# Non-Homogeneous Robin Boundary conditions and Interpretations of Signs

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• ConicalDrupe

#### ConicalDrupe

TL;DR Summary
When dealing with the advection-diffusion equation with robin conditions. How may we control the direction and magnitude of flux in a variety of situations?
I have been solving the constant coefficient 1D advection-diffusion equation ##\frac{\partial c}{\partial t} + v\frac{\partial c}{\partial x} = D\frac{\partial^2 c}{\partial x^2}## on ##0<x<L,t>0## with a variety of robin BC's.
Namely $$vc + D\frac{\partial c}{\partial x} = J^f ~~at~~ x=L$$ $$vc - D\frac{\partial c}{\partial x} = J^b ~~at~~ x=0$$
When ##J^{f/b}=0##, we have a perfect insulating boundary. Wikipedia states the reasoning behind the signs of the diffusive flux term in the BC's. Positive at x=L because the normal points in the positive direction, and negative at x=0 because the normal points in the negative direction.

I have two main questions coming from this scenario.
(1) What is the physical interpretation of changing the sign of the diffusive flux term in the BC? For ex. ##vc - D\frac{\partial c}{\partial x} = J^f ~~at~~ x=L ## and ##J^f## is positive, does this mean we have an influx or efflux at x=L? Is there a way of understanding these cases in terms of direction using normal vectors?
(2) How can we control the direction of the flux if ##J^{f/b}## is a parameter of our choosing? Obviously choosing ##J^{f/b} \neq 0## gives us control on flux passing at the boundary, but does ##J^{f/b}>0## or ##J^{f/b}<0## have a predicted effect on the direction flux travels?

I have run a few numerical experiments in MATLAB using the Crank-Nicolson method with the two types of robin bc's above. One BC set with positive diffusive flux at x=L and negative at x=0. The other has negative diffusive flux at both x=0 and x=L. The results are confusing to me, and could be due to numerical issues. My goal is to have ##J^{f/b}>0## and have influx at x=L and efflux at x=0. None of my experiments so far have shown this result.

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