MHD Flow of Fluid over a Rotating Inclined Permeable Plate with Variable Reactive Index

MHD free convection, heat and mass transfer flow over a rotating inclined permeable plate with the influence of magnetic field, thermal radiation and chemical reaction of various order has been investigated numerically. The governing boundary-layer equations are formulated and transformed into a set of similarity equations with the help of similarity variables derived by lie group transformation. The governing equations are solved numerically using the Nactsheim-Swigert Shooting iteration technique together with the Runge-Kutta six order iteration schemes. The simulation results are presented graphically to illustrate influence of magnetic parameter   M , porosity parameter    , rotational parameter   R , Grashof number   r G , modified Grashof number   m G , thermal conductivity parameter   c T , Prandtl number   r P , radiation parameter   R , heat source parameter   Q , Eckert number   c E , Schmidt number   c S , reaction parameter    Original Research Article Wahiduzzaman et al.; PSIJ, 6(3): 144-162, 2015; Article no.PSIJ.2015.041 145 andorder of chemical reaction   n on the all fluid velocity components, temperature and concentration distribution as well as Skin-friction coefficient, Nusselt and Sherwood number at the plate.


INTRODUCTION
Coupled heat and mass transfer problems in the presence of chemical reactions are of importance in many processes and have, therefore, received considerable amount of attention of researchers in recent years. Chemical reactions can occur in processes such as drying, distribution of temperature and moisture over agricultural fields and groves of fruit trees, damage of crops due to freezing, evaporation at the surface of a water body, energy transfer in a wet cooling tower and flow in a desert cooler. Chemical reactions are classified as either homogeneous or heterogeneous processes. A homogeneous reaction is one that occurs uniformly throughout a given phase. On the other hand, a heterogeneous reaction takes a restricted area or within the boundary of a phase. Analysis of the transport processes and their interaction with chemical reactions is quite difficult and closely related to fluid dynamics. Chemical reaction effects on heat and mass transfer has been analyzed by many researchers over various geometries with various boundary conditions in porous and nonporous media. Symmetry groups or simply symmetries are invariant transformations that do not alter the structural form of the equation under investigation which is described by Bluman and Kumei [1]. MHD boundary layer equations for power law fluids with variable electric conductivity is studied by Helmy [2]. In the case of a scaling group of transformations, the group-invariant solutions are nothing but the well known similarity solutions which is studied by Pakdemirli and Yurusoy [3]. Symmetry groups and similarity solutions for free convective boundary-layer problem was studied by Kalpakides and Balassas [4]. Makinde [5] investigated the effect of free convection flow with thermal radiation and mass transfer past moving vertical porous plate. Seddeek and Salem [6] investigated the Laminar mixed convection adjacent to vertical continuously stretching sheet with variable viscosity and variable thermal diffusivity. Ibrahim, Elaiw and Bakr [7] studied the effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction. El-Kabeir, El-Hakiem and Rashad [8] studied Lie group analysis of unsteady MHD three dimensional dimensional by natural convection from an inclined stretching surface saturated porous medium. Rajeswari, Jothiram and Nelson [9] studied the effect of chemical reaction, heat and mass transfer on nonlinear MHD boundary layer flow through a vertical porous surface in the presence of suction. Chandrakala [10] investigated chemical reaction effects on MHD flow past an impussively started semi-infinite vertical plate. Joneidi, Domairry and Babaelahi [11] studied analytical treatment of MHD free convective flow and mass transfer over a stretching sheet with chemical reaction. Muhaimin, Kandasamy and Hashim [12] studied the effect of chemical reaction, heat and mass transfer on nonlinear boundary layer past a porous shrinking sheet in the presence of suction. Rahman and Salahuddin [13] studied hydromagnetic heat and mass transfer flow over an inclined heated surface with variable viscosity and electric conductivity. Very recently a number of studies of MHD heat and mass transfer boundary layer fluid flow with the various effects of physical parameter reported in the literature [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. As per standard text and works of previous researchers, the radiative flow of an electrically conducting fluid and heat and mass transfer situation arises in many practical applications such as in electrical power generation, astrophysical flows, solar power technology, space vehicle re-entry, nuclear reactors.
The objective of this study is to present a similarity analysis of boundary layer flow past a rotating inclined permeable plate with the influence of magnetic field, thermal radiation, thermal conductivity and chemical reaction of various orders.

MATHEMATICAL MODEL OF THE FLOW AND GOVERNING EQUATIONS
Steady two dimensional MHD heat and mass transfer flow with chemical reaction and radiation over an inclined permeable plate 0 y  in a rotating system under the influence of transversely applied magnetic field is considered. The sketch of the physical configuration and coordinate system are shown in Fig. 1. The xaxis is taken in the upward direction and y-axis is normal to it. Again the plate is inclined at an angle  with the x-axis. The flow takes place at 0 y  , where y is the coordinate measured normal to the x-axis. Initially we consider the plate as well as the fluid is at rest with the same Also it is assumed that the fluid and plate is at rest after that the whole system is allowed to rotate with a constant angular velocity   0, , 0 R   about the y-axis and then the temperature and species concentration of the plate are raised to respectively, which are thereafter maintained constant, where w T and w C is the temperature and concentration respectively at wall and T  and C  is the temperature and concentration respectively far away from the plate.
The electrical conductivity is assumed to vary with the velocity of the fluid and have the form The applied magnetic field strength is considered, as follows [13] The temperature dependent thermal conductivity is assumed to vary linearly, as follows [6]     Where k  is the undisturbed thermal conductivity and c is the constant depending on the properties of the fluid.
The governing equations for the continuity, momentum, energy and concentration in laminar MHD incompressible boundary-layer flow is presented follows and the boundary conditions for the model is k  for generative reaction, n (integer) is the order of chemical reaction, r q is the chemical reaction parameter, w T and w C is the temperature and concentration respectively at wall and T  and C  is the temperature and concentration respectively far away from the plate.

Method of Solution
Introducing the following dimensionless variables Now, by using equation (7), the equations (1), (2), (3), (4) and (5) are transformed to using equation (7), the boundary condition (6) becomes, Where, In order to deal with the problem, we introduce the stream function  (since the flow is The mathematical significance of using equation (14) is that the continuity equation (8) is satisfied automatically.
by equation (14), equations (9), (10), (11) and (12) transformed as follows, and the boundary conditions (13) become, Finding the similarity solution of the equations (15) to (18) Here,   0   is the parameter of the group and ' s  are arbitrary real numbers whose interrelationship will be determined by our analysis. Equations (20) Again, the following relations are obtained from the boundary conditions (19), Solving the system of linear equations (21) and (22), the following relationship are obtained, In terms of differentials Solving the equation (24) the following similarity variables are introduced, By using the above mentioned variables, equations (15), (16), (17) and (18) The corresponding boundary conditions (19) become where primes denote differentiation with respect to  only and the parameters are defined as is the reaction parameter and n (integer) is the order of chemical reaction

Skin-friction Coefficients, Nusselt and Sherwood Number
The physical quantities of the skin-friction coefficients, the reduced Nusselt number and reduced Sherwood number are calculated respectively by the following equations, is the Reynolds number.

RESULTS AND DISCUSSION
Heat and mass transfer problem associated with laminar flow past an inclined plate of a rotating system are studied in this work. In order to investigated the physical representation of the problem, the numerical values of primary velocity, secondary velocity, temperature and species concentration from equations (25), (26), (27) and (28)

. Effect of rotational parameter on a) primary velocity b) secondary velocity c) temperature d) concentration profiles
In Figs. 6a-6c, we see that with the increase of Grashof number, primary velocity profiles increases but secondary velocity and temperature profiles decreases. Figs. 7a-7c show that with the increase of modified Grashof number, primary velocity profiles increases but secondary velocity and concentration profiles decreases. With the increase of Prandtl number, primary velocity profiles increases but temperature decreases (Figs. 8a and 8b). Fig. 9a, It is observe that the primary velocity profile increases with the increase of Eckert number. In Fig. 9b, temperature profile increases with the increase of Thermal conductivity parameter.
In Fig. 10a, concentration profiles decreases with the increase of Schmidt number. Fig. 10b represents no reaction    Figs. 13a and 13b represent that primary shear stress decreases but secondary shear stress increases with the increase ofporosity parameter.
In Fig. 14a and Fig. 14b, primary shear stress increases with the increase ofGrashof number and modified Grashof number respectively.
In Fig. 15a, primary shear stress decreases with the increase ofinclination angle. In Fig. 15b, the heat transfer rate increases with the increase ofthermal conductivity parameter. Fig. 16a, the heat transfer rate decreases with the increase ofPrandtl number. Fig. 16b, the heat transfer rate increases with the increase ofheat source parameter.
In Fig. 17a -17b, the heat transfer rate increases with the increase ofEckert number andradiation parameter. Fig. 18a -Fig. 18b, the mass transfer rate decreases with the increase ofSchmidt number and reaction parameter. Fig. 19 represents that the mass transfer rate increases with the increase oforder of chemical reaction parameter.

CONCLUSION
Primary velocity profiles decreases and primary shear stress with the increase of magnetic parameter, rotational parameter, but reverse effect is found for the secondary velocity profiles and secondary shear stress. Primary shear stress decreases due to increase of magnetic parameter where as the reverse effect is found for secondary shear stress.
Temperature and concentration boundary layer thickness increases due to increase of rotational parameter.
The primary velocity profiles and primary shear stress decreases due to increase of permeability of the porous medium and inclination angle but reverse effect is found for the secondary velocity profiles and secondary shear stress. Temperature and concentration boundary layer thickness are increases due to increase of permeability of the porous medium.
The primary velocity profiles and primary shear stress increases due to increase of Grashof number where as the reverse effect is found for the secondary velocity profiles. Also the temperature boundary layer thickness is decreases due to increase of Grashof number.
The primary velocity profiles and primary shear stress increases due to increase of modified Grashof number where as the reverse effect is found for the secondary velocity profiles. Also the concentration boundary layer thickness decreases due to increase of modified Grashof number.
The primary velocity profiles increases due to increase of Prandtl number. The thermal boundary layer thickness as well as the heat transfer rate at the plate decreases as the Prandtl number increases.
The heat transfer rate at the plate as well as the primary velocity is increases due to increase of Eckert number and thermal conductivity parameter.
The heat transfer rate at the plate increases due to increase of heat source parameter and radiation parameter.
The concentration boundary layer thickness as well as the mass transfer rate at the plate decreases due to increase of Schmidt number, no reaction and destructive reaction.
The mass transfer rate at the plate increases due to increase of order of chemical reaction.