- #1

anvoice

- 16

- 3

I have a certain diffusion problem I am trying to solve. Admittedly, I'm further behind on my math than I'd like, and have trouble setting it up properly, and no luck finding an exact analogue in the literature.

I would like to solve for the time-dependent concentration profile given a point sink in an infinite plane. The diffusion problem has the general form:

dC(x,y,z,t)/dt = ∇(D*∇C(x,y,z,t))

where C is the concentration, t is the time, D is the diffusion coefficient, and ∇ is the partial differential operator. Assuming constant D, and that the problem is two-dimensional as well as radially symmetric, we can simplify the equation to:

dC(r,t)/dt = D*∇

^{2}C(r,t)

where r is the radius. At this point I'm stuck. I have seen several solutions in the literature to similar problems involving the error function, and that seems to be the right path to take. However those were typically given for diffusion through an infinite rod, etc., so it seems like the solution here should not be the same. At least it seems logical that the concentration profile will look like the right half of a sigmoidal at any given point in time. I would appreciate any help or helpful resources on this.