Boundary condition problem for diffusion equation

[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we cannot assume that the chemical is not dispersed at the end of the channel- so the boundary condition c(a,t)=0 is no longer valid?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?

 

[mfb]

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

 

[Chestermiller]

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.

 

[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 $\partial c/\partial x = 0$

 

[PJC01]

The boundary condition problem for the diffusion equation is an important consideration in solving any diffusion problem, as it determines the behavior of the solution at the boundaries of the domain. In this case, the boundary condition c(a,t)=0 assumes that the chemical is not dispersed at the end of the channel. However, in a real-world scenario, this assumption may not hold true, as the chemical may continue to disperse even at the outlet of the channel.

To address this issue, we can use a different boundary condition that takes into account the dispersion of the chemical at the outlet. One possible option is to use a Neumann boundary condition, where the flux of the chemical at the boundary is specified instead of the concentration. This boundary condition can be derived from the physical understanding that the flux of the chemical should be zero at the outlet, as it has fully dispersed into the surrounding environment.

Another option is to use an outflow boundary condition, where the concentration at the outlet is allowed to vary based on the concentration in the neighboring cells. This approach is commonly used in computational fluid dynamics simulations, where the flow and concentration fields are solved simultaneously.

In conclusion, the choice of boundary condition for the diffusion equation depends on the specific problem and the assumptions made. It is important to carefully consider the physical behavior of the system and select an appropriate boundary condition to accurately capture the diffusion process.