Mathematical Problems in Engineering

Volume 2018, Article ID 7946078, 12 pages

https://doi.org/10.1155/2018/7946078

## Analytical and Numerical Study of Soret and Dufour Effects on Double Diffusive Convection in a Shallow Horizontal Binary Fluid Layer Submitted to Uniform Fluxes of Heat and Mass

^{1}Faculty of Sciences Semlalia, Department of Physics, LMFE, BP 2390, Marrakesh, Morocco^{2}National School of Applied Sciences, Physics Department, LMFE, BP 575, Marrakesh, Morocco^{3}High School of Technology, Cadi Ayyad University, LMFE, Essaouira, Morocco

Correspondence should be addressed to M. Hasnaoui; am.ca.acu@iuoansah

Received 30 August 2017; Accepted 17 December 2017; Published 12 February 2018

Academic Editor: Filippo de Monte

Copyright © 2018 A. Lagra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Combined Soret and Dufour effects on thermosolutal convection induced in a horizontal layer filled with a binary fluid and subject to constant heat and mass fluxes are investigated analytically and numerically. The thresholds marking the onset of supercritical and subcritical convection are predicted analytically and explicitly versus the governing parameters. The present investigation shows that different regions exist in the -Du plane corresponding to different parallel flow regimes. The number, the extent, and the locations of these regions depend on whether or . Conjugate effects of cross-phenomena on thresholds of fluid flow and heat and mass transfer characteristics are illustrated and discussed.

#### 1. Introduction

Great interest in the study of thermosolutal convection in fluid and porous media has been motivated by its presence in many engineering applications, such as in hydrology, petrology, geophysics, and material processing technology where melting and solidification of binary alloys are involved [1]. More specifically, such flows are encountered in nature (lakes, solar ponds, and atmosphere). In the industrial field, examples include food processing, chemical processes, crystal growth, energy storage, material processing, and many other examples. Important experimental, analytical, and numerical results on convective heat and mass transfer are documented in earlier books of Nield and Bejan [2] and De Groot and Mazur [3]. Most studies on this topic are concerned with double diffusive convection in vertical/horizontal cavities for which the flows induced by the buoyancy forces result from the imposition of both thermal and solutal boundary conditions on the vertical/horizontal walls [4–6].

The diffusion of mass due to temperature gradient is called Soret or thermodiffusion effect that often counts among the main drivers of various convective phenomena occurring within thermal stratified media. Recently, Rahman and Saghir [7] proposed a detailed historical review of works, focusing on different aspects of Soret effect. Examples of interesting phenomena resulting from the coupling between double diffusive convection and Soret effect are available in the paper by Bourich et al. [8] who investigated analytically and numerically Soret-driven thermosolutal convection within a shallow porous or fluid layer subject to a vertical gradient of temperature, using a Brinkman-Hazen-Darcy model in its transient form. The critical Rayleigh numbers for the onset of subcritical, oscillatory, and stationary convection were determined explicitly as functions of the governing parameters for infinite and finite layers.

Generally, Soret and Dufour effects are assumed negligible in problems related to double diffusive convection. However, such effects could be of significant effect when density differences exist in the flow regime. In fact, energy flux can be generated by composition of gradients (Dufour or diffusion-thermoeffect). Similarly, mass fluxes can be created by temperature gradients (Soret or thermodiffusion effect). In an earlier study, Malashetty [9] investigated the effect of anisotropic thermoconvective currents, in the presence of Soret and Dufour effects, on the critical Rayleigh number for both marginal and overstable motions. Mortimer and Eyring [10] used the elementary transition state approach to obtain a simple model theory for the Soret and Dufour effects. They found that the results of the theory conform to the Osanger reciprocal relationship. Gaikwad et al. [11] studied the onset of double diffusive convection in a two-component couple stress fluid layer with Soret and Dufour effects using both linear and nonlinear stability analysis. The effects of Soret and Dufour parameters together with the couple stress parameter on the stationary and oscillatory convection are graphically illustrated and discussed. In the presence of Soret and Dufour effects, Nithyadevi and Yang [12] presented numerical results on natural convection in a square enclosure filled with water, partially heated from one vertical wall, and totally cooled from the opposite vertical wall. The study was conducted around the maximum density for three different combinations of the heating element location. Makinde et al. [13] described a theoretical study used to analyze the hydromagnetic flow and mass diffusion of chemical reactive species with first- and higher-order reactions of an electrically conducting fluid over a moving vertical plate. The study was conducted in the presence of Soret and Dufour effects with convective heat exchange at the plate surface. Pal and Mondal [14] considered the problem of steady laminar, hydromagnetic two-dimensional mixed convection flow due to stretching sheet in the presence of Soret and Dufour effects. Cheng [15] examined the Dufour and Soret effects on the steady boundary layer flow due to natural convection heat and mass transfer over a vertical cone embedded in a porous medium with constant wall temperature and concentration. The results presented show that the effects of Dufour and Soret parameters on the local surface temperature are increased by increasing the Lewis number. Tsai and Huang [16] investigated heat and mass transfer from natural convection flow along a vertical surface with variable heat fluxes embedded in a porous medium due to Soret and Dufour effects. They concluded that Soret and Dufour effects could play a significant role. Soret and Dufour effects have been also considered by Hayat et al. [17] who studied mixed convection boundary layer flow about a linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid and, more recently, by Wang et al. [18] who studied the onset of double diffusive convection in a horizontal cavity.

The main purpose of the present investigation is to study analytically and numerically the combined effects of Soret and Dufour parameters on double diffusive convection developed in a horizontal layer filled with a binary fluid. This paper is an extended version of preliminary results presented in a conference [19]. Analytical predictions are developed and validated numerically for shallow enclosures. The Dufour parameter effects on thresholds of stationary convection, subcritical convection, flow structure, and heat and mass transfer are also discussed.

#### 2. Mathematical Formulation

The system under study is a two-dimensional shallow cavity of length and height , filled with a binary fluid. The vertical end-walls of the layer are adiabatic and impermeable to mass transfer while its horizontal walls are subject to uniform fluxes of heat, , and mass, The flow is assumed to obey the Boussinesq approximation. Using the vorticity and the stream function formulation and taking into account the cross-phenomena (Soret and Dufour effects), the dimensionless governing equations are obtained as follows:

The associated hydrodynamic, thermal, and solutal boundary conditions are

The parameters governing the problem are the thermal Darcy-Rayleigh number, , the Prandlt number, , the Lewis number, , the buoyancy ratio, , and the aspect ratio of the cavity, *.* The parameters and are, respectively, the Soret and Dufour parameters expressed as and , with being the thermal diffusivity, being cross-diffusion due to solute concentration component, being cross-diffusion due to temperature component, and being solute diffusivity.

In the presence of the cross-diffusion phenomena, the Nusselt and Sherwood numbers are defined as follows:where and are the temperature and concentration differences, evaluated at with the origin of the coordinate system being taken at the center of the cavity.

#### 3. Methods

##### 3.1. Numerical Solution

The numerical solution of (1) to (3) was obtained using a finite-difference method, described in detail by Bourich et al. [8]. A second-order scheme was used for the discretization of the spatial derivatives. Equations (2)-(3) were marched in time using the Alternating Direction Implicit (ADI) method. The stream function equation (4) was solved at each time step with the Point Successive Overrelaxation (PSOR) method with an optimum overrelaxation coefficient calculated for the used grid. In addition, a convergence criterion was adopted for the stream function to satisfy a variation by less than 10^{−5} for each time step. A second criterion was used to check the convergence of the numerical code. Here, stands for any of the variables , , or . The superscripts and () indicate the iterations numbers, and the subscripts and indicate locations in the grid system. For large aspect ratio enclosures, nonuniform grid was used in the -direction near the short walls to capture the flow details near the enclosures end-walls and also in the -direction to obtain a finer grid in the close vicinity of the horizontal walls.

##### 3.2. Analytical Solution

For a shallow enclosure with constant heat and mass flux boundary conditions, an approximate analytical solution based on the parallel flow concept is possible, which renders the problem amenable to a parametric study while retaining the essential physics of the problem. The analytical solution is developed for steady-state flows using the parallel flow approximation (see, e.g., Bourich et al. [8]), which leads to the following simplifications (justified by examining the streamlines, isotherms, and iso-concentration lines obtained numerically). These simplifications lead to , , and , where and are, respectively, unknown constant temperature and concentration gradients in the -direction (the direction of the long sides of the cavity). Using these approximations together with the boundary conditions (5a) and (5b), (1)–(3) reduce to a set of ordinary differential equations for which the solution is obtained as follows:where is the stream function value at the midheight of the layer, given byThe analytical expressions of and were determined by using thermal and solutal balances in the layer, which leads to the following expressions:where , , and

Introducing the expressions of and into (8) yields a fourth-order polynomial in terms of for which the following solutions are obtained:withwhere and .

From a mathematical point of view, (10) may exhibit, in addition to the rest state solution, two types of bifurcations depending on the sign within the square root. Although it should be mentioned that several numerical tests were performed, only the solutions corresponding to positive sign were obtained numerically. Hence, the convective solution with the negative sign within the square root is termed “unstable solution” and the convective solution corresponding to positive sign within the square root is similarly qualified “stable solution.”

The new expressions of Nusselt and Sherwood numbers are obtained as follows:

Foremost, it may be remarked from (10) that the parallel flow solutions exist only when the following two conditions are satisfied:

The resolution of the inequalities of (13) is performed in the -Du plane with Sr, Le, and as parameters. Depending of the sign of (parameter in the expression of given before), two main cases are possible.

###### 3.2.1. The Parameter

This condition is satisfied if . Depending on the signs of and , four cases are to be distinguished.

*Case 1 ( and ). *For this case, and , which means that the parallel flow solution is not existing regardless the values of Sr, Le and . In Figure 1, the domain in the -Du plane where the convective parallel flow is not possible is denoted as region 1. This region is defined by and .