NUMERICAL APPROACH OF HEAT AND MASS TRANSFER OF MHD CASSON FLUID UNDER RADIATION OVER AN EXPONENTIALLY PERMEABLE STRETCHING SHEET WITH CHEMICAL REACTION AND HALL EFFECT

In this paper, heat and mass transfer of MHD Casson fluid under radiation over an exponentially permeable stretching sheet with chemical reaction and Hall Effect investigated numerically. Suitable similarity transformations are used to convert the governing partial differential equations to nonlinear ordinary differential equations. Using a numerical technique named Keller box method the equations are then solved. Study of various effects such as chemical reaction, hall effect, suction /injection on magneto hydrodynamic Casson fluid along with radiation the heat source parameter, chemical reaction parameter, Schmidt number are tabulated for various parameters. Also local parameters are calculated and compared with previous literature the results are found to be in good agreement. The velocity, temperature, concentration visual representations are plotted for various parameters using matlab. Skin friction coefficient, Nusselt number and Sherwood number are calculated in both cases of Newtonian and non-Newtonian it is observed that the friction factor and the heat and mass transfer rates reduces for increase in magnetic parameter. Also for progressive values of radiation parameter, thermal grashof number, concentration grashof number and hall parameter, skin friction coefficient, heat and mass transfer rates increases where as they decreases for chemical reaction parameter, Schmidt number.


INTRODUCTION
Experience shows that due to several industrial applications of Non Newtonian fluids such as polymer industry and mining industry attracted many researchers to study about non Newtonian fluids. Also fluid flow over a stretching sheet is extensively used in manufacturing process because there is an association between stretching sheet and fluid that flows on it. Dutta et.al. (1985) analyzed fluid flow on a stretching sheet with uniform heat flux and observed that with increase in Prandtl number first wall temperature increases quickly and then decreases gradually. Masahide and Tadashi (1988) proved that with the non-Newtonian property of blood, flow speed decreases along the stenosis. James and Milivoje (1990a) provided theoretical and practical study of heat transfer effects in both the cases of Newtonian and Non Newtonian fluid. Lai (1990b) derived a closed form of solutions for a special case of Lewis number. Lin and Wu (1995) studied boundary layer flow in case of a vertical plate and the effects of buoyancy ratio and Lewis number on Heat and mass transfer are observed. Magyari and Keller (1999) observed that boundary layer thickness increases on enhancing the value of wall temperature distribution by fixing Prandtl number also by keeping wall temperature fixed and enhancing Prandtl number. Swathi Mukhopadhyay et al. (2005) observed that by expanding the length of the stretching sheet velocity of the fluid decreases by reducing thickness of the fluid. Later, Subhas et al. (2010) analyzed non Newtonian fluid flow through a porous medium along with the effect of suction concluded that wall temperature will be decreased with the effect of viscous dissipation. Vedavathi et.al (2015) used Runge-Kutta fourth order and concluded that suction effect maintains fixed growth of the thermal, concentration, hydrodynamic boundary layers. After that Vedavathi et.al (2017) concluded that on increasing radiation absorption parameter velocity profiles will be enhanced and increasing Prandtl number skin friction number decreases. Talla, Kumari and Sridhar (2018a) studied MHD Casson fluid flow over a exponentially stretching surface and observed that with increase in Casson parameter velocity reduces and concentration increases. Reddy and Krishna (2018b) observed that as the Soret number increases velocity boundary layer, thermal boundary layer, concentration boundary layer diminishes. Chandra Sekhar (2018c) observed that temperature increases with increase of chemical reaction parameter. Konda Reddy et al. (2018d) studied MHD mixed convection flow of a Casson Nano fluid over a nonlinear stretching sheet temperature, concentration enhanced and velocity diminished with increase in Casson parameter. Charan Kumar et al. (2018e) used RK-Fehlberg method to study the effects of joule heating and chemical reaction effects on fluid flow over a stretching sheet along with radiation and porous medium. Anki Reddy and Suneetha (2018f) concluded that temperature of the fluid and thermal relaxation time are oppositely related. Ghiasi and Saleh (2018g) witnessed that casson fluid along with suction effect reduces heat, mass transfer rates. Flilihi et al. (2019a) concluded that fluid temperature raises with increase in dissipation parameter. Dharmaiah et al. (2019b) used perturbation method and observed that chemical reaction parameter dominates the concentration profiles. Sampath and Pai (2019c) found that enhancing magnetic parameter skin friction decreases. Nagaraju et al.(2019d) used HAM technique and concluded that raise in magnetic parameter enhances velocity also energy dissipation step ups causing increase in temperature, later Huang pin et al.(2019e) used Keller box method and observed that suction increases the Nusselt number and Sherwood number. Blowing reduces them. Ganapathirao et al. (2019f) found that skin friction and coefficient of heat transfer raises with enhancing buoyancy parameter. Ibrahim et al. (2019g) observed that velocity increases with increasing buoyancy parameter. Raghunandana Sai and Ramana Murthy (2019h) studied the impact of various parameters on velocity and temperature profiles and observed that with increase in Prandtl number and time temperature increases. Ravi Kumar et al. (2019i) noticed that with

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Available at www.ThermalFluidsCentral.org 2 increase in slip parameter, Grashof's number velocity increases. Kavitha and Naikoti (2019j) used quasi-linearization technique to study the power law fluid flow under the influence of radiation. Swamy et al. (2019k) studied MHD flow in a porous channel with suction and observed that as the intensity of the magnetic field increases fluid velocity reduces. Vijaya and Reddy (2019l) analyzed the MHD Casson fluid flow on a vertical porous plate using Perturbation technique method. Manjula and Chandra Sekhar (2019m) observed Soret and heat generation effects on Casson fluid flow concluded that raise in Soret number values increases buoyancy force hence velocity increases. Sivaiah et al. (2019n) concluded that due to thermal radiation parameter, temperature decreases and with the presence of Eckert number temperature increases. Dharmaiah. et al. (2019o) implemented perturbation technique and noticed that concentration of the fluid reduces with the presence of chemical reaction. Balamurugan et al. (2020a) concluded that entropy generation increases for higher values of magnetic parameter or thermal radiation parameter. Later Vijaya et al. (2020b) used bvp4c technique and noted that for increasing values of magnetic parameter velocity of fluid improved. Dharmaiah. et al. (2020c) studied the influence of hall and ion slip on a Nano fluid and concluded that skin friction coefficient raises with magnetic parameter.Nagalaksmi and vijaya(2020d) used Runge Kutta method and observes for progressive values of Prandtl number thermal boundary layer thickness diminishes. Ibrahim.et al. (2020e) used Homotopy analysis method to study the influence of various parameters like radiation, chemical reaction of MHD casson fluid flow over a stretching sheet.
The study of flow above an exponentially stretching sheet has significant applications in various industries and technological processes for example fluid film condensation process, cooling process of metallic sheets, design of chemical processing equipment, polymer industries. Also the radiation effect on MHD boundary layer flow has various applications in manufacturing industries like glass-fiber production, paper production etc. Hall effect has some industrial applications like automotive safety, fluid monitoring, building automation, personal electronics like disk drives, power supply protectors .with this interest in the present paper, Keller Box method was implemented to study the influence of hall effect in presence of radiation and chemical reaction of MHD Casson fluid flow on an exponentially permeable stretching sheet.

FORMULATION OF THE PROBLEM
In this study a steady, two dimensional, incompressible, radiative, MHD Casson fluid flow over an exponentially permeable stretching sheet under the influence of Hall Effect is considered. u, v represents velocity components in x and y directions. The exponentially stretching sheet is assumed to be at y=0. Also flow is assumed to be above x-axis only. Magnetic field is applied externally normal to stretching sheet. The magnetic Reynolds number is very small because of the magnetic field. Here we used external heat source, Hall Effect, chemical reaction.   is critical value of this product.
Supporting to the above assumptions the guiding partial differential equations are taken as below.
By observing the equations here we introduce the similarity transformations, The partial differential equations are transformed to

NUMERICAL PROCEDURE
The moderate implicit finite difference method called Keller Box method to convert equations (7)-(9) into first order. By using this method, the resultant equations are linearized and then converted into matrix form by introducing Newton's method. Finally, the tri-diagonal elimination method is applied to solve the linear system of equations Cebeci et al. (1988).
We linearize the system of equations given in (15) Using Newton's method for that we introduce Substitute them in above system (15) we get the system of equations as     Where j= 2, 3, n-1

RESULTS AND DISCUSSION
Graphs are plotted for various parameters like Hartman's number, exponential parameter, Thermal Grashof's number, concentration Grashof's number, radiation parameter, Prandtl number, heat source parameter, Eckert number, Schmidt number, chemical reaction parameter, Suction parameter, Hall parameter using MATLAB.      Fig. 2, 3, 4 respectively indicates that with increase in the Hartmann number velocity decreases and temperature, concentration profiles increases due to an opposing force called Lorentz force. It is noticed that temperature plots rise adequately in case of Casson fluid when compared with Newtonian fluid. Fig. 5, 6, 7 respectively indicates that with increase in exponential parameter velocity, temperature and concentration profiles decreases because rise in exponential parameter causes reduction in momentum, thermal and concentration boundary layers. Fig. 8, 9 respectively indicates that with increase in thermal Grashof's number, raise in velocity profiles and decreasing tendency observed in temperature because developing buoyancy force suppress the value of contaminant concentration within the boundary layer regime normal to the barrier. Fig. 10, 11 indicates that with increase in concentration Grashof's number, raise in velocity profiles and fall in concentration profiles is observed. With increase in concentration Grashof's number momentum boundary layer thickness increases concentration boundary layer thickness decreases. Fig. 12 indicates that increase in Radiation parameter temperature enhancement is observed for both fluids. This happens due to the reason that raise in radiation heat energy with that effect temperature increases. Fig. 13 indicates that increase in Prandtl number temperature decreases for both fluids. For Large values of Prandtl number heat spreads slowly from surface when  Fig. 14 indicates that increase in heat source parameter temperature diminishes for both the fluids due to decrease in thermal boundary layer thickness. Fig. 15 indicates that enhancement in Eckert number (Ec) raise in thermal conductivity of the fluid is observed so temperature of fluid increases. Fig.16 indicates that with increase in Schmidt number mass transfer increases so concentration profiles decreases. Fig. 17 indicates that increase in chemical reaction parameter, concentration profiles decreases. Fig. 18, 19 indicates that with increase in hall parameter, velocity increases and reverse trend is observed for temperature. Fig.20, 21, 22 indicates that increase in S fluid will come nearer to the surface which causes reduction in thermal, momentum and concentration boundary layer thickness diminishes so velocity and concentration profiles decreases. To

CONCLUSIONS
It is evident from the tables that 1. Raise in the Hartman number reduces the friction factor and the heat and mass transfer rates. 2. Increase in the radiation parameter, thermal Grashof number, and concentration Grashof number increases the skin friction coefficient and the heat and mass transfer rates. 3. The skin friction coefficient declines and the Nusselt number and the Sherwood number grow with increasing suction parameter. 4. As the Prandtl number and the heat source parameter increase, the heat transfer rate grows and the skin friction coefficient and mass transfer rate decrease. 5. Increase in the Eckert number enhances the skin friction coefficient and the mass transfer rate but depreciates the heat transfer rate. 6. As the chemical reaction parameter and Schmidt number rise, the mass transfer rate grows and the skin friction coefficient and the heat transfer rate decrease. 7. Increase in hall parameter, skin friction, heat transfer rate, and mass transfer rate increases. Ec-Eckert number Q-Heat generation parameter Sc-Schmidt number m-hall parameter -density of the fluid k-thermal conductivity of the fluid -chemical reaction rate S-Suction parameter