Natural Convection in Bounded Domains

In a rigid, porous medium, gravitational forces and density differences due to temperature gradients can set a viscous fluid in motion. Attempts to describe this motion have always met serious difficulties. Supposing that the skeleton is fixed, we can accept that the velocity of the fluid is far lower than the acoustic velocity, and thus that the motion has little effect on the pressure; therefore we neglect variations of the thermodynamic quantities resulting from pressure changes. Moreover, we assume that the temperature differences are small enough to use the Boussinesq approximation; that the density of the gravitational force varies affinely with the temperature. We adopt the model proposed in Chapter 1, which was obtained by a homogenization process for which we have proved the convergence; we refer to it sometimes as the Darcy-Boussinesq system.