MHD CASSON FLUID FLOW WITH CHEMICAL REACTION AND SUCTION IN A POROUS MEDIUM OVER AN EXPONENTIALLY STRETCHING SHEET.

MHD Casson fluid flow with chemical reaction and suction in a porous medium over an exponentially stretching sheet was studied. The mathematical problem was solved numerically using shooting technique with fourth order Runge-Kutta method after converting the resulting partial differential equations to a system of ordinary differential equations and applying similarity transformation on the latter. The effects of some fluid parameters on the momentum and concentration profiles are examined through graphs using MATLAB. The results from the graphs show that increase in the values of Casson parameter decreases the velocity profiles and causes increase in the concentration profiles. The momentum boundary layers decrease while the solute boundary layers increase with increasing values of magnetic parameter and the permeability parameter. The concentration profile reduces with increasing values of reaction parameters.


Introduction
There has been growing interest in the study of fluid flow over stretching sheets, particularly Newtonian and non-Newtonian fluids, due to their use in various industrial processes such as polymer processing and glass fiber production.The study of magnetic field effect on electrically conducting fluid is important and has various industrial applications, including magnetic materials processing, crude oil purification, and magneto hydrodynamic power generation.(Crane, 1970) performed the first study on steady boundary layer flow caused by a linear stretching sheet.(Aly and Ebaid, 2016) examined the exact analysis for the effect of heat transfer of MHD, and radiation Marangoni boundary layer nano fluid flow past a surface embedded in a porous medium.(Turkyilmazoglu, 2016) explored the flow of a micro polar fluid due to a porous stretching sheet and heat transfer.(Jain and Choudhary, 2015) investigated the effects of MHD on boundary layer flow in a porous medium due to exponentially shrinking sheet with slip.(Reddy, 2016) discussed the MHD flow of a Casson fluid over an exponentially inclined permeable stretching surface with thermal radiation and chemical reaction.(Ganesh et al., 2018) explored Darcy-Forchheimer flow of hydromagnetic nano fluid over a stretching or shrinking sheet in a thermal stratified porous medium with secondorder slip, viscous and ohmic dissipations effect.Meanwhile, (Abel et al., 2016) analysed MHD viscoelastic boundary layer flow and heat transfer past a convectively heated radiation stretching or shrinking sheet with temperature-dependent heat source or sink.(Bhattacharyya et al., 2014) (Alam et al.,2016) analyzed the effects of variable fluid properties and thermophoresis on an unsteady forced convective boundary layer flow along a permeable stretching or shrinking wedge with variable Prandtl and Schmidt numbers.(Chaudhary and Chaudhary, 2016) studied the heat and mass transfer by MHD flow near the stagnation point over a stretching or shrinking sheet in a porous medium.(Ali et al., 2015) reported on analytical solution for fluid flow over an exponentially stretching porous sheet with surface heat flux in a porous medium by means of homotopic analysis method.(Ekang et al., 2021) examined MHD heat and mass flow of Nano-fluid over a non-linear permeable stretching sheet.(Senge et al., 2020) investigated the influence of radiation on magnetohydrodynamics flow over an exponentially stretching sheet embedded in a thermally stratified porous medium in the presence of heat source.(Nabeela and Rashid, 2013) found solutions for non-similarity boundary layer flow in porous medium.(Mat et al., 2015) studied boundary layer stagnation-point slip flow and heat transfer towards a stretching or shrinking cylinder over a permeable surface.(Chakrabarti and Gupta,1979) investigated hydromagnetic flow and heat transfer over a stretching sheet.In this study, we analyze MHD Casson fluid flow with chemical reaction and suction in a porous medium over an exponentially stretching sheet.The source terms for magnetic field and porous medium are introduced on the flow.

Mathematical Formulation
Let us consider the flow of an incompressible viscous fluid past a flat sheet coinciding with the plane  = 0.The fluid flow is confined to  > 0. Two equal and opposite forces are applied along the x axis so that the wall is stretched keeping the origin fixed.The standard rheological equation of state for an isotropic and incompressible flow of a Casson fluid is where  =     and   is the ( , )th component of the deformation rate,  is the product of the deformation rate with itself,  is the critical value of the product of the component of the rate of strain tensor with itself,   is the plastic dynamic viscosity of casson fluid and   is the yield stress of the fluid.
The positive  coordinate is measured along the sheet and the positive  coordinate is measured perpendicular to the sheet.
The following assumptions are made to derive the governing equations; fluid suction and magnetic field are imposed at the plate surface.Let   be the concentration of the fluid at the surface which is assumed to be greater than the ambient concentration of the fluid,  ∞ , given by where  ∞ , and  0 are constants.By these assumptions the governing fluid equations are as follows; (2) where ,  are the velocity respectively in the ,  directions,  is the kinematic viscosity, ⍴ is the fluid density (assumed  2) and (7), the governing equations reduce to: The transformed boundary conditions are: ′ (0) = 1, (0) = 0, (0) = 1,   = 0,  ′ () → 0, () → 0,   → ∞ (9) where the prime denotes differentiation with respect to  and the dimensionless parameters are as follows: is the reaction rate parameter, where  > 0,  = 0,  < 0 is the destructive reaction, absence of reaction and generative reaction respectively.Hence the dimensionless form of Skin friction (  ) and the Local Nusselt number (  ) are given by; where   =    is the local Reynolds number.

Numerical Solution
The dimensionless equations ( 7) and ( 8) subject to the boundary conditions ( 9) are nonlinear equations, hence solving numerically, we convert them to a system of equations by setting the following: Let where satisfying where,  1 and  2 are guessed such that These equations are solved numerically using shooting technique with fourth order Runge-Kutta method.The boundary condition as  → ∞ was replaced by a finite value in accordance with standard practice in the boundary layer analysis.We set  ∞ = 10

Results and Discussion
The effects of the dimensionless governing parameters namely: Casson parameter (), Schmidt number (Sc), magnetic parameter (), permeability parameter () and Reaction rate parameter () on the velocity and temperature distribution profiles are analyzed numerically using the method mentioned in the previous section.To do this, we set the fluid properties to be constant.Numerical values are generated and using MATLAB, the following graphs are plotted by varying the fluid properties with basics at:  = 2.0,  = 0.5,  = 0.2,  = 1.0,  = 0 investigated dual solutions in boundary layer flow of Maxwell fluid over a porous shrinking sheet.(Chaudhary et al., 2016) obtained numerical solution for MHD stagnation point flow towards a stretching or shrinking surface in a saturated porous medium.
be constant),  =  √ 2    is the parameter of the Casson fluid,  is the diffusion coefficient of the diffusing species in the fluid,  =  0    is the exponential reaction rate;  0 is a constant.The boundary conditions are:   = 0,  = ,  = 0,  =     → ∞,  → 0,  →  ∞(4)Here,  =  0    is the stretching velocity,   is the concentration at the sheet,  0 and  0 are the reference velocity and reference concentration, respectively.Now, we introduce the stream function (, ), defined by; equations into equations ( Figure 1 depicts the effect of Casson parameter  on velocity profiles.The velocity profile decreases as the Casson parameter () value increases.The Casson parameter produces a resistance in the fluid flow and consequently the boundary layer thickness decreases for higher value of Casson parameter.Figures 2, 3, 4 show the effect of Casson parameter  on concentration profiles.Increasing the values of the Casson parameter  increases the concentration profiles and the solute boundary layer thickness.It is noted that the effect of  is much more pronounced in the case of constructive chemical reaction.

Figures 7
Figures7 and 8show the effect of Schmidt number on concentration in the absence of chemical reaction (when  = 0) and the presence of destructive chemical reaction (when  > 0).The concentration profiles decrease with increasing values of Schmidt number.The reaction rate parameter is a decelerating agent and as a result the solute boundary layer becomes thinner.Figure9shows the nature of the concentration profiles for different values of Schmidt number in the presence of constructive reaction (when  < 0).An increase in Schmidt number increases the solute boundary layer thickness.Here, it is observed that the concentration profiles increase as the values of the Schmidt number increase.Figures 10 and 11 examine the effect of magnetic parameter, , and the permeability parameter, , on the concentration profiles respectively.Increasing the values of magnetic parameter, , and the permeability parameter, , increases the concentration profiles and the solute boundary layers.Figure12displays the rate of reaction  against concentration profiles.It is found that as  increases, concentration reduces.It is noted that the reaction rate parameter is a decelerating agent, as a result, the concentration boundary layer decreases.ConclusionThe effects of the fluid parameters on the heat transfer are examined with the help of graphs.From the graph we can see that increase in the values of Casson parameter, decreases the velocity profiles and causes increase in the concentration profiles.The momentum boundary layers decreases while the solute boundary layers increase with increasing values of magnetic parameter and the permeability parameter.The concentration profile reduces with increasing values of reaction rate parameters.The solute boundary layers decrease with increasing values of Schmidt number in the absence of chemical reaction and the presence of destructive chemical reaction while they increase in the presence of constructive chemical reaction.