Integration of exiting flow using PDEPE

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Dom 2018-10-3

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https://ww2.mathworks.cn/matlabcentral/answers/422062-integration-of-exiting-flow-using-pdepe

评论： Dom 2018-10-9

Hi, i am currently working on possible solutions for a system of two components A and B, with A slowly diffusing out of the inner circle and B diffusing into it, resulting in an equally mixed solution inside and outside. So far, this has been my approach:

 function [pl,ql,pr,qr] = pdexbc(xl,ul,xr,ur,t)
 global ur_development
 global t_development
 global n_bulk_development
 if numel(t_development)>1
 while numel(t_development)>=1 && t<=t_development(end) 
    t_development = t_development(:,1:end-1); 
    n_bulk_development = n_bulk_development(:,1:end-1);
 end
 end
 setup = initialize_setup();
 t_development = [t_development,t];
 n_bulk = setup.n_bulk;
 if length(t_development(1,:))>=2
    n_bulk=n_bulk_development(:,end);
    n_bulk_propagation=n_bulk+(t_development(:,end)-t_development(:,end- 1)).*((-n_bulk./setup.V_bulk+ur).*setup.D_i.*(xr^2*pi*4/3));
    n_bulk=n_bulk_propagation;
 end
 n_bulk_development =[n_bulk_development,n_bulk];
 pr = -n_bulk./setup.V_bulk+ur; 
 qr = zeros(setup.index_molecules(end),1);
 end

So far, i tried to implement the changning amount of substance in the outer area(n_bulk) by using a discrete approach: n_bulk(t)=n_bulk(t-1)+delta_t*(ur-c_bulk)*D*A in order to change the concentration at the outer boundary, which has not worked, the way i hoped. So my question would be, if you have an alternative idea, for example by using qr for the diffusion, while integrating over the outgoing flux.

Any help would be greatly appreciated.

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Torsten 2018-10-4

Please supply the model equations, initial and boundary conditions in a mathematical notation.

Dom 2018-10-9

在 MATLAB Online 中打开

Hi Torsten, thank your for your reply,

 c = [1;1];
 f = D*DuDx; % D being a constant diffusion coefficient
 s = 0;

for the ic:

 u1=1;
 u2=0;

i hope this is sufficient for my purpose. My big issues have been the outer bcs pr and qr. I try to implement two equations.

1: ur=c_bluk

with c_bulk being the concentration of an adjacent area, which has the following ic:

 ic_bulk: u1_bulk=0;
          u2_bulk=1;

My second condition is a constant flux going from the inner area to the outer area and vice versa.

f=-D*DuDx % Du=c_bulk-ur

Unfortunatly my mathematical understanding is not sufficient to put this into a single bc and make it look like i want it to look like.

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Integration of exiting flow using PDEPE

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