Nonlinear Stability for Thermal Convection in a Brinkman Porous Material with Viscous Dissipation

Abstract

We investigate nonlinear stability in a model for thermal convection in a saturated porous material using Brinkman theory, taking into account viscous dissipation effects. There are (at least) two models for viscous dissipation available, and we include a derivation of one of these by assuming that the flow in the porous medium may be described by a theory for a mixture of an elastic solid and a linearly viscous fluid. A fully nonlinear stability result is provided when either of the viscous dissipation functions is taken into account, and it is shown that from the nonlinear energy stability viewpoint both models are, in a sense, equivalent.

Onset of Thermal Instabilities in the Plane Poiseuille Flow of Weakly Elastic Fluids: Viscous Dissipation Effects

“…The effects of viscous dissipation were ignored in all the aforementioned investigations. Recent studies concerning fluid and porous media [16][17][18][19][20][21][22][23][24][25][26][27] have shown that viscous dissipation, often considered a negligible effect, can induce instabilities even without external forcing. In other words, differently from the classical Rayleigh-Bénard (RB) system where the vertical temperature gradient is a consequence of the prescribed boundary conditions, viscous dissipation is an internal heat generation mechanism that can also lead to unstable temperature gradients.…”