Non-Homogeneous Robin Boundary conditions and Interpretations of Signs

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• ConicalDrupe
In summary, the BC's with positive diffusive flux at x=L and negative diffusive flux at x=0 produce an influx at x=L and efflux at x=0, but it is unclear whether this is due to numerical issues or a desired effect of the BC's.
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.

Last edited:
You really need to use LateX for these equations, for me personally. I think others will be similarly disposed. FYI
It will only hurt for a while, but it is a pain..

hutchphd said:
You really need to use LateX for these equations, for me personally. I think others will be similarly disposed. FYI
It will only hurt for a while, but it is a pain..
Thanks for the response, I completely agree! I had great difficulty with mathJax and my browser, it is slightly less painful the second and third time

What are Non-Homogeneous Robin Boundary Conditions?

Non-homogeneous Robin boundary conditions are a type of boundary condition in mathematical models that describe the behavior of a system at the boundaries. They involve specifying both the value of the function and its derivative at the boundary, rather than just one or the other.

How are Non-Homogeneous Robin Boundary Conditions used in scientific research?

Non-homogeneous Robin boundary conditions are commonly used in mathematical models of physical systems, such as heat transfer, fluid dynamics, and quantum mechanics. They allow for more accurate and realistic descriptions of the behavior of a system at its boundaries, which is crucial for understanding and predicting the behavior of complex systems.

What is the interpretation of the signs in Non-Homogeneous Robin Boundary Conditions?

The signs in Non-homogeneous Robin boundary conditions have different interpretations depending on the specific problem being modeled. In general, the sign of the constant in front of the derivative term determines whether the boundary condition is considered "inflow" or "outflow." The sign of the constant in front of the function term can represent either a source or a sink of the quantity being modeled.

Can Non-Homogeneous Robin Boundary Conditions be simplified to Homogeneous Boundary Conditions?

In some cases, Non-homogeneous Robin boundary conditions can be simplified to Homogeneous boundary conditions. This is typically done by transforming the problem into a different coordinate system or by using a change of variables. However, this is not always possible and depends on the specific problem being modeled.

What are some examples of systems that use Non-Homogeneous Robin Boundary Conditions?

Non-homogeneous Robin boundary conditions are commonly used in a variety of physical systems, including heat transfer in buildings, fluid flow in pipes, and quantum mechanical systems. They are also used in computer simulations and modeling to study the behavior of complex systems in fields such as engineering, physics, and biology.

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