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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.

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  • $\begingroup$ Look at this. $\endgroup$ – Deep Oct 24 '18 at 6:17
  • $\begingroup$ In eq (1), the right-hand side would be $D \nabla^2 \rho + k \nabla \cdot \rho$ at $x = 0$. Naïvely, this looks like you're adding a scalar to a vector. Can you clarify what you mean by editing the question? $\endgroup$ – Michael Seifert Oct 24 '18 at 12:53
  • $\begingroup$ Also, see this old answer of mine for a cautionary tale on setting a "boundary at $r = 0$" in cylindrical coordinates. The same argument would apply to a spherical coordinate system. $\endgroup$ – Michael Seifert Oct 24 '18 at 12:57
  • $\begingroup$ @MichaelSeifert You are right, I should have added the depletion term to $\partial{\rho}/\partial{t}$, i.e. $\partial{\rho}/\partial{t} = D\nabla^2 \rho - k \rho$ at $x=0$ $\endgroup$ – Dimitri Oct 24 '18 at 13:21
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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 $$\frac{(c_{\infty}-c)}{(c_{\infty}-c_0)}=\frac{r_0}{r}$$

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  • $\begingroup$ Thank you. I have a follow-up question: What would the solution look like if the sphere was moving at a constant velocity? $\endgroup$ – Dimitri Oct 24 '18 at 13:49
  • $\begingroup$ This is a much more complicated problem involving both fluid mechanics and convective/diffusive mass transfer. Are you sure you want to get into this? I'm sure that this problem has been solved in the literature. Do you feel that you want to solve this on your own? $\endgroup$ – Chet Miller Oct 24 '18 at 14:38
  • $\begingroup$ No, actually I would rather prefer a reference to the literature instead of trying to work this out on my own. You would not happen to know some nice references? $\endgroup$ – Dimitri Oct 24 '18 at 15:40
  • $\begingroup$ OK. Well, first of all, you need to change the frame of reference so that the sphere is stationary and the fluid is moving past at constant velocity (in the far field). I'm sure you can find an analysis of this problem in Transport Phenomena by Bird, Stewart, and Lightfoot (either of the analog heat transfer problem or of the actual mass transfer problem). At the very least, they will give the mass transfer coefficient between the sphere and the fluid. $\endgroup$ – Chet Miller Oct 24 '18 at 17:25

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