Effects of Non-uniform Temperature Gradients on Triple Diffusive Surface Tension Driven Magneto Convection in a Composite Layer

Abstract The problem of triple diffusive surface tension driven magneto convection in a composite layer is investigated for linear, parabolic and inverted parabolic temperature profiles. The corresponding thermal Marangoni numbers are obtained analytically depending on various physical parameters of interest. The parameters which are effective to enhance convection and to control convection are determined. From the investigation it is found that the linear profile is suitable for fluid layer dominant composite layer where as parabolic and inverted parabolic profiles are conducive for the porous layer dominant composite layer.


Introduction
There are many fluid systems containing more than two components occurring in nature. The subject of systems having multi components in porous and fluid layers has attracted many researchers due to its importance in the study of crystal growth, geothermally heated lakes, earth core, solidification of molten alloys, underground water flow, acid rain effects, natural phenomena such as contaminant transport, warming of stratosphere, magmas and sea water etc. In many situations particularly in geophysics, astrophysics and in some industrial problems maintaining a uniform temperature gradient is a limitation and non-uniform temperature/concentration gradients is a reality. In that case, the stability or instability of a fluid in the presence of a nonlinear temperature profile is of practical importance and has not been given much attention.
For the single fluid layer, Melviana Johnson Fu et al. [1] have investigated the effect of a non-uniform basic temperature gradient and magnetic field on the onset of Marangoni convection in a horizontal micro polar fluid layer using the Rayleigh-Ritz technique. They found that the micro polar fluid layer heated from below is more stable compared to the classical fluid layer. Using linear stability analysis, Isa et al. [2,3,4] have explored the impact of non-uniform temperature gradient and magnetic field on Marangoni and Benard-Marangoni convection in a horizontal fluid layer heated from below and cooled from above. Norihan Md. Arifin et al. [5] have studied the effect of a non-uniform basic temperature gradient and magnetic field on the onset of Benard-Marangoni convection in a horizontal micro polar fluid layer using Rayleigh-Ritz technique. The combined effects of vertical magnetic field and non-uniform temperature profiles on the onset of steady Marangoni convection in a horizontal layer of micro polar fluid are investigated by Mahmud et al. [6]. The effects of electric field and non-uniform basic temperature gradient on the onset of Rayleigh-Benard convection in a micro polar fluid are studied by Pranesh and Riya Baby [7] using the Galerkin technique. The onset of convection is discussed for six nonlinear temperature profiles. Pranesh and Kiran [8] have studied the effect of non-uniform temperature gradient on the onset of Rayleigh-Benard magneto convection in a micropolar fluid with Maxwell-Cattaneo law using the Galerkin technique. The onset of convection is discussed for six non-linear temperature profiles. Thadathil Varghese Joseph et al. [9] studied effect of non-uniform basic temperature gradient on the onset of Rayleigh-Benard-Marangoni electro-convection in a micropolar fluid. Recently, Azmi and Idris [10] have applied the linear stability analysis to study the effects of non-uniform basic temperature gradients on the onset of Rayleigh-Benard-Marangoni magneto convection in a micro polar fluid by using the Galerkin technique. They considered three temperature profiles and their comparative influence on onset of convection is discussed. Using linear stability analysis, Sandhya and Sangeetha George [11] have studied the different temperature gradients and rotation on the onset of Marangoni convection in a fluid which has suspended particles in it and confined Universal Journal of Mechanical Engineering 7 (6): 398-410, 2019   399 between an upper free, adiabatic and lower rigid, isothermal boundaries under micro gravity condition.
The literature cited above is concerned with convection in the presence of a non-uniform temperature gradient in the fluid layer. However, many engineering problems involve porous media and in such cases it is essential to control convection. This can be achieved by using a non-uniform temperature gradient in the porous layer. In view of this, Shivakumara [12] has studied the onset of convection in a layer of couple-stress fluidsaturated porous medium is investigated for different types of basic temperature gradients using the Galerkin technique. It is observed that the critical thermal depth decreases marginally with an increase in the couple-stress parameter. Shivakumara et al. [13] studied the simultaneous effect of local thermal non equilibrium, vertical heterogeneity of permeability and nonuniform basic temperature gradient on the criterion for the onset of Darcy-Benard convection. The eigenvalue problem is solved numerically using the Galerkin method. Shivakumara et al. [14] have investigated the effect of six different temperature gradients on the onset of convection in a couple stress fluid saturated porous medium using the Galerkin technique. They found that parabolic and inverted parabolic basic temperature profiles have the same effect on the onset of convection.
For the composite layers, Manjunatha and Sumithra [15,16,17] are investigated the combined effects of magnetic field and non-uniform basic temperature gradients on two and three component convection in two layer system.
In this paper, the lower rigid surface of the porous layer and the upper free surface are considered to be insulating to temperature, insulating to both solute 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. Effects of variation of different physical parameters on the thermal Marangoni numbers for the profiles are compared.

Mathematical formulation
We consider a three different diffusing components with different molecular diffusivities, electrically conducting fluid layer of thickness d horizontally isotropic sparsely packed porous layer of saturated with same fluid of thickness d m in the presence of magnetic field H 0 in the vertical z− direction. The lower surface of the porous layer is considered to be rigid and the upper surface of the fluid layer is free at which the surface tension effects depending on temperature and both the species concentrations is considered. Both the boundaries are kept at different constant temperatures and salinities. A Cartesian coordinate system is chosen with the origin at the interface between porous and fluid layers and the z− axis, vertically upwards. The basic equations for fluid and porous layer respec- .
Here − → q = (u, v, w) is the velocity vector, − → H is the magnetic field, ρ 0 is the fluid density, t is the time, µ is the fluid viscosity, P = p + is the total pressure, T is the temperature, κ is the thermal diffusivity of the fluid, κ 1 and κ 2 are the solute1 and solute2 diffusivity of the fluid in the fluid layer, C 1 and C 2 are the concentration1 and concentration2 for the fluid in the fluid layer, ν m = 1 µpσ is the magnetic viscosity, K is the permeability of the porous medium, A = (ρ0Cp)m (ρCp) f is the ratio of heat capacities, C p is the specific heat, ε is the porosity , µ m is the effective viscosity of the fluid in the porous layer, κ m1 and κ m2 is the solute1 and solute2 diffusivity of the fluid in porous layer, C m1 and C m2 are the concentration1 and concentration2 for the fluid in porous layer, ν em = νm ε is the effective magnetic viscosity and the subscripts 'm' and 'f' refer to the porous medium and the fluid respectively.
The equations (1) to (14) have a basic steady solution for 400 Effects of Non-uniform Temperature Gradients on Triple Diffusive Surface Tension Driven Magneto Convection in a Composite Layer fluid and porous layer respectively and they are Where h(z) and h m (z m ) are temperature gradients in fluid and porous layers respectively and the subscript 'b' denotes the basic state. The interface temperature is T 0 = κd m T u + κ m dT l κd m + κ m d and the interface concentrations are C 10 = To examine the stability of the system, we give a small perturbation to the system as Where the primed quantities are the dimensionless ones. Equations (23) & (24) are substituted into the (1) to (14), apply curl twice to eliminate the pressure term from (3) & (10) and then variables are nondimensionalised for fluid and porous layer.
To render the equations nondimensional, we choose different scales for the two layers ( Chen and Chen [18], Nield [19]), so that both layers are of unit length such that Omitting the primes for simplicity, we get in 0 ≤ z ≤ 1 and 0 ≤ z m ≤ 1 respectively Here, for the fluid layer P r = ν κ is the Prandtl number, the diffusivity ratio for fluid layer, τ 1 = κ 1 κ and τ 2 = κ 2 κ are the ratios of solute1 and solute2 diffusivity to thermal diffusivity fluid in fluid layer. For the porous layer, Here a and a m are the nondimensional horizontal wave numbers, n and n m are the frequencies. Since the dimensional horizontal wave numbers must be the same for the fluid and porous layers, we must have a d = a m d m and hence a m = da.
Introducing Eqs. (35) and (36) into the Eqs. (25) to (34) then we get an Eigen value problem consisting of the following differential equation in 0 ≤ z ≤ 1 and 0 ≤ z m ≤ 1 respectively Assume that the principle of exchange of instabilities holds for present problem, hence we take n = n m = 0. Eliminating the magnetic field in Eqs. (41) and (46). The Eigen value problem becomes, in 0 ≤ z ≤ 1 and 0 ≤ z m ≤ 1 respectively

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

Method of Solution
From Eqs. (47) and (51), we get velocity distributions for fluid and porous layer respectively and A i s(i = 1, 2, ....8) are arbitrary constants are obtained by using velocity boundary conditions of (55). The expressions for W (z) and W m (z m ) are appropriately written as We get the species concentration for fluid layer S 1 , S 2 from (49) and (50) also from (53) and (54) we get the species concentration for porous layer S m1 , S m2 using the species concentration boundary conditions of (55) as where , , ], ∆ 50 = a S 1 sinh a cosh a m + a m cosh a sinh a m , ], ∆ 62 = a S 2 sinh a cosh a m + a m cosh a sinh a m ,

Linear temperature profile
Here taking Substituting equation (64) into (48) and (52), we get temperature distributions for fluid and porous layers using temperature boundary conditions of (55) and they are Where  The thermal Marangoni number obtained from (55) and is found to be For this model Where Λ 1 = δ 2 cosh δ + a 1 δ 2 sinh δ + a 2 ζ 2 cosh ζ + a 3 ζ 2 sinh ζ,

Parabolic temperature profile
We consider the profile as following Sparrow et al. [20], Substituting equation (68) into (48) and (52), we get temperature distributions for fluid and porous layers using temperature boundary conditions of (55) and they are where Effects of Non-uniform Temperature Gradients on Triple Diffusive Surface Tension Driven Magneto Convection in a Composite Layer T (a 10 cosh a m + a 11 sinh a m ) + ∆ 35 , a 9 = 1 a (a 10 a m sinh a m + a 11 a m cosh a m + ∆ 36 ), [(a 3 + a 2 ζ) sinh ζ + (a 2 + a 3 ζ) cosh ζ], The thermal Marangoni number for this model obtained from (55) and is found to be where Λ 5 = [a 8 cosh a + a 9 sinh a + R 6 + R 7 ],

Inverted parabolic temperature profile
We have Substituting equation (72) into (48) and (52), we get temperature distributions for fluid and porous layers using temperature boundary conditions of (55) and they are Where ∆ 400 = 2(ζ 2 + a 2 ) (ζ 2 − a 2 ) 2 (a 3 sinh ζ + a 2 cosh ζ), The thermal Marangoni number for this model obtained from (55) and is found to be Where

Results and Discussion
The thermal Marangoni numbers M 1 for linear, M 2 for parabolic and M 3 for inverted parabolic temperature profiles are obtained for diffrent physical parameters and the constraints are drawn against the depth The