How do I plot Streamlines of velocity components of spherical coordinate system?

I solved Navier stokes in Spherical coordinates and I got velocity field inside a sphere i.e

If I plot contours using the code below its working. But, The same technique is not working for streamlines, instead I'm getting blank.

The streamlines are look like this

nr  = 21;
nth = 21;
L = 1;

r = linspace(0,1,nr);
th = linspace(0,pi,nth);

[R,Th] = meshgrid(r,th);
Xi = R.*cos(Th); Yi = R.*sin(Th);

ui = (R.^2-1)/2/(L+1).*cos(Th);
vi = (1-2*R.^2)/2/(L+1).*sin(Th);

figure;contourf(Xi,Yi,ui,100,'LineStyle','none');axis image

figure;contourf(Xi,Yi,vi,100,'LineStyle','none');axis image

figure;streamline(Xi,Yi,ui,vi)

I'm getting streamlines as empty, can anyone help me in this regard?

Thanks in advance.

Answer 1

[VBBV]

verts = stream2(Xi,Yi,ui,vi,R,Th);
streamline(verts);

[Jagadeesh Korukonda]

Thanks for responding @VBBV

when i do the same as you mentioned, I'm getting the following error

Error using matlab.internal.math.interp1

Sample points must be unique.

Error in interp1 (line 188)

VqLite = matlab.internal.math.interp1(X,V,method,method,Xqcol);

Error in stream2 (line 63)

syi=interp1(yy(:),1:szu(1),sy(k));

[VBBV]

use streamslice instead as below

clc 
clear all
close all
nr  = 21;
nth = 21;
L = 1;
r =linspace(-1,1,10)
th =linspace(-pi/4,pi/4,10)

[R,Th] = meshgrid(r,th);

Xi = (R).*cos(Th); 
Yi = R.*sin(Th);

ui = (R.^2-1)/2./(L+1).*cos(Th);
vi = (1-2*R.^2)/2./(L+1).*sin(Th);

% figure;contourf(Xi,Yi,ui,100,'LineStyle','none');axis image
% figure;contourf(Xi,Yi,vi,100,'LineStyle','none');axis image
 
streamslice(Xi,Yi,(ui),(vi));
ylim([0 1])

[Jagadeesh Korukonda]

Thanks for responding @VBBV. we can't change limits of r and th. Here we have analytical solution but I mostly work on numerical soluiton, so I have u,v as matrices.

limits of r and th comes from physics.

[VBBV]

Ok, it seems your equations are in spherical coordinates, so use sph2cart function

clc 
clear all
close all
nr  = 21;
nth = 21;
L = 1;
r =linspace(-1,1,nr);
th =linspace(0,pi,nth);

[R,Th] = meshgrid(r,th);
[x y z] = sph2cart(cos(Th),sin(Th),R);

Xi = (R).*cos(Th); 
Yi = R.*sin(Th);

ui = ((R.^2-1)/(2*(L+1))).*cos(Th);
vi = ((1-2*R.^2)/(2*(L+1))).*sin(Th);

% figure;contourf(Xi,Yi,ui,100,'LineStyle','none');axis image
% figure;contourf(Xi,Yi,vi,100,'LineStyle','none');axis image
 
streamslice(x,y,ui,vi);

[Jagadeesh Korukonda]

Thank you @VBBV

I've made small modification to your code. now its working fine. since my r-domian is [0 1] only.

clc
clear all
close all
nr  = 5;
nth = 5;
L = 1;
r =linspace(0,1,nr);
th =linspace(0,pi,nth);
[R,Th] = meshgrid(r,th);
[x y z] = sph2cart(cos(Th),sin(Th),R);
Xi = (R).*cos(Th);
Yi = R.*sin(Th);
ui = ((R.^2-1)/(2*(L+1))).*cos(Th);
vi = ((1-2*R.^2)/(2*(L+1))).*sin(Th);
UI = [fliplr(ui(:,2:end)) ui];
VI = [fliplr(vi(:,2:end)) vi];
X1 = [fliplr(-x(:,2:end)) x];
Y1 = [fliplr(-y(:,2:end)) y];
streamslice(X1,Y1,UI,VI); axis image