Massa, Giuliana (2022) Recent Hydrodynamic Stability Results for Single and Double Porosity Materials. [Tesi di dottorato]

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Abstract

In this doctoral thesis, the onset of buoyancy-driven thermal convection in horizontal layers of single and double porosity media uniformly heated from below is investigated. Bi-disperse porous media are dual porosity materials characterized by two types of pores - macropores and micropores - and are composed of clusters of large particles which are themselves agglomerations of smaller particles. Recently, the attention of many researchers is heading toward the onset of convection in dual porosity materials due to the growing need for engineered materials for industrial applications. In particular, it was demonstrated that in dual porosity materials convective heat transfer is delayed and so dual porosity materials are better suited for insulation problems and thermal management problems. The mathematical model describing convective fluid motions is a dynamical system of partial differential equations. With the aim of improving the mathematical model describing bi-disperse convection and obtaining even more realistic mathematical models useful for designing man-made materials and engineered systems, the first part of the thesis is devoted to anisotropic bi-disperse porous materials - for which macro-permeability and micro-permeability are symmetric second-order tensors - since anisotropy is a powerful tool to optimise heat transfer. In particular, the interactions between anisotropy, inertial effects, high porosities and Coriolis effects are analysed, in order to determine how these physical aspects affect the onset of convection and the type of arising convective cells, in the cases of single and double fluid mixtures. The instability analysis of the stationary state for each mathematical model has been performed by solving differential eigenvalue problems via numerical schemes, such as the shooting method, which have been carefully implemented for the specific problems. Moreover, the stability analysis of the stationary state for each mathematical model has been performed via the energy method, introducing peculiar functionals for the problems at stake, to obtain (i) a stability threshold as close as possible to the instability threshold and (ii) global stability (i.e. without additional restrictions on the initial data). The final Chapters of this dissertation are devoted to the description of new stability results related to porous convection problems. In particular, the thermodynamic consistency of the Oberbeck-Boussinesq approximation is deeply discussed and, to the best of our knowledge, the Darcy-Bénard problem for an extended-quasi-thermal-incompressible fluid is studied for the first time. A more realistic constitutive equation for the fluid density is employed - in body force term due to gravity - in order to obtain more thermodynamic consistent instability results. Finally, we perform linear instability and weakly nonlinear stability analyses of the throughflow solution for a horizontal layer of fluid-saturated porous medium heated from below and subject to a downward vertical net mass flow: this physical set-up is known as the Sutton Problem. The goal is to determine for which values of the velocity of the throughflow the transition from supercritical to subcritical instability happens.

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