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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.

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I will try to answer as best as I can your three questions.

1) Your approach is quite classical in that you are considering the particles to be an active scalar. What most people would usually do is to consider that the particle concentration affects the viscosity (via a model such as the Kreiger-Dougherty model for particle suspension) instead of penalizing the velocity via a Darcy term like you are doing. Generally this gives a more appropriate result for the velocity gradient and models the viscous dissipation that occurs at the particle scale. Furthermore, your model does not consider the particle-particle interaction and the diffusive fluxes that occur due to the particle-particle interaction. You might want to look at models like the Phillips model ( see http://aip.scitation.org/doi/abs/10.1063/1.858498) which consider a diffusive flux due to particle-particle and particle-fluid interaction. The authors of that paper also implement a viscosity model instead of a Darcy term.

2) it is admissible to decouple the equations, but you might want to solve them iteratively inside a single time step. For instance, solve for velocity, then solve concentration, recalculate viscosity and solve again for velocity, etc. until convergence. It is best not to put the concentration and the (velocity,pressure) in the same system from my experience. Be careful about the equation for the particle concentration, as you will most likely need to do some form of upwinding.

3) You can model this phenomena using concentration values at the nodes (or cell centers) depending on the numerical model you are using. There is no need for multiple meshes or for a level set approach.

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  • $\begingroup$ I had a question about 2). Can't we just treat the concentration explicitly in the Navier-Stokes equations? We would not have to do that convergence procedure. Also, what do you mean with not putting the concentration in the same system? The equations are coupled, it is a system... $\endgroup$ – balborian May 10 '17 at 16:07
  • $\begingroup$ You can treat the concentration explicitly, but you will be bound by a CFL condition on the particle concentration advection. Since the particle concentration greatly affects the viscosity, this can sometimes lead to strong coupling between the two equation. When I said putting the two equations in the same system I meant linear system of equation, you have a system of equations, but when you write your discretized system (either via FVM or FEM), I would suggest to solve Navier-Stokes and concentration separately and not couple the two equations in the same linear system. $\endgroup$ – BlaB May 10 '17 at 18:26

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