Combined effects of lateral heating and thermo-diffusion on convection bifurcations in a porous enclosure subject to vertical thermal and solutal gradients

doi:10.1016/j.ijmecsci.2020.105597

International Journal of Mechanical Sciences

Volume 178, 15 July 2020, 105597

https://doi.org/10.1016/j.ijmecsci.2020.105597 Get rights and content

Highlights

•
The case of lateral heat flux balanced by induced Soret flux is considered.
•
Linear stability analysis is used to find the thresholds for onset of convections.
•
The flow structure is monocellular or multicellular depending on lateral heating.
•
Universal correlations are derived for flow intensity and heat transfer.
•
Numerical simulations are used to calibrate the correlations.

Abstract

The present study deals with double diffusive convection within a horizontal well packed porous layer of finite aspect ratio where thermo-diffusion has a predominant effect on the convective flow stability. The porous layer is subject to vertical thermal and solutal gradients and a lateral heating flux. The investigation is focused on a special situation where the lateral heat flux is balanced by the horizontally induced Soret mass flux, which allows a possible equilibrium state (motionless state) that becomes unstable under certain conditions. The aim of the present investigation is to study the flow stability and to develop some universal correlations for the flow intensity and heat transfer, which is valid for any aspect ratio independently of the governing parameters. The correlations were constructed from the parallel flow assumption valid for an infinite aspect ratio enclosure but can be used for a finite aspect ratio. To this end, numerical simulations are performed to support the investigation and to calibrate the correlations. By using a linear stability analysis, based on a finite element method, the onset conditions for the stationary convection and over stabilities are investigated with the focus on the effect of the lateral heating magnitude on the stability thresholds. In general, the thermo-diffusion problem balanced by a lateral heating effect is found to exhibit a rich variety of different bifurcation phenomena and complex unsteady flow patterns near criticality.

Graphical abstract

Introduction

Thermo-diffusion effect on double diffusion convection has received a growing attention through the decades due to its presence in many industrial and natural processes. Details concerning the applications of the Soret effect in science and industry were given by Platten [1]. Many theoretical studies were conducted in the past to predict the thermo-diffusion effect on the onset of the convective flows in rectangular porous enclosures. The investigation conducted by Marcoux et al. [2] was devoted to the study of the onset of thermo-gravitational diffusion within a vertical porous cavity subject to horizontal thermal gradients in the case of opposing and equal thermal and solutal buoyancy forces. Their numerical results, based on a linear stability analysis, showed different flow structures and the existence of time-periodic oscillatory solutions. Bahloul et al. [3] investigated analytically and numerically the Soret effect on thermosolutal convection within a shallow horizontal porous layer subject to a vertical uniform heat flux. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities were determined in terms of the governing parameters using linear and nonlinear stability analyses. The existence of subcritical convection was predicted for negative values of the Soret parameter. A similar problem was studied by Bourich et al. [4] by performing a comparative study for the limiting cases of Darcy porous and clear fluid media. Soret effect on thermosolutal convection induced in a horizontal porous layer subject to constant heat and mass fluxes was investigated analytically and numerically by the same authors [5]. This investigation showed the existence of different regions in the (buoyancy ratio N, Lewis number Le) plane that correspond to different parallel flow regimes. The number and the locations of these regions depended on the Soret parameter. Charrier-Mojtabi et al. [6] investigated two-dimensional thermosolutal natural convection with Soret effect under the simultaneous action of vibrational and gravitational accelerations. The authors found that, for both the stationary and the Hopf bifurcation, the vertical vibration had a stabilizing effect while the horizontal vibration had a destabilizing effect on the onset of convection. Natural convection in a porous layer heated and salted from below and subject to a horizontal heat flux was investigated by Mansour et al. [7]. They considered the particular case, for which an equilibrium state existed between the horizontal heat flux and the induced Soret mass flux. It was found that five distinct regions describing different flow behaviors were possible. Charrier-Mojtabi et al. [8] performed an analytical and numerical stability analysis of Soret-driven-convection in a horizontal porous cavity saturated by a binary fluid. They found that the equilibrium solution lost its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio characterizing the Soret effect and the normalized porosity of the medium. The case of a horizontal layer subject to constant vertical fluxes of heat and solute and filled with a porous matrix, whose permeability varies exponentially with the depth, was studied by Alloui et al. [9]. The onset of supercritical, subcritical and Hopf convections was predicted in this study. Er-Raki et al. [10] studied analytically and numerically the Soret effect on double diffusive natural convection in a horizontal Darcy porous layer subject to lateral heat and mass fluxes. The work focused on the particular situation where the solutal to thermal buoyancy forces ratio, N, and the Soret parameter, S_P, were related such that N = 1/(S_P −1). For this particular situation, the rest state was a solution of the problem. Only the subcritical convection was found possible for this case and its threshold was determined analytically. Stability analysis of double-diffusive convection for a visco-elastic fluid in the presence of Soret effect was investigated by Wang and Tan [11] using a modified-Maxwell–Darcy model. They used the linear stability analysis to investigate how the Soret parameter and the relaxation time of the visco-elastic fluid affect the onset of convection. It was found that the Soret effect destabilized the system for oscillatory convection. Gaikwad and Kamble [12] studied the combined effect of rotation and anisotropy, in the presence of Soret effect, on the double diffusive convection in a horizontal sparsely packed porous layer by using linear stability analysis. The same authors [13] investigated Soret effect on double diffusive convection in a horizontal porous layer bounded from above and below by two impermeable and free boundaries. The Rayleigh numbers for both stationary and oscillatory convection were derived using a linear stability analysis. Yacine et al. [14] conducted an analytical and numerical study of Soret-driven convection in a horizontal layer subject to uniform cross heat fluxes. For a cell heated from below without lateral heating, the linear stability analysis showed that the equilibrium solution lost its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. Rebhi et al. [15] studied numerically natural convection in a tall porous enclosure filled with a binary fluid. The Darcy–Dupuit model was adopted to describe the flow in the porous medium. Based on the linear stability theory, the onset of motion from the rest state was predicted for both double diffusive and Soret convection. Al-Mudhaf et al. [16] considered the Soret and Dufour effects on the unsteady double-diffusive natural convection, inside a trapezoidal enclosure filled with a porous medium with exponential variation of boundary conditions. The authors found that the increase of the Soret number reduced the average Sherwood number and increased the average Nusselt number.

Roy and Murthy [17] studied the impact of Soret parameter on the linear stability of a double-diffusive convection due to viscous dissipation in a horizontal porous channel. They showed that the Soret parameter had significant effect on convective instability. The same authors [18] used the linear stability to study double-diffusive convection in a horizontal porous layer with an open upper boundary. A horizontal temperature gradient was applied along this boundary. It was assumed that the viscous dissipation and Soret effect were significant in the medium. They found that the Soret parameter had a significant effect on the stability of the flow when the upper boundary was at constant pressure. Deepika [19] examined the onset of double-diffusive convection in a horizontal fluid-saturated porous layer by considering the Soret effect. In both linear and nonlinear stability theories, the onset criterion for all possible modes was determined analytically. These authors observed that the effect of stabilization or destabilization caused by the Soret parameter was significant for the Soret parameters lower than 2.

The linear stability analysis of vertical throughflow of a power-law fluid induced in a porous channel in the presence of Soret effect was investigated by Kumari and Murthy [20]. The upper and lower boundaries were assumed to be permeable, isothermal and iso-solutal. Their results indicated that the Soret parameter had a significant influence on convective instability of the power-law fluid. Very recently, the same authors [21] studied the onset of double-diffusive convective instability in a horizontal porous layer saturated with a power-law fluid subjected to concentration based internal heat source and Soret effect. They found that the instability of the base flow appeared in the form of oblique rolls, which could be transformed into longitudinal rolls or transverse rolls. Stationary instability was seen for the longitudinal rolls.

The mathematical modeling of non-stationary double diffusion induced in an enclosure with heat-conducting and impermeable walls of finite thickness, in the presence of Soret and Dufour effects and local heat and mass sources, was carried out by Sheremet [22] by taking into account convective-radiative heat exchange with the environment. Results were obtained over wide ranges of the governing parameters. Umavathi et al. [23] studied the onset of three-dimensional double-diffusive convection in a horizontal porous layer saturated with a nanofluid in the presence of Soret and Dufour effects. Both linear and nonlinear stability analyses were used to examine the governing parameters effects on stationary and oscillatory modes. Using the same approach, Umavathi and Sheremet [24] examined the onset of convection of a sparsely packed micropolar fluid in a porous layer saturated by a nanofluid. The Darcy–Brinkman–Forchheimer model was adopted for flow modelling within the porous medium. The critical Rayleigh number, wave number for stationary and oscillatory modes and frequency of oscillations were derived analytically. Grosan et al. [25] studied numerically double diffusive natural convection in a differentially heated wavy cavity under thermophoresis effect. They showed that the effect of thermophoresis could be quite significant in appropriate situations. The number of undulations could essentially modify the heat transfer rate and fluid flow intensity.

The objective of the present work is to examine the combined effect of the lateral heating and thermodiffusion on the natural convection developed within a horizontal porous layer subject to vertical uniform fluxes of heat and mass. Attention is focused on the situation where the lateral imposed heating flux is balanced by the horizontally induced Soret mass flux, which leads to an equilibrium state that becomes unstable under certain conditions. Using a linear stability analysis, the onset thresholds of stationary and Hopf convections are determined. The critical Rayleigh number and the corresponding wavenumber for the exchange of stability and over stability are obtained. Based on the parallel flow approximation, some universal correlations, valid for any aspect ratio independently of the governing parameters, are developed for flow intensity and heat transfer. Numerical simulations are performed to calibrate the correlations.

Section snippets

Mathematical formulation

The flow configuration considered in the present work is a finite aspect ratio porous enclosure saturated with a binary mixture, as shown in Fig. 1. To investigate thoroughly the effect of a lateral heating on the convective flow instabilities, a simple rectangular flow configuration is chosen for the ease of the mathematical and numerical modeling. The horizontal walls of the cavity are subject to uniform fluxes of heat, q’, and mass, j’, while the vertical ones are impermeable to mass but

Numerical solution

For finite amplitude convection, the numerical method used for solving the full governing equations is based on a second order finite-difference scheme. The temperature and concentration equations, Eqs. (2)-(3), are solved iteratively using the alternate direction implicit method. Nodal values of the stream function are obtained, from Eq. (1), via a point successive-over-relaxation method. Details concerning the validation of the present code were reported by Bourich et al. [28] and its

Analytical solution

Towards developing some new interesting semi-empirical formula, that could be used for quick understanding of the convective flow behavior, the limiting case of an infinite layer, which allows to derive an exact steady state analytical solution, was considered. It is well known that for the same flow structure (same number of convective cells), whether in confined or infinite layer, the flow behavior and the heat transfer rate trend remains roughly the same. Therefore, the convective solutions

Linear stability analysis of the rest state solution

For any set of the governing parameters values, the rest state solution given by Ψ = 0 can be realized when the horizontally prescribed heat flux is balanced by the induced Soret mass flux within the porous layer (MN = 1), which nullifies the resulting buoyancy forces. For this situation, the rest diffusive state solution is given by: $T_{c} (x, y) = - ax - y and S_{c} (x, y) = aMx + (M - 1) y$

In practical situations, as it is well known from the classical configuration heated from below, the equilibrium state may be

Onset of oscillatory convection

For a given set of values of the governing parameters, Le, M, a and A_r, the procedure for searching the onset of the oscillatory convection (over stabilities) is performed by varying the Rayleigh number R_T while examining the real part of the maximum eigenvalue. The critical Rayleigh number, $R_{T C}^{o v e r}$ that characterizes the onset of overstability (also known as the threshold for Hopf's bifurcation) is determined when the real part of the eigenvalue p switches from negative to positive value (i.e.

Onset of stationary convection

For stationary convection, p = 0, after introducing the boundary conditions, the equations in (21) are combined and simplified to yield the following eigenvalue problem: $\begin{matrix} \begin{matrix} {[K]}_{ψ} {ψ} = \frac{R_{T}}{M} [B] (M {θ} + {ϕ}) \\ {[B]}_{θ} {ψ} = [K] {θ} \\ Le (1 - M) {[B]}_{ϕ} {ψ} = [K] (M {θ} + {ϕ}) \end{matrix}} \end{matrix}$

The above equations can be combined to yield the following eigenvalue problem: $\begin{matrix} [[E] - λ [I]] {ψ} = 0 \\ [E] = {[K]}_{ψ}^{- 1} [B] {[K]}^{- 1} {[B]}_{ϕ} \\ λ = \frac{M}{R_{T} L e (1 - M)} \end{matrix}}$

The solution of Eq. (27) yields m eigenvalues. The maximum eigenvalue λ_max gives the critical Rayleigh number for the onset of

Results and discussion

For a finite aspect ratio enclosure, the general solution given by Eq. (27) shows that a supercritical bifurcation (onset of convection through zero amplitude convection, i.e. Ψ₀ = 0) occurs at a supercritical Rayleigh number given by: $R_{T C}^{s u p} = \frac{M}{L e (1 - M)} R^{s u p}$ where R^sup = 1/λ_max (with $λ_{\max} = \frac{M}{R_{T} Le (1 - M)}$ ).

As it can be seen from the governing equations, R^sup depends on the aspect ratio of the enclosure, A_r, and the side-heating parameter a_n.

At a first glance, it appears from Eq. (28) that the onset of

Correlations

Numerical tests show that the flow characteristics behavior is not affected by the aspect ratio of the enclosure, regardless of whether it is finite or infinite. In other words, for any finite aspect ratio of the enclosure, the flow intensity and the Nusselt number are described, respectively, by the solution of Eq. (15) and the expression given by Eq. (16). For a given aspect ratio of the enclosure, the expressions of Ψ₀ and Nu remain unchanged provided that the Rayleigh number is normalized

Conclusion

Double diffusive natural convection in a horizontal porous layer subject to vertical fluxes of heat and mass and to a lateral heating was studied analytically (parallel flow approximation) and numerically in the presence of the Soret effect. Attention was focused on the case for which the effect of the lateral heating was balanced by the Soret effect. For this case, the rest state is a solution of the problem. By using a finite element method, a numerical linear stability analysis was performed

Declaration of Competing Interest

The authors declare that there is no conflict of interest.

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View full text

Combined effects of lateral heating and thermo-diffusion on convection bifurcations in a porous enclosure subject to vertical thermal and solutal gradients

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Mathematical formulation

Numerical solution

Analytical solution

Linear stability analysis of the rest state solution

Onset of oscillatory convection

Onset of stationary convection

Results and discussion

Correlations

Conclusion

Declaration of Competing Interest

Int Comm Heat Mass Transf

Int J Heat Mass Transf

Int J Heat Fluid Flow

Int J Therm Sci

Int J Heat Mass Transf

Int J Mech Sci

Int J Heat Mass Transf

Eur J Mech B/Fluids

Energy Convers Manage

The Soret effect: a review of recent experimental results

J Appl Mech

Naissance de la thermogravitation dans un mélange binaire imprégnant un milieu poreux

Entropie

Double-diffusive and Soret-induced convection in a shallow horizontal porous layer

J Fluid Mech

Soret effect inducing subcritical and Hopf bifurcations in a shallow enclosure filled with a clear binary fluid or a saturated porous medium: a comparative study

Phys Fluids

Influence of vibration on Soret-driven convection in porous media

Num Heat Transf Part A