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Issue Date: 4-May-2011
Authors: Falsaperla, Paolo
Title: Stability in Convection Problems for Fluids and Flows in Porous Media with General Boundary Conditions
Abstract: We investigate the onset of thermal convection in fluid layers and layers of fluids saturating a porous medium, when the temperature field is subject to the so called "natural" or Newton-Robin boundary conditions. Special attention is devoted to the limit case of fixed heat flux boundary conditions, corresponding to Neumann conditions on the temperature. Several interesting results, both from a physical and a mathematical point of view, appear when such conditions, coupled with one or more stabilizing fields (rotation, solute and magnetic field), are considered. The transition from fixed temperatures to fixed heat fluxes is shown to be destabilizing in all cases. In Part I, we study fluids modeled by the Oberbeck-Boussinesq equations. When further fields are considered, additional equations and terms are included. In Part II, flows in porous media are described by the Darcy law, with and without the inclusion of an inertial term. Even for this system some stabilizing effects are considered. In Chapter 3 we study the rotating Benard problem. The wave number of the critical perturbations is shown to be zero, but only up to a *threshold of rotation speed* (depending on the kinetic boundary conditions). In Chapter 4 the Benard problem for a binary fluid is investigated. In this case, the stabilizing effect of a gradient of solute is *totally lost* for fixed heat fluxes. In Chapter 5 it is shown that, when the same system is rotating, the solute field is again stabilizing, and the critical wave number is positive in some regions of values of rotation and concentration gradient. A more complex interaction, between rotation and magnetic field, is briefly discussed in Chapter 6. Again, zero and non-zero wave numbers appear for different values of the parameters. The solute Benard problem is also investigated in Chapter 7, along the lines of the classical book of Chandrasekhar, in a fully algebraic way. Here, finite slip boundary conditions, and Robin conditions on the solute field are also taken into account. Flow in a rotating porous layer, and flow of a binary mixture in a porous layer, are studied in Chapters 8 and 9, with results qualitatively similar to those obtained for the Benard system. The effects of inertia and rotation are investigated in Chapter 10, with a detailed analysis of the region in parameter space corresponding to the onset of stationary convection or overstability. For most of the above systems, an asymptotic analysis for vanishing wave numbers was also performed, providing support to numerical calculations and some explicit analytic results.
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