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In this work, the Micropolar fluid flow and heat and mass transfer past a horizontal nonlinear stretching sheet through porous medium is studied including the Soret-Dufour effect in the presence of suction. A uniform magnetic field is applied transversely to the direction of the flow. The governing differential equations of the problem have been transformed into a system of non-dimensional differential equations which are solved numerically by Nachtsheim-Swigert iteration technique along with the sixth order Runge-Kutta integration scheme. The velocity, microrotation, temperature and concentration profiles are presented for different parameters. The present problem finds significant applications in hydromagnetic control of conducting polymeric sheets, magnetic materials processing, etc.

The natural convection processes involving the combined mechanism of heat and mass transfer are encountered in many natural and industrial transport processes such as hot rolling, wire drawing, spinning of filaments, metal extrusion, crystal growing, continuous casting, glass fiber production, paper production, and polymer processing, etc. Ostrach [

Micropolar fluids, distinctly non-Newtonian in nature, are referred to those that contain micro-constituents belonging to a class of complex fluids with nonsymmetrical stress tensor. These fluids respond to micro-rotational motions and spin inertia, and therefore can support couple stress and distributed body torque which are not achievable with the classical Navier-Stokes equations or the viscoelastic flow models. The Micropolar fluid models are developed to make an analysis of the flow characteristics of physiological fluids (blood containing corpuscles), colloidal suspensions, paints, liquid crystal suspensions, concentrated silica particle suspensions, oils containing very fine suspensions, industrial contaminants containing toxic chemicals, lubricants, organic/inorganic hybrid nano-composites and clay which are fabricated by melt intercalation etc. Eringen [

Crane [

An important matter is that the final product depends to a great extent on the rate of cooling. By drawing such strips in an electrically conducting fluid subjected to a magnetic field, the rate of cooling can be controlled and a final product of desired characteristics can be achieved. The study of heat and mass transfer is necessary for determining the quality of the final product. Sparrow [

The boundary layer models for steady or unsteady micropolar fluids in various geometries (stationary or moving surface, linear or nonlinear stretching surface etc.) with/or without heat transfer considering various flow conditions (no slip or slip, suction/injection at the surface) and thermal boundary conditions (constant/variable surface temperature or heat flux) have extensively been studied by numerous researchers [7-16].

Moreover, the thermal-diffusion (Soret) effect, for instance, has been utilized for isotope separation, and in mixtures between gases with very light molecular weight (Hz, He) and of medium molecular weight (Nz, air) the diffusion-thermo (Dufour) effect was found to be of a considerable magnitude such that it cannot be ignored, described by Eckert and Drake [

From the above referenced work and the numerous possible industrial applications of the problem, it is of paramount interest in this study in order to clarify the parametric behavior of magneto-hydrodynamic flow of free convection of a micropolar fluid over a nonlinear stretching sheet in the presence of dynamic effects of suction, thermal-diffusion and diffusion-thermo.

We consider the isothermal, steady, laminar, hydromagnetic free convection flow of an incompressible micropolar fluid flowing past a nonlinear stretching sheet coinciding with the plane, the flow being confined in the region. The flow under consideration is also subjected to a strong transverse magnetic field with a constant intensity along the y-axis.

Two equal and opposite forces are introduced along the x-axis so that the surface is stretched keeping the origin fixed. The flow configurations and the coordinate system are shown in

In Equation (2) the Darcian porous drag force term is defined by the term, , which is linear in terms of the translational velocity, u. With, the micropolar effects disappears and this term reduces to the conventional Newtonian Darcy drag force i.e..

The micro-rotation component, N, is coupled to the linear momentum Equation (2) via the angular velocity gradient term,. Very strong coupling exists between the translational velocity components, u and v, in Equation

(3) via the convective acceleration terms, and. Furthermore, there is a second coupling term linking the angular velocity with the x-direction velocity gradient, in Equation (3),. The microrotation viscosity (or spin-gradient viscosity) is defined by (Rahman [_{p} the specific heat at constant pressure, is the chemical molecular diffusivity, is the Thermophoretic constant, is the Mean fluid temperature and c_{s} is the Concentration susceptibility.

The appropriate boundary conditions suggested by the physical conditions are:

1) on the plate surface at:

2) matching with the quiescent free stream as:

where the subscripts w and refer to the wall and boundary layer edge, respectively. The relationship between the microrotation function N and the surface shear

is chosen for investigating the effect of different surface conditions for the microrotation of the micropolar fluid elements. The conditions are generally of importance in micropolar boundary layer analysis. When microrotation parameter, we obtain which represents no-spin condition i.e. the microelements in a concentrated particle flow-close to the wall are not able to rotate (Rahman [

The partial differential Equations (1) to (5) are transformed into non-dimensional form by mean of following dimensionless variables

Implementing Equation (8) into Equations (1) to (5) produces the following ordinary differential equations:

and corresponding boundary conditions are reduce to:

where the primes denote differentiation with respect to

(non-dimensional y-coordinate) and is the vortex viscosity parameter, is the local Grashof number, is the local magnetic parameter and is the magnetic field, is the Darcy number, is the micro-inertia density parameter, is the Prandtl number, is the Eckert number,

is the Dufour number, is the Schmidt number, is the Soret number and

is the suction parameter.

The parameters of engineering interest for the present problem are the skin friction coefficient, plate couple stress, local Nusselt number and Sherwood number which indicate physically the wall shear stress, couple stress, the rate of heat transfer and the local surface mass flux respectively. The dimensionless skin-friction coefficient, Couple stress, Nusselt number and Sherwood number for impulsively started plate are given by

where is the Reynolds number. And hence the values proportional to the skin-friction coefficient, couple stress, Nusselt number and Sherwood number are and respectively.

The numerical solutions of the non-linear differential Equations (9) to (12) under the boundary conditions (13) have been performed by applying a shooting method namely Nachtsheim and Swigert [^{−6} in all cases. The value of has been found to each iteration loop by. The maximum value of to each group of parameters and has been determined when the values of the unknown boundary conditions at not change to successful loop with error less than 10^{−6}. In order to verify the effects of the step size, we have run the code for our model with three different step sizes as Δη = 0.01, Δη = 0.005 and Δη = 0.001, and in each case we have found excellent agreement among them shown in Figures 2-5.

For the purpose of discussing the results of the flow field represented in the

Due to free convection problem positive large values of is chosen. The value of and. However, numerical computations have been carried out for different values of the vortex viscosity parameter, surface nonlinearity parameter, Eckert number, constant parameter, Dufour number, Soret number and suction parameter. The numerical results for the velocity, microrotation, temperature and concentration profiles are displayed in Figures 6-33.

The effects of the surface nonlinearity constant n are characterized in the Figures 10-13. At the beginning the velocity profiles decrease with the increase of the value of but far away from the plate they increase after displayed in

Figures 14-17 exhibit the velocity, microrotation, temperature and concentration profiles for the different values of the Eckert number Ec (0.03, 0.1, 0.5 and 1.0).

Figures 18-21 represent the influence of the constant parameter g for the values. All the profiles except microrotation profiles decrease with the increase of g. The effects of g are very significant smooth on the distributions. The microrotation profiles increase with the increase of the value of g.

It is observed from the

of the value of Du the velocity profiles occur higher. The effect of Du on the microrotation profiles is insignificant illustrated in

Figures 26-29 display the effects of the Soret number Sr on the velocity, microrotation, temperature and concentration profiles respectively. It is observed that Sr has very negligible effect on the velocity, microrotation and temperature profiles.

number Sr influences the concentration profiles to a great extent. Quantitatively when and Du increases from 0.5 to 1, there is 23.08% decrease in the concentration value, whereas the corresponding decrease is 23.07% when Du increases from 2 to 3.

_{w} has strong effect on the velocity profiles. With the increase of the value of fw the velocity profiles decrease. Elaborately when and Du decreases from 0 to 0.5, there is 176.79% decrease in the concentration value, whereas the corresponding decrease is 21.47% when Du increases from 1 to 3. It is observed that, when suction f_{w} increases, the microrotation increase monotonically seen in