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Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: Linear stability analysis

Abstract : In the present work, we study the onset of double-diffusive convective regimes in a tilted rectangular cavity, filled with a porous medium, saturated by a binary fluid. Two opposite walls are maintained at different but uniform temperatures and concentrations while the two other walls are impermeable and adiabatic. When the thermal and solutal buoyancy forces are comparable in intensity but have opposite signs, the motionless double-diffusive regime with linear temperature and concentration profiles is a solution of the problem. The first part of the study consists of a linear stability analysis of the motionless regime. We determine the critical thermal Rayleigh number for the onset of stationary and oscillatory convection. Indeed, we point out that there exist primary Hopf bifurcations for the studied problem in porous medium, while in the same configuration with a fluid medium only primary stationary bifurcations exist. When the first primary bifurcation creates a steady state branch of solutions, the bifurcation is either transcritical or pitchfork depending on the aspect ratio, A and the tilt, φ of the cavity. The onset of oscillatory convection (Hopf bifurcation) depends not only on A and φ but also on the Lewis number, Le and the normalized porosity, ε. Then, we determine the parts of the (Le, ε) parameter space for which the first primary bifurcation is stationary or oscillatory. In particular, it is found that in the case Le⩾1 and for ε Le2<1 the first primary bifurcation is always a Hopf bifurcation for any A and φ except for φ=+90°. For ε Le2>1 only stationary primary bifurcations exist. In the case Le<1, zones where stationary and oscillatory primary bifurcations exist are separated by a curve depending on A and φ. The last part of this work consists of a series of numerical simulations. The onset of stationary and oscillatory convection is obtained numerically at the critical Rayleigh number predicted by linear analysis. We also verified the frequency of oscillations for several sets of dimensionless parameters. The numerical simulations show multiple subcritical solutions.
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Mohammad Karimi-Fard, Marie-Catherine Charrier-Mojtabi, Abdelkader Mojtabi. Onset of stationary and oscillatory convection in a tilted porous cavity saturated with a binary fluid: Linear stability analysis. Physics of Fluids, American Institute of Physics, 1999, 11 (6), pp.1346-1358. ⟨10.1063/1.870000⟩. ⟨hal-01946085⟩

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