# Deposition model in laminar flow

I have a chamber full with a fluid flowing horizontally in laminar regime from one side to the other. It carries a suspension with concentration $c$. This suspension also falls to the bottom of the chamber with settling velocity $\mathbf{v_s}$.

When it reaches a critical concentration $c_{max}$ at the bottom, it starts building up as a deposition. This blocks the influx of particles, hence the function $\phi(c)$, this nonlinear term is similar to the one in the Burgers equation for the traffic problem. Now the region with concentration $c_{max}$ can grow upwards.

This deposition is big enough that it can disturb the flow. To account for this, I added the term $\psi(c)\mathbf{u}$ in the Navier-Stokes equations below. This term stops the flow in regions where the concentration is $c_{max}$. The equations are:

$\frac{\partial c}{\partial t} = \nabla (\phi(c,c_{max})(\mathbf{v_s} + \mathbf{v}))$

$\frac {\partial \mathbf {u} }{\partial t}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \nabla ^{2}\mathbf{u} + \nabla p + \psi(c,c_{max})\mathbf{u}= \mathbf{f}$

$\nabla \cdot \mathbf{u} = 0$

\psi(c, c_{max}) = \left\{ \begin{align} 0 && c < c_{init} \\ 10000 (c - c_{init})^3 && else \end{align} \right.

$c_{init}$ is the initial concentration in the entire chamber, it only increases at the bottom, when it starts building up. Therefore $\psi>0$ only at the bottom.

I am not 100% confident with this model, and hence, my question. I believe that the penalization term $\psi$ can restrict my step size.

• Is there a better approach to model a flow with a growing deposition that can disturb it?
• Is it admissible to decouple the equations by treating $\mathbf{u}$ in the first equation and $c$ in the NS equations explicitly?
• I am solving both equations on the same mesh. So far I haven't done a proper error estimation, but I believe that I would need a high refinement near the deposition layer. Can I model this phenomena with level set methods and save costs in refinement?

I have not been able to find literature on similar physics, I would appreciate any reference.