The Bénard problem for quasi-thermal-incompressible materials: A linear analysis

In this paper we apply the ideas introduced with the so-called extended-quasi-thermal-incompressible (EQTI) model, recently proposed by Gouin and Ruggeri (Int. J. Non-Linear Mech. 47 (2012) 688-693) [12]. In particular, in the Oberbeck-Boussinesq approximation we consider the more realistic constitutive equation compatible with the thermodynamical stability by putting in the buoyancy term a density which depends not only by the temperature but also on the pressure. The equation for the pressure is then modified by an extra dimensionless parameter β^ which is proportional to the positive compressibility factor β. The 2-D linear instability of the thermal conduction solution in horizontal layers heated from below (Bénard problem) is investigated. It is shown that for any β^: (i) the rest state pressure profile is different from the parabolic one; (ii) if convection arises, then it first arises via a stationary state and the strong principle of exchange of stability holds; for small β^: (iii) convection certainly arises provided Ra is sufficiently large; (iv) the related critical Rayleigh number coincides -in the limit of vanishing β^ - with the classical one, and decreases as β^ increases.