Gianfrani, Jacopo Alfonso (2022) Convection problems in porous media: the effect of vertical throughflow and local thermal nonequilibrium. [Tesi di dottorato]

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Abstract

In the present thesis, we address qualitative studies of some fluid dynamics problems. Specifically, we investigate the type of instability occurring in a porous medium subject to a vertical downward throughflow and the onset of thermal convection in fluid-saturated porous media uniformly heated from below in the regime of local thermal nonequilibrium (LTNE). Throughout the whole thesis, the study undertaken involves linear and nonlinear stability analyses of the steady conduction solution of the different models considered. The former provides a sufficient condition for the onset of instability and gives information about the fate of small-amplitude perturbations. The latter, instead, provides a sufficient condition for the persistence of the conduction solution. Moreover, the two analyses determine threshold values for the characteristic dimensionless parameter (Rayleigh number). The best result that one can get is the coincidence between thresholds from the two analyses. In that case, necessary and sufficient condition for the stability of the conduction solution is obtained. Otherwise, if they do not coincide, a so-called subcritical instability region arises, where we do not know what the fate of a small perturbation is. Conditions for the existence of this region may be determined, as done in the second chapter, where a weakly nonlinear analysis of the effect of vertical throughflow on Darcy-Bénard convection is performed. When modelling fluid dynamics problems in a porous medium involving high-velocity fluids or a solid matrix with thermal conductivity much different from the fluid one, two different temperatures need to be defined (one for the solid matrix, one for the fluid phase). Hence, according to the LTNE assumption, two energy balance equations are involved in the model to allow for heat exchanges between the phases and get more realistic results. Qualitative comprehension of the behaviour of a fluid in presence of a porous medium heated from below in LTNE is very important in several industrial and engineering processes involving quick heat transfer, metal foams, heat exchangers and tube refrigerators. For each mathematical model introduced, the qualitative study is followed by numerical simulations. Moreover, some differential eigenvalue problems are solved via numerical schemes such as the Chebyshev-tau method and the shooting method, which have been carefully implemented for the specific problems.

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