Profile of a large drop

In class we derived the profile $z(x)$ of a large drop as being:

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

where we had defined $e$ as being the maximum height of the drop:

$e=2\kappa^{-1}\sin{\frac{\theta_e}{2}}$

and $\kappa=\sqrt{\frac{\rho g}{\sigma_{lv}}}$ 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 (see below), 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 $\theta_e$ 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

$2\kappa^{-1}\approx 3.5 \text{mm}$

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

dropshapeODE.m

function df = dropshapeODE(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);


dropshape.m (the parameter to change here is "angle")

global angle;

angle = 100;
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);

y0 = [0];
[Xout,Yout] = ode45('dropshapeODE',[0 5*kappa^(-1)],y0);

dydx = zeros(size(Xout)-1);
if theta > pi/180 * 90
dydx = (Yout(2:end)-Yout(1:end-1))./(Xout(2:end)-Xout(1:end-1));
[p q] = max(dydx);
for i = 1:q
Xout(i) = 2*Xout(q)-Xout(i);
end
Xout = Xout-Xout(1);
end

figure(1);
clf;
hold on;
plot(Xout,Yout);
axis equal;
A = axis;
axis([A(1) A(2) 0 A(4)]);
box on;