Difference between revisions of "Profile of a large drop"
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+ | Zach Wissner-Gross | ||
+ | |||
+ | ==Discussion== | ||
+ | [[Image:Dropshape1.jpg|300px|thumb|right|Figure 1: Drop profiles for various contact angles and using the physical constants given in class.]] | ||
+ | |||
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: | ||
Line 9: | Line 14: | ||
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. | + | 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 | 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 | ||
− | <center><math>2\kappa\approx 3.5 \text{mm}</math></center> | + | <center><math>2\kappa^{-1}\approx 3.5 \text{mm}</math></center> |
Nevertheless, the bottom left corner of the 179 degree profile does resemble a quarter-circle. | 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. | ||
+ | |||
+ | <pre> | ||
+ | % 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); | ||
+ | </pre> | ||
+ | |||
+ | <pre> | ||
+ | % 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; | ||
+ | </pre> |
Latest revision as of 18:23, 18 February 2009
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 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 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;