# Simulate Fick's 1st and 2nd laws of mass diffusion.

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JAEYEON il 19 Mag 2024
Risposto: Torsten il 19 Mag 2024
Dear a person who is looking this.
I am trying to simulate the mass diffusion based on Fick's laws on 2D plane.
The initial condition is that the concentration on the left wall is 1.
But the result seems to be odd as below.
I guess the problem is from the code for the 2nd law. So I have tried to modify that like
C(i,j) = C(i,j) ...
- dt * ((Jx(i, j) - Jx(i-1,j))/(dx) ...
+ (Jy(i, j) - Jy(i,j-1))/(dy));
But the result was the same.
How can I solve this problem. I really want some help..
======Here is my code (you can operate by just copy and paste)=======
% Parameters
Nx = 20; % Number of grid points in x-direction
Ny = 20; % Number of grid points in y-direction
D = 0.1; % Diffusion coefficient
dx = 1.0; % Spatial step in x-direction
dy = 1.0; % Spatial step in y-direction
dt = 0.1; % Time step
T = 200.0; % Total simulation time
C = zeros(Nx, Ny); % Initial concentration
Jx = zeros(Nx, Ny);
Jy = zeros(Nx, Ny);
C(:, 1) = 1.0; % Initial condition: concentrated mass at the center
% Time loop
for t = 0:dt:T
% Create a copy of the concentration array to hold the new values
% C_new = C;
% Iterate over each interior point
% flux
% Fick's 1st law: J = -D * del c
for i = 2:Nx-1
for j = 2:Ny-1
Jx(i,j) = - D * (C(i,j) - C(i-1,j))/dx - D * (C(i+1,j) - C(i,j))/dx ;
Jy(i,j) = - D * (C(i,j) - C(i,j-1))/dy - D * (C(i,j+1) - C(i,j))/dy ;
end
end
% concentration
% Fick's 2nd law: partial dc/dt = - div J
for i = 2:Nx-1
for j = 2:Ny-1
C(i,j) = C(i,j) ...
- dt * ((Jx(i+1, j) - Jx(i-1,j))/(2*dx) ...
+ (Jy(i, j+1) - Jy(i,j-1))/(2*dy));
end
end
% Optionally plot the concentration at each time step
slice_view = squeeze(C(:,:)); % View a slice in the middle of the z-plane
imagesc(slice_view);
colorbar;
colormap('jet');
title(sprintf('Time = %.2f', t));
drawnow;
end
% Final plot
slice_view = squeeze(C(:,:)); % View a slice in the middle of the z-plane
imagesc(slice_view);
colorbar;
title('Final Concentration');
xlabel('X');
ylabel('Y');
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### Risposta accettata

Torsten il 19 Mag 2024
The correct update for inner grid points is
C_new(i,j) = C_old(i,j) + dt * (D_x*(C_old(i-1,j)-2*C_old(i,j)+C_old(i+1,j))/dx^2 + D_y*(C_old(i,j-1)-2*C_old(i,j)+C_old(i,j+1))/dy^2 )
And since you don't change the values for C for i=1,i=Nx,j=1,j=Ny, they always have a value of 0. Thus you boundary condition is C = 0 at all the four edges.
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