# Boundary condition problem for diffusion equation

1. Oct 24, 2012

### brambram

BOUNDARY CONDITION PROBLEM
I have came up with matrix for numerical solution for a problem where chemical is introduced to channel domain, concentration equation:

δc/δt=D*((δ^2c)/(δx^2))-kc

assuming boundary conditions for c(x,t) as : c(0,t)=1, c(a,t)=0. Where a is channel's length.

What if domain is not-infinite (as it is above) and we cannot assume that the chemical is not dispersed at the end of the channel- so the boundary condition c(a,t)=0 is no longer valid?

What boundary condition can we use when we cannot assume that the chemical is not dispersed at the outlet of the channel? Can you state it or not? How about computational fluid dynamics boundary conditions options?

2. Oct 25, 2012

### Staff: Mentor

$\lim_{a\to\infty} c(a,t)=0$?

3. Oct 25, 2012

### Staff: Mentor

At the two ends of the sample, you can have a wide variety of different combinations of boundary conditions:

Specified concentration c = C(t)
Zero flux: δc/δx = 0
Specified flux: -D ∂c/∂x = $\phi$(t)
Convective mass transfer: -D ∂c/∂x = k (c - c)

There are an infinite number of others also, but these are the ones you run into most often in practice.

4. Oct 25, 2012

### AlephZero

If there is no flux crossing a boundary, then the gradient of the flux normal to the boundary is 0.

For a 1-D problem like yours, that means $\partial c/\partial x = 0$