# Flux boundary condition in solute transport

I have a pretty naive question, though important to me. Usually when solving the following PDE in solute transport:

$$\frac{{\partial C}}{\partial t } = \nabla. (D\nabla C -vC )=0,$$

one can be asked to use a third-type boundary condition (called flux in software like Comsol or Phreeqc):

$$(D\nabla C -vC )=vCo,$$

where the left-hand side is assumed to be evaluated at the center of the cell, and the right-hand side represents the inflow or outflow flux (Am I right about this last sentence?)

In some software (Comsol) you do not have to specify Co, therefore I assume that in such cases the boundary condition is reduced to:

$$D\nabla C = 0.$$

Is that true? Can such boundary condition be considered constant flux boundary conditon?

Thanks.

## 1 Answer

The correct boundary condition is in fact $$(D\nabla C - vC) \cdot n = g.$$ That is, it is not the vector-valued quantity in parentheses that you can describe, but only the normal component, where $n$ is the normal vector to the boundary.

The term in the parentheses is called the flux. It describes the amount of material (solute) that moves around, and the entire left side of the equation above is the amount of material that flows across the boundary. Specifically, if you at an inflow boundary condition, and the concentration of your solute on the outside of the domain is $C_0$, then $g=(vC_0)\cdot n$ is the correct description.

If, in Comsol, you can't specify a right hand side $g=vC_0$, then I my interpretation would be that that implies that $g=0$, which would mean $$(D\nabla C - vC) \cdot n = 0.$$ In other words, no material flows across the boundary.