Difference between revisions of "Profile of a large drop"

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In class we derived the profile <math>z(x)</math> of a large drop as being:
 
In class we derived the profile <math>z(x)</math> of a large drop as being:
  
<center><math>\sigma_{lv}(\frac{1}{\sqrt{1+\dot{z}^2}}-\cos\theta_e)=\frac{1}{2}\rho g(2ez-z^2)</math>,</center>
+
<center><math>\sigma_{lv}(\frac{1}{\sqrt{1+\dot{z}^2}}-\cos\theta_e)=\frac{1}{2}\rho g(2ez-z^2)</math></center>
  
 
where we had defined <math>e</math> as being the maximum height of the drop:
 
where we had defined <math>e</math> as being the maximum height of the drop:
  
<center><math>e=2\kappa^{-1}\sin{\frac{\theta_e}{2}}</math>,</center>
+
<center><math>e=2\kappa^{-1}\sin{\frac{\theta_e}{2}}</math></center>
  
 
and <math>\kappa=\sqrt{\frac{\rho g}{\sigma_{lv}}}</math> was the inverse capillary length.
 
and <math>\kappa=\sqrt{\frac{\rho g}{\sigma_{lv}}}</math> was the inverse capillary length.
 +
 +
To visualize what the edge of a large drop then looks like, you can use the following two MATLAB programs, which will solve and plot the above differential equation. Figure 1 also shows what some of these plots look like.
 +
 +
In class we had a brief discussion as to the nature of the drop's periphery on superhydrophobic surfaces. In the superhydrophobic limit (i.e., as <math>\theta_e</math> approaches <math>180^{\circ}</math>

Revision as of 18:00, 18 February 2009

In class we derived the profile <math>z(x)</math> of a large drop as being:

<math>\sigma_{lv}(\frac{1}{\sqrt{1+\dot{z}^2}}-\cos\theta_e)=\frac{1}{2}\rho g(2ez-z^2)</math>

where we had defined <math>e</math> as being the maximum height of the drop:

<math>e=2\kappa^{-1}\sin{\frac{\theta_e}{2}}</math>

and <math>\kappa=\sqrt{\frac{\rho g}{\sigma_{lv}}}</math> was the inverse capillary length.

To visualize what the edge of a large drop then looks like, you can use the following two MATLAB programs, which will solve and plot the above differential equation. Figure 1 also shows what some of these plots look like.

In class we had a brief discussion as to the nature of the drop's periphery on superhydrophobic surfaces. In the superhydrophobic limit (i.e., as <math>\theta_e</math> approaches <math>180^{\circ}</math>