## Preprint 2014-25

## Raimund Bürger, Sarvesh Kumar, Ricardo Ruiz-Baier:

### Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation

### Abstract:

The sedimentation-consolidation and flow processes of a mixture of small particles dispersed in a viscous fluid at low Reynolds numbers can be described by a nonlinear transport equation for the solids concentration coupled with the Stokes problem written in terms of the mixture flow velocity and the pressure field. Here both the viscosity and the forcing term depend on the local solids concentration. A continuous in time discontinuous finite volume element (DFVE) discretization for this model is proposed. The numerical method is constructed on a baseline finite element family of linear discontinuous elements for the approximation of velocity components and concentration field, whereas the pressure is approximated by piecewise constant elements. The unique solvability of both the nonlinear continuous problem and the semidiscrete DFVE scheme is discussed, and optimal convergence estimates in several spatial norms are derived. Properties of the model and the predicted space accuracy of the proposed formulation are illustrated by detailed numerical examples, including flows under gravity with changing direction, a secondary settling tank in an axisymmetric setting, and batch sedimentation in a tilted cylindrical vessel.

This preprint gave rise to the following definitive publication(s):

**Raimund BüRGER, Sarvesh KUMAR, Ricardo RUIZ-BAIER: ***Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation*. Journal of Computational Physics, vol. 299, pp. 446-471, (2015).