of Hall currents along a vertical plate beneath the combined results of buoyancy force, species and thermal diffusion in transverse magnetic subject. The resultant conditions are fathomed by fourth-order R-K strategy with the application of little Reynolds **number**. Joule warming and dissemination impacts of thickness on shaky MHD stream from a vertical plate inserted in permeable media beneath the impact of warm source. Ion-slip and hall current is analyzed by Emad M. Abo-Eldahab et al. [3]. The overseeing conditions are fathomed by utilizing explicit finite difference method. Salem and Abd El-Aziz [4] have displayed the impact of chemical response and hall currents on MHD stream over extending surface with warm era or assimilation. A detailed study of the unsteady Magneto Hydrodynamic flow of an Casson fluid between parallel insulating porous plates in state of viscous dissipation, heat transfer and the Hall Effect were studied by Hazem Ali Attia et al. [5]. Sudhakar et al. [6] analyzed lobby impact on the MHD stream along a permeable level plate with, soret impact, duffer impact and chemical response by utilizing Galerkin finite element strategy. Afikuzzaman et al [7] investigated the impacts of Lobby streams on the insecure coutte stream of Casson liquid by keeping up steady temperatures on the plates beneath transverse attractive field. The Magneto Hydro Dynamic move of a Casson liquid beyond a vertical plate with swaying and constant temperature..The liquid which is passing through a permeable media and has conduction was examined by Asam Khalid et al. [8].Vasundhara, Sarojamma [9] investigated the concept

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5.2.From this figure, 5.2 (a) illustrate the temperature field increases with increasing values of Nanofluid parameter (Nt). Fig.5.2 (b) illustrate the temperature field increases with increasing values of Heat generation or absorption coefficient (S) and Fig.5.2 (c) illustrate the temperature field decreases while the increase of **Dufour** **Number** (D f ).

Figure 8 shows the Soret and **Dufour** effect on the thermalsolutal convection. It can be noted average Nusselt and Sherwood **number** grow congruously with Soret as shown in Fig. 8 (a), at a **Dufour** **number** of 0.1. Although weaken **Dufour** effect on heat and mass transfer is constant, Soret effect become more intense with increasing Soret **number**. That is because the effect of thermal buoyancy on thermalsolutal convection rises up. And the **Dufour** effect on the thermalsolutal convection is similar with the Soret effect on that. It should be known that there is no significant difference in the streamline, isotherm and isosolute. Consequently, the growth rate of Nusselt and Sherwood **number** are calculated. As Soret effect become intense, heat transfer rate is increased by 21% and mass transfer 35% in Fig. 8 (a). However, the slope of the Nusselt is steeper in the Fig. 8 (b). As **Dufour** effect become intense, heat transfer rate is increased by 25% and mass transfer 34%.Therefore, both Soret numbers due to the temperature gradient and **Dufour** **number** due to the concentration gradient enhance the heat and mass transfer. But concentration difference and temperature difference have great impact on mass transfer. Also, mass diffusion is more exacerbated at temperature difference and concentration difference than thermal convection.

The influence of Soret and **Dufour** on unsteady mixed convection flow of Casson fluid over a nonlinearly stretch- ing sheet under the influence of viscous dissipation and heat generation/absorption was investigated numerically. The governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equa- tions by employing similarity transformations. The resulting equations were solved numerically by an implicit finite difference scheme. In order to check the accuracy and validate the present method, results are compared with the results of existing literature. An excellent agreement is noticed with those results. Influence of local unsteadiness parameter A, Casson fluid parameter β, nonlinear stretching sheet parameter n, magnetic parameter M, thermal Grashof **number** Gr, mass Grashof **number** Gm, Prandtl **number** Pr, temperature ratio parameter θ w , Eckert **number** Ec, heat generation/absorption parameter λ 1 , **Dufour** **number** D f , Schmidt **number** Sc, thermo-

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The objective of this paper is to analyze the effects of heat and mass transfer in the presence of thermal radiation, internal heat generation and **Dufour** effect on an unsteady magneto-hydrodynamic mixed convection stagnation point flow towards a vertical plate embedded in a porous medium. The non-linear partial differential equations governing the flow are transformed into a set of ordinary differential equations using suitable similarity variables and then solved numerically using shooting method together with Runge- Kutta algorithm. The effects of the various parameters on the velocity, temperature and concentration profiles are depicted graphically and values of skin- friction coefficient, Nusselt **number** and Sherwood **number** for various values of physical parameters are tabulated and discussed. It is observed that the temperature increases for increasing values of the internal heat generation, thermal radiation and the **Dufour** **number** and hence thermal boundary layer thickness increases.

Figure 5 shows the variation of Nusselt Sherwood numbers according to **Dufour** and Soret numbers. Fig. 5a, for Sr = 2, shows that the Nusselt **number** increases with increasing **Dufour** **number**, whereas for its lower values, Nu decreases with increasing the **number** of **Dufour**. This reveals that there, there is a value in the **number** of Soret, for which the Nusselt **number** remains invariant, regardless of Df. This corresponds to the values of Soret **number** of neighboring 1.48 on both sides of this issue. The Nusselt **number** increases almost linearly with an increase with the Soret **number**. This trend is more nuanced for high values of the **Dufour** **number**. For Sr 1.5 the Nusselt decreases linearly with the increase **Dufour** **number** and gives the opposite effect when Sr > 1.5 . Fig.5b shows the variations of the Sherwood **number** for different values of **Dufour** and Soret numbers. This figure is the dimensionless concentration profile, represented by the Sherwood **number**, based on the **number** of Soret and **Dufour** numbers. Of these graphs, it appears that the influence of thermodifusion has no significant influence on mass transport, while that the Soret effect is very significant. No matter how many Soret, the Sherwood **number** increases with the increase **Dufour** **number**. The Sherwood **number** increases with the decrease in Soret. We observe the Sherwood **number** decreases with the increase in Sr whatever the values of Df.

The Soret and **Dufour** effects on unsteady MHD mixed convection flow past an infinite radiative vertical porous plate embedded in a porous medium in the presence of chemical reaction have been studied. A uniform magnetic field acts perpendicular to the porous surface. The Rosseland approximation has been used to describe the radiative heat flux in energy equation. The governing equations are solved numerically by applying explicit finite difference Method. The effects of various parameters on the velocity, temperature and concentration fields have been examined with the help of graphs.

The MHD free convective heat and mass transfer past a moving vertical plate with Soret and **Dufour** effect in the presence of suction/injection parameters are analyzed by Olanrewaju et al. (2011). The governing equations are solved numerically by shooting iteration technique together with a sixth order Runge–Kutta integration scheme. It is found that the skin friction coefficient and Nu decreases with increasing Du and decreasing Sr. A numerical model to examine the combined effects of Sr and Du on mixed convection MHD heat and mass transfer in a Darcian porous medium in the presence of thermal radiation and Ohmic dissipation developed by Pal et al. (2011). The influence of Schmidt **number** (Sc), Da, Pr, Sr, Du, magnetic parameter, on the velocity, temperature, and concentration fields are studied. Magyari et al. (2011) studied the double-diffusive natural convection past a vertical plate embedded in a fluid-saturated porous medium in the boundary-layer and Boussinesq approximation, considering that the Soret–**Dufour** cross- diffusion effects are significant. They found that when the Soret effect is more dominating than **Dufour** effect the Sherwood **number** influences the flow with thermos-solutal symmetry than Nusselt **number**. The **Dufour** effects influence the temperature profile in the thermal boundary layer. The three dimensional numerical analysis of laminar mixed convection and entropy generation in a cubic lid-driven cavity is carried out by Lioua et al. (2011). The results were obtained for different Richardson **number** ranging from 0.01 to 100. The findings were, for low values of Ri the entropy generation is symmetrical in the cavity whereas it is insignificant for higher Ri.

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4.1.4. Effect of Soret **number** : The thermophoresis factor has been calculated from molecular interaction potentials derived from known molecular models. The ratio of diffusion coefficient is small compared to thermo diffusion coefficient, and hence the values of in our discussion is greater than one. Thus increase in thermo diffusion brings about increase in concentration. As in the case for thermal distribution, increase in Soret **number** decreases temperature constitution of the flow. It could be confirm from figure 9 that effect of Soret **number** on temperature is significant when and slightly significant when outside which the temperature becomes insignificant. Soret **number**, increases the velocity with maximum velocity in the body of the fluid close to the surface as shown in figure 15. The maximum for each of Soret **number** tilt to the right of as Soret **number** increases.

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An exact solution provided by Deka and Paul [7] for unsteady natural convection flow with thermal stratification effect over a vertical cylinder. An influence of electrophoresis and chemical reaction on unsteady convection doubly stratified flow past a vertical plate in the presence of a heat source/sink is presented by Ganesan and Suganthi [8]. Soret and **Dufour** effects of free convection heat and mass transfer in the presence of doubly stratified Darcy porous medium, analyzed and presented by Lakshmi Narayana and Murthy [9]. Recently Srinivasacharya and Upendar [4] analyzed and presented, mixed convection in MHD doubly stratified micropolar fluid.

The physical situation defined in all the above investigations is connected to the process of uniform temperature and concentration over Newtonian fluid flows (Kumar, 2009; Ali, 1995; Elbashbeshy, 1998; Andersson et al. 1992). Whereas, the influence of power law form of temperature and concentration on non-Newtonian fluid flows has received a little attention even though it has huge applications in many industrial and engineering sciences. This motivates the present work to explore the Soret and **Dufour** effects on MHD electrically conducting non-Newtonian Jeffrey fluid over a linear stretching sheet in the presence of chemical reaction and heat source. After using similarity transformations, the governing equations are converted to a system of non-linear ordinary differential equations. The resulting equations are then solved numerically by using fourth order Runge-Kutta method along with shooting technique. The effect of different parameters on velocity, temperature and concentration profiles are shown with the help of graphs and tables. Further, the skin friction coefficient, Nusselt and Sherwood numbers are computed and analyzed.

Unsteady micropolar fluid flow on streatching surface with heat and mass transfer in presence of magnetic field with Soret and **Dufour** effects has been considered taking viscosity and thermal conductivity as time dependent variable. In the light of above numerical results, an analysis of the graph is being conducted and conluded as follows :

But in case of heated semi-infinite horizontal plate facing upward the buoyancy force has no component along its length. In this case, heat is absorbed by the surrounding fluid progressively, thus inducing a horizontal temperature gradient within the fluid, which in turn give rise to a favourable pressure gradient that leads to the formation of boundary layer flow at the surface of the plate due to indirect natural convection provided that a suitably defined Grashof **number** is large. Mathematically, above the surface of the horizontal plate, the temperature is everywhere T so that, as in the static field, there exist a pressure distribution having pressure gradient p y g where origin is taken at one of the leading edge. In the boundary region adjacent to the plate dimensional temperature T w is larger than T and so the density of the fluid is lower than . Decreased in pressure gradient

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Figure 21 dispicts the effect of Schmidt **number** on the concentration distribution. It is seen that concentration decreases with increasing values of Schmidt **number** for both small ( =10) and large ( =20) rotations of the system. Figure 22 displays the effect of reaction parameter on the concentration distribution. It is shows that concentration decreases with increasing values of reaction parameter for both small ( =10) and large ( =20) rotations of the system. Figure 23 exhibites the effect of Soret **number** on the concentration distribution. It is observed that concentration increases with increasing values of Soret **number** for both small ( =10) and large ( =20) rotations of the system. Figure 24 illustrates the effect of Prandtl **number** on the concentration distribution. It is perceived that the concentration increases with increasing values of Prandtl **number** for both small ( =10) and large ( =20) rotations of the system. Figure 25 presents the effect of viscoellastic parameter on the concentration distribution. It shows that concentration increases with increasing values of viscoelastic parameter for both small ( =10) and large ( =20) rotations of the system.

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generation parameter Q. Figure 15 and 16 display the velocity and concentration profiles for different values of the Soret **number** it can be seen that an increase in Soret **number** results in an increase in the velocity and concentration within the boundary layer. This is because of the mass flux created by temperature gradient is inversely proportional to the mean temperature., this causes the concentration of the fluid increases due to the thermal diffusion rate is increasing with the velocity .

The numerical solutions are simulated for different values of the Prandtl **number** (Pr), Schmidt **number** (Sc), Eckert **number** (Ec), thermal Grashof **number** (Gr), mass Grashof **number** (Gc), **Dufour** **number** (Du), radiation (N), magnetic field (M), porosity (K), thermal conductivity ( ), suction ( ), reaction order (n), chemical reaction (Kr) and heat source (S), parameters. The following parameters values are fixed throughout the calculations except where otherwise stated, Pr = 0.71, Sc = 0.62, Ec = 0.01, M = 1, K = 1, Gr = 1, Gc = 1, N = 0.1, S = 1, = 1, = 0.1, Kr = 0.1, n = 1, Du = 1.

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pressure, C s is the concentration susceptibility, k T is the thermal diffusion, T m is the mean fluid temperature, C is the mass concentration, y is distance, q r is the radiative heat flux, 0 is the magnetic field, D is the molecular diffusivity and D m is the coefficient of mass diffusivity. The mathematical formulation is an extension of Anwar Beg and Ghosh [5], **Dufour** and Soret effects were added to the existing equations. The fluid is assumed to have constant properties except for the influence of the density variations with temperature and concentration, which are considered only in the body force term. under the above assumptions the physical variables are functions of y and t.

In this paper, the **Dufour** and Soret effects on unsteady MHD free convection heat and mass transfer flow past an infinite vertical porous plate in the presence of radiation are provided. The dimensionless governing equations of the flow have been solved numerically by applying Galerkin FEM. It is shown that the flow characteristics are influenced by the material parameters involved in the problem. It has been found that the fluid velocity and temperature decrease when the Prandtl **number** increases whereas an increase in the radiation parameter enhances the fluid velocity and temperature in the boundary layer. The fluid velocity increases with increasing **Dufour** and Soret numbers. An increase in the Schmidt **number** leads to a decrease in the fluid velocity and concentration. Influence of the magnetic parameter decelerates the fluid velocity. Further, the velocity, temperature and concentration increase with increasing time parameter. The present study of the physics of the fluid flow could be useful in the scientific and engineering applications.

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cross-diffusion impacts are peculiar to nanofluids, and at the same time investigating the interaction between these effects when the base fluid of the nanofluid is itself a binary fluid like salty water. Recently, Loganathan et al. [12] observed that the local heat transfer rate enhances with an enhancement in **Dufour** and Soret numbers for both air and water by analyzing the problem of the cross-diffusion effects on the unsteady natural convection and Srinivasacharya and Surender [13] studied and presented the effect of double stratification along with **Dufour** and Soret type diffusivities on mixed convection boundary layer flow of a nanofluid past a vertical plate in a porous medium.

In the past few decades, great observations of double- diffusive convection have been rapidly investigated by many scientists because of their applications in various physical phenomena, such as oceanography, crystal production, metallurgy, and astrophysics. Double-diffusive convection is a process of interaction of two fluid components that diffuse at imbalance rates. In thermosolutal convection, buoyancy forces can arise not only from density differences due to variation in temperature gradient but also from those due to variation cross diffusive coefficients viz., Soret and **Dufour** effects. In [1] studied oceanographical phenomena where they found that the ascending (or descending) water in the tube would exchange heat but not salinity with the ambient ocean and would be accelerated due to its deficit in salt and density relative to fluid at the same level outside the tube. In [2] investigated the thermohaline convection and discussed the stability characteristics in laminar flow. In [3] analyzed the effect of the thermal and solutal gradient in thermosolutal convective instability in a double-diffusive fluid layer where both effects either can stabilize or destabilize the stationary and oscillatory mode. In [4] demonstrated the soret-driven