I want to solve 2 coupled PDEs with inner region boundary conditions.
Diffuse equation
There are 2 kinds of particles in my system, $p$ and $a$. Their diffusion equations are as follows: $\hat{D}=\begin{pmatrix} D_T& 0& 0 \\ 0 & D_T &0\\0 &0 &D_r \end{pmatrix}\\ \quad$
$D_T$: transportation coefficient
$D_r$: rotation coefficient
A:
$a = a (x,y,\theta)$
$\frac{\partial a }{\partial t}=\nabla\cdot \vec{j_A}$
$\vec{j_A}=\hat{D}\nabla a -\vec{v} a $
$\vec{v}=(v_0 \cos\theta,v_0 \sin\theta,0) $
P can be seen as A whose $v_0=0$:
$p = p (x,y,\theta)$
$\frac{\partial p }{\partial t}=\nabla\cdot \vec{j_P}$
$\vec{j_P}=\hat{D}\nabla p $
Boundary condition
P will change to A at circle 1, and A will transfer to P at circle 2.
$\hat{n}=\vec{r}/|r|$
On the wall $r=R$
$\hat{n}\cdot \nabla p (x,y,\theta)=0$
On the circle2 ($r=\sqrt{x^2+y^2}=r_2$):
$\hat{n}\cdot (\nabla p (x,y,\theta)-(\vec{j_A}+\vec{j_P}))=0$
$a =0$
On the circle1 ($r=\sqrt{x^2+y^2}=r_1$):
$\hat{n}\cdot (\nabla a (x,y,\theta)-(\vec{j_A}+\vec{j_P}))=0$
$ p =0$
On the center($r=\sqrt{x^2+y^2}=0$):
$\rho(x,y,\theta)=1$
code
I want to get a stable solution.
I know how to solve single PDE using Mathematica, but i don't know how to combine $ a $ and $ p $.
This program assumes that $a(x,y,\theta)==1$ on circle 1, in fact it should be solved by coupling $a(x,y,\theta)$ and $p(x,y,\theta)$.
{R, r1, r2, v0, Difft, Diffr} = {29.5, 15, 25, 0.1, 1/29.5^2, 0.01};
Needs["NDSolveFEM"]
[CapitalOmega] =
RegionDifference[Cylinder[{{0, 0, 0}, {0, 0, 2 [Pi]}}, r2/R],
Cylinder[{{0, 0, 0}, {0, 0, 2 [Pi]}}, r1/R]];
cylinderMesh = ToElementMesh[[CapitalOmega], MaxCellMeasure -> 0.001];
c = {{Difft, 0, 0}, {0, Difft, 0}, {0, 0, Diffr}};
v = {v0/R Cos[[Theta]], v0/R Sin[[Theta]], 0};
With[{a = a[x, y, [Theta]]},
op = 0 ==
Div[c . Grad[a, {x, y, [Theta]}] - v a, {x, y, [Theta]}]];
res = NDSolveValue[{op,
DirichletCondition[a[x, y, [Theta]] == 1, x^2 + y^2 == (r1/R)^2],
DirichletCondition[a[x, y, [Theta]] == 0, x^2 + y^2 == (r2/R)^2]},
a, {x, y, [Theta]} [Element] cylinderMesh]
I guess that I should decompose the disk into 3 regions($r<r_1,r_1<r<r_2,r>r_2$) and respectively solved them.
How shall I solve it? thank you!
