Stability analysis of the Rayleigh–Bénard convection for a fluid with temperature and pressure dependent viscosity

Abstract

The classical problem of thermal-convection involving the classical Navier-Stokes fluid with a constant or temperature dependent viscosity, within the context of the OberbeckBoussinesq approximation, is one of the most intensely studied problems in fluid mechanics. In this paper, we study thermal-convection in a fluid with a viscosity that depends on both the temperature and pressure, within the context of a generalization of the Oberbeck-Boussinesq approximation. Assuming that the viscosity is an analytic function of the temperature and pressure we study the linear as well as the non-linear stability of the problem of RayleighBénard convection. We show that the principle of exchange of stability holds and the Rayleigh numbers for the linear and non-linear stability coincide.

Stability analysis of Rayleigh–Bénard convection in a porous medium

“…Lemma 1. Let the disturbances u, P 1 , θ satisfy the initial-boundary value problem (11)- (10). Then, if…”

“…When the material parameters depend only on the temperature, the results established by Rajagopal et al [9] reduces to the classical Oberbeck-Boussinesq approximation. Using this approximation, Rajagopal et al [10] have studied the problem of Rayleigh-Bénard convection and assuming that the viscosity is an analytic function of the temperature and pressure they studied both the linear as well as the non-linear stability corresponding to the Rayleigh-Bénard problem. They showed that the principle of exchange of stabilities holds and that the critical Rayleigh numbers for the linear and non-linear stability coincide.…”

Global Stability for Thermal Convection in a Couple-Stress Fluid With Temperature and Pressure Dependent Viscosity

“…When the material parameters depend only on the temperature, the result established by Rajagopal et al [47] reduces to the classical Oberbeck-Boussinesq approximation. Using this approximation, Rajagopal et al [48] studied the problem of Rayleigh-Bènard convection and assuming that the viscosity is an analytic function of the temperature and pressure they studied both the linear as well as the nonlinear stability corresponding to the RayleighBènard problem. They showed that the principle of exchange of stabilities holds and that the critical Rayleigh numbers for the linear and nonlinear stability coincide.…”

Unconditional nonlinear stability for double‐diffusive convection with temperature‐ and pressure‐dependent viscosity

“…In the limiting case, it was shown that when material parameters are the function of only temperature, the results reduce to the classical Oberbeck–Boussinesq approximation. With this extended approximation, Rajagopal et al 26 analyzed the standard Rayleigh–Bénard problem by including variable viscosity. Using the linear and nonlinear theories, they observed the effect of variation of viscosity by calculating the critical Rayleigh number and showed that results of linear and nonlinear theories coincide.…”