Stokes stream function for spherical system
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Can anybody see where I am going wrong? Trying to produce a contour plot of the Stokes stream function with radial and polar velocity of u1 and u2.
Number_of_grid_points = 500;
scale = 200;
xx0 = -1; %minimum
xx = 1; %maximum
dx = (xx-xx0)/Number_of_grid_points; %step
a = 1; %radius
W = 1; %uniform velocity W along z axis
The = xx0:dx:xx;
R = The;
[the, r] = meshgrid(The,R);
u1 = (W * cos(the)).*(1 + (a^3)/(2*r^3) - (3*a)/(2*r)); %radial velocity
u2 = -(W * sin(the)).*(1 - (a^3)/(4*r^3) - (3*a)/(4*r)); %polar velocity
Z1 = u1 + u2 %velocity of fluid
contour(Z1,scale);
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AHMAD TALAEI
le 18 Oct 2019
Modifié(e) : AHMAD TALAEI
le 18 Oct 2019
Use the following code. I found it online but did a few modifications to plot spherical stream function. For more info look at: https://www.mathworks.com/matlabcentral/fileexchange/4674-streamlines-magnus-and-cp-aroud-a-cylinder-section
close all
clear all
clc
V_0 = 1; % flow veloicty (m/s)
a = 1; % radius (m)
b = a*2;
c = -a*2;
n = a*15; % number of intervals
% the spherical coordinate solution for a falling sphere looking only at x-y plane (looking from side not top)
[x,y]=meshgrid((c:(b-c)/n:b),(c:(b-c)/n:b)');
% Preliminar DATA & purification
for i = 1:length(x)
for k = 1:length(x)
if sqrt(x(i,k).^2 + y(i,k).^2) < a
x(i,k) = 0;
y(i,k) = 0;
end
end
end
% definition of polar variables
r = sqrt(x.^2 + y.^2);
theta = atan2(y,x);
% creation of the streamline function
z = V_0.*sin(theta).*r.*(1-(a.^2./(r.^2))); % laminar stream function in polar coordinate
% stream line Plot
figure(1), contour(x,y,real(z),25,'Linewidth',2); hold on; axis square;
% circle plot
r = ones(1,n+1)*a;
t = (0:2*pi/n:2*pi);
polar(t,r,'-k');
% creation of vectors around the circle
figure(2), contour(x,y,abs(z),10); axis square;
% recreation of x,y
n = a*15; % number of intervals
[x,y] = meshgrid((c:(b-c)/n:b),(c:(b-c)/n:b)');
for i = 1:length(x)
for k = 1:length(x)
if sqrt(x(i,k).^2 + y(i,k).^2) < a
x(i,k) = 0;
y(i,k) = 0;
end
end
end
r = sqrt(x.^2 + y.^2);
theta = atan2(y,x);
u_r = V_0 .* (1 - (3/2).*(a./r) + (1/2).*(a./r).^3) .* cos(theta); % laminar stream function in spherical coordinate
u_theta = - V_0 .* (1 - (3/4).*(a./r) - (1/4).*(a./r).^3) .* sin(theta); % laminar stream function in spherical coordinate
% velocities in x-y cordinates
u = u_r .* cos(theta) - u_theta .* sin(theta);
v = u_r .* sin(theta) + u_theta .* cos(theta);
% vectors plotting
figure(2); hold on
quiver(x,y,u,v)
% creating The Filled Circle
t_r = (0:.1:2*pi);
xxx = a*cos(t_r);
yyy = a*sin(t_r);
fill(xxx,yyy,'y'); axis square
2 commentaires
Harshit Anand
le 20 Mai 2021
Hi, can you tell me how to plot the streamlines vertically? In my problem statement ball is falling vertically. So the streamline and vector field shoud be in upward direction.
AHMAD TALAEI
le 20 Mai 2021
Modifié(e) : AHMAD TALAEI
le 20 Mai 2021
Just contourplot z, which is the stream function. Also, notice that stream function is a scalar quantity, so it has no vertical component:
contour(x,y,real(z),'LineStyle','--','LineWidth',1.5,'LineColor',[0.5 0.5 0.5],'LevelStep',0.4);
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/624573/image.png)
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