RADIATION AND HEAT SOURCE/SINK EFFECTS ON MHD CASSON FLUID FLOW OVER A STRETCHING SHEET WITH SLIP CONDITIONS

An analysis has been carried out on the study of the Radiation impact of MHD Casson fluid flow, over an exponentially stretching sheet embedded in a porous medium, with slip effects in the presence of the heat source /sink and viscous dissipation. The governing partial differential equations (PDEs) are simplified to a set of non-linear ordinary differential equations (ODEs) by using appropriate similarity transformations and numerical solutions are obtained by using the MATLAB in built solver bvp4c package. The impacts of the nondimensional parameters on the velocity, temperature, and concentration distributions are discussed graphically and the Nusselt rate of heat and mass transfer coefficients values are tabulated to elucidate the effect of the numerous physical parameters.


INTRODUCTION
The analytical review pertaining to laminar flow, along with phenomenon of heat transfer allowed fluid through stretching sheet assumes larger area of importance for the research in fluid mechanics owing to wider scale of applications in different product processing's across the industry sectors, like polymer sheet extrusions, wire drawing, hot rolling, manufacturing of paper, plastic films, and oil production. An extensive study was pursued on Newtonian fluids. The foremost one was crane [1] who made elaborate study in the direction over a linear liquid surface.
The computational solutions were investigated by Mukhopadhyay and Gorla [2] for steady boundary layer and radioactive heat transfers flows over an exponentially stretched surface with partial step at the boundary. Sajid and Hayat [3] pursued to know the effect of radiation on the flow of boundary layer, and transfer of heat of a viscous fluid across the stretched sheet. Al-odat et al. [4] made an investigation of computational solutions in the functional aspect of magnetic field for the thermal boundary layer of stretched surface with variation in temperature. Pramanik [5] has studied extensively heat flow while blowing or through suctioning at the surface. heat transfer with slip effects of casson has been duly studied by Saidulu and Venkata Lakshmi [6]. Jawad Raza [7] investigated the impact cumulatively in terms of thermal radiation and velocity step on a layer that is convectively extended. The behaviour of magneto hydrodynamic (MHD) beside the stagnation point has been explained by Mackinde and Ibrahim [8]. They investigated the impact of slip and convection on MHD stagnant point flows and heat transfer due to casson nanofluid part stretching layer. Ibrahim et al. [9] briefs the impact of chemical reaction and heating source on MHD mixed convection of flow of a casson nanofluid on non-linear stretched sheet. The focus on this was pinned by many of the scholars and have formulated relative solutions, namely Hari Singh Naik et al. [10], Magyari and Keller [11], Bidin and Nazara [12], Shankar Goud [13], Ishak [14], Sumitra et al. [28] and others. A brief narration of boundary layer flow and transferring of heat passed over a porous exponentially stretched sheet set in presence of an uniform magnetic field has been offered by Mukhopadhyay [15]. Harinath Reddy et al. [16] observed the radiation absorption and studied the impact of chemical reaction on MHD flow of 6543 RADIATION AND HEAT SOURCE/SINK EFFECTS ON MHD CASSON FLUID FLOW heat generating casson fluid past oscillating vertical porous plate. MHD free convective non-Newtonian flow over a stretched sheet enclosed in a porous medium with slip condition effects are extensively pursued by ImranUllah et al. [17]. Besides analysing the properties of the viscous dissipation, the flow of casson fluid into stagnation center area to a stretching sheet also was made out through report by Meraz Mustafa et al. [18]. B.S.Goud and et al. [19-22, 24, 26] pursued study on the moving plate with sort and radiation effects, behaviour of thermal radiation on MHD stagnation point flow, Hall effect and impact on inclined parabolic accelerated plate, radiation effect on exponentially stretched sheet, sourcing heat effect. Sriramulu et al. [23] has investigated the heat transfer effect on stretching sheet over a porous medium steady flow of a viscous incompressible fluid flow. The laminar MHD boundary flow of nano-fluid past and exponentially permeable stretch layer was verified by Kishan et al. [25]. Ibrahim et al. [27] reviewed a conceptual model analytically, which has been improvised to further analyze the blended convection over a non-linear permeable stretched layer on thermal radiation, viscous dissipation, heat sources and heat sinks. The objective of this analysis is to find numerical approximations for the steady state of MHD casson fluid over an exponentially stretched sheet with a steady and continuous viscous dissipation, suction/blowing and thermal radiation.

MATHEMATICAL FORMULATION
While considering 2D as an electrically conducting incompressible fluid flow of casson with respect to exponentially stretched sheet, whereas the x -axis and motion of the fluid are taken along with the stretching surface, y -axis is drawn and is normal to the surface in terms of referring to velocity 0 U , temperature 0 T , length as L and x -axis is supported with equally opposite powers in such manner that the wall is extended to maintain its origins. The flow presumably has been manifested by stretching sheet of elastic limit a slit with which a strong force is inclined therein to show that limit plate velocity of directional flow coordinate x has an exponential order.
In the context, the governing equations can be stated as: The boundary conditions in the present flow are ( ) Now introducing the similarity transformation The system can be transformed to 0 Pr The substantial components and skin friction coefficient and local Nusselt number given by 2 1 Re (1 ) (0) The above skin friction coefficient and Nusselt numbers displays that its deviations depends on the changes of the aspects (0) f  and respectively.

NUMERICAL METHOD
The set of ODEs (8) -(9) which has been transformed and the exact boundary conditions (10) have been formulated numerically, by adopting the bvp4c shooting techniques. Program pattern comprises the below given steps. Boundary conditions may be applied or used to ascertain three of the new (factors). Before establishing the required accuracy, the procedure will be to reduce Eqs (8)  The following first order set of equation are as The associated initial conditions are ( )   With the increase in the magnetic parameter, there will be reduction in the rate of velocity on the surface. In contrast the transverse magnetic field demonstrates the fluid flow and crucially decreases the rate of transportation. This change can be attributed to the creation of Lorentz force by the magnetic field. It can be noticed that, the surface temperature has increased with respect to increase in parameter along the boundary layer. We understand that due to excessive heating, the latent temperature inside thermal boundary layer rises to an extent that the magnetic field becomes adaptive of monitoring the flow functioning.
.     Suction has found to have presumably higher impact on thermal boundary layer thickness. The net result is a slowdown flow process to determine the rate of velocity and temperature. The blowing parameter will be just opposite and it assumes that the suction can easily be used to cool the sheet rapidly.  The influence of the Eckert number on the temperature profile is depicted in the Fig.12. It is seen that the thermal boundary layer thickness increases with an increase of the viscous dissipation parameter.   The influence of heat source/sink parameter on the temperature field is shown in Fig.13. The temperature profile increases with increased quantum of the sources. Fig.15

CONCLUSION
The following conclusions can be drawn from the above analysis.
• Velocity decreases with an increase of , M  and a reverse effect in temperature profile is duly observed.
• An increase of , will result in a decrease of velocity and the opposite effect is observed in the temperature profile.
• Velocity and temperature decreases with an increase of . s • The temperature decreases with increase of Pr, however we can also observe an increase of ,.
c EQ • The Skin friction parameter increases and wall temperature gradient decreases besides the increased value of . s Pr R