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 compres- sibility 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.

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

RUGGERI, TOMMASO ANTONIO
2014

Abstract

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 compres- sibility 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11585/519941
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