MHD mixed convection on an inclined stretching plate in Darcy porous medium with Soret effect and variable surface conditions

Abstract This work is concerned with a steady 2D laminar MHD mixed convective flow of an electrically conducting Newtonian fluid with low electrical conductivity along with heat and mass transfer on an isothermal stretching semi-infinite inclined plate embedded in a Darcy porous medium. Along with a strong uniform transverse external magnetic field, the Soret effect is considered. The temperature and concentration at the wall are varying with distance from the edge along the plate, but it is uniform at far away from the plate. The governing equations with necessary flow conditions are formulated under boundary layer approximations. Then a continuous group of symmetry transformations are employed to the governing equations and boundary conditions which determine a set of self-similar equations with necessary scaling laws. These equations are solved numerically and similar velocity, concentration, and temperature for various values of involved parameters are obtained and presented through graphs. The momentum boundary layer thickness becomes larger with increasing thermal and concentration buoyancy forces. The flow boundary layer thickness decreases with the angle of inclination of the stretching plate. The concentration increases considerably for larger values of the Soret number and it decreases with Lewis number. The skin friction coefficient increases for increasing angle of inclination of the plate, magnetic and porosity parameters, however it decreases for rise of thermal and solutal buoyancy parameters. In this double diffusive boundary layer flow, Nusselt and Sherweed numbers increase for rise of thermal and solutal buoyancy parameters, Prandtl number, but they behave opposite nature in case of angle of inclination of the plate, magnetic and porosity parameters. The Sherwood number increases for increasing Lewis number but it decreases for increasing Soret number.


Introduction
Similar to the natural convective ow, the mixed convective ow on stretching surface attracts focus of researchers for its vital applications in many realistic systems of serious interest such as thermal energy storage; geothermal energy utilization, recoverable systems, petroleum reservoirs, metallurgy, electro-chemistry, polymer processing, geophysics, etc. The ow of viscous Newtonian uid in boundary layer for the motion of a at sheet was initiated by Sakiadis [1]. Rather, there are many articles of boundary layer ow in a linearly as well as non-linearly at stretching sheet [2][3][4] with various conditions. In a non-Newtonian Willamson uid, Khan and Khan [5] pioneered the boundary layer ow.
Heat transfer is the energy transfer due to temperature di erence in a medium or between two or more media. Heat can transfer in three ways, viz., conduction, convection, and radiation. Mass transfer is the movement of some other liquid mass in the uid motion from one location to another. It occurs in many processes, such as absorption, evaporation, drying, etc. In a micropolar uid, MHD boundary layer ow with heat and mass transfer was investigated by Chamkha et al. [6] and Khedr et al. [7]. Takhar et al. [8] analyzed unsteady MHD three dimensional boundary layer ow over a bidirectional linear stretching surface.
Darcy law [9] is a basic law for uid ow through a porous medium. This law is valid when the porous mate-rial is saturated with uid. Darcy's law with surface can be applied in various ow problems but there is some limitations, such as, saturated and unsaturated ow, transient ow, fractured rocks and granular media ow, aquitards and aquifers ow, homogenious and heterogeneous system ow, etc. There were many investigations done about boundary layer Newtonian uid ow with convective heat and mass transfer on a permeable surface [10][11][12][13][14] with various physical conditions. Boundary layer nano uid ow with heat transfer over a non-linear stretching sheet with porous medium was derived by Prasannakumara et al. [15]. Chamkha et al. [16,17] obtained MHD boundary layer ow of a Newtonian uid with convective heat transfer over an inclined porous medium. Chen et al. [18] discussed the MHD ow with natural convective heat and mass transfer from a permeable inclined surface for variable surface temperature and concentration.
When heat transfer and mass transfer simultaneously happens between uxes, the ow causing potential is of more complicated type, as energy ux may be generated by composition gradient. The energy ux causes due to temperature gradient is known as the Dufour e ect or di usion-thermo e ect. Whereas, temperature gradient can also produce muss uxes, which is Soret e ect or thermo-di usion e ect. Soret and Dufour e ects may be ignored on the basis that they are of lesser order of magnitude compared to the e ects illustrated by Fourier's and Fick's laws with some exceptions. Eckert and Drake [19] prescribed many cases when these e ects can't be neglected. Out of which, the Soret e ect is an important one, which may be applicable to isotope separation and in a mixture of gases with very lesser molecular weight (H , He, etc.). The Dufour e ect is considerable for medium molecular weight (N , air, etc.). There were some works regarding the Soret and Dufour e ects on MHD boundary layer ow of Newtonian uid [20][21][22][23][24][25][26]. The Soret-Dufour e ects on the boundary layer ow for viscoelastic uid was investigated by Hayat et al. [27] and Gbadeyan et al. [28]. Saritha et al. [29], pioneered boundary layer ow with heat and mass transfer of a power-law uid over a porous at plate taking Soret and Dufour e ects. In a trapezoidal enclosures, Mudhaf et al. [30] obtained unsteady double diffusive natural convection taking Soret and Dufour e ect. In a convective nano uid ow, Soret and Dufour e ect was considered by Reddy et al. [31,32].
In many laminar boundary layer ows, the symmetries are inherently present. Under symmetry transformations, the equations along with the boundary conditions and their solutions are expressed in the form of new variables. Using dimensional analysis, scaling invariances are found for many such ows and this process requires the knowledge about physical characters of the involved variables. The method, Lie symmetry analysis is a systematic one where there is no need to have prior assumptions and knowledge of physical characters of the considered equations and their invariant solutions. One of the key points of the aforesaid method is that it gives all possible symmetries of the system and it may determine whether a system of di erential equations is solvable or not by known numerical or analytical techniques. Importantly, it provides physical perceptions about the system of equations. Many cases of uid motion where invariant solutions are obtained using Lie symmetry can be found in Cantwell [33], Layek [34], Blumen and Kumei [35]. Remarkably, Layek and Sunita [36] and She et al. [37] used the above method to obtain scaling laws of mean velocity in turbulent boundary layer ows and pipe ows, respectively. Lie-group theoretic technique was applied in laminar boundary layer ows with heat and mass transfer by various researchers [38][39][40][41][42][43][44]. Sivasankaran et al. [45] studied recently heat and mass transfer in natural convection.
The group invariant solutions for MHD mixed convective boundary layer ow of Newtonian uid on a stretching inclined porous plate in a porous medium with Soret e ect and variable temperature and concentration at the wall are obtained. This type of treatment on the considered problem is original and novel. Special type magnetic eld is required to determine similar equations of MHD double-di usive system. The problem has wide application in engineering processes, such as, isotope separation, solar collectors, geothermal energy utilization ow in a desert cooler, etc. The obtained self-similar equations are solved numerically and all the variation in physical characteristics of the ow, heat and mass transfer are illustrated in this study.

Formulation of problem
Consider the mixed convective laminar boundary layer steady two-dimensional ow of an incompressible viscous electrically conducting Newtonian uid with low electrical conductivity along an isothermal stretching semiin nite inclined porous plate in a Darcy porous medium at an acute angle α with the vertical. A specially generated strong variable magnetic eld B(x) is applied along y-axis, which is perpendicular to the plate. Here, the induced magnetic eld e ect is neglected due to low conductivity of the liquid and it is very small in comparison to the externally applied magnetic eld. Here, there is no interaction of charge particles and the velocity of the uid is very small in compare to the velocity of light. Hence, the Hall effect is very small and is neglected in this analysis. But the electro-magnetic force, the Lorentz force is taken into account which generally reduces the growth of velocity components and other ow quantities. Both wall temperature Tw and wall concentration Cw are taken as variables with the distance from the leading edge along the stretching plate. The uniform ambient temperature and concentration of the ow are T∞ and C∞, respectively. Also, the Soret e ect is which is important for liquid is considered. However, the Dufour e ect which is prominent in gases and so its e ect is not considered here. Under the usual boundary layer approximations, the governing equations for mass, momentum, energy, and concentration for the steady ow may be stated as: along with the boundary conditions where x-direction is taken along the plate, y-direction is perpendicular to the plate, u and v are the velocity components along the x-and y-directions, respectively, g is the acceleration due to gravity, υ is the kinematic viscosity, β * is the coe cient of expansion with concentration, β is the coe cient of thermal expansion, T is the temperature, C is the concentration, ρ is the constant uid density, σ is the constant electrical conductivity of the uid, κ is the thermal conductivity of the uid, k is the permeability of the porous medium, Dm is the coe cient of mass di usivity, cp is the speci c heat at constant pressure, K T is the thermal di usion ratio, Tm is the mean uid temperature, the parameters m, n, p are related to the stretching speed, temperature change and concentration change along the plate respectively. Moreover, a(> ), b and c(> ) are constants with b > for heated stretching plate (Tw > T∞) and b < for a cooled stretching plate (Tw < T∞). A physical sketch of the aforesaid problem along with the coordinate system is given in Figure 1.
The boundary conditions (5)-(6) are transformed into the following forms:

Symmetry analysis of the governing equations with boundary conditions
We now demonstrate symmetry analysis which determines invariant solutions via invariance principles of continuous group of transformations operating on the independent and dependent variables of equations (7), (8), (9) along with the boundary conditions (10) and (11). (See the books Cantwell [17], Layek [18], Blueman and Kumei [19]). There are two independent variables in the equations, x, y and four dependent variables ψ(x, y),T(x, y), C(x, y),B(x). According to the group theory the in nitesimal generator has the form The equations are of 3 rd order, and so the above in nitesimal generator can extend three times and after extension the generator may be written in the following standerd form The coe cients are written as follows: The in nitesimals (ξ ,ξ , η , η , η , η ) are obtained by solving the de ning equations X [ ] F | F= = ,with the help of Mathematica package IntroToSymmetry.m (Cantwell [18]) as where α is a polynomial function of x only. Now we will nd the similarity variables for selfsimilar solutions of the equations in (7), (8) and (9) along with boundary conditions in (10) and (11). The boundary value problem, the equations as well as boundary conditions remain invariant under the group of transformations. We now apply the invariant criteria X [ ] F F= = to the boundary conditions. The invariance of the boundary condition y = gives α(x) = , T = Tw(x) = T∞ + bx n gives b = , n = , and T∞ = − b b . Also, C = Cw(x) = C∞ + cx p gives p = . So, the in nitesimals are transformed into Therefore, the equations (7), (8) and (9) along with boundary conditions in (10) and (11) admit two-parameter symmetry group of transformations and corresponding generators may be written as Solving the Lie equations dx dε = ξ i (x,ū) (where i = , ) and dū dε = n j (x,ū) (where j = , , , ) with the initial con- Hence, we are left with a single dilatational group G b and a single translational groupG b . As two parameters Lie group is always solvable, therefore the Eqns. (7), (8) and (9) with boundary conditions in (10) and (11) are surely solvable. So, the corresponding characteristic equations can be written as Substituting these, the equations (7), (8) and (9) are transformed into and The invariance boundary conditions are and To non-dimensional the variables, we apply the transformations Under these transformations, the equations (19), (20), (21) with boundary conditions in (22), (23) are transformed to

Quantities of engineering interest
The physical quantities of interest for engineering of the prescribed problem are the local skin friction coe cient C f , the local Nusselt number Nu and the local Sherwood number Sh.
The boundary layer phenomena is analyzed from the physical quantity, namely, the skin friction coe cient (C f ), which is dimensionless form of wall shear stress and is expressed as C f = τw

Validation of numerical scheme
To verify the accuracy of our present numerical scheme, we have compared wall shear stress −f ( ) for some porosity parameter and wall temperature gradient −θ ( ) for some values of Prandtl number Pr taking λ = λ * = M = K = with published papers Yih [10] and Hayat et al. [27] in Table 1 and Table 2, respectively and those data are in decent agreements.

Results and discussion
The computed numerical solutions are depicted graphically in the form of dimensionless velocity f (η), temperature θ(η), and concentration ϕ(η) for various values of parameters. The involved physical parameters are selected suitably, where Pr = . corresponds to air at 20 • C, Pr = . corresponds to electrolyte solution, such as, salt-water and Pr = . corresponds to pure water.
At rst, the most important aspect of investigation, i.e., the in uences of thermal buoyancy parameter λ and local solutal buoyancy parameter λ * on velocity, temperature and concentration pro les are presented in Figures  2-7. It is clear that λ > corresponds to assisting ow (heated plate), λ < corresponds to opposing ow (cooled plate) and λ = corresponds to forced convection ow respectively. Also λ * > and λ * < corresponds to mass transfer from the plate to the uid and reverse respectively. λ * = corresponds to the case where the mass transfer is absent. It can be noticed that for an increase of λ or λ * (taking anyone xed and others vary) the velocity pro le f (η) along the plate increases but the temperature pro le θ(η) and the concentration pro le ϕ(η) (at a xed η) along the plate decrease keeping other parameters unchanged. Consequently, the momentum boundary layer thickness increases but the thermal and concentration boundary layer thickness decrease for increasing λ and λ * .
Next, Figures 8-10 exhibit the self-similar velocity, temperature, and concentration pro les for di erent values of angle of inclination of the plate α. It is worth noting that if the plate is rotated from vertical to horizontal with xed origin without changing other parameters, the velocity pro le along the plate decreases, but the temperature pro le and the concentration pro le increase. The    uid has higher velocity when the surface is vertical than when inclined because of the fact that buoyancy force due to gravity and hence its e ect decreases as the inclination of the plate becomes larger. So, the momentum boundary layer thickness reduces but the thermal and concentration boundary layer thicknesses increase with increasing angle of inclination of the plate α.
The e ects of magnetic parameter M and porosity parameter K on the velocity, temperature and concentration pro les are depicted in Figures 11-16. It is obvious that for an increase of M or K (taking anyone constant and others vary) the self-similar velocity along the plate decreases but the temperature pro le and the concentration pro le increase (at a xed η) at constant other parameters. So, the uid has higher velocity if the magnetic eld is absent (i.e.M = ) and the medium is non-porous (i.e., K=0) but in this case heat and mass transfer decrease. The magnetic eld and the porosity of the medium therefore have a tendency to resist the momentum transport and to increase the thermal and the concentration boundary layers.          Table 3. It is found that the value of −f ( ) increases for increasing α, M, K, Pr, Le, however it decreases for rise of λ, λ * , Sr. We have noticed that the magnitude of −θ ( ) and −ϕ ( ) increase for rise of λ, λ * , Pr, but they behave opposite nature in case of α, M, K. So the magnitude of −θ ( ) and −ϕ ( ), the similar e ect occurs in case of parameters λ, λ * , Pr, α, M, K. But for the parameters Le, Sr the opposite behavior occurs for the magnitude of −θ ( ) and −ϕ ( ). The values of −ϕ ( ) increases for increasing Le, but it decreases for increasing Sr.

Concluding remarks
The double-di usive boundary layer ow of an electrically conducting uid along a stretching inclined plate embedded in a Darcy porous medium is investigated. The governing equations are formulated and appropriate boundary conditions are prescribed. By adopting Lie group of continuous transformations, we obtain scaling laws and similarity equations. The self-similar equations are then solved numerically by using shooting method applicable to nonlinear ODEs. The results are plotted graphically. We can summarize the following important points: 1. The momentum boundary layer thickness increases with increasing thermal buoyancy parameter and solutal buoyancy parameter but it decreases with increasing angle of inclination of the plate, magnetic parameter, and porosity parameter. 2. Thermal boundary layer thickness decreases with increasing thermal buoyancy parameter, solutal buoyancy parameter and Prandtl number but it increases with increasing angle of inclination of the plate, magnetic parameter and porosity parameter. 3. The concentration boundary layer thickness decreases with increasing thermal buoyancy parameter, solutal buoyancy parameter and Prandtl number and Lewis number, but it increases with increasing angle of inclination of the plate, magnetic parameter and porosity parameter. 4. Though there are not many variations in temperature and velocity pro les due to Soret e ect, but it causes an increment in concentration inside the boundary layer considerably. 5. The skin friction coe cient increases for increasing angle of inclination of the plate, magnetic and porosity parameters, however it decreases for rise of thermal and solutal buoyancy parameters.
6. Nusselt and Sherwood numbers increase for rise of thermal and solutal buoyancy parameters, Prandtl number, but they behave oppositely in case of increments of angle of inclination of the plate, magnetic and porosity parameters. Lewis number has tendency to increase the values of Sherwood number, but Soret number has opposite tendency.
ρ density of the uid θ dimensionless temperature ϕ dimensionless concentration Superscript di erentiation with respect to η Subscript w condition at the surface ∞ condition at in nity