Mario Álvarez, Bryan Gomez-Vargas, Ricardo Ruiz-Baier, James Woodfield:
Stability and finite element approximation of phase change models for natural convection in porous media
In this paper we study a phase change problem for non-isothermal incompressible viscous flows. The underlying continuum is modelled as either as a viscous Newtonian fluid where the change of phase is encoded in the viscosity itself, or using a Brinkman-Boussinesq approximation where the solidication process influences the drag directly. We address these and other modelling assumptions and their consequences in the simulation of differentially heated cavity flows of diverse type. A second order finite element method for the primal formulation of the problem in terms of velocity, temperature, and pressure is constructed, and we provide conditions for its stability. We finally present several numerical tests corroborating the accuracy of the numerical scheme as well as illustrating key properties of the model.