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International Heat Transfer Conference 12

ISSN: 2377-424X (online)
ISSN: 2377-4371 (flashdrive)

Onset of Soret-driven convection in porous medium under vertical vibration

Olivier Sovran
IMFT, UMR CNRS/INP/UPS N°5502, UFR MIG, Université Paul Sabatier, 118r oute de Narbonne, 31062, Toulouse Cedex, France

Marie-Catherine Charrier-Mojtabi
Laboratoire PHASE E.A. 3028, Université Paul Sabatier, France

Mejdi Azaiez
IMFT, UMR CNRS/INP/UPS N°5502, UFR MIG, Université Paul Sabatier, 118r oute de Narbonne, 31062, Toulouse Cedex, France

Abdelkader Mojtabi
Institut de Mecanique des Fluides, UMR CNRS-INP-UPS №5502 Universite Paul Sabatier, 118 route de Narbonne, F 31062 Toulouse, Cedex France.

DOI: 10.1615/IHTC12.4750
6 pages

Abstract

In the present work, we study the influence of vertical high frequency vibration on the onset of Soret-driven convection, in an infinite horizontal porous layer saturated by a binary fluid, and heated from below or from above. We consider the case of high frequency, small amplitude vibrations, so that we may adopt a formulation using time averaged equations.
In the first part, we develop a linear stability analysis of the mechanical equilibrium solution, using the Galerkin method. The problem depends on five non-dimensional parameters: the thermal Rayleigh number, Ra, the modified vibrational Rayleigh number, Rv , the Lewis number, Le, the normalized porosity, ε*, and the separation ratio, Ψ. For a given set of parameters ( Rv, Le and ε*), we determine the bifurcation diagrams, Rac = f(Ψ) and kc = f(Ψ), where Rac and kc are respectively the critical thermal Rayleigh number and the critical wave number in the infinite horizontal direction. For a layer heated from below, Rac increases with Rv , whereas kc decreases with Rv, either for stationary or Hopf bifurcations (the latter occurring only for Ψ < 0). For stationary bifurcations, when Rv increases, the value Ψ1 of the separation ratio beyond which the critical wave number vanishes (i.e. kcs = 0), decreases. Using a regular perturbation method, in the case of long wave disturbances (i.e. k = 0) , we show that, for Ψ > Ψ1, Racs = 12/(Le Ψ), ∀ Rv . For a layer heated from above, and Ψ < 0 , the first primary bifurcation is a stationary one and the set of critical parameters is: Racs = 12/(Le Ψ), kcs = 0, ∀ Rv; for Ψ > 0, the equilibrium solution is infinitely linearly stable.
The second part concerns 2D direct numerical simulations of the governing equations using a spectral method. The simulations allow us to corroborate the results obtained with the linear stability analysis both for stationary and Hopf bifurcations.

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