# Profile of a large drop

Zach Wissner-Gross

## Discussion

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

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 (see below), 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 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

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

## MATLAB code

To visualize your own large drop profiles, copy and paste the following two MATLAB scripts into two corresponding files. Then run dropshape.m -- it will automatically plot the profile. You can adjust the contact angle by variying the parameter "angle" in dropshape.m.

dropshapeODE.m

% 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

% dropshape.m 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;