The numerical analysis of the effect of Grashof number, modified Grashof number and chemical reaction on the non-Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet in a porous medium

Received: 11/May/2019, Accepted: 02/Jun/2019, Online: 30/Jun/2019 AbstractIn this paper, we have studied the numerical analysis of the effect of Grashof number , modified Grashof number and chemical reaction on the non-Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet in a porous medium. In the mathematical model, using similarity variables, the momentum , energy and concentration equations are transformed to non-dimensional ordinary differential equations.. And then these are solved numerically using bvp4c method, a Matlab inbuilt bvp4c-programm. A discussion for the effects of the parameters involved on the boundary layer regions and the magnitude of the velocity, temperature and concentration and Local skin friction , Local Nusselt Number and Local Sherwood Number have been done graphically and numerically using figures and tables.


I. INTRODUCTION
Fluid flow over a stretching sheet has many important applications: in polymer sheet manufacturing, in chemical engineering, and in metal processing in metallurgy etc. Crane [1] first initiated the study of flow of Newtonian viscous incompressible fluid over a linearly stretching sheet. He investigated the flow of viscous incompressible fluid along a stretching plate whose velocity is proportional to the distance from the slit; such situation occurs in drawing of plastic films. The study was extended to non-Newtonian fluids by many researchers. Rajagopal [2] studied the flow of viscous incompressible fluid on moving (stretching) surface in the boundary layer region. Ishak et al. [3] investigated the MHD flow of viscous incompressible fluid along a moving wedge under the condition of suction and injection.
Non -Newtonian types of flow occurs in the drawing of plastic films and artificial fibres . The moving fibre produces a boundary layer in the medium. Surrounding medium of the fibre is of technical importance; in that it governs the rate at which the fibre is cooled and this in turn affects the final properties of the yarn. Some of the studies on non-Newtonian fluid are as follows.
Siddappa et al. [4] investigated the flow of visco-elastic fluid (a non-Newtonian fluid) of 'Walters's liquid B Model' for the boundary layer flow past a stretching plate. Andersson [5] investigated the flow of viscoelastic fluid along a stretching sheet in the presence of transverse magnetic field. Dandapat [6] investigated the effect of transverse magnetic field on the stability of flow of viscoelastic fluid over a stretching sheet. Fang [7] studied that variable transformation method can be used to get the solution of extended Blasius equation from original Blasius equation. Mamaloukas et al. [8] have discussed some alike nature of free-parameter method and separation of variable method and have found exact solution of equation representing flow of two-dimensional viscoelastic second grade fluid over a stretching sheet. Khidir [9] used spectral homotopy perturbation method and successive linearization method to solve Falker-Skan equation (A nonlinear boundary value problem).
Bataller [10] investigated the flow in the boundary layer of the viscous incompressible fluid under two situations: one about a moving plate in a quiescent ambient fluid (Sakiadis flow) and another uniform free stream flow over a resting flat-plate (Blasius flow).
Motsa, et al. [11] investigated the MHD boundary layer flow of upper-convected Maxwell (UCM) fluid over a porous stretching surface. Motsa et al. [12] had analysed the MHD flow of viscous incompressible fluid over a nonlinearly stretching sheet. Rosca [13] discussed the flow of viscous electrically conducting fluid over a shrinking surface in the presence of transverse magnetic field. Nadeem et al. [14] investigated the MHD boundary layer flow of Williamson fluid over a stretching sheet. Mukhopadhyay [15] analysed the axis symmetric boundary layer flow of viscous incompressible fluid along a stretching cylinder in the presence of uniform magnetic field and under partial slip conditions. Akbar et al. [16] investigate the MHD boundary layer flow of Carreau fluid over a permeable shrinking sheet. © 2019, IJSRPAS All Rights Reserved 51 Nadeem et al. [17] investigated the MHD boundary layer flow of a Casson fluid over an exponentially shrinking sheet. Biswas et al. [18] studied the effects of radiation and chemical reaction on MHD unsteady heat and mass transfer of Casson fluid flow past a vertical plate. Ahmmed et al. [19] analysed the unsteady MHD free convection flow of nanofluid through an exponentially accelerated inclined plate embedded in a porous medium with variable thermal conductivity in the presence of radiation. Biswas et al. [20] investigated the effects of Hall current and chemical reaction on MHD unsteady heat and mass transfer of Casson nanofluid flow through a vertical plate.
Noor et al. [21] investigated the hall current and thermophoresis effects on MHD mixed convective heat and mass transfer thin film flow. Sharada et al. [22] studied MHD mixed convection flow of a Casson fluid over an exponentially stretching surface with the effects of soret, dufour, thermal radiation and chemical reaction. Mukhopadhyay et al. [23] investigated exact solutions for the flow of Casson fluid over a stretching surface with transpiration and heat transfer effects. Pandya et al. [24] have discussed combined effect of variable permeability and variable magnetic field on mhd flow past an inclined plate with exponential temperature and mass diffusion with chemical reaction through porous media.
Kala [25] studied the analysis of non-Darcy MHD flow of a Casson fluid over a non-linearly stretching sheet with partial slip in a porous medium.
This work is the extension of the work [25] in which analysis of non-Darcy MHD flow of a Casson fluid over a non-linearly stretching sheet with partial slip in a porous medium is studied.
This work deals with the numerical analysis of the effect of Grashof number , modified Grashof number and chemical reaction on the non-Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet in a porous medium.

II. MATHEMATICAL MODELLING
We consider steady two-dimensional laminar boundary-layer flow of viscous, incompressible, electrically conducting non-Newtonian Casson fluid in a saturated homogeneous non-Darcy porous medium caused by nonlinearly stretching sheet, which is inclined with an acute angle ( ) to the vertical , placed at the bottom of the porous medium. The xaxis is taken along the stretching surface in the direction of the motion while the y-axis is normal to the surface. A Cartesian coordinate system is used. The -axis is along the direction of the continuous stretching surface (the sheet) and y-axis is normal to the -axis. The sheet is assumed to be stretched along the -axis, keeping the position of the origin unaltered and stretching velocity varies nonlinearly with the distance from the origin. A uniform magnetic field of strength B is applied normal to the sheet.
It is assumed that the fluid is optically dense, non-Newtonian, and without phase change. Flow region is in non-Darcy porous medium. This integrates a linear Darcian drag for low velocity effects (bulk impedance of the porous matrix at low Reynolds numbers) and a quadratic (second order) resistance, the Forchheimer drag force, for high velocity flows, as may be come across in chemical engineering systems operating at higher velocities. Brinkman's equation takes into account the boundary effects (the viscous force).
It is assumed that the induced magnetic field, the external electric field and the electric field due to polarization of charges are negligible in comparison to the applied magnetic field. So, all of the Hall effects and Joule heating effects are neglected.   The equation of continuity: The Equation of Momentum: The Equation of Energy: The Equation of Mass concentration: (4) where, the sign  refers to the cases of assisting and opposing flow (here we shall consider the case of assisting flow which is shown by positive sign), The strength of the magnetic field is assumed to vary spatially by The sheet is assumed to move with power law velocity, and varies nonlinearly in spatial coordinates with some index, in the boundary layer region, so that relevant velocity boundary conditions for equations (1) to (4) are as follows: . , , , 0 : where u w is the surface velocity of the sheet with Using equations (6), equations (1) to (4) can be written as (9) And boundary conditions (5) as , Here prime denotes differentiation with respect to  .
The parameters occurring in equations (7) to (10) are defined as follows: stands for generative chemical reaction. The local Skin-friction coefficient is Local Sherwood number:

III. METHOD OF NUMERICAL SOLUTION
The numerical solutions are obtained using the above equations for some values of the governing parameters, namely, the Magnetic parameter ( ), the Permiability porosity parameter (Kp), the Forchhemier parameter(Fs), inclination parameter(

IV. RESULT AND ANALYSIS
In order to validate the method used in this study and to judge the accuracy of the present analysis, comparisons with available results of Andersson [4] ,Mahdy [16] and Ahmed [17] ,corresponding to the skin-friction coefficient when are presented in Table 1. As it can be seen, there are excellent agreements between the results. so we are confident that the present numerical method works very efficiently.
For drawing from figures 2 to 40 and from tables 3 to 20 following common parameter values are considered:                           . Figure 9 shows velocity boundary layer thickness and the magnitude of the velocity increase and this increase speeds up as the modified Grashof parameter  increases.   . Figure 11 shows thermal boundary layer thickness and the magnitude of the temperature decrease and this decrease speeds up as the modified Grashof parameter  increases.                                     Table 9 Local skin friction ) 0 ( ' ' f with respect to variation in  and Pr .   Table 11 Local Sherwood Number with respect to variation in  and Pr . Table 12 Local skin friction with respect to variation in and Sc .  ) with respect to variation in  and n .

V. CONCLUSION
In this paper, we have studied numerically the effect of Grashof number , modified Grashof number and chemical reaction on the non-Darcy MHD flow of a Casson fluid over a nonlinearly stretching sheet in a porous medium. In the mathematical model, using similarity variables, the momentum , energy and concentration equations are transformed to non-dimensional ordinary differential equations. These equations are solved numerically using bvp4c method, a Matlab in-built bvp4c-programm. A discussion for the effects of the parameters involved on the boundary layer regions and the magnitude of the velocity, temperature and concentration and local Skin friction , Local Nusselt number and Local Sherwood number have been done graphically and numerically using figures and tables. From this investigation, we have drawn the following conclusions: From the graphs we have following conclusion: As the Prandtl number Schmidth number the Chemical reaction parameter, the Casson parameter increases, velocity boundary layer thickness and magnitude of velocity decreases. (ii) As the Chemical reaction parameter, Schmidth number the Casson parameter increases thermal boundary layer thickness and magnitude of temperature increases. (iii) As the Prandtl number, the Schmidth number , the Casson parameter, the stretching index parameter increases, concentration boundary layer thickness and magnitude of concentration increases. (iv) As the Grashof parameter, modified Grashof parameter , the stretching index parameter increases, velocity boundary layer thickness and magnitude of velocity increases. (v) As the Grashof parameter , modified Grashof parameter , the Prandtl number, the stretching index parameter increases, the thermal jump(thermal slip) parameter thermal boundary layer thickness and magnitude of temperature decreases. (vi) As the Grashof parameter , modified Grashof parameter, the Schmidth number, the Chemical reaction parameter increases concentration boundary layer thickness and magnitude of concentration decreases. From the tables we have following conclusion: