Boundary conditions for the heat equation when solving a mass density gradient

Question

I'm working with a mass density gradient with length $L$ and I'm trying to solve the heat equation in 1-D (mass diffusion equation, $\partial_t\rho(t,x)=D\Delta\rho(t,x)$), but I'm not sure which boundary conditions should I use and what would they physically mean.

The starting mass density profile ($\rho(0,x)=f(x)$) is a step-function, where the lower half has density $\rho_1$ and the upper half has $\rho_2$.

$f(x)=\begin{array}{ll} \rho_1 & 0<x<L/2 \\ \rho_2 & L/2<x<L \\ \end{array}$

Considering the boundary conditions, I find more difficult to interpret them. The mass inside the sample's volume is kept constant during the experiment as the sample is isolated from surroundings.

For this purpose, would Neumann Boundary Conditions be appropriate?

$$\partial_x \rho(t,0)=0=\partial_x \rho(t,L) $$

I'm unsure because all the examples I found do not treat the heat equation/diffusion equation as function of mass density ($\rho$). Would Neumann BC suggest No Mass Transfer between the sample's volume and the surroundings?

Answer 1

From a physics point of view, you have some kind of mass-diffusion problem. It is true that a mathematician will name it "heat equation" because the classical problem with the operator $\partial_t - \Delta$ is the heat equation.

Homogeneous Neumann BC is indeed appropriate to model a "no flux" condition at the boundaries, be it a flux of heat or of mass. A Dirichlet BC here would mean that your domain is in contact with a large (infinite) reservoir of fixed density there.