Effects of Non-Uniform Temperature Gradients on Triple Diffusive Marangoni Convection in a Composite Layer

The problem of triple diffusive Marangoni convection is investigated in a composite layer comprising an incompressible three component fluid saturated, sparsely packed porous layer over which lies a layer of the same fluid. The lower rigid surface of the porous layer and the upper free surface are considered to be insulating to temperature, insulating to both salute concentration perturbations. At the upper free surface, the surface tension effects depending on temperature and salinities are considered. At the interface, the normal and tangential components of velocity, heat and heat flux, mass and mass flux are assumed to be continuous. The resulting eigenvalue problem is solved exactly for linear, parabolic and inverted parabolic temperature profiles and analytical expressions of the thermal Marangoni number are obtained. The effects of variation of different physical parameters on the thermal Marangoni numbers for the profiles are compared.

For the fluid layer, Chand [8] has applied the linear stability analysis and a normal mode analysis to study the triple-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below. Suresh Chand [9] has investigated the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. Shivakumara and Naveen Kumar [10] have investigated the effect of couple stresses on linear and weakly nonlinear stability of a triply diffusive fluid layer using a modified perturbation technique.
Kango et al. [11] have studied the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. Vivek Kumar and Mukesh Kumar Awasthi [12] have considered the problem of triple-diffusive convection in a horizontal nanofluid layer heated and salted from below using linear stability theory and normal mode technique. A linear stability analysis is carried out for triple diffusive convection in Oldroyd-B liquid and rotating couple stress liquid by Sameena Tarannum and Pranesh [13] [14].
In porous medium, Suresh Chand [15] has obtained closed-form of solution for the rotation in a magnetized ferrofluid with internal angular momentum, heated and soluted from below saturating a porous medium and subjected to a transverse uniform magnetic field. Salvatore Rionero [16] have studied a triple convective-diffusive fluid mixture saturating a porous horizontal layer, heated from below and salted from above and below. Kango et al. [17] have studied the triple-diffusive convection in Walters (Model B') fluid with varying gravity field is considered in the presence of uniform vertical magnetic field in porous medium. Khan et al. [18] investigated the steady triple diffusive boundary layer free convection flow past a horizontal flat plate embedded in a porous medium filled by a water-based nanofluid and two salts. Moli Zhao et al. [19] have investigated the linear stability of triply diffusive convection in a binary Maxwell fluid saturated porous layer using modified Darcy-Maxwell model. The triply diffusive convection in a Maxwell viscoelastic fluid is mathematically investigated in the presence of uniform vertical magnetic field through porous medium studied by Pawan Kumar Sharma et al. [20] using linearized stability theory and normal mode analysis. Jyoti Prakash et al. [21] [22] have studied the magnetohydrodynamic triply diffusive convection with one of the components as heat, with diffusivity and sparsely distributed porous medium using the Darcy-Brinkman model. Rana et al. [23] have studied the triple-diffusive convection in a horizontal layer of nanofluid heated from below and salted from above and below. Goyal et al. [24] have studied the triple diffusive natural convection under Darcy flow Open Journal of Applied Sciences over an inclined plate embedded in a porous medium saturated with a binary base fluid containing nanoparticles and two salts using group theory transformations. Patil et al. [25] studied a numerical investigation on steady triple diffusive mixed convection boundary layer flow past a vertical plate moving parallel to the free stream in the upward direction. A linear stability analysis is performed for the onset of triple-diffusive convection in the presence of internal heat source in a Maxwell fluid saturated porous layer studied by Mukesh Kumar Awasthi et al. [26]. Raghunatha et al. [27] have investigated the weakly nonlinear stability of the triple diffusive convection in a Maxwell fluid saturated porous layer. For the composite layers, Sumithra [28] has studied the triple-diffusive Marangoni convection in a two layer system and obtained the analytical expression for the Thermal Marangoni Number. The combined effects of magnetic field and non uniform basic temperature gradients on two component convection in two layer system is investigated by Manjunatha and Sumithra [29] [30].
This paper investigates the triple diffusive Marangoni convection in a composite layer and studies the effects of the linear, parabolic and inverted parabolic temperature gradients on the corresponding thermal Marangoni numbers.

Mathematical Formulation
is the velocity vector, 0 ρ is the fluid density, t is the time, µ is the fluid viscosity, P is the pressure for fluid layer, T is the temperature, κ is the thermal diffusivity of the fluid, 1 κ and 2 κ is the solute1 and solute2 diffusivity of the fluid in the fluid layer, 1 C is the concentration1 or the salinity field1 for the fluid, 2 C is the concentration2 or the salinity field2 for the fluid in the fluid layer, m P is the pressure for porous layer, K is the permeability of the porous medium, is the ratio of heat capacities, p C is the specific heat, ε is the porosity,  (11) ( ) ( ) ( ) ( )  (12) ( )  (19) ( ) ( ) ( ) (20) where the primed quantities are the dimensionless one. Introducing (19) & (20) are substituted into the (1) to (10), apply curl twice to eliminate the pressure term from (2) to (7)  To render the equations nondimensional, we choose different scales for the two layers (Chen and Chen [31], Nield [32]), so that both layers are of unit Omitting the primes for simplicity, we get in 0 Here, for the fluid layer, Pr ν κ = is the Prandtl number, We apply normal mode expansion on dependent variables as follows: Introducing Equation (29) and Equation (30) into the Equations (21) to (28) and then we get an eigenvalue problem consisting of the following ordinary differential equations in 0 1 z ≤ ≤ and 0 1 It is known that the principle of exchange of instabilities holds for triple diffusive convection in both fluid and porous layers separately for certain choice of parameters. Therefore, we assume that the principle of exchange of instabilities holds even for the composite layers. In other words, it is assumed that the onset of convection is in the form of steady convection and accordingly we take 0 m n n = = .
We get, in 0

Boundary Conditions
The boundary conditions are nondimensionlised then subjected to normal mode analysis and finally they take the form

Method of Solution
From Equation (39) and Equation (43) We get the species concentration for fluid layer 1 S , 2 S from (41) and (42) also from (45)

Parabolic Temperature Profile
We consider the profile as following (Sparrow et al. [ 3  2  2 3  2  1  3  3  1  6  3  3   2  3  3  2  3  3  sinh  cosh ,  6  6 a z z a aa z a z z a aa z R az az a a

Results and Discussion
The Thermal Marangoni numbers        For all the profiles, it is evident from the graph that an increase in the value of a, the thermal Marangoni number increases and its effect is to stabilize the system. which is physically reasonable, as there is more way for the fluid to move. So, the system is destabilized.

Conclusions
1) The inverted parabolic temperature profile is the most suitable for the situations demanding the control of Marangoni convection, whereas the linear and parabolic profile is suitable for the situations where the convection is needed.
2) By increasing the values of  3) In the manufacture of pure crystal growth, our work can be useful. The people who are manufacturing crystals can refer this paper. This can give them an initial insight into the effects of parameters in the multicomponent crystal growth problems.