Difference between revisions of "Profile of a large drop"

From Soft-Matter
Jump to: navigation, search
(MATLAB code)
 
(17 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
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
 +
 
 +
<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.
 +
 
 +
==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
  
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
+
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

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;