http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=Fritz&feedformat=atomSoft-Matter - User contributions [en]2020-06-07T10:33:23ZUser contributionsMediaWiki 1.24.2http://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13992Soft lubrication2009-12-05T09:00:08Z<p>Fritz: /* Conclusion */</p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This in turn creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal force.<br />
<br />
== Understanding the physics ==<br />
<br />
The fact that a maximal normal force exists is quite surprising. Let's first imagine the nonelastic case where a cylinder moves steadily over a flat surface, with a very small gap between the two. If we move in the reference frame of the cylinder and position our coordinate system to the point of minimal distance, then the geometry of the problem is symmetric for all times. The governing equations, the lubrication approximation is fully time reversible which implies a perfect anti-symmetric pressure distribution and thus the normal force has to be always zero.<br />
<br />
To understand how the elastic layer changes the picture, let's have a look at figure 1. Let's imagine that we start out exactly where we began, with a steadily translating cylinder over a flat surface creating an antisymmetric pressure distribution. This antisymmetric pressure distribution will now deform the elastic layer, pushing it down in front of the moving cylinder and sucking it up behind it. This in turn breaks the geometry, which means the pressure distribution will no longer by perfectly antisymmetric. This leads to a finite force upwards.<br />
<br />
== An exemplary scaling ==<br />
<br />
A simple way to derive the scaling for the pressure as a function of the gap height is to balance the only two forces existing in the layer according to the lubrication approximation: pressure and viscosity. This means <math>p/l \approx \mu V / h^2</math> or<br />
<br />
<math>p \approx \frac{\mu V R^{1/2}}{h^{3/2}}</math><br />
<br />
for definitions of the quatities, refer to figure 1.<br />
<br />
It is possible to split up this pressure into an "inelastic" component that cannot contribute to the lift as outlined before and an "elastic" component that will. The lift itself will then scale like<br />
<br />
<math>F \approx \frac{\mu V R}{h_o^2} \Delta h</math> <br />
<br />
where <math>\Delta h</math> is the solution of an elasticity problem. For the geometry drawn in figure 1, a thin compressible layer the normal strain scales like <math>\Delta h/H_l \approx p/G</math> where ''G'' is the elasticity of the layer. We can plug this into our general scaling for the lift to get<br />
<br />
<math>F_{t,c} = \frac{\mu^2 V^2}{G} \frac{H_l R^{3/2}}{h_0^{7/2}}</math><br />
<br />
== Conclusion ==<br />
<br />
In a very similar manner a large class of other geometries and elastic models can be treated, which are summarized in the table in figure 2. The scalings in this table govern in principle all elastic lubrication problems with non-conforming contacts, a class of problems with a truly astounding breadth. A possible extension to this problem formulation is however still possible. In a large class of biological systems the goal is not create the maximal normal force possible, but to create a high enough normal force while keeping the viscous forces on the particle small. The framework presented in this paper would certainly allow such an extension.<br />
<br />
[[Image:Fritz_lubrication2.png|thumb||left|1200px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13991Soft lubrication2009-12-05T08:59:28Z<p>Fritz: /* Conclusion */</p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This in turn creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal force.<br />
<br />
== Understanding the physics ==<br />
<br />
The fact that a maximal normal force exists is quite surprising. Let's first imagine the nonelastic case where a cylinder moves steadily over a flat surface, with a very small gap between the two. If we move in the reference frame of the cylinder and position our coordinate system to the point of minimal distance, then the geometry of the problem is symmetric for all times. The governing equations, the lubrication approximation is fully time reversible which implies a perfect anti-symmetric pressure distribution and thus the normal force has to be always zero.<br />
<br />
To understand how the elastic layer changes the picture, let's have a look at figure 1. Let's imagine that we start out exactly where we began, with a steadily translating cylinder over a flat surface creating an antisymmetric pressure distribution. This antisymmetric pressure distribution will now deform the elastic layer, pushing it down in front of the moving cylinder and sucking it up behind it. This in turn breaks the geometry, which means the pressure distribution will no longer by perfectly antisymmetric. This leads to a finite force upwards.<br />
<br />
== An exemplary scaling ==<br />
<br />
A simple way to derive the scaling for the pressure as a function of the gap height is to balance the only two forces existing in the layer according to the lubrication approximation: pressure and viscosity. This means <math>p/l \approx \mu V / h^2</math> or<br />
<br />
<math>p \approx \frac{\mu V R^{1/2}}{h^{3/2}}</math><br />
<br />
for definitions of the quatities, refer to figure 1.<br />
<br />
It is possible to split up this pressure into an "inelastic" component that cannot contribute to the lift as outlined before and an "elastic" component that will. The lift itself will then scale like<br />
<br />
<math>F \approx \frac{\mu V R}{h_o^2} \Delta h</math> <br />
<br />
where <math>\Delta h</math> is the solution of an elasticity problem. For the geometry drawn in figure 1, a thin compressible layer the normal strain scales like <math>\Delta h/H_l \approx p/G</math> where ''G'' is the elasticity of the layer. We can plug this into our general scaling for the lift to get<br />
<br />
<math>F_{t,c} = \frac{\mu^2 V^2}{G} \frac{H_l R^{3/2}}{h_0^{7/2}}</math><br />
<br />
== Conclusion ==<br />
<br />
In a very similar manner a large class of other geometries and elastic models can be treated, which are summarized in the table in figure 2. The scalings in this table govern in principle all elastic lubrication problems with non-conforming contacts, a class of problems with a truly astounding breadth. A possible extension to this problem formulation is possible. In a large class of biological problems the goal is not create the maximal normal force possible, but to create a high enough normal force while keeping the viscous forces on the particle small. The framework presented in this paper would certainly allow such an extension.<br />
<br />
[[Image:Fritz_lubrication2.png|thumb||left|1200px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13990Soft lubrication2009-12-05T08:58:57Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This in turn creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal force.<br />
<br />
== Understanding the physics ==<br />
<br />
The fact that a maximal normal force exists is quite surprising. Let's first imagine the nonelastic case where a cylinder moves steadily over a flat surface, with a very small gap between the two. If we move in the reference frame of the cylinder and position our coordinate system to the point of minimal distance, then the geometry of the problem is symmetric for all times. The governing equations, the lubrication approximation is fully time reversible which implies a perfect anti-symmetric pressure distribution and thus the normal force has to be always zero.<br />
<br />
To understand how the elastic layer changes the picture, let's have a look at figure 1. Let's imagine that we start out exactly where we began, with a steadily translating cylinder over a flat surface creating an antisymmetric pressure distribution. This antisymmetric pressure distribution will now deform the elastic layer, pushing it down in front of the moving cylinder and sucking it up behind it. This in turn breaks the geometry, which means the pressure distribution will no longer by perfectly antisymmetric. This leads to a finite force upwards.<br />
<br />
== An exemplary scaling ==<br />
<br />
A simple way to derive the scaling for the pressure as a function of the gap height is to balance the only two forces existing in the layer according to the lubrication approximation: pressure and viscosity. This means <math>p/l \approx \mu V / h^2</math> or<br />
<br />
<math>p \approx \frac{\mu V R^{1/2}}{h^{3/2}}</math><br />
<br />
for definitions of the quatities, refer to figure 1.<br />
<br />
It is possible to split up this pressure into an "inelastic" component that cannot contribute to the lift as outlined before and an "elastic" component that will. The lift itself will then scale like<br />
<br />
<math>F \approx \frac{\mu V R}{h_o^2} \Delta h</math> <br />
<br />
where <math>\Delta h</math> is the solution of an elasticity problem. For the geometry drawn in figure 1, a thin compressible layer the normal strain scales like <math>\Delta h/H_l \approx p/G</math> where ''G'' is the elasticity of the layer. We can plug this into our general scaling for the lift to get<br />
<br />
<math>F_{t,c} = \frac{\mu^2 V^2}{G} \frac{H_l R^{3/2}}{h_0^{7/2}}</math><br />
<br />
== Conclusion ==<br />
<br />
In a very similar manner a large class of other geometries and elastic models can be treated, which are summarized in the table in figure 2. This scalings in this table govern in principle all elastic lubrication problems with non-conforming contacts, a class of problems with a truly astounding breadth. A possible extension to this problem formulation is possible. In a large class of biological problems the goal is not create the maximal normal force possible, but to create a high enough normal force while keeping the viscous forces on the particle small. The framework presented in this paper would certainly allow such an extension.<br />
<br />
[[Image:Fritz_lubrication2.png|thumb||left|1200px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13986Soft lubrication2009-12-05T08:53:28Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This in turn creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal force.<br />
<br />
== Understanding the physics ==<br />
<br />
The fact that a maximal normal force exists is quite surprising. Let's first imagine the nonelastic case where a cylinder moves steadily over a flat surface, with a very small gap between the two. If we move in the reference frame of the cylinder and position our coordinate system to the point of minimal distance, then the geometry of the problem is symmetric for all times. The governing equations, the lubrication approximation is fully time reversible which implies a perfect anti-symmetric pressure distribution and thus the normal force has to be always zero.<br />
<br />
To understand how the elastic layer changes the picture, let's have a look at figure 1. Let's imagine that we start out exactly where we began, with a steadily translating cylinder over a flat surface creating an antisymmetric pressure distribution. This antisymmetric pressure distribution will now deform the elastic layer, pushing it down in front of the moving cylinder and sucking it up behind it. This in turn breaks the geometry, which means the pressure distribution will no longer by perfectly antisymmetric. This leads to a finite force upwards.<br />
<br />
== An exemplary scaling ==<br />
<br />
A simple way to derive the scaling for the pressure as a function of the gap height is to balance the only two forces existing in the layer according to the lubrication approximation: pressure and viscosity. This means <math>p/l \approx \mu V / h^2</math> or<br />
<br />
<math>p \approx \frac{\mu V R^{1/2}}{h^{3/2}}</math><br />
<br />
for definitions of the quatities, refer to figure 1.<br />
<br />
It is possible to split up this pressure into an "inelastic" component that cannot contribute to the lift as outlined before and an "elastic" component that will. The lift itself will then scale like<br />
<br />
<math>F \approx \frac{\mu V R}{h_o^2} \Delta h</math> <br />
<br />
where <math>\Delta h</math> is the solution of an elasticity problem. For the geometry drawn in figure 1, a thin compressible layer the normal strain scales like <math>\Delta h/H_l \approx p/G</math> where ''G'' is the elasticity of the layer. We can plug this into our general scaling for the lift to get<br />
<br />
<math>F_{t,c} = \frac{\mu^2 V^2}{G} \frac{H_l R^{3/2}}{h_0^{7/2}}</math><br />
<br />
<br />
<br />
== Conclusion ==<br />
<br />
In a very similar manner a large class of other geometries and elastic models can be treated, which are summarized in the table in figure 2. <br />
<br />
<br />
[[Image:Fritz_lubrication2.png|thumb||left|1200px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13985Soft lubrication2009-12-05T08:36:56Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal force.<br />
<br />
== Understanding the physics ==<br />
<br />
The fact that a maximal normal force exists is quite surprising. Let's first imagine the nonelastic case where a cylinder moves steadily over a flat surface, with a very small gap between the two. If we move in the reference frame of the cylinder and position our coordinate system to the point of minimal distance, then the geometry of the problem is symmetric for all times. The governing equations, the lubrication approximation is fully time reversible which implies a perfect anti-symmetric pressure distribution and thus the normal force has to be always zero.<br />
<br />
To understand how the elastic layer changes the picture, let's have a look at figure 1. Let's imagine that we start out exactly where we began, with a steadily translating cylinder over a flat surface creating an antisymmetric pressure distribution. This antisymmetric pressure distribution will now deform the elastic layer, pushing it down in front of the moving cylinder and sucking it up behind it. This in turn breaks the geometry, which means the pressure distribution will no longer by perfectly antisymmetric. This leads to a finite force upwards.<br />
<br />
<br />
== An exemplary scaling ==<br />
<br />
== Conclusion ==<br />
<br />
<br />
[[Image:Fritz_lubrication2.png|thumb||left|1200px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13984Soft lubrication2009-12-05T08:28:08Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal<br />
force.<br />
<br />
== An exemplary scaling ==<br />
<br />
== Conclusion ==<br />
<br />
<br />
[[Image:Fritz_lubrication2.png|thumb||left|1200px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13983Soft lubrication2009-12-05T08:24:15Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz_lubrication1.png|thumb|400px|Fig.1 Schematic illustrating the geometry and physics of soft lubrication. The thin non-conforming interface creates an anti-symmetric pressure distribution. This pressure distribution leads to a deformation of the elastic layer. This creates a different gap geometry and thus a new pressure distribution.]]<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal<br />
force.<br />
<br />
== An exemplary scaling ==<br />
<br />
== Conclusion ==<br />
<br />
[[Image:Fritz_lubrication2.png|thumb|800px|Fig.2 A table summarizing the scalings for all cases considered in this paper.]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fritz_lubrication2.png&diff=13980File:Fritz lubrication2.png2009-12-05T08:21:19Z<p>Fritz: </p>
<hr />
<div></div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fritz_lubrication1.png&diff=13979File:Fritz lubrication1.png2009-12-05T08:19:13Z<p>Fritz: </p>
<hr />
<div></div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13972Soft lubrication2009-12-05T08:10:05Z<p>Fritz: /* Summary */</p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming surfaces moving tangentially to each other, where a thin elastic layer covers one or both of them. The main result is that an optimal combination of material and geometric properties exists, which maximizes the normal<br />
force.<br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13971Soft lubrication2009-12-05T08:08:46Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
Nature is full of examples for soft lubrication. From the ejection of fungal spores over the flow of red blood cells in arteries to the movement of joints, the concept of reducing wear and adhesion between two surfaces with a thin film of water is ubiquitous. This gives rise to a very interesting problem that couples fluid mechanics and the elasticity of soft (e.g. biological) materials. This paper examines a model problem for symmetric nonconforming where a thin elastic layer covers one or both surfaces moving tangentially to each other. The main result is that there exists an optimal combination of material and geometric properties which maximizes the normal<br />
force.<br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13968Fluid rope trick investigated2009-12-05T08:00:07Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament as an experimental check of the predictions.]]<br />
<br />
What sets the speed <math>\Omega</math> with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous torque scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The inertial forces scale like <math>f_i \approx T\rho \Omega r^2 R^3</math> and the torque due to this force is <math>f_i \ r^2 R^2</math>. A balance of the two torques together with the continuity equation leads to<br />
<br />
<math>\Omega \approx Q^{4/3} r^{-10/3} (\mu/\rho)^{-1/3}</math><br />
<br />
Experimental results, shown in figure 2, agree very well with this scaling, indicating that all the other neglected effects, like surface tension, air drag and non-Newtonian material properties are in fact not important in this problem.<br />
<br />
== Conclusion ==<br />
<br />
Soft matter is everywhere. And even for apparently very complicated phenomena there is sometimes an elegant scaling that gives the desired answer without involved numerical simulations. The beauty of this result is that it is testable on a daily basis. We can derive three quantitative predictions that can be tested during the next weekend breakfast over toast and honey:<br />
# The coiling starts for a critical height for the falling filament of honey<br />
# Above this height a further increase of height has no influence on the rotation speed<br />
# An increase in the flow rate should lead to a marked increase in coiling velocity.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13967Fluid rope trick investigated2009-12-05T07:59:01Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament as an experimental check of the predictions.]]<br />
<br />
What sets the speed <math>\Omega</math> with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous torque scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The inertial forces scale like <math>f_i \approx T\rho \Omega r^2 R^3</math> and the torque due to this force is <math>f_i \ r^2 R^2</math>. A balance of the two torques together with the continuity equation leads to<br />
<br />
<math>\Omega \approx Q^{4/3} r^{-10/3} (\mu/\rho)^{-1/3}</math><br />
<br />
Experimental results, shown in figure 2 agree very well with this scaling, indicating that all the other neglected effects, like surface tension, air drag and non-Newtonian effects are infact not important in this problem.<br />
<br />
== Conclusion ==<br />
<br />
Soft matter is everywhere. And even for apparently very complicated phenomena there is sometimes an elegant scaling that gives the desired answer without involved numerical simulations. The beauty of this result is that it is testable on a daily basis. We can derive three quantitative predictions that can be tested during the next weekend breakfast over toast and honey:<br />
# The coiling starts for a critical height for the falling filament of honey<br />
# Above this height a further increase of height has no influence on the rotation speed<br />
# An increase in the flow rate should lead to a marked increase in coiling velocity.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13960Fluid rope trick investigated2009-12-05T07:48:28Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament as an experimental check of the predictions.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous torque scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The inertial forces scale like <math>f_i \approx T\rho \Phi r^2 R^3</math> and the torque due to this force is <math>f_i \ r^2 R^2</math>. A balance of the two torques together with the continuity equation leads to<br />
<br />
<math>\Phi</math><br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13958Fluid rope trick investigated2009-12-05T07:45:45Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament as an experimental check of the predictions.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous force scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The torque due to this force is <math>f_v \ r^2 R^2</math><br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13957Fluid rope trick investigated2009-12-05T07:45:20Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament as an experimental check of the predictions.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous force scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The torque due to this force is <math> \left f_v r^2 R^2</math><br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13956Fluid rope trick investigated2009-12-05T07:44:44Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament as an experimental check of the predictions.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. The viscous force scales like <math>f_v = \int \sigma r dA \approx \mu U r^4 / R^2</math> where R is a characteristic radius of curvature, which is approximately the radius of the coil structure on the toast. The torque due to this force is <math>f_v r^2 R^2</math><br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13953Fluid rope trick investigated2009-12-05T07:38:03Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to a buckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on fluid density <math>\rho</math>, the viscosity <math>\mu</math>, the<br />
flow rate ''Q'', the gravity constant ''g'', the filament radius ''r'', and the height ''h'' from which the filament is falling. Unfortunately these are too many to reach the desired result directly by dimensional analysis. We have to make use of physical arguments to arrive at the solution.<br />
<br />
Observations tell us that the filament of honey starts to rotate once it impacts the toast from a height that is big enough to create coils on the toast. We would thus assume that the rotation is due to an instability to buckling in the filament very close to where it impacts on the toast. It has been previously shown that the onset of a buckling instability of a falling jet is determined by the competition of two effects, gravity and viscosity. We can compare the two time scales associated with them, the ratio of which gives us something like a Reynolds number for this problem: <math>Re = \frac{g r^3 \rho^2}{\mu^2}</math>. Only when this parameter fall below a critical value will the filament start to rotate (an instability exist).<br />
<br />
Once the fluid filament starts to rotate the dominant balance is on of torques where inertial effects are compensated for by bending torque due to viscous stresses. <br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13952Fluid rope trick investigated2009-12-05T07:19:53Z<p>Fritz: </p>
<hr />
<div>[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to abuckling instability.]]<br />
<br />
Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on <br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13951Fluid rope trick investigated2009-12-05T07:19:00Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
[[Image:Fritz honey1.png|thumb|400px|Fig.1 A slightly more reproducable version of the honey-toast problem: silicon oil filament coiling due to abuckling instability.]]<br />
<br />
This paper studies a very rich problem at the interface between fluid dynamics and classical mechanics that can be observed in thousands of household every morning. If honey is poured from a sufficient height, it approaches the morning toast as a thin<br />
filament which twists and whirls steadily even if the pouring hand is completely static. This can be explained by the theory of buckling with very simple scaling laws.<br />
<br />
== The Math ==<br />
<br />
[[Image:Fritz honey2.png|thumb|400px|Fig.2 Plot of normalized coiling frequency over the radius of the falling filament.]]<br />
<br />
What sets the speed with which the honey rotates? We could imagine up to six parameters that could have an influence on the rotation speed. On an intuitive basis it could depend on <br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fritz_honey2.png&diff=13950File:Fritz honey2.png2009-12-05T07:18:14Z<p>Fritz: </p>
<hr />
<div></div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fritz_honey1.png&diff=13949File:Fritz honey1.png2009-12-05T07:16:23Z<p>Fritz: </p>
<hr />
<div></div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Instability&diff=13948Instability2009-12-05T07:05:12Z<p>Fritz: </p>
<hr />
<div>In the most general sense instability in any system can be described by the idea that one parameter describing the system grows without bound for a small change in another parameter.<br />
<br />
For the purpose of soft matter we are mostly concerned with two more specific formulations of instabilities. The first one is originally derived from structural engineering of solid materials. A structure is unstable when a small change in applied load applied leads to a large deflection. This is due to a positive feedback loop. Above a critical stress, the deflection due to the applied stress itself increases the stress. This is a very general mechanism that lies at the heart of many instability phenomenon. For solid structures the instability is very often [[Buckling|buckling]].<br />
<br />
The other type of instabilities occurs in fluids and is usually associated with the creation of discontinuities at originally smooth interfaces. The most common governing effects are due to surface tension or viscosity. Examples include the Kelvinâ€“Helmholtz, the Rayleighâ€“Taylor, or the Plateau-Rayleigh instability</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Instability&diff=13947Instability2009-12-05T07:01:07Z<p>Fritz: </p>
<hr />
<div>In the most general sense instability in any system can be described by the idea that one parameter describing the system grows without bound for a small change in another parameter.<br />
<br />
For the purpose of soft matter we are mostly concerned with two more specific formulations of instabilities. The first one is originally derived from structural engineering of solid materials. A structure is unstable when a small change in applied load applied leads to a large deflection. This is due to a positive feedback loop. Above a critical stress, the deflection due to the applied stress itself increases the stress. This is a very general mechanism that lies at the heart of many instability phenomenon. For solid structures the instability is very often [[Buckling|buckling]].</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Instability&diff=13946Instability2009-12-05T07:00:33Z<p>Fritz: </p>
<hr />
<div>In the most general sense instability in any system can be described by the idea that one parameter describing the system grows without bound for a small change in another parameter.<br />
<br />
For the purpose of soft matter we are mostly concerned with two more specific formulations of instabilities. The first one is originally derived from structural engineering of solid materials. A structure is unstable when a small change in load applied to it leads to a large deflection. This is due to a positive feedback loop. Above a critical stress, the deflection due to the applied stress itself increases the stress. This is a very general mechanism that lies at the heart of many instability phenomenon. For solid structures the instability is very often [[Buckling|buckling]].</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Instability&diff=13945Instability2009-12-05T06:53:24Z<p>Fritz: </p>
<hr />
<div>In the most general sense instability in any system can be described by the idea that one parameter describing the system grows without bound for a small change in another parameter.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Lubrication_theory&diff=13944Lubrication theory2009-12-05T06:49:52Z<p>Fritz: </p>
<hr />
<div>Lubrication theory refers to a simplification of the Navier-Stokes equations which assumes that one dimension of the problem is significantly smaller than the others.<br />
<br />
It is in most cases formulated for two dimensions, where the governing equations to first order then become <br />
<br />
:<math>\frac{\partial p}{\partial z} = 0</math><br />
:<math>\frac{\partial p}{\partial x} = \frac{\partial^2 u}{\partial z^2}</math><br />
<br />
where ''p'' is pressure, ''u'' is the velocity component in the x direction and ''z'' is the small dimension in the problem.<br />
<br />
The term derives from the tremendous importance the areas of lubrication of machinery and fluid bearings had when these equations where first formally developed.<br />
<br />
Today these equations finds application in a very wide range of fields: from free films over biological flows to the study of [[Soft lubrication| elastohydrodynamic interactions]] or the [[Precursors to droplet splashing on a solid surface| splashing of water drops]].</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Lubrication_theory&diff=13943Lubrication theory2009-12-05T06:48:59Z<p>Fritz: </p>
<hr />
<div>Lubrication theory refers to a simplification of the Navier-Stokes equations which assumes that one dimension of the problem is significantly smaller than the others.<br />
<br />
It is in most cases formulated for two dimensions, where the governing equations to first order then become <br />
<br />
:<math>\frac{\partial p}{\partial z} = 0</math><br />
:<math>\frac{\partial p}{\partial x} = \frac{\partial^2 u}{\partial z^2}</math><br />
<br />
where ''p'' is pressure and ''u'' is the velocity component in the x direction.<br />
<br />
The term derives from the tremendous importance the areas of lubrication of machinery and fluid bearings had when these equations where first formally developed.<br />
<br />
Today these equations finds application in a very wide range of fields: from free films over biological flows to the study of [[Soft lubrication| elastohydrodynamic interactions]] or the [[Precursors to droplet splashing on a solid surface| splashing of water drops]].</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Lubrication_theory&diff=13942Lubrication theory2009-12-05T06:47:21Z<p>Fritz: </p>
<hr />
<div>Lubrication theory refers to a simplification of the Navier-Stokes equations which assumes that one dimension of the problem is significantly smaller than the others.<br />
<br />
It is in most cases formulated for two dimensions, where the governing equations to first order then become <br />
<br />
:<math>\frac{\partial p}{\partial z} = 0</math><br />
:<math>\frac{\partial p}{\partial x} = \frac{\partial^2 u}{\partial z^2}</math><br />
<br />
The term derives from the tremendous importance the areas of lubrication of machinery and fluid bearings had when these equations where first formally developed.<br />
<br />
Today these equations finds application in a very wide range of fields: from free films over biological flows to the study of [[Soft lubrication| elastohydrodynamic interactions]] or the [[Precursors to droplet splashing on a solid surface| splashing of water drops]].</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Joerg_Fritz&diff=13941Joerg Fritz2009-12-05T06:35:28Z<p>Fritz: </p>
<hr />
<div>Definitions:<br />
<br />
[[Universality]]<br />
<br />
[[Sphere Packing]]<br />
<br />
[[Buckling]]<br />
<br />
[[Particle rafts]]<br />
<br />
[[Random walk]]<br />
<br />
[[Splashing]]<br />
<br />
[[Colloidal Crystal]]<br />
<br />
[[Plateau's laws]]<br />
<br />
[[Instability]]<br />
<br />
[[Lubrication theory]]<br />
<br />
Weekly wiki entries:<br />
<br />
[[Evaporation Driven assembly of colloidal particles]]<br />
<br />
[[The Self Assembly of Flat Sheets into Closed Surfaces]]<br />
<br />
[[Dynamics of surfactant-driven fracture of particle rafts]]<br />
<br />
[[Taylor expansions for random-walk polymers]]<br />
<br />
[[Precursors to droplet splashing on a solid surface]]<br />
<br />
[[Slip, yield, and bands in colloidal crystals under oscillatory shear]]<br />
<br />
[[Dynamics of foam drainage]]<br />
<br />
[[Fluid rope trick investigated]]<br />
<br />
[[Soft lubrication]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Lubrication_theory&diff=13940Lubrication theory2009-12-05T06:35:04Z<p>Fritz: New page: Lubrication theory is used to describe the flow of fluids (liquids or gases) in a geometry where one dimension is significantly smaller than the others. ...working on it...</p>
<hr />
<div>Lubrication theory is used to describe the flow of fluids (liquids or gases) in a geometry where one dimension is significantly smaller than the others.<br />
<br />
...working on it...</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13939Soft lubrication2009-12-05T06:33:16Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [[Poroelasticity]], [[Adhesion]]<br />
<br />
== Summary ==<br />
<br />
working on it<br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Soft_lubrication&diff=13938Soft lubrication2009-12-05T06:33:03Z<p>Fritz: New page: Original entry by Joerg Fritz, AP225 Fall 2009 == Source == [http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] J.M. Skotheim and L. Mahadevan: ''Physical Re...</p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/2004-16.pdf "Soft lubrication"] <br />
<br />
J.M. Skotheim and L. Mahadevan: ''Physical Review Letters'', 2004, 92, pp 245509-1 to 245509-4<br />
<br />
== Keywords ==<br />
<br />
[[Lubrication theory]], [[Elsticity]], [[Soft interface]], [Poroelasticity]], [Adhesion]]<br />
<br />
== Summary ==<br />
<br />
working on it<br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Joerg_Fritz&diff=13937Joerg Fritz2009-12-05T06:29:37Z<p>Fritz: </p>
<hr />
<div>Definitions:<br />
<br />
[[Universality]]<br />
<br />
[[Sphere Packing]]<br />
<br />
[[Buckling]]<br />
<br />
[[Particle rafts]]<br />
<br />
[[Random walk]]<br />
<br />
[[Splashing]]<br />
<br />
[[Colloidal Crystal]]<br />
<br />
[[Plateau's laws]]<br />
<br />
[[Instability]]<br />
<br />
Weekly wiki entries:<br />
<br />
[[Evaporation Driven assembly of colloidal particles]]<br />
<br />
[[The Self Assembly of Flat Sheets into Closed Surfaces]]<br />
<br />
[[Dynamics of surfactant-driven fracture of particle rafts]]<br />
<br />
[[Taylor expansions for random-walk polymers]]<br />
<br />
[[Precursors to droplet splashing on a solid surface]]<br />
<br />
[[Slip, yield, and bands in colloidal crystals under oscillatory shear]]<br />
<br />
[[Dynamics of foam drainage]]<br />
<br />
[[Fluid rope trick investigated]]<br />
<br />
[[Soft lubrication]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Instability&diff=13936Instability2009-12-05T06:23:18Z<p>Fritz: New page: working on it</p>
<hr />
<div>working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13935Fluid rope trick investigated2009-12-05T06:22:04Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Instability]]<br />
<br />
== Summary ==<br />
<br />
working on it<br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Joerg_Fritz&diff=13934Joerg Fritz2009-12-05T06:21:48Z<p>Fritz: </p>
<hr />
<div>Definitions:<br />
<br />
[[Universality]]<br />
<br />
[[Sphere Packing]]<br />
<br />
[[Buckling]]<br />
<br />
[[Particle rafts]]<br />
<br />
[[Random walk]]<br />
<br />
[[Splashing]]<br />
<br />
[[Colloidal Crystal]]<br />
<br />
[[Plateau's laws]]<br />
<br />
[[Instability]]<br />
<br />
Weekly wiki entries:<br />
<br />
[[Evaporation Driven assembly of colloidal particles]]<br />
<br />
[[The Self Assembly of Flat Sheets into Closed Surfaces]]<br />
<br />
[[Dynamics of surfactant-driven fracture of particle rafts]]<br />
<br />
[[Taylor expansions for random-walk polymers]]<br />
<br />
[[Precursors to droplet splashing on a solid surface]]<br />
<br />
[[Slip, yield, and bands in colloidal crystals under oscillatory shear]]<br />
<br />
[[Dynamics of foam drainage]]<br />
<br />
[[Fluid rope trick investigated]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Fluid_rope_trick_investigated&diff=13933Fluid rope trick investigated2009-12-05T06:16:26Z<p>Fritz: New page: Original entry by Joerg Fritz, AP225 Fall 2009 == Source == [http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] L. Mahadevan, W.S. Ryu, A.D.T...</p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://www.seas.harvard.edu/softmat/downloads/pre2000-11.pdf "Fluid rope trick investigated"] <br />
<br />
L. Mahadevan, W.S. Ryu, A.D.T. Samuel: ''Nature'', 1998, 392, pp 140 to 141<br />
<br />
== Keywords ==<br />
<br />
[[Buckling]], [[Viscosity]], [[Surface tension]], [[Stability]]<br />
<br />
== Summary ==<br />
<br />
working on it<br />
<br />
== Conclusion ==<br />
<br />
working on it</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Joerg_Fritz&diff=13932Joerg Fritz2009-12-05T06:11:36Z<p>Fritz: </p>
<hr />
<div>Definitions:<br />
<br />
[[Universality]]<br />
<br />
[[Sphere Packing]]<br />
<br />
[[Buckling]]<br />
<br />
[[Particle rafts]]<br />
<br />
[[Random walk]]<br />
<br />
[[Splashing]]<br />
<br />
[[Colloidal Crystal]]<br />
<br />
[[Plateau's laws]]<br />
<br />
Weekly wiki entries:<br />
<br />
[[Evaporation Driven assembly of colloidal particles]]<br />
<br />
[[The Self Assembly of Flat Sheets into Closed Surfaces]]<br />
<br />
[[Dynamics of surfactant-driven fracture of particle rafts]]<br />
<br />
[[Taylor expansions for random-walk polymers]]<br />
<br />
[[Precursors to droplet splashing on a solid surface]]<br />
<br />
[[Slip, yield, and bands in colloidal crystals under oscillatory shear]]<br />
<br />
[[Dynamics of foam drainage]]<br />
<br />
[[Fluid rope trick investigated]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Joerg_Fritz&diff=13931Joerg Fritz2009-12-05T06:02:50Z<p>Fritz: </p>
<hr />
<div>Definitions:<br />
<br />
[[Universality]]<br />
<br />
[[Sphere Packing]]<br />
<br />
[[Buckling]]<br />
<br />
[[Particle rafts]]<br />
<br />
[[Random walk]]<br />
<br />
[[Splashing]]<br />
<br />
[[Colloidal Crystal]]<br />
<br />
[[Plateau's laws]]<br />
<br />
Weekly wiki entries:<br />
<br />
[[Evaporation Driven assembly of colloidal particles]]<br />
<br />
[[The Self Assembly of Flat Sheets into Closed Surfaces]]<br />
<br />
[[Dynamics of surfactant-driven fracture of particle rafts]]<br />
<br />
[[Taylor expansions for random-walk polymers]]<br />
<br />
[[Precursors to droplet splashing on a solid surface]]<br />
<br />
[[Slip, yield, and bands in colloidal crystals under oscillatory shear]]<br />
<br />
[[Dynamics of foam drainage]]<br />
<br />
[[Non-stick water]]</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Slip,_yield,_and_bands_in_colloidal_crystals_under_oscillatory_shear&diff=13926Slip, yield, and bands in colloidal crystals under oscillatory shear2009-12-05T05:49:08Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://scitation.aip.org.ezp-prod1.hul.harvard.edu/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000097000021215502000001&idtype=cvips&gifs=yes "Slip, Yield, and Bands in Colloidal Crystals under Oscillatory Shear"] <br />
<br />
Itai Cohen, Benny Davidovitch, Andrew B. Schofield, Michael P. Brenner, and David A. Weitz: ''Physical Review Letters'', 2006, 97, pp 215502-1 to 215502-4<br />
<br />
<br />
== Keywords ==<br />
<br />
[[Colloidal Crystal]], [[Shear stress]], [[Suspension]], [[Nonlinear rheology]], [[Shear band]], [[Confocal microscopy]]<br />
<br />
== Summary ==<br />
<br />
Rheological measurments for complex fluids are notoriously difficult. One reason for this is the formation of shear bands where the material separates into bands with significantly different strain and flow rates. The paper describes a new experimental approach to this problem where confocal microscopy is used to investigate the behavior of colloidal crystals under oscillatory shear. For large shear rates the suspension forms shear bands that exhibit a harmonic response to the applied forcing. This creates an interesting theoretical problems. Nonlinear models of rheology, which are usually used to describe shear bands are ruled out by this observation. Instead the authors present a linear model that accounts for all the phenomena seen in the experiments.<br />
<br />
== Experiments ==<br />
<br />
[[Image:Fritz Shear1.png|thumb|400px|Fig.1 Maximal displacement over the height of the experimental setup for two different strain rates and frequency varied as a parameter from 0.02 to 60 Hz.]]<br />
<br />
[[Image:Fritz_colloidal_1.jpg|thumb|400px|Fig.2 Plots of speed over time for a specific choice of parameters. The curves correspond to flows in the crystalline (<math>z=0.04</math>), transition (<math>z=0.59</math>) and liquid (<math>z=0.96</math>)<br />
regions.]]<br />
<br />
The study uses a dense suspension of polymethyl methacrylate particles marked with rhodamine dye and stabilized by a hydrostearic acid suspended in a mixture of cyclohexyl bromide and decalin. The mixture can be chosen in such a way as to match particle density and index of refraction, which makes 3-dimensional imaging of the dyed colloids possible. This suspension is contained in a shear cell which simple consists of a movable microscope slide and a fixed glass plate. This setup allows to control the suspension thickness, the shear frequency and the strain amplitude.<br />
<br />
After preparing the samples to create a uniform random crystal (rhcp) by shearing it at a selected frequency for over an hour measurments were started. The maximal particle displacement ''u'' was measured at different heights ''z'' above the oscillating plate, for a large range of frequencies and strain rates. Figure 1 shows a plot for two different strain rates where frequency is used as a parameter and varied from 0.02 to 60 Hz.<br />
<br />
The plots show quite surprisingly that for higher frequencies a region close to the upper static plate exhibits a significantly larger strain than that near the lower oscillating plate. We have a very nice visual representation of shear banding. A comparison of both plots shows that for increased strain rate the upper region gets bigger until the high strain region stretches the full domain.<br />
<br />
Another interesting behavior can be observed at the top plate (<math>z=0</math>). The displacement has been normalized by the amplitude of the cover slip, but th measured displacements never reach 1, which indicates slip at this surface for all frequencies and strain rates.<br />
<br />
Since normal explanations for the formation of shear banding rely on non-linear characteristics of the materials the author also test for any aharmonic contributions for the variation of the displacement field over time for one selected position ''z''. Figure 2 shows the results at three different positions between the two plates. Surprisingly, the displacements in all three regions can be fitted perfectly to a harmonic oscillation, thus making an explanation along the lines of non-linear material characteristics impossible.<br />
<br />
== A linear model ==<br />
<br />
These observations motivate the development of a model that is linear in its response to forcing. We can note that at rest<br />
the suspension forms a rhcp crystal. This phase occupies the entire gap for very small strains. Previous studies have shown that near equilibrium the crystal stress can be modeled as a sum of viscous and elastic components with effective viscosity and shear modulus. The authors extend this idea to still be valid in the initially larger area when the shear band begin to form. In the second phase, that only appears for larger strains, the colloidal sheets flow freely over each other, which can be interpreted as a vanishing shear modulus. This allows to model this phase as a Newtonian fluid with effective viscosity. For a given frequency the stress in both phases is linearly proportional to the strain. This automatically creates a harmonic response to oscillatory displacements, which is consistent with the flow patters that can be observed in figure 2.<br />
<br />
This model can be used to calculate a general non-dimensional displacement field as a function of position. By using realistic parameters the solid lines in figure 1 can be generated. The agreement with the experimental results is quite striking, indicating that the model used by the authors in fact captures all of the colloid behavior exhibited in this experiment.<br />
<br />
== Conclusion ==<br />
<br />
Colloidal crystals, just as polydisperse suspensions of colloids exhibit highly interesting and sometimes hard to understand material properties. But this example shows in a nice way that unexpected experimental results not always require complex nonlinear rheology as an explanation.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Slip,_yield,_and_bands_in_colloidal_crystals_under_oscillatory_shear&diff=13925Slip, yield, and bands in colloidal crystals under oscillatory shear2009-12-05T05:45:20Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://scitation.aip.org.ezp-prod1.hul.harvard.edu/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000097000021215502000001&idtype=cvips&gifs=yes "Slip, Yield, and Bands in Colloidal Crystals under Oscillatory Shear"] <br />
<br />
Itai Cohen, Benny Davidovitch, Andrew B. Schofield, Michael P. Brenner, and David A. Weitz: ''Physical Review Letters'', 2006, 97, pp 215502-1 to 215502-4<br />
<br />
<br />
== Keywords ==<br />
<br />
[[Colloidal Crystal]], [[Shear stress]], [[Suspension]], [[Nonlinear rheology]], [[Shear band]], [[Confocal microscopy]]<br />
<br />
== Summary ==<br />
<br />
Rheological measurments for complex fluids are notoriously difficult. One reason for this is the formation of shear bands where the material separates into bands with significantly different strain and flow rates. The paper describes a new experimental approach to this problem where confocal microscopy is used to investigate the behavior of colloidal crystals under oscillatory shear. For large shear rates the suspension forms shear bands that exhibit a harmonic response to the applied forcing. This creates an interesting theoretical problems. Nonlinear models of rheology, which are usually used to describe shear bands are ruled out by this observation. Instead the authors present a linear model that accounts for all the phenomena seen in the experiments.<br />
<br />
== Experiments ==<br />
<br />
[[Image:Image:Fritz Shear1.png|thumb|400px|Fig.1 Maximal displacement over the height of the experimental setup for two different strain rates and frequency varied as a parameter from 0.02 to 60 Hz.]]<br />
<br />
[[Image:Fritz_colloidal_1.jpg|thumb|400px|Fig.2 Plots of speed over time for a specific choice of parameters. The curves correspond to flows in the crystalline (<math>z=0.04</math>), transition (<math>z=0.59</math>) and liquid (<math>z=0.96</math>)<br />
regions.]]<br />
<br />
The study uses a dense suspension of polymethyl methacrylate particles marked with rhodamine dye and stabilized by a hydrostearic acid suspended in a mixture of cyclohexyl bromide and decalin. The mixture can be chosen in such a way as to match particle density and index of refraction, which makes 3-dimensional imaging of the dyed colloids possible. This suspension is contained in a shear cell which simple consists of a movable microscope slide and a fixed glass plate. This setup allows to control the suspension thickness, the shear frequency and the strain amplitude.<br />
<br />
After preparing the samples to create a uniform random crystal (rhcp) by shearing it at a selected frequency for over an hour measurments were started. The maximal particle displacement ''u'' was measured at different heights ''z'' above the oscillating plate, for a large range of frequencies and strain rates. Figure 1 shows a plot for two different strain rates where frequency is used as a parameter and varied from 0.02 to 60 Hz.<br />
<br />
The plots show quite surprisingly that for higher frequencies a region close to the upper static plate exhibits a significantly larger strain than that near the lower oscillating plate. We have a very nice visual representation of shear banding. A comparison of both plots shows that for increased strain rate the upper region gets bigger until the high strain region stretches the full domain.<br />
<br />
Another interesting behavior can be observed at the top plate (<math>z=0</math>). The displacement has been normalized by the amplitude of the cover slip, but th measured displacements never reach 1, which indicates slip at this surface for all frequencies and strain rates.<br />
<br />
Since normal explanations for the formation of shear banding rely on non-linear characteristics of the materials the author also test for any aharmonic contributions for the variation of the displacement field over time for one selected position ''z''. Figure 2 shows the results at three different positions between the two plates. Surprisingly, the displacements in all three regions can be fitted perfectly to a harmonic oscillation, thus making an explanation along the lines of non-linear material characteristics impossible.<br />
<br />
== A linear model ==<br />
<br />
These observations motivate the development of a model that is linear in its response to forcing. We can note that at rest<br />
the suspension forms a rhcp crystal. This phase occupies the entire gap for very small strains. Previous studies have shown that near equilibrium the crystal stress can be modeled as a sum of viscous and elastic components with effective viscosity and shear modulus. The authors extend this idea to still be valid in the initially larger area when the shear band begin to form. In the second phase, that only appears for larger strains, the colloidal sheets flow freely over each other, which can be interpreted as a vanishing shear modulus. This allows to model this phase as a Newtonian fluid with effective viscosity. For a given frequency the stress in both phases is linearly proportional to the strain. This automatically creates a harmonic response to oscillatory displacements, which is consistent with the flow patters that can be observed in figure 2.<br />
<br />
This model can be used to calculate a general non-dimensional displacement field as a function of position. By using realistic parameters the solid lines in figure 1 can be generated. The agreement with the experimental results is quite striking, indicating that the model used by the authors in fact captures all of the colloid behavior exhibited in this experiment.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=File:Fritz_Shear1.png&diff=13921File:Fritz Shear1.png2009-12-05T05:29:48Z<p>Fritz: </p>
<hr />
<div></div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Plateau%27s_laws&diff=13909Plateau's laws2009-12-05T05:05:03Z<p>Fritz: </p>
<hr />
<div>Plateu's laws are a set of rules, formulated in the 19th century by the Belgian physicist Joseph Plateau from experimental observations, that describe the structure of soap films in foams.<br />
<br />
They can be split into two observations that are true for bubble arrangements in general<br />
<br />
* Soap films are made of entire smooth surfaces.<br />
* The average curvature of a portion of a soap film is always constant on any point on the same piece of soap film.<br />
<br />
and two further ones which are true if three bubbles are touching each other<br />
*Soap films always meet in threes, and they do so at an angle of cos<sup>&minus;1</sup>(&minus;1/2) = 120 degrees forming an edge called a Plateau Border.<br />
* These Plateau Borders meet in fours at the tetrahedral angle to form a vertex.<br />
<br />
These rules were later examined with the formalisms of geometric measure theory by Jean Taylor. He showed that these experimental rules have a close relation to the stability of the structures.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Plateau%27s_laws&diff=13907Plateau's laws2009-12-05T05:04:34Z<p>Fritz: </p>
<hr />
<div>Plateu's laws are a set of rules, formulated in the 19th century by the Belgian physicist Joseph Plateau from experimental observations, that describe the structure of soap films in foams.<br />
<br />
They can be split into two observations that are true for bubble arrangements in general<br />
<br />
# Soap films are made of entire smooth surfaces.<br />
# The average curvature of a portion of a soap film is always constant on any point on the same piece of soap film.<br />
<br />
and two further ones which are true if three bubbles are touching each other<br />
# Soap films always meet in threes, and they do so at an angle of cos<sup>&minus;1</sup>(&minus;1/2) = 120 degrees forming an edge called a Plateau Border.<br />
# These Plateau Borders meet in fours at the tetrahedral angle to form a vertex.<br />
<br />
These rules were later examined with the formalisms of geometric measure theory by Jean Taylor. He showed that these experimental rules have a close relation to the stability of the structures.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Dynamics_of_foam_drainage&diff=13905Dynamics of foam drainage2009-12-05T04:59:48Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://prola.aps.org.ezp-prod1.hul.harvard.edu/abstract/PRE/v58/i2/p2097_1 "Dynamics of foam drainage"] <br />
<br />
S. A. Koehler, H. A. Stone, M. P. Brenner, and J. Eggers: ''Physical Review E'', 1998, 58, pp 2097 to 2106<br />
<br />
== Keywords ==<br />
<br />
[[Liquid foam]], [[Surface tension]], [[Viscosity]], [[Drainage]], [[Wetting]], [[Plateau Borders]], [[Plateau's laws]]<br />
== Summary ==<br />
<br />
[[Image:Fritz Drainage1.png|thumb|400px|Fig.1 Image of an aluminum foam, where dark regions indicate aluminum and light regions air. Note the increased aluminum concentration at the bottom due to drainage of air.]]<br />
<br />
[[Image:Fritz_foams_1.jpg|thumb|400px|Fig.2 An aluminum foam exhibiting the typical structure of Plateau borders.]]<br />
<br />
[[Image:Fritz_Drainage3.png|thumb|400px|Fig.3 Simulation results showing the average area of the plateau regions over the height of the foam for different times.]]<br />
<br />
In a large class of applications, from the reduction of explosion impact to dish-washing by hand, uniformity of a foam is highly desirable. One pathway that can lead to non-uniformity (and that is important in both examples stated above) is the drainage of the fluid component due to gravity. This process is governed by the interplay of three forces: surface tension, viscosity and gravity. The authors develop a PDE that describes this phenomenon and apply it to several experimentally accessible geometries.<br />
<br />
== Foam Drainage Equation ==<br />
<br />
Foams usually exhibit a very particular geometry, where the spaces between three adjacent bubbles are filled and connected by thin liquid-filled channels, which are termed Plateau borders (see figure 2). The primary dependent variable for the model will be the average cross-sectional area ''A'' of these Plateau borders, which will in general vary with position z and time t. The foam drainage equation simply results from a combination of mass conservation and the dynamic equation for Stokes flow, keeping in mind that capillary effects also produce a pressure gradient<br />
<br />
<math>\frac{\partial A}{\partial t} + \frac{\rho g}{\eta} \ \frac{\partial A^2}{\partial z} - \frac{\gamma \delta^{1/2}}{2 \eta} \nabla \cdot ( A^{1/2} \nabla A)</math><br />
<br />
here <math>\rho</math> is the density of the fluid, <math>g</math> is gravitational acceleration, <math>\eta</math> is an effective visosity which to a certain extent masks the lack of understanding of viscosity for this problem and <math>\gamma</math> is the surface tension. <br />
<br />
A very interesting feature of this equation is that the bubble radius, does not appear explicitly, in contrast to what might have been expected on dimensional grounds.<br />
<br />
== One Dimensional Drainage ==<br />
<br />
The paper in detail examines three different cases of the foam drainage problem in one dimension:<br />
# free drainage: an initially uniform foam that evolves towards a steady-state profile under the influence of gravity<br />
# wetting of a foam: a very dry foam comes in contact with a liquid bath after which a spreading front forms in the foam and again evolves toward a steady-state profile<br />
# pulsed drainage: an initially nearly homogeneous dry foam is disturbed by the addition of a finite amount of liquid, which is redistributed through a combination of gravitational and surface tension-driven flows. This can to a certain level be regarded as a combination of the two earlier effects.<br />
<br />
As intuition tells us examples one and two both evolve towards a steady state. The first step is thus to determine the steady-state profile of ''A'' for a given concentration <math>A_0</math> at one of the boundaries. If we pick the lower boundary and denote the height of the foam as ''L'' we can solve for the distribution<br />
<br />
<math>A(z) = A_0 \left ( 1 + \frac{\rho g A_0^{1/2}}{\gamma \delta^{1/2}} (L-z) \right)^{-2}</math><br />
<br />
In all three of the cases our strategy will be to non-dimensionalize the evolution equation for the average Plateau area and then look for similarity solutions. The main results for the three cases considered are<br />
# In the free drainage problem there are asymptotically two regimes. The border between these two regimes is when the transient distribution of Plateau-area first shows a non-zero gradient at the top boundary (<math>\tau \approx 2</math> in figure 3). In the first regime the drainage profile <math>\partial A / \partial z</math> varies as <math>t^{-1}</math>. In the second regime the solution starts to forget about the initial conditions and a generic similarity solution shows very good agreement with the results from a numerical simulation<br />
# For the wetting of a dry foam the profile <math>\partial A / \partial z</math> simply spreads as <math>t^{1/3}</math><br />
# For pulsed drainage the profile itself exhibits three distinct regions. An advancing nose which shows similar behavior as forced wetting (describe din earlier papers) that spreads as <math>t^{1/4}</math>, a middle region with maximum that spreads as <math>t^{1/2}</math> and decreases as <math>t^{-1/2}</math> and an advancing rear that shows similarities to the dry wetting case and thus scales with time as <math>t^{1/3}</math><br />
<br />
The results, especially of the numerical simulations solving exactly for the general foam drainage equation, agree very well with the few experimental results available.<br />
<br />
== Drainage in Higher Dimensions ==<br />
<br />
In higher dimensions the problem only remains tractable only if we ignore the influence of gravity. This is reasonable in many microscale problems. We can then look at the equivalent of the pulsed drainage problem described earlier. To visualize this we can imagine a dry foam with a localized spherical wet domain. In this case it makes sense to work not in terms of an area of the Plateau borders but in terms of the local volume fraction <math>\epsilon</math> of liquid in the foam. The two can be related to each other by an effective bubble radius ''R''. Note that by doing this we have now the bubble size as an additional parameter, just as one would expect from a casual observation of the problem. The evolution equation in terms of volume fraction of liquid in radial coordinates then becomes<br />
<br />
<math>\frac{\partial \epsilon}{\partial t} = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta r^{2}} \frac{\partial}{\partial r} \left( \epsilon^{1/2} r^2 \frac{\partial \epsilon}{\partial t} \right )</math><br />
<br />
where <math>c_n</math> and <math>\delta</math> are constants describing the geometry. <br />
<br />
This is essentially a non-linear diffusion equation with the diffusivity <br />
<br />
<math>D = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta}</math><br />
<br />
We can thus see that our solution is not only valid for an initial spherical wet domain but for larger times asymptotically for any initial domain form.<br />
<br />
Guided by this observation we can again find a (very complicated) similarity solution. Its main features are that the pulse spreads over time with <math>t^{2/7}</math> and its amplitude is reduced with time like <math>t^{-6/7}</math>.<br />
<br />
Given that no experimental results in this parameter regime exist the authors state that they plan to perform experiments to test these predictions themselves. <br />
<br />
== Conclusion ==<br />
<br />
The drainage in foams is in certain ways a typical problem for the area of soft matter. On the one hand we have a very interesting phenomenon with applications in many fields. But on the other we lack both a detailed physical understanding of what is going on (in this case mostly related to the effects of viscosity) and also detailed experiments investigating specifically this particular phenomenon. The paper discussed above shows that we can get remarkably far reaching results, that can explain most features that have been experimentally observed, by determining simple similarity solutions to a simplified governing equation.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Dynamics_of_foam_drainage&diff=13904Dynamics of foam drainage2009-12-05T04:58:58Z<p>Fritz: /* Summary */</p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://prola.aps.org.ezp-prod1.hul.harvard.edu/abstract/PRE/v58/i2/p2097_1 "Dynamics of foam drainage"] <br />
<br />
S. A. Koehler, H. A. Stone, M. P. Brenner, and J. Eggers: ''Physical Review E'', 1998, 58, pp 2097 to 2106<br />
<br />
== Keywords ==<br />
<br />
[[Liquid foam]], [[Surface tension]], [[Viscosity]], [[Drainage]], [[Wetting]], [[Plateau Borders]], [[Plateau's laws]]<br />
== Summary ==<br />
<br />
[[Image:Fritz Drainage1.png|thumb|400px|Fig.1 Image of an aluminum foam, where dark regions indicate aluminum and light regions air. Note the increased aluminum concentration at the bottom due to drainage of air.]]<br />
<br />
[[Image:Fritz_foams_1.jpg|thumb|400px|Fig.2 An aluminum foam exhibiting the typical structure of Plateau borders.]]<br />
<br />
[[Image:Fritz_Drainage3.png|thumb|400px|Fig.3 Simulation results showing the average area of the plateau regions over the height of the foam for different times.]]<br />
<br />
In a large class of applications, from the reduction of explosion impact to dish-washing by hand, uniformity of a foam is highly desirable. One pathway that can lead to non-uniformity (and that is important in both examples stated above) is the drainage of the fluid component due to gravity. This process is governed by the interplay of three forces: surface tension, viscosity and gravity. The authors develop a PDE that describes this phenomenon and apply it to several experimentally accessible geometries.<br />
<br />
== Foam Drainage Equation ==<br />
<br />
Foams usually exhibit a very particular geometry, where the spaces between three adjacent bubbles are filled and connected by thin liquid-filled channels, which are termed Plateau borders (see figure 2). The primary dependent variable for the model will be the average cross-sectional area ''A'' of these Plateau borders, which will in general vary with position z and time t. The foam drainage equation simply results from a combination of mass conservation and the dynamic equation for Stokes flow, keeping in mind that capillary effects also produce a pressure gradient<br />
<br />
<math>\frac{\partial A}{\partial t} + \frac{\rho g}{\eta} \ \frac{\partial A^2}{\partial z} - \frac{\gamma \delta^{1/2}}{2 \eta} \nabla \cdot ( A^{1/2} \nabla A)</math><br />
<br />
here <math>\rho</math> is the density of the fluid, <math>g</math> is gravitational acceleration, <math>\eta</math> is an effective visosity which to a certain extent masks the lack of understanding of viscosity for this problem and <math>\gamma</math> is the surface tension. <br />
<br />
A very interesting feature of this equation is that the bubble radius, does not appear explicitly, in contrast to what might have been expected on dimensional grounds.<br />
<br />
== One Dimensional Drainage ==<br />
<br />
The paper in detail examines three different cases of the foam drainage problem in one dimension:<br />
# free drainage: an initially uniform foam that evolves towards a steady-state profile under the influence of gravity<br />
# wetting of a foam: a very dry foam comes in contact with a liquid bath after which a spreading<br />
front forms in the foam and again evolves toward a steady-state profile<br />
# pulsed drainage: an initially nearly homogeneous dry foam is disturbed by the addition of a finite amount of liquid, which is redistributed through a combination of gravitational and surface tension-driven flows. This can to a certain level be regarded as a combination of the two earlier effects.<br />
<br />
As intuition tells us examples one and two both evolve towards a steady state. The first step is thus to determine the steady-state profile of ''A'' for a given concentration <math>A_0</math> at one of the boundaries. If we pick the lower boundary and denote the height of the foam as ''L'' we can solve for the distribution<br />
<br />
<math>A(z) = A_0 \left ( 1 + \frac{\rho g A_0^{1/2}}{\gamma \delta^{1/2}} (L-z) \right)^{-2}</math><br />
<br />
In all three of the cases our strategy will be to non-dimensionalize the evolution equation for the average Plateau area and then look for similarity solutions. The main results for the three cases considered are<br />
# In the free drainage problem there are asymptotically two regimes. The border between these two regimes is when the transient distribution of Plateau-area first shows a non-zero gradient at the top boundary (<math>\tau \approx 2</math> in figure 3). In the first regime the drainage profile <math>\partial A / \partial z</math> varies as <math>t^{-1}</math>. In the second regime the solution starts to forget about the initial conditions and a generic similarity solution shows very good agreement with the results from a numerical simulation<br />
# For the wetting of a dry foam the profile <math>\partial A / \partial z</math> simply spreads as <math>t^{1/3}</math><br />
# For pulsed drainage the profile itself exhibits three distinct regions. An advancing nose which shows similar behavior as forced wetting (describe din earlier papers) that spreads as <math>t^{1/4}</math>, a middle region with maximum that spreads as <math>t^{1/2}</math> and decreases as <math>t^{-1/2}</math> and an advancing rear that shows similarities to the dry wetting case and thus scales with time as <math>t^{1/3}</math><br />
<br />
The results, especially of the numerical simulations solving exactly for the general foam drainage equation, agree very well with the few experimental results available.<br />
<br />
== Drainage in Higher Dimensions ==<br />
<br />
In higher dimensions the problem only remains tractable only if we ignore the influence of gravity. This is reasonable in many microscale problems. We can then look at the equivalent of the pulsed drainage problem described earlier. To visualize this we can imagine a dry foam with a localized spherical wet domain. In this case it makes sense to work not in terms of an area of the Plateau borders but in terms of the local volume fraction <math>\epsilon</math> of liquid in the foam. The two can be related to each other by an effective bubble radius ''R''. Note that by doing this we have now the bubble size as an additional parameter, just as one would expect from a casual observation of the problem. The evolution equation in terms of volume fraction of liquid in radial coordinates then becomes<br />
<br />
<math>\frac{\partial \epsilon}{\partial t} = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta r^{2}} \frac{\partial}{\partial r} \left( \epsilon^{1/2} r^2 \frac{\partial \epsilon}{\partial t} \right )</math><br />
<br />
where <math>c_n</math> and <math>\delta</math> are constants describing the geometry. <br />
<br />
This is essentially a non-linear diffusion equation with the diffusivity <br />
<br />
<math>D = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta}</math><br />
<br />
We can thus see that our solution is not only valid for an initial spherical wet domain but for larger times asymptotically for any initial domain form.<br />
<br />
Guided by this observation we can again find a (very complicated) similarity solution. Its main features are that the pulse spreads over time with <math>t^{2/7}</math> and its amplitude is reduced with time like <math>t^{-6/7}</math>.<br />
<br />
Given that no experimental results in this parameter regime exist the authors state that they plan to perform experiments to test these predictions themselves. <br />
<br />
== Conclusion ==<br />
<br />
The drainage in foams is in certain ways a typical problem for the area of soft matter. On the one hand we have a very interesting phenomenon with applications in many fields. But on the other we lack both a detailed physical understanding of what is going on (in this case mostly related to the effects of viscosity) and also detailed experiments investigating specifically this particular phenomenon. The paper discussed above shows that we can get remarkably far reaching results, that can explain most features that have been experimentally observed, by determining simple similarity solutions to a simplified governing equation.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Dynamics_of_foam_drainage&diff=13902Dynamics of foam drainage2009-12-05T04:58:31Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://prola.aps.org.ezp-prod1.hul.harvard.edu/abstract/PRE/v58/i2/p2097_1 "Dynamics of foam drainage"] <br />
<br />
S. A. Koehler, H. A. Stone, M. P. Brenner, and J. Eggers: ''Physical Review E'', 1998, 58, pp 2097 to 2106<br />
<br />
== Keywords ==<br />
<br />
[[Liquid foam]], [[Surface tension]], [[Viscosity]], [[Drainage]], [[Wetting]], [[Plateau Borders]], [[Plateau's laws]]<br />
== Summary ==<br />
<br />
[[Image:Fritz Drainage1.png|thumb|400px|Fig.1 Image of an aluminum foam, where dark regions indicate aluminum and light regions air. Note the increased aluminum concentration at the bottom due to drainage of air.]]<br />
<br />
[[Image:Fritz_foams_1.jpg|thumb|400px|Fig.2 An aluminum foam exhibiting the typical structure of Plateau borders.]]<br />
<br />
[[Image:Fritz_Drainage3.png|thumb|400px|Fig.3 Simulation results showing the average area of the plateau regions over the height of the foam for different times.]]<br />
<br />
In a large class of applications, from the reduction of explosion impact to dish-washing by hand, uniformity of a foam is highly desirable. One pathway that can lead to non-uniformity (and that is important in both examples stated above) is the drainage of the fluid component due to gravity. This process is governed by the interplay of three forces, surface tension, viscosity and gravity. The authors develop a PDE that describes this phenomenon and apply it to several experimentally accessible geometries. <br />
<br />
== Foam Drainage Equation ==<br />
<br />
Foams usually exhibit a very particular geometry, where the spaces between three adjacent bubbles are filled and connected by thin liquid-filled channels, which are termed Plateau borders (see figure 2). The primary dependent variable for the model will be the average cross-sectional area ''A'' of these Plateau borders, which will in general vary with position z and time t. The foam drainage equation simply results from a combination of mass conservation and the dynamic equation for Stokes flow, keeping in mind that capillary effects also produce a pressure gradient<br />
<br />
<math>\frac{\partial A}{\partial t} + \frac{\rho g}{\eta} \ \frac{\partial A^2}{\partial z} - \frac{\gamma \delta^{1/2}}{2 \eta} \nabla \cdot ( A^{1/2} \nabla A)</math><br />
<br />
here <math>\rho</math> is the density of the fluid, <math>g</math> is gravitational acceleration, <math>\eta</math> is an effective visosity which to a certain extent masks the lack of understanding of viscosity for this problem and <math>\gamma</math> is the surface tension. <br />
<br />
A very interesting feature of this equation is that the bubble radius, does not appear explicitly, in contrast to what might have been expected on dimensional grounds.<br />
<br />
== One Dimensional Drainage ==<br />
<br />
The paper in detail examines three different cases of the foam drainage problem in one dimension:<br />
# free drainage: an initially uniform foam that evolves towards a steady-state profile under the influence of gravity<br />
# wetting of a foam: a very dry foam comes in contact with a liquid bath after which a spreading<br />
front forms in the foam and again evolves toward a steady-state profile<br />
# pulsed drainage: an initially nearly homogeneous dry foam is disturbed by the addition of a finite amount of liquid, which is redistributed through a combination of gravitational and surface tension-driven flows. This can to a certain level be regarded as a combination of the two earlier effects.<br />
<br />
As intuition tells us examples one and two both evolve towards a steady state. The first step is thus to determine the steady-state profile of ''A'' for a given concentration <math>A_0</math> at one of the boundaries. If we pick the lower boundary and denote the height of the foam as ''L'' we can solve for the distribution<br />
<br />
<math>A(z) = A_0 \left ( 1 + \frac{\rho g A_0^{1/2}}{\gamma \delta^{1/2}} (L-z) \right)^{-2}</math><br />
<br />
In all three of the cases our strategy will be to non-dimensionalize the evolution equation for the average Plateau area and then look for similarity solutions. The main results for the three cases considered are<br />
# In the free drainage problem there are asymptotically two regimes. The border between these two regimes is when the transient distribution of Plateau-area first shows a non-zero gradient at the top boundary (<math>\tau \approx 2</math> in figure 3). In the first regime the drainage profile <math>\partial A / \partial z</math> varies as <math>t^{-1}</math>. In the second regime the solution starts to forget about the initial conditions and a generic similarity solution shows very good agreement with the results from a numerical simulation<br />
# For the wetting of a dry foam the profile <math>\partial A / \partial z</math> simply spreads as <math>t^{1/3}</math><br />
# For pulsed drainage the profile itself exhibits three distinct regions. An advancing nose which shows similar behavior as forced wetting (describe din earlier papers) that spreads as <math>t^{1/4}</math>, a middle region with maximum that spreads as <math>t^{1/2}</math> and decreases as <math>t^{-1/2}</math> and an advancing rear that shows similarities to the dry wetting case and thus scales with time as <math>t^{1/3}</math><br />
<br />
The results, especially of the numerical simulations solving exactly for the general foam drainage equation, agree very well with the few experimental results available.<br />
<br />
== Drainage in Higher Dimensions ==<br />
<br />
In higher dimensions the problem only remains tractable only if we ignore the influence of gravity. This is reasonable in many microscale problems. We can then look at the equivalent of the pulsed drainage problem described earlier. To visualize this we can imagine a dry foam with a localized spherical wet domain. In this case it makes sense to work not in terms of an area of the Plateau borders but in terms of the local volume fraction <math>\epsilon</math> of liquid in the foam. The two can be related to each other by an effective bubble radius ''R''. Note that by doing this we have now the bubble size as an additional parameter, just as one would expect from a casual observation of the problem. The evolution equation in terms of volume fraction of liquid in radial coordinates then becomes<br />
<br />
<math>\frac{\partial \epsilon}{\partial t} = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta r^{2}} \frac{\partial}{\partial r} \left( \epsilon^{1/2} r^2 \frac{\partial \epsilon}{\partial t} \right )</math><br />
<br />
where <math>c_n</math> and <math>\delta</math> are constants describing the geometry. <br />
<br />
This is essentially a non-linear diffusion equation with the diffusivity <br />
<br />
<math>D = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta}</math><br />
<br />
We can thus see that our solution is not only valid for an initial spherical wet domain but for larger times asymptotically for any initial domain form.<br />
<br />
Guided by this observation we can again find a (very complicated) similarity solution. Its main features are that the pulse spreads over time with <math>t^{2/7}</math> and its amplitude is reduced with time like <math>t^{-6/7}</math>.<br />
<br />
Given that no experimental results in this parameter regime exist the authors state that they plan to perform experiments to test these predictions themselves. <br />
<br />
== Conclusion ==<br />
<br />
The drainage in foams is in certain ways a typical problem for the area of soft matter. On the one hand we have a very interesting phenomenon with applications in many fields. But on the other we lack both a detailed physical understanding of what is going on (in this case mostly related to the effects of viscosity) and also detailed experiments investigating specifically this particular phenomenon. The paper discussed above shows that we can get remarkably far reaching results, that can explain most features that have been experimentally observed, by determining simple similarity solutions to a simplified governing equation.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Dynamics_of_foam_drainage&diff=13894Dynamics of foam drainage2009-12-05T04:34:30Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://prola.aps.org.ezp-prod1.hul.harvard.edu/abstract/PRE/v58/i2/p2097_1 "Dynamics of foam drainage"] <br />
<br />
S. A. Koehler, H. A. Stone, M. P. Brenner, and J. Eggers: ''Physical Review E'', 1998, 58, pp 2097 to 2106<br />
<br />
== Keywords ==<br />
<br />
[[Liquid foam]], [[Surface tension]], [[Viscosity]], [[Drainage]], [[Wetting]], [[Plateau Borders]], [[Plateau's laws]]<br />
== Summary ==<br />
<br />
[[Image:Fritz Drainage1.png|thumb|400px|Fig.1 Image of an aluminum foam, where dark regions indicate aluminum and light regions air. Note the increased aluminum concentration at the bottom due to drainage of air.]]<br />
<br />
[[Image:Fritz_foams_1.jpg|thumb|400px|Fig.2 An aluminum foam exhibiting the typical structure of Plateau borders.]]<br />
<br />
[[Image:Fritz_Drainage3.png|thumb|400px|Fig.3 Simulation results showing the average area of the plateau regions over the height of the foam for different times.]]<br />
<br />
In a large class of applications, from the reduction of explosion impact to dish-washing by hand, uniformity of a foam is highly desirable. One pathway that can lead to non-uniformity (and that is important in both examples stated above) is the drainage of the fluid component due to gravity. This process is governed by the interplay of three forces, surface tension, viscosity and gravity. The authors develop a PDE that describes this phenomenon and apply it to several experimentally accessible geometries. <br />
<br />
== Foam Drainage Equation ==<br />
<br />
Foams usually exhibit a very particular geometry, where the spaces between three adjacent bubbles are filled and connected by thin liquid-filled channels, which are termed Plateau borders (see figure 2). The primary dependent variable for the model will be the average cross-sectional area ''A'' of these Plateau borders, which will in general vary with position z and time t. The foam drainage equation simply results from a combination of mass conservation and the dynamic equation for Stokes flow, keeping in mind that capillary effects also produce a pressure gradient<br />
<br />
<math>\frac{\partial A}{\partial t} + \frac{\rho g}{\eta} \ \frac{\partial A^2}{\partial z} - \frac{\gamma \delta^{1/2}}{2 \eta} \nabla \cdot ( A^{1/2} \nabla A)</math><br />
<br />
here <math>\rho</math> is the density of the fluid, <math>g</math> is gravitational acceleration, <math>\eta</math> is an effective visosity which to a certain extent masks the lack of understanding of viscosity for this problem and <math>\gamma</math> is the surface tension. <br />
<br />
A very interesting feature of this equation is that the bubble radius, does not appear explicitly, in contrast to what might have been expected on dimensional grounds.<br />
<br />
== One Dimensional Drainage ==<br />
<br />
The paper in detail examines three different cases of the foam drainage problem in one dimension:<br />
# free drainage: an initially uniform foam that evolves towards a steady-state profile under the influence of gravity<br />
# wetting of a foam: a very dry foam comes in contact with a liquid bath after which a spreading<br />
front forms in the foam and again evolves toward a steady-state profile<br />
# pulsed drainage: an initially nearly homogeneous dry foam is disturbed by the addition of a finite amount of liquid, which is redistributed through a combination of gravitational and surface tension-driven flows. This can to a certain level be regarded as a combination of the two earlier effects.<br />
<br />
As intuition tells us examples one and two both evolve towards a steady state. The first step is thus to determine the steady-state profile of ''A'' for a given concentration <math>A_0</math> at one of the boundaries. If we pick the lower boundary and denote the height of the foam as ''L'' we can solve for the distribution<br />
<br />
<math>A(z) = A_0 \left ( 1 + \frac{\rho g A_0^{1/2}}{\gamma \delta^{1/2}} (L-z) \right)^{-2}</math><br />
<br />
In all three of the cases our strategy will be to non-dimensionalize the evolution equation for the average Plateau area and then look for similarity solutions. <br />
<br />
== Drainage in Higher Dimensions ==<br />
<br />
In higher dimensions the problem only remains tractable only if we ignore the influence of gravity. This is reasonable in many microscale problems. We can then look at the equivalent of the pulsed drainage problem described earlier. To visualize this we can imagine a dry foam with a localized spherical wet domain. In this case it makes sense to work not in terms of an area of the Plateau borders but in terms of the local volume fraction <math>\epsilon</math> of liquid in the foam. The two can be related to each other by an effective bubble radius ''R''. Note that by doing this we have now the bubble size as an additional parameter, just as one would expect from a casual observation of the problem. The evolution equation in terms of volume fraction of liquid in radial coordinates then becomes<br />
<br />
<math>\frac{\partial \epsilon}{\partial t} = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta r^{2}} \frac{\partial}{\partial r} \left( \epsilon^{1/2} r^2 \frac{\partial \epsilon}{\partial t} \right )</math><br />
<br />
== Conclusion ==<br />
<br />
The drainage in foams is in certain ways a typical problem for the area of soft matter. On the one hand we have a very interesting phenomenon with applications in many fields. But on the other we lack both a detailed physical understanding of what is going on (in this case mostly related to the effects of viscosity) and also detailed experiments investigating specifically this particular phenomenon. The paper discussed above shows that we can get remarkably far reaching results, that can explain most features that have been experimentally observed, by determining simple similarity solutions to a simplified governing equation.</div>Fritzhttp://soft-matter.seas.harvard.edu/index.php?title=Dynamics_of_foam_drainage&diff=13890Dynamics of foam drainage2009-12-05T04:33:00Z<p>Fritz: </p>
<hr />
<div>Original entry by Joerg Fritz, AP225 Fall 2009 <br />
<br />
== Source ==<br />
[http://prola.aps.org.ezp-prod1.hul.harvard.edu/abstract/PRE/v58/i2/p2097_1 "Dynamics of foam drainage"] <br />
<br />
S. A. Koehler, H. A. Stone, M. P. Brenner, and J. Eggers: ''Physical Review E'', 1998, 58, pp 2097 to 2106<br />
<br />
== Keywords ==<br />
<br />
[[Liquid foam]], [[Surface tension]], [[Viscosity]], [[Drainage]], [[Wetting]], [[Plateau Borders]], [[Plateau's laws]]<br />
== Summary ==<br />
<br />
[[Image:Fritz Drainage1.png|thumb|400px|Fig.1 Image of an aluminum foam, where dark regions indicate aluminum and light regions air. Note the increased aluminum concentration at the bottom due to drainage of air.]]<br />
<br />
[[Image:Fritz_foams_1.jpg|thumb|400px|Fig.2 An aluminum foam exhibiting the typical structure of Plateau borders.]]<br />
<br />
[[Image:Fritz_Drainage3.png|thumb|400px|Fig.3 Simulation results showing the average area of the plateau regions over the height of the foam for different times.]]<br />
<br />
In a large class of applications, from the reduction of explosion impact to dish-washing by hand, uniformity of a foam is highly desirable. One pathway that can lead to non-uniformity (and that is important in both examples stated above) is the drainage of the fluid component due to gravity. This process is governed by the interplay of three forces, surface tension, viscosity and gravity. The authors develop a PDE that describes this phenomenon and apply it to several experimentally accessible geometries. <br />
<br />
== Foam Drainage Equation ==<br />
<br />
Foams usually exhibit a very particular geometry, where the spaces between three adjacent bubbles are filled and connected by thin liquid-filled channels, which are termed Plateau borders (see figure 2). The primary dependent variable for the model will be the average cross-sectional area ''A'' of these Plateau borders, which will in general vary with position z and time t. The foam drainage equation simply results from a combination of mass conservation and the dynamic equation for Stokes flow, keeping in mind that capillary effects also produce a pressure gradient<br />
<br />
<math>\frac{\partial A}{\partial t} + \frac{\rho g}{\eta} \ \frac{\partial A^2}{\partial z} - \frac{\gamma \delta^{1/2}}{2 \eta} \nabla \cdot ( A^{1/2} \nabla A)</math><br />
<br />
here <math>\rho</math> is the density of the fluid, <math>g</math> is gravitational acceleration, <math>\eta</math> is an effective visosity which to a certain extent masks the lack of understanding of viscosity for this problem and <math>\gamma</math> is the surface tension. <br />
<br />
A very interesting feature of this equation is that the bubble radius, does not appear explicitly, in contrast to what might have been expected on dimensional grounds.<br />
<br />
== One Dimensional Drainage ==<br />
<br />
The paper in detail examines three different cases of the foam drainage problem in one dimension:<br />
# free drainage: an initially uniform foam that evolves towards a steady-state profile under the influence of gravity<br />
# wetting of a foam: a very dry foam comes in contact with a liquid bath after which a spreading<br />
front forms in the foam and again evolves toward a steady-state profile<br />
# pulsed drainage: an initially nearly homogeneous dry foam is disturbed by the addition of a finite amount of liquid, which is redistributed through a combination of gravitational and surface tension-driven flows. This can to a certain level be regarded as a combination of the two earlier effects.<br />
<br />
As intuition tells us examples one and two both evolve towards a steady state. The first step is thus to determine the steady-state profile of ''A'' for a given concentration <math>A_0</math> at one of the boundaries. If we pick the lower boundary and denote the height of the foam as ''L'' we can solve for the distribution<br />
<br />
<math>A(z) = A_0 \left ( 1 + \frac{\rho g A_0^{1/2}}{\gamma \delta^{1/2}} (L-z) \right)^{-2}</math><br />
<br />
In all three of the cases our strategy will be to non-dimensionalize the evolution equation for the average Plateau area and then look for similarity solutions. <br />
<br />
== Drainage in Higher Dimensions ==<br />
<br />
In higher dimensions the problem only remains tractable only if we ignore the influence of gravity. This is reasonable in many microscale problems. We can then look at the equivalent of the pulsed drainage problem described earlier. To visualize this we can imagine a dry foam with a localized spherical wet domain. In this case it makes sense to work not in terms of an area of the Plateau borders but in terms of the local volume fraction <math>\epsilon</math> of liquid in the foam. The two can be related to each other by an effective bubble radius ''R''. Note that by doing this we have now the bubble size as an additional parameter, just as one would expect from a casual observation of the problem. The evolution equation in terms of volume fraction of liquid in radial coordinates then becomes<br />
<br />
<math>\frac{\partial \epsilon}{\partial t} = \frac{(c_n \rho)^{1/2} R \gamma}{2 \eta r^{2} \frac{\partial}{\partial r} \left( \epsilon^{1/2} r^2} \frac{\partial \epsilon}{\partial t} \right )</math><br />
<br />
== Conclusion ==<br />
<br />
The drainage in foams is in certain ways a typical problem for the area of soft matter. On the one hand we have a very interesting phenomenon with applications in many fields. But on the other we lack both a detailed physical understanding of what is going on (in this case mostly related to the effects of viscosity) and also detailed experiments investigating specifically this particular phenomenon. The paper discussed above shows that we can get remarkably far reaching results, that can explain most features that have been experimentally observed, by determining simple similarity solutions to a simplified governing equation.</div>Fritz