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

[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?

 

[HallsofIvy]

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
$$\nabla^2 f= \frac{\partial^2 f}{\partial r^2}+ \frac{1}{r}\frac{\partial f}{\partial r}+ \frac{\partial^2 f}{\partial z^2}$$

So your equation says
$$\frac{\partial C}{\partial t}= D\left(\frac{\partial^2 C}{\partial r^2}+ \frac{1}{r}\frac{\partial C}{\partial r}+ \frac{\partial^2 C}{\partial z^2}\right)$$

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.

 

[nazmulislam]

Thanks Hallsoflvy for your nice explanation.