Non-Darcian-Bènard Double Diffusive Magneto-Marangoni Convection in a Two Layer System with Constant Heat Source/Sink

The problem of non-Darcian-Bènard double diffusive magneto-Marangoni convection is considered in a horizontal infinite two layer system. The system consists of a two-component fluid layer placed above a porous layer, saturated with the same fluid with a constant heat sources/sink in both the layers, in the presence of a vertical magnetic field. The lower porous layer is bounded by rigid boundary, while the upper boundary of the fluid region is free with the presence of Marangoni effects. The system of ordinary differential equations obtained after normal mode analysis is solved in a closed form for the eigenvalue and the Thermal Marangoni Number (TMN) for two cases of Thermal Boundary Combinations (TBC); these are type (i) Adiabatic-Adiabatic and type (ii) Adiabatic-Isothermal. The corresponding two TMNs are obtained and the impacts of the porous parameter, solute Marangoni number, modified internal Rayleigh numbers, viscosity ratio, and the diffusivity ratios on the non-Darcian-Bènard double diffusive magneto Marangoni convection are studied in detail.


Introduction
Double diffusive convection (DDC) is a type of convection, which consists of double density gradients diffusing at varied rates. DDC commonly occurs in natural processes, like those taking place in sea water and the mantle flow in the Earth's crust, and has lots of engineering geothermal applications; for example, contaminant transport in saturated soils, food processing, and spread of toxins. It also appears in solar ponds and crystal growth industries. In crystal growth industries, double diffusive Marangoni convection (DDMC) plays an important role in the production of pure crystals. Applying magnetic field on DDMC can exert outstanding outcomes in crystal growth industries. Steady conjugate double-diffusive natural convective heat and mass transfer in a twodimensional variable porosity layer sandwiched between two walls was studied numerically by Al-Farhany and Turan [1]. The linear and nonlinear stability of double diffusive convection in a layer of couple stress fluid-saturated porous medium was theoretically investigated by Shivakumara et al. [2]. A numerical study of double-diffusive natural convective heat and mass transfer in an inclined rectangular cavity filled with a porous medium was conducted by Al-Farhany and Turan [3]. Mehmood et al. [4] explored the unsteady flow of viscous nanofluid driven by an inclined stretching sheet for the effect of a ISSN: 0067-2904 Manjunatha and Sumithra Iraqi Journal of Science, 2021, Vol. 62, No. 11, pp: 4039-4055 4040 non-uniform heat source/sink in a thermally and solutally stratified magnetonanofluid, using the RK4 method with the shooting technique. Al-Farhany and Turan [5] experimentally investigated the mixed convection in a square enclosure partitioned in two layers. The results showed that the effect of cylinder rotation was exerted around the cylinder only. Mixed convection heat transfer of nanofluid in a lid-driven porous medium square enclosure with several pairs of heat source-sinks was numerically simulated by Munshi et al. [6] using the finite element method. Ahmadpour et al. [7] studied the natural convection of a non-Newtonian ferrofluid in a porous elliptical enclosure in the presence of a non-uniform magnetic field. The results showed that, by applying the magnetic field via a wire, the overall heat transfer rate increased significantly. The onset of double diffusive magneto Marangoni convection in a two-layer system, comprising incompressible two components, was studied by Komala and Sumithra [8] using the regular perturbation technique for uniform and nonuniform salinity gradients. Recently, the effect of heat source/sink on MHD free convection flow in a channel filled with nanofluid in the existence of induced magnetic field was studied by Jha and Samaila [9]. Kannan and Pullepu [10] studied the effects for chemical reaction, the heat source/sink, Schmidt number, and Prandtl number on double diffusion natural convective flow along a vertically inclined infinite plate. The governing non-dimensional equations were solved using iterative tri-diagonal implicit finite-difference scheme. Anurag et al. [11] studied the influence of Newtonian heating/cooling in the presence of heat source/sink on laminar free convective flow in a vertical annular permeable region. The closed-form analytical solutions of the governing equations were obtained for two different cases of internal heat generation/absorption. Sumithra et al. [12,13] and Manjunatha and Sumithra [14] studied the effects of constant heat source / sink and temperature gradients on composite layer with and without magnetic field. They obtained the closed form of solution to thermal Marangoni number for three different temperature gradients. Sumithra and Venkatraman [15] studied the problem of Bènard Marangoni convection in a composite layer, comprised of an incompressible couple stress fluid, for adiabatic and isothermal boundaries using an exact technique. In this paper, the problem of non-Darcy-Bènard double diffusive magneto Marangoni convection is investigated in a horizontally infinite two layer system. The system consists of a two-component fluid layer above a porous layer, saturated with the same fluid, for Darcy-Brinkmann model with constant heat sources in both the layers under microgravity condition. The lower boundary of the porous region is rigid, while the upper boundary of the fluid region is free with the presence of Marangoni effects. The system of ordinary differential equations obtained after normal mode analysis is solved in a closed form for the eigenvalue and TMN for two types of TBC; type (i) Adiabatic-Adiabatic and type (ii) Adiabatic-Isothermal. The corresponding two thermal Marangoni numbers heat sources m  and  , respectively. The lower surface of the porous layer is rigid and the upper surface of the fluid layer is free with the presence of surface tension effects, depending on temperature and concentration. A Cartesian coordinate system is chosen that originates at the interface between the porous and fluid layers and the z-axis, in the vertically upward direction. The basic equations for the fluid and porous layers, respectively, governing such a system are: Manjunatha and Sumithra Iraqi Journal of Science, 2021, Vol. 62, No. 11, pp: 4039-4055 . .
where for the fluid layer, V is the velocity vector, 0  is the fluid density, t is time,  is the fluid viscosity, is the total pressure, H is the magnetic field, T is the temperature,  is the thermal diffusivity of the fluid, is the magnetic viscosity, and p  is the magnetic permeability. For the porous layer,  is the porosity, m  is the effective viscosity of the fluid in the porous layer, K is the permeability of the porous medium, A is the ratio of heat capacities, m  is the thermal diffusivity, em     is the effective magnetic viscosity, and the subscript 'm' denotes the quantities in the porous layer. The aim of this paper is to investigate the stability of a quiescent state to infinitesimal perturbations superposed on the basic state. The basic state of the liquid being quiescent is described by Fluid layer: (20) Following the standard linear stability analysis procedure and assuming that the principle of exchange of stability holds (Manjunatha and Sumithra [14]), we arrive at the following stability equations: in 01 z  In the above equations, Q , * is the depth ratio.

Boundary Conditions
The suitable velocity boundary conditions are nondimensionalized and then subjected to normal mode analysis; they are is the solute Marangoni number, and t  is the surface tension.

Method of Solution
The solutions of () Wz and () mm Wz are obtained by solving (21) and (24) using the velocity boundary condition (28), as follows , is the solutal ratio. The concentration distributions () Sz and () mm Sz are obtained using the concentration boundary conditions (31), as follows

Thermal Marangoni number Type (i): Adiabatic-Adiabatic (A-A) Boundaries
We solve equations (22) and (25) for the temperature distributions () z  and () mm z  using the following temperature boundary conditions, where both the boundaries are adiabatic and the heat and heat flux are continuous at the interface: ( The temperature distributions () z  and () mm z  are obtained by using the temperature boundary condition (34), as follows 1

Type (ii): Adiabatic-Isothermal (A-I) Boundaries
We solve equations (22) and (25) for the temperature distributions () z  and () mm z  using the following temperature boundary conditions, where the upper boundary of the fluid layer is adiabatic and the lower boundary of the porous layer is isothermal, and at the interface, heat and heat flux are continuous: (  A comparison of the TMNs for types (i) and (ii) TBC is explained by Figure 1, for where TMN, t M is the dependent variable and the thermal ratio T is the independent variable. From Figure 1, it is evident that for smaller values of thermal ratio, TMN for Adiabatic-Adiabatic is higher than that for Adiabatic-Isothermal, indicating that, for ˆ2 T  , the TBC type (i) Adiabatic-Adiabatic is suitable for situations where the convection has to be controlled. Whereas for ˆ2 T  , the type (ii) Adiabatic-Isothermal is suitable for the same situations. The other important observation is that in the type (i) Adiabatic-Adiabatic, the TMN increases rapidly as the value of thermal ratio T increases, whereas in type (ii) Adiabatic-Isothermal, the TMN slowly increases, reaches a peak value, then decreases with the thermal ratio. The TBCs play an important role in convection, hence choosing containers with appropriate TBC is very important during these processes.
The variation of porous parameter  on TMN is shown in Figures 2a and 2b for two types respectively, for 0.1,1.0,10.0,50.0,100.0   . From Figure 2a, it is observed that the TMN for type (i) Adiabatic-Adiabatic simply increases as the value of thermal ratio is increased. Also, for a fixed value ofT , increasing the value of the porous parameter beta, increases the TMN. Thus, the rise in the porous parameter  is favouring the stability of the system. Hence non-Darcian-Bènard double diffusive magneto-Marangoni convection can be postponed by increasing the porous parameter that is by choosing the size of the pores in the porous layer of the two layer system.   Figure 2b depicts the TMN results for type (ii) Adiabatic-Isothermal TBC. The TMN value increases slowly, reaches its peak, and falls again as the thermal ratio value increases. Also, for a fixed value of thermal ratioT , as the  value increases from 0.1 to 100, the TMN decreases. The T value at which the peak of TMN also shifts towards smallerT . It is very important to note that increasing permeability means more space for the fluid to move in the porous layer, which is stabilizing for type (i) but destabilizing for the type (ii) TBC. Thus, by merely changing the thermal boundary condition at the upper boundary, one can reverse the effects on the non-Darcian-Bènard double diffusive magneto-Marangoni convection.

Manjunatha and Sumithra
Iraqi Journal of Science, 2021, Vol. 62, No. 11, pp: 4039-4055 4052 The effects of viscosity ratio  on the TMN is shown in Figure 4 for ˆ0. Im R  . For both types of TBCs, the curves are diverging, which indicates that the impact of  is stronger for higher thermal ratios. For a fixed value of thermal ratio, the increase in the ratio of the effective viscosity of the fluid in the porous region to that in the fluid region leads to an increase in the TMN value. Hence, the non-Darcian-Bènard double diffusive magneto-Marangoni convection is postponed. That is, as the effective viscosity of the fluid in the porous layer is increased, the two layer system is getting stabilized for higher thermal ratio values.
The effects of Chandrasekhar number Q on the TMN is shown in Figure 5  Im R  . For both types of TBCs, the curves are diverging, which indicates that the impact of magnetic field, i.e. Chandrasekhar number, is stronger for higher thermal ratios. For a fixed value of thermal ratio, the increase in the ratio of Q leads to a decrease in the TMN. Hence, the non-Darcian-Bènard double diffusive magneto-Marangoni convection is preponed. The applied magnetic field, which is usually thought as stabilizing, is destabilizing in this type. This may be due to the presence of a second diffusing component.  The core of heat source/sink in the fluid region reflects the significance of the modified internal Rayleigh number * I R =-2, -1, 0, 1, 2 as shown in Figure 6. of thermal ratios. For a fixed thermal ratio, the increase in the value of modified internal Rayleigh number, which occurs from the sink to the source in the porous layer, leads to an increase in the TMN. Hence, larger numbers of * Im R are suitable for the situations controlling the non-Darcian-Bènard double diffusive magneto-Marangoni convection for type (ii) TBC. For a fixed thermal ratio, the increase in the value of modified internal Rayleigh number, which occurs from the sink to the source in the porous layer, leads to a decrease in the TMN. Hence destabilize the system.
The effects of the solute-diffusivity ratio  on the non-Darcian-Bènard double diffusive Marangoni convection is displayed in Figure 8,  Im R  . The effect of this parameter is uniform for all thermal ratios for both types of TBCs. For a fixed depth ratio, the increase in the value of diffusivity ratio leads to a decrease in the thermal Marangoni number. Hence, it is an important parameter, the high value of which can accelerate the non-Darcian-Bènard double diffusive magneto Marangoni convection.

Conclusions
The present study has reached several findings. The non-Darcian-Bènard double diffusive magneto-Marangoni convection can be controlled or amplified by choosing the appropriate TBC at the boundaries. For type (i), i.e. Adiabatic-Adiabatic, TBC, the higher values of porous parameter, solute Marangoni number, viscosity ratio, and internal Rayleigh number in the fluid layer, along with the lower values of Chandrasekhar number and the modified internal Rayleigh number in the porous layer, are conducive to the situations that provide more control to the non-Darcian-Bènard double diffusive magneto-Marangoni convection. For type (ii), i.e. Adiabatic-Isothermal, TBC, the lower values of porous parameter, Chandrasekhar number, and modified internal Rayleigh number in the fluid layer, as well as the higher values of solute Marangoni number, viscosity ratio, and modified internal Rayleigh number in the porous layer are conducive to the situations that provide more control to the non-Darcian-Bènard double diffusive magneto-Marangoni convection. Heat source/sink plays an important role in the convection; by choosing an appropriate strength of heat source, the onset of the non-Darcian-Bènard double diffusive magneto-Marangoni convection can be controlled or amplified.