />
Equation 6 from [1]:<br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
<br />
[[Image:FluidVelocity.png|400px|thumb|left|Figure 3 from [1]]]<br />
<br />
New Theory: "channel-slip theory"<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
surface evolver <br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13462Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T01:52:12Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a dry foam (much more air than liquid) are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers are actually interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage." They introduce fluid to the top of a column of foam at a controlled rate <math>Q</math> and observe the front of wetter foam penetrate through the column of dry foam at velocity <math>\nu_f</math>. The experimenters looked at three sizes of bubbles.<br />
<br />
Old Theory: Equation 6 "rigid channel walls" <br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
*the exponent of 1/2 is what the experimental data disagree with<br />
<br />
[[Image:FluidVelocity.png|400px|thumb|left|Figure 3 from [1]]]<br />
<br />
New Theory: "channel-slip theory"<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
surface evolver <br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13457Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T01:45:45Z<p>Perry: /* Soft Matter Details */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a foam are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers were really interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage."<br />
<br />
The experimenters were able to change the bubble size and the amount of fluid added to the top of the foam tube. They recorded two velocities using fluorescing markers.<br />
<br />
Old Theory: Equation 6 "rigid channel walls" <br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
*the exponent of 1/2 is what the experimental data disagree with<br />
<br />
[[Image:FluidVelocity.png|400px|thumb|left|Figure 3 from [1]]]<br />
<br />
New Theory: "channel-slip theory"<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
surface evolver <br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13456Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T01:45:22Z<p>Perry: /* Summary */</p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, experimental results, and new theory more closely matching the experimental results.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The network of channels in a foam are the liquid layers in between polyhedral air bubbles. Along the edges of the polyhedra, there are thin "Plateau borders" or channels which meet at vertices of the polyhedra in volumes the authors call nodes. See Figure 1 for a diagram of one network unit.<br />
<br />
The researchers were really interested in the flow through the channels and nodes, but rather than struggle to image very small channels, they decide to learn from bulk flow through the foam driven by gravity: "forced drainage."<br />
<br />
The experimenters were able to change the bubble size and the amount of fluid added to the top of the foam tube. They recorded two velocities using fluorescing markers.<br />
<br />
Old Theory: Equation 6 "rigid channel walls" <br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
*the exponent of 1/2 is what the experimental data disagree with<br />
<br />
[[Image:FluidVelocity.png|400px|thumb|left|Figure 3 from [1]]]<br />
<br />
New Theory: "channel-slip theory"<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Liquid_Flow_through_Aqueous_Foams:_The_Node-Dominated_Foam_Drainage_Equation&diff=13450Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation2009-12-01T01:34:05Z<p>Perry: </p>
<hr />
<div>== Overview ==<br />
* [1] Koehler, S., Hilgenfeldt, S., & Stone, H. Physical Review Letters. '''82''', 21. 4232-4235 (1999).<br />
* '''Keywords:''' Foam, Drainage, Plateau Border, [[Tetrakaidecahedron (Kelvin Cell)]]<br />
<br />
== Summary ==<br />
Koehler, Hilgenfeldt, and Stone write about fluid flow through the network of channels in a soap foam. The article presents an existing theory, a new theory, and experimental evidence supporting their new theory.<br />
[[Image:FoamStructure.png|300px|thumb|left|Figure 1 from [1]]]<br />
<br />
The first hurdle the researchers crossed was figuring out how to quantify such a complex flow. They decided to look at the bulk flow driven by gravity rather than attempt to observed the flow in any one channel.<br />
<br />
<br />
The experimenters were able to change the bubble size and the amount of fluid added to the top of the foam tube. They recorded two velocities using fluorescing markers.<br />
<br />
Old Theory: Equation 6 "rigid channel walls" <br />
<math>\nu_f=(V_0^{rigid}V_s)^{1/2}</math>, <math>V_0^{rigid}=\frac{\delta_a\rho gL^2}{3\delta_\epsilon\delta_\mu \mu}</math><br />
*the exponent of 1/2 is what the experimental data disagree with<br />
<br />
[[Image:FluidVelocity.png|400px|thumb|left|Figure 3 from [1]]]<br />
<br />
New Theory: "channel-slip theory"<br />
<math>\nu_f=((V_0^{slip})^2V_s)^{1/3}</math>, <math>V_0^{slip}=\frac{2\delta_a\rho gL^2}{\mu\delta_\epsilon^{1/2}I}</math><br />
*I is dimensionless, viscous forces in nodes<br />
<br />
== Soft Matter Details ==<br />
'''Experimental Methods:'''<br />
<br />
Question about constant foam generation<br />
<br />
Testing that the theory is robust<br />
<br />
Deciding which term dominates in equation 4<br />
<br />
Dimensional analysis equation 7<br />
<br />
Foam- further applications?</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13429Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:39:26Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedra as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use tetrakaidecahedra in their paper, ''[[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]]''.<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface area, space-filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. Architects used the Weaire-Phalen geometry in the design of the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open question [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13428Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:33:25Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedra as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use tetrakaidecahedra in their paper, ''[[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]]''.<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface area, space-filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. Architects used the Weaire-Phalen geometry in the design of the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13427Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:31:42Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedra as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use tetrakaidecahedra in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface area, space-filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. Architects used the Weaire-Phalen geometry in the design of the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13426Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:28:28Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedra as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use tetrakaidecahedra in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface area, space-filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13425Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:26:29Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface area, space-filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13424Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:20:35Z<p>Perry: /* Definition */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which improved upon the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13423Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:19:30Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which was an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing National Aquatics Center, "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13422Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:15:59Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which was an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube" [3]. <br />
<br />
Whether or not anyone has discovered the space-filling shape with the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13421Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:13:35Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which was an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example of a space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube" [3]. <br />
<br />
The proof of which space-filling shape gives the absolute minimum surface area is still an open problem [2]!<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13420Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:08:11Z<p>Perry: /* Definition */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of the bubbles which make up a foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which was an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube" [3].<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13419Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:06:13Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which was an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example space-filling geometry with even less surface area in a Philosophical Magazine Letters article [2]. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube" [3].<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13418Tetrakaidecahedron (Kelvin Cell)2009-11-30T23:04:11Z<p>Perry: /* Definition */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces which was an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use the packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example space-filling geometry with even less surface area in a Philosophical Magazine Letters article. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube." [3]<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13384Tetrakaidecahedron (Kelvin Cell)2009-11-30T16:19:06Z<p>Perry: /* Applications */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved which is an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use the packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron remained the best contender for a minimal surface-area, space filling shape from 1887 until 1994. In 1994, Weaire and Phalen presented a counter-example space-filling geometry with even less surface area in a Philosophical Magazine Letters article. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube." [3]<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
<br />
[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13383Tetrakaidecahedron (Kelvin Cell)2009-11-30T16:12:07Z<p>Perry: /* Definition */</p>
<hr />
<div><br />
== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
<br />
[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
<br />
In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of foam [1]. Kelvin's proposed tetrakaidecahedron actually had some slightly curved which is an improvement over the typical flat-faced polyhedron pictured above [2].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
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[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
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== Applications ==<br />
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Scientists still use the packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
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The tetrakaidecahedron filled space with the least amount of surface area from 1887 to 1994. In 1994, Weaire and Phalen presented a space-filling geometry with even less surface area in a Philosophical Magazine Letters article. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube." [3]<br />
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== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
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[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
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[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perryhttp://soft-matter.seas.harvard.edu/index.php?title=Tetrakaidecahedron_(Kelvin_Cell)&diff=13382Tetrakaidecahedron (Kelvin Cell)2009-11-30T15:54:13Z<p>Perry: /* Applications */</p>
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== Definition ==<br />
The '''tetrakaidecahedron''' (shown below) is a polyhedron studied in conjunction with foams and minimal surface area, space-filling shapes. The tetrakaidecahedron has 14 faces (6 quadrilateral and 8 hexagonal) and 24 vertices [1].<br />
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[[Image:Tetrakaidecahedron.png|300px|center|thumb|From Wikimedia Commons.]]<br />
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In 1887, Lord Kelvin proposed that the tetrakaidecahedron was the best shape for packing equal-sized objects together to fill space with minimal surface area. Kelvin thought about this problem in the context of foam. Kelvin's proposed tetrakaidecahedron actually had some slightly curved faces in contrast to the typical flat-faced polyhedron pictured above [1].<br />
<br />
[http://mathworld.wolfram.com/Tetradecahedron.html Click here for other 14-sided poyhedra.]<br />
<br />
[http://zapatopi.net/blog/?post=200407047160.make_your_own_kelvin_cells Click here to print a template to make your own tetrakaidecahedron!]<br />
<br />
== Applications ==<br />
<br />
Scientists still use the packed tetrakaidecahedrons as a model for regular, monodisperse foam. For example, Koehler, Hilgenfeldt, and Stone use the tetrakaidecahedron in their paper [[Liquid Flow through Aqueous Foams: The Node-Dominated Foam Drainage Equation]].<br />
<br />
The tetrakaidecahedron filled space with the least amount of surface area from 1887 to 1994. In 1994, Weaire and Phalen presented a space-filling geometry with even less surface area in a Philosophical Magazine Letters article. The Weaire-Phalen geometry was used as the structure for the Beijing Aquatic Center "the water cube." [3]<br />
<br />
== References ==<br />
[1] Weaire, Denis. "Kelvin and Ireland: Kelvin's Ideal Foam Structure," Journal of Physics: Conference Series '''158''' (2009).<br />
<br />
[2] Weisstein, Eric. [http://mathworld.wolfram.com/KelvinsConjecture.html "Kelvin's Conjecture."] From MathWorld--A Wolfram Web Resource.<br />
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[3] Rogers, Peter. [http://www.guardian.co.uk/science/2004/may/06/research.science1 "Welcome to the WaterCube, the Experiment that Thinks it's a Swiming Pool,"] The Guardian (5 May 2004).</div>Perry