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Start-up Poiseuille Flow

Start-up Poiseuille Flow

Start-up Poiseuille Flow

(OP)
Hi,
I'm trying to solve start-up flow in a hollow cylinder with matlab and finite difference method. I'm still new in matlab and I have this code:

clear;clc;
h = 1;
Dr = 0.01; Dt = 0.01;
nmax = 4000; epsilon = 0.0001;
r  = [0:Dr:h];
t = [0:Dt:h];
beta=0.1;
n1 = fix(h/Dr)+1;
m1 = fix(h/Dt)+1;
% Initialize solution matrices
velocity = zeros(n1,m1)
velocityN =velocity
% Iterative process for the solution
for k = 1:nmax
   % boundary conditions:
   velocityN(:,n1)        = zeros(n1,1);
   
   % Calculate psiN in interior points
   for i = 2:n1-1
      for j = 1:m1
         velocityN(i,j+1)= (beta*(1-(1/(2*i)))*velocity(i-1,j)+(1-2*beta)*velocity(i,j)+beta*(1+1/(2*i))*velocity(i+1,j));     
      end;
   end;
   % Check convergence
   nfail = 0;
   for i = 1:n1
      for j = 1:m1
         if abs(velocityN(i,j)-velocity(i,j))>epsilon
            nfail = nfail+1;
         end;
      end;
   end;
   
   if nfail > 0
      velocity = velocityN;
   else
      return;
   end;
end;
 fprintf('No convergence after %i iterations.',k);
[X,Y] = meshgrid(r,t);
contour(X,Y,velocity',10)

however, for r=0 I have dv/dr=0 as the second boundary condition. How do I incorporate this into the program. dV/dr should be:
velocityN(0,j+1)=((1-4*beta)*velocity(0,j)+4*beta(velocity(1,j)))

 

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