The Bénard Problem for Slightly Compressible Materials: Existence and Linear Instability
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We study a generalization of the Oberbeck–Boussinesq system, which consists in a buoyancy term where the density depends also on the pressure. A new pressure equation is introduced, which is deduced from the divergence-free condition on the velocity; such an equation cannot be decoupled from the system and is studied under Robin’s boundary conditions. Then, the existence of regular periodic solutions is proved for the full system. In Bénard’s problem, the two-dimensional linear instability of the solution depends on a dimensionless parameter that is proportional to the compressibility factor: the related critical Rayleigh number decreases as it increases.
KeywordsIncompressible fluids Oberbeck–Boussinesq approximation Bénard problem instability
Mathematics Subject Classification76R10 76D07 76E06
The first author is a member of “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA) of the “Istituto Nazionale di Alta Matematica” (INdAM) and acknowledges support from this institution.