THE SIMULTANEOUS EFFECT OF ECKERT NUMBER AND MAGNETIC PRANDTL NUMBER ON CASSON FLUID FLOW

This study includes the observation of electrically conducting non-Newtonian fluid flow through a vertical porous plate considering the effect of the induced magnetic field. Our approach is numerical to investigate how the variation of magnetic Prandtl number and Eckert number effect the flow profiles. Influence of Casson parameter and Hartmann number in the profiles is also observed and depicted in graphs. The rate of heat transfer, the rate of mass transfer and the skin friction are calculated and presented in tables. A significant effect of magnetic Prandtl number and Eckert number is observed. We compared our results with some previous analytical and numerical works.


INTRODUCTION
The study of the flow of Casson fluids in a porous medium has a significant influence on industrial, agricultural as well as medical research. Since the viscosity of the Casson fluid is dependable on the yield stress, various circumstances of blood flow can be studied with the help of Casson model. In agricultural research, the flow of the mixture of manure in the soil can be studied assuming it as the flow of Casson fluid model. Casson introduced this model in 1959 to describe a flow equation for pigment oil-suspensions of printing ink. Farther, this model has been studied by various researchers in theoretical as well as in experimental field to explore the different characteristics of shear-thinning fluid flow. Kenjeres (2008) contributed to the study for delivery of the drug in the affected area. (Mukhopadhyay et al., 2013) investigated the effect of Casson parameter on an unsteady flow over a stretching sheet. (Completo et al., 2014)investigated experimentally and theoretically the blood destructive phenomena and compared various Newtonian and non-Newtonian fluids. (Khalid et al., 2015) approached an analytical investigation to observe the Casson fluid flow past an oscillating vertical plate.
Casson nanofluid flow is another emerging area of fluid dynamics. (Sulochana et al., 2016) performed similarity solution compared the 3D MHD Newtonian and Casson Nano Fluid flow. (Ullah et al., 2017) investigated the effect of slip condition on a Casson fluid flow. (Siddiqa et al., 2018) investigated the Casson fluid flow in a wavy cone with the radiating surface. (Reddy et al., 2018a) observed the stable and convergent numerical results of Casson fluid flow through an oscillating sheet. (Hsua et al., 2019) investigated numerically the MHD flow of non-Newtonian fluid over a vertical plate embeded in a porous medium.
Most of the research of Casson fluid flow conducted by avoiding the effect of the induced magnetic field. Magnetic field plays an impor- † Corresponding author. Email: gsilpisikha@gmail.com tant rule as the controller of the rate of cooling of a machine and helps to obtain the required product. Moreover considering the presence of a magnetic field has a significant influence on MHD energy generator systems, thermo magneto aerodynamics and nuclear reactors. Ibrahim (2016) investigated the effect of the induced magnetic field on fluid flow in a stagnation point. (Hayat et al., 2018) observed the effect of melting heat transfer and the induced magnetic field in fluid flow. Arora and Gupta (1972) investigated the effect of the radial magnetic field on the magnetohydrodynamic flow between two rotating coaxial cylinders. M. Kumari and Nath (1990) studied the effect of the induced magnetic field on the heat transfer over a stretching field. (Mekheimer et al., 2012) approached analytically to observe the peristaltic flow of second-order fluid in the presence of an induced magnetic field in a channel. (Jafar et al., 2013) observed the MHD boundary layer flow in the presence of the induced magnetic field. (Raju et al., 2016)  In this study, we observed the Casson fluid flow by considering the effect of the induced magnetic field. The assumed two-dimensional configuration is placed in a porous medium with the flow of viscous, elec-trically conducting fluid. The influence of involved physical parameters on the velocity, temperature, concentration, rate of heat transfer, rate of mass transfer and skin friction was depicted in graphs and tables. It is a extended work of (Reddy et al., 2018b) by introducing induced magnetic field to it. The results appeared to be great agreement with the previous works.

MATHEMATICAL FORMULATION
The physical transformation of the problem is assumed to be in a two-dimensional system with vertical porous plate through which an electrically conducting, incompressible, non-Newtonian and viscous fluid is passing with constant velocity along y direction. The problem is visualised in a Cartesian coordinate system, where x axis is considered along with the plate while y is placed perpendicular to the plate. A constant magnetic field B0 is applied normal to the plate. A comparatively high process is assumed under free convection. The rheological equations are stated below, 2πc eij if π < πc Here, π = eijeij, the product of component of deformation rates controls the flow. As the plate grows infinitely in vertical direction, all the variable varies along y * axis only. therefore the continuity equation can be stated as, ∂v * ∂y * = 0 (1) v = U is considered as the uniform velocity along y axis. This non-Newtonian unsteady fluid flow is governed by the following equations, Momentum equation: Energy equation: Following are the required initial and boundary conditions for the respective flow profiles, The thermal radiation heat flux gradient can be assumed as, We assume the negligible temperature difference to get T * as a linear function. Neglecting the higher orders of the Taylor's series we get, With the help of following non-dimensional quantities we convert the above partial differential equations into non-dimensional form, The non-dimensional form of the governing equations are given below, Non-dimensional boundary conditions are, u = 1, θ = 1 + e nt , φ = 1 + e nt , H = h at y = 0 Our model has been configured with this set of partial differential equations and boundary conditions. DOI: 10.5098/hmt.16.12 Global Digital Central ISSN: 2151-8629

SOLUTION OF THE PROBLEM
Our approach is numerical to obtain the fluid profiles from the above set of equations. Finite Difference Method(FDM) is one of the most appropriate method for unsteady flow to get the approximate results. The used FDM technique is given below, Where, Z represents the non-dimensional flow characteristic. Using the above technique we convert the set of PDE to a set of algebraic equations. These equations were solved in MATLAB to get the required results.

RESULTS AND DISCUSSION
We observe the simultaneous effect of magnetic Prandtl number and Eckert number in the fluid profiles. The high value of both the parameters leads to a zigzag motion in the temperature profile far away from the plate. For large Ec.P r energy dissipation is getting charged actively for which the influence of the kinetic energy plays the determining role in temperature distribution and heat transfer. For higher value Ec, also the effect of suction occurs. For physical relevance of the problem, we consider the P r value within 0.7-1 which corresponds to air. The value of Sc is considered as 0.22 which corresponds to Hydrogen in 25 0 C and 1 atmospheric pressure.  Fig. 1-4 depict the changes in velocity, induced magnetic field, temperature and concentration profiles respectively concerning Casson parameter. Increasing Casson parameter decelerates velocity. Increase in non-Newtonian Casson parameter decreases yield stress which leads to an increase in plastic viscosity. This phenomenon resists fluid flow. Increasing γ strengthens induced magnetic field vector. Increase in Casson parameter leads to increase in viscosity of the fluid due to which the fluid getting steadier that can justify the increase in induced magnetic field. The temperature profile increases with increasing γ . Increasing γ leads to enhancing viscocity of the fluid which imply the increase in temperature. At the free stream concentration increases with increasing γ. Increase in viscosity makes the fluid thicker which leads to increase   (2013). Table 1 depicts the changes in skin friction, Nusselt number and Sherwood number for the various values of γ. The changes appeared to be in good agreement with the previous work of ( Shateyi et al., 2017),  and Mukhapadhyay (2013).  5-7 replicate the influence of Eckert number in the flow profiles. Velocity increases with increasing Ec. Increased Ec implies increased in kinetic energy of the flow which leads to an increase in velocity. Induced magnetic field decreases with an increasing value of Ec. By controlling the magnetic Prandtl number to a diminutive value, we observe an increasing θ far away from the plate with increasing Ec. Ec can be expressed as the ratio of the dynamic temperature to the temperature, i. e. accelerating Ec can lead to an increase in dynamic temperature, which can explain the increasing phenomena of θ.  Table 2 indicates the changes in skin friction, Nusselt number and Sherwood number for the various values of Ec. It is observed that the direction of changes agrees with Chamkha (2004).  8-12 show the variations in the fluid profiles according to the increasing value of magnetic Prandtl number. With the increasing value of P m velocity decreases.P m is the ratio of viscous diffusion rate to the magnetic diffusion rate. i. e. increase in P m can increase the viscous diffusion rate, which can be a cause of the decrease in velocity. Induced magnetic field vector decreases with increasing P m. Whereas, the temperature is decreasing and concentration is increasing for increasing Ec. Increase of viscous diffusion rate for an increase in P m can be a reason for these phenomena.

CONCLUSION
Our foremost motive was to observe the controlling power of Ec and P m on characteristic profiles. We can conclude by justifying the established result that Ec and P m has a stable influence specifically on the temperature profile if both are inversely proportionate, e.i. Ec.P m = constant. Casson fluid flow in a porous medium has its influence on many industrial purposes. Further, this problem can be investigated by observing the effect of chemical reaction parameter and radiation.