Marie-Catherine Charrier-Mojtabi

Analytical and numerical stability analysis of Soret-driven convection in a horizontal porous layer

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the monocellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, characterizing the Soret effect and the normalized porosity, are investigated theoretically and numerically. From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio >0(Le,), while for 0, while a stationary or an oscillatory bifurcation occurs if mono the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and <0, the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters.

Double-diffusive convection in an annular verticalporous layer

Abstract This paper reports an analytical and numerical study of double-diffusive naturalconvection through a fluid-saturated, vertical and homogeneous porous annulus subjected touniform fluxes of heat and mass from the side. The influence of each leading parameter andespecially curvature, a major parameter of this geometry, has been numerically investigated.Solutions are presented for 1 ⩽ A ⩽ 10, 1 ⩽ RaT ⩽ 200, 1 ⩽ Le ⩽ 20, 0.1 ⩽ N ⩽ 10 and 0 ⩽ γ ⩽ 10 where A, RaT, Le, N and γ denote the aspect ratio, thermal Rayleigh number, Lewis number, buoyancy ratio andcurvature parameter, respectively. For the steady state, an analytical solution valid for stratifiedflow in slender enclosures is presented. A good agreement is observed between the analyticalpredictions and the numerical simulation for sufficiently high aspect ratios.

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

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 ...

Thermal Vibrational Convection in a Porous Medium Saturated by a Pure or Binary Fluid

What is Thermal Vibration? The importance of mechanical vibration, as a source of pattern generating mech-anism on the surface of a container filled with liquid, was recognized as early as the beginning of the 19th century by Faraday (1831). However, its importance as a mechanism controlling the convective motion has only been recognized during the 20th century. Originally, mechanical vibration was used in mathematical modeling aimed at increasing the stability threshold of thermo-fluid system (Gershuni et al. 1970, Gresho and Sani 1970). The space exploration and especially the benefits expected from material production in space stations accelerated its development (Alexander 1994). Formally, the thermo-vibrational convection studies concern the form of a mean flowin a confined cavity filled with a fluid presenting temperature non-homogeneities. Compared to the gravity-induced convection, this type of convection presents the advantage that it may exist under weightlessness condition. Under micro-gravity conditions, the gravitational force is reduced drastically. However this situation may cause other forces, which under earth conditions are of minor importance, to be more significant.

Naissance de la convection thermosolutale dans une cellule rectangulaire poreuse soumise à des flux de chaleur et de masse

Abstract We studied the onset of double-diffusive convection in a rectangular cell filled with a porous medium saturated with a binary fluid. Uniform heat and mass fluxes are applied to the vertical walls while the horizontal walls are impermeable and adiabatic. When the ratio of resulting solutal to thermal buoyancy forces is equal to −1, we obtain a purely diffusive regime (motionless). We demonstrate that this regime is linearly stable while the thermal Rayleigh number is lower than a critical value, Ra c , depending of the aspect ratio of the cell, A , and the Lewis number, Le . For a square cavity, we obtained Ra c | Le −1| = 209.84 and for an infinite layer, the critical parameters are found to satisfy Ra c | Le −1| = 105.33 with a wavenumber k c = 2.51. These analytical results are in good agreement with direct numerical simulations. These numerical simulations show that subcritical solutions disappear for a thermal Rayleigh number R 0 Ra c , which depends on the aspect ratio of the cell and the Lewis number. For thermal Rayleigh numbers lower than R 0 , only the purely diffusive solution is stable.

Naissance de regimes de convection thermosolutale dans une cellule rectangulaire poreuse

The onset of thermosolutal convection in a rectangular cell filled with a porous medium saturated by a binary fluid is studied. The vertical walls are maintained at different, but uniform, temperatures and concentrations. The horizontal walls are impermeable and adiabatic. When the ratio of resulting solutal and thermal buoyancy forces is equal to −1, we obtain an equilibrium solution corresponding to a purely diffusive regime. We demonstrate that this regime is linearly stable up to a critical thermal Rayleigh number, Rac1, depending on the aspect ratio of the cell and the Lewis number. These analytical results are in good agreement with numerical direct simulations which allow us to describe the double-diffusive convective regimes taking place when the equilibrium regime loses its stability.

Onset of Free Convection in Solutions with Variable Soret Coefficients

Abstract It is not unusual for a Soret coefficient to change sign with temperature. We develop the theory for the onset of convection in such systems, heated from below or from above, provided that the mean temperature is precisely that at which the change of sign occurs. We also consider the realistic case of rigid, conducting, impervious boundaries for later comparison with laboratory experiments.

A NEW PROCESS FOR SPECIES SEPARATION IN A BINARY MIXTURE USING MIXED CONVECTION

In this paper, a numerical and analytical analysis is performed in order to improve the species separation process in a binary fluid mixture by decoupling the thermal gradient from the convective velocity. The configuration considered is a horizontal rectangular cavity, of large aspect ratio, filled with a binary fluid. A constant tangential velocity is applied to the upper horizontal wall. The two horizontal impermeable walls are maintained at different and uniform temperatures T1 and T2 with ΔT = T1 − T2 Species separation is governed by two control parameters, the temperature difference ΔT and the velocity of the upper plate Uex. The intensity of the thermodiffusion is controlled by the temperature, while the velocity Uex controls the convective flow. This problem depends on six dimensionless parameters, namely, the separation ratio, ψ, the Lewis number, Le, the Prandtl number Pr, the aspect ratio of the cell, A and two control parameters: the thermal Rayleigh number, Ra and the Peclet number Pe. In this study, the formulation of the separation (mass fraction difference between the two ends of the cell) as a function of the Peclet number and the Rayleigh number is obtained analytically. For a cell heated from below, the optimal separation m = √42/15 is obtained for Pe = √42/Le and Ra = 540/(Leψ). 2D numerical results, obtained by solving the full governing equations, are in good agreement with the analytical results based on a parallel flow approach.

A Dynamical Model for Studying the Stability of Thermo-Solutal Convection Under the Effect of Vertical Vibration

This paper concerns the thermal stability analysis of porous layer saturated by a binary fluid under the influence of mechanical vibration. The linear stability analysis of this thermal system leads us to study the following damped coupled Mathieu equations: B H + B(π2 + k2) + 1 H +(π2 + k2) − k2k2 + π2 RaT(1 + R sin ω* t*) H =k2k2 + π2 (NRaT)(1 + R sin ω* t*) Fe* B F + B π2 + k2Le + e* F +π2 + k2Le − k2k2 + π2 NRaT(1 + R sin ω* t*) F =k2k2 + π2 RaT(1 + R sin ω* t*) H where RaT is thermal Rayleigh number, R is acceleration ratio (bω2 /g), Le is the Lewis number, k is the dimensionless wave-number, e* is normalized porosity and N is the buoyancy ratio (H and F are perturbations of temperature and concentration fields). In the follow up, the non-linear behavior of the problem is studied via a generalization of the Lorenz model (five coupled non-linear differential equations with periodic coefficients). In the presence or absence of gravity, the stability limit for the onset of stationary as well as Hopf bifurcations is determined.Copyright

Active Control of the Onset of Convection in Porous Medium by Mechanical Vibration

Theoretical studies of Rayleigh-Benard convection subjected to sinusoidal acceleration modulations have been conducted by several workers. Linear and weakly nonlinear stability analyses have been developed by Gresho and Sani [4], Clever et al. [2] and by the Russian workers Gershuni and Lyubimov [3]. As shown by these studies, the Rayleigh number for the onset of synchronous convection increases with the frequency of vibration for a layer oscillating with constant vertical amplitude until a certain frequency of modulation is reached and at which the onset is in the form of subharmonic motions.