Solution of diffusion equation with spherical sink

Question

I hope this question is not too basic, but I have no experience with partial differential equations and would like to ask for some hints on how to solve the following problems:

The visual idea is to describe the diffusion of some dilute chemical around a spherical sink or a sink at some point.

It would be nice to obtain a time evolution when starting with a uniform density (this is only possible in problems 1) and 5)), but I would already be satisfied with a "nice" steady-state solution.

1) The simplest case would be a diffusion equation in one dimension with a sink at $x=0$, i.e. $$\frac{\partial{\rho}(x)} {\partial{t}} = - \nabla \cdot J(x), \qquad x \in \mathbb{R}$$ with $J(x) = -D \nabla \rho(x)$ for $x \neq 0$ and $J(x) = -D \nabla \rho(x) -k\rho(x)$ for $x = 0$ with $k$ some large depletion constant.

Moreover, I would like to require that $\rho(x) \rightarrow c$ for $|x| \rightarrow \infty$ for some constant $c >0$.

The initial conditions are not so important, let's say $\rho(x;t=0) = c_0$ for all $x \in \mathbb{R}$. Actually, I am not as much interested in the time evolution as in finding a nice steady-state solution, which does not diverge for $x \rightarrow \infty$. Thus, some more appropriate initial conditions could be chosen.

2) Alternatively, One could just require the boundary condition $\rho(x=0;t) = 0$ for all $t$ in the above situation. Then the initial condition needs to be adjusted accordingly.

3) The problem in 3D is the same: to solve the diffusion equation with $x \in \mathbb{R}^3$ and boundary conditions $\rho(x) = 0$ for $x = 0$ and $\rho(x) \rightarrow c$ for $|x| \rightarrow \infty$ with some $c >0$. I guess with spherically symmetric initial conditions, this case is completely equivalent to 2).

4) Now I would like to have a real sphere in $\mathbb{R}^3$ of radius $r >0$, i.e. the boundary conditions $\rho(x) = 0$ for $|x| < r$ and $\rho(x) \rightarrow c$ for $|x| \rightarrow \infty$ with some $c >0$.

5) Maybe 4) is simpler when formulated with a finite depletion rate $k$ as in 1) instead of the boundary condition around $0$. Physically, this is very similar anyway.

Any literature suggestions are also highly appreciated.

Answer 1

For steady state diffusion in an unbounded region toward a spherical sink of radius $r_0$, we have $$4\pi r^2 D\frac{dc}{dr}=Q$$where Q is the diffusion rate of the species toward the sink (a constant). The solution to this equation is $$c=c_{\infty}-\frac{Q}{4\pi rD}$$where $c_{\infty}$ is the concentration far from the sphere. Applying the boundary condition $c=c_0$ at $r=r_0$ for the surface of the sphere, we have: $$Q=4\pi r_0D(c_{\infty}-c_0)$$Therefore, the concentration profile is $$\frac{(c_{\infty}-c)}{(c_{\infty}-c_0)}=\frac{r_0}{r}$$