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:

<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:


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.

Figure 1: Drop profiles for various contact angles and using the physical constants given in class.

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 180 degrees), will the edge of the drop have a semicircular shape? The shape is apparently not semicircular at 179 degrees, since the horizontal range of the drop is only 1 mm, while it vertically extends to a height of

<math>2\kappa\approx 3.5 \text{mm}</math>

Nevertheless, the bottom left corner of the 179 degree profile does resemble a quarter-circle.

function df = altruismODE(dt,y)

global angle;
theta = pi/180 * angle;
sigma = 30*10^(-3);
rho = 10^3;
g = 9.8;
kappa = sqrt(rho*g/sigma);
E = 2/kappa*sin(theta/2);

X = cos(theta) + kappa^2/2*(2*E*y-y^2);
df = sqrt(1/X^2-1);