Difference between revisions of "Profile of a large drop"

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Zach Wissner-Gross
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==Discussion==
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[[Image:Dropshape1.jpg|300px|thumb|right|Figure 1: Drop profiles for various contact angles and using the physical constants given in class.]]
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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:
  
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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.
  
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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.
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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.
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[[Image:Dropshape.jpg|frame|Drop profiles for various contact angles and using the physical constants given in class.]]
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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
  
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<center><math>2\kappa\approx 3.5 \text{mm}</math></center>
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<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.
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==MATLAB code==
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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.
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<pre>
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% dropshapeODE.m
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function df = dropshapeODE(dt,y)
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global angle;
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theta = pi/180 * angle;
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sigma = 30*10^(-3);
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rho = 10^3;
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g = 9.8;
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kappa = sqrt(rho*g/sigma);
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E = 2/kappa*sin(theta/2);
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X = cos(theta) + kappa^2/2*(2*E*y-y^2);
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df = sqrt(1/X^2-1);
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</pre>
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<pre>
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% dropshape.m
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global angle;
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angle = 100;
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theta = pi/180 * angle;
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sigma = 30*10^(-3);
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rho = 10^3;
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g = 9.8;
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kappa = sqrt(rho*g/sigma);
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E = 2/kappa*sin(theta/2);
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y0 = [0];
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[Xout,Yout] = ode45('dropshapeODE',[0 5*kappa^(-1)],y0);
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dydx = zeros(size(Xout)-1);
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if theta > pi/180 * 90
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    dydx = (Yout(2:end)-Yout(1:end-1))./(Xout(2:end)-Xout(1:end-1));
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    [p q] = max(dydx);
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    for i = 1:q
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        Xout(i) = 2*Xout(q)-Xout(i);
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    end
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    Xout = Xout-Xout(1);
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end
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figure(1);
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clf;
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hold on;
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plot(Xout,Yout);
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axis equal;
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A = axis;
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axis([A(1) A(2) 0 A(4)]);
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box on;
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</pre>

Latest revision as of 18:23, 18 February 2009

Zach Wissner-Gross

Discussion

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:

<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 (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

<math>2\kappa^{-1}\approx 3.5 \text{mm}</math>

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;
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