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A mixture model for a dilute dispersion of gas in a liquid flow: Mathematical analysis and application to aluminium electrolysis

Marco Picasso, Jacques Rappaz, Emile Tryphon Pierre Soutter

A mixture model to take into account the flow of small carbon dioxide bubbles dissolved in a liquid is presented. The model describes the evolution of the velocity fields (mixture and gas), the pressure and the volume fraction of gas. The system of equations is derived from mass and momentum conservation of the mixture and gas. Well-posedness is proved for a simplified problem when the volume fraction of gas is known and small. A priori error estimates are proved for a stabilized finite element approximation. An industrial application pertaining to aluminium electrolysis is presented. Numerical results indicated that the effect of gas bubbles on the flow cannot be neglected.

2021

On Von Karman Modeling For Turbulent Flow Near A Wall

Jacques Rappaz, Jonathan Rochat

Mixing-length models are often used by engineers in order to take into account turbulence phenomena in a flow. This kind of model is obtained by adding a turbulent viscosity to the laminar one in Navier-Stokes equations. When the flow is confined between two close walls, von Karman's model consists of adding a viscosity which depends on the rate of strain multiplied by the square of distance to the wall. In this short paper, we present a mathematical analysis of such modeling. In particular, we explain why von Karman's model is numerically ill-conditioned when using a finite element method with a small laminar viscosity. Details of analysis can be found in [1], [2].

2019

On Some Weighted Stokes Problems. Application on Smagorinsky Models

Jacques Rappaz, Jonathan Rochat

In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characteri- zation of these considered equations is that the viscosity depends on the strain rate of the velocity field with a weight being a positive power of the distance to the boundary of the domain. These non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman’s on which the viscosity depends on the square of the distance to the boundary of the domain. Numerical experiments conclude the work and show prop- erties from the theory.

Springer2016