Solution of diffusion equation with three independent variable (r,z,t)

1. Sep 8, 2014

nazmulislam

Hi,

I want to solve the following diffusion equation:

(d/dt) C(r,z,t)=D*∇^2 C(r,z,t)

where C is the concentartion and D is the coefficient of diffusivity (constant)

with initial condition C(r,z,0)=C0 (constant)
and boundary condition dc/dr=0 at r=0; (dc/dz) at z=-L equal to (dc/dz) at z=L

where I have considered axisymmetric tube of length L.

Can anybody help to solve the above mentioned problem for the concentration C?

2. Sep 8, 2014

HallsofIvy

Staff Emeritus
Since you are using "symmetric tube" you will probably want to use "cylindrical coordinates", suppressing the $\theta$ dependence. In cylindrical coordinates the Laplacian is
$$\nabla^2 f= \frac{\partial^2 f}{\partial r^2}+ \frac{1}{r}\frac{\partial f}{\partial r}+ \frac{\partial^2 f}{\partial \theta^2}+ \frac{\partial^2 f}{\partial z^2}$$

Assuming "axially symmetric" so f does not depend on $\theta$, that is
$$\nabla^2 f= \frac{\partial^2 f}{\partial r^2}+ \frac{1}{r}\frac{\partial f}{\partial r}+ \frac{\partial^2 f}{\partial z^2}$$

$$\frac{\partial C}{\partial t}= D\left(\frac{\partial^2 C}{\partial r^2}+ \frac{1}{r}\frac{\partial C}{\partial r}+ \frac{\partial^2 C}{\partial z^2}\right)$$
A standard method of solving that is to look for "separable solutions". That is, look for solutions of the from $C(r,z,t)= R(r)Z(z)T(t)$. That will separate the equation into ordinary differential equations for R, Z, and T separately. Depending on the boundary and initial conditions, the solution can be written as a sum of such "separated" solutions.