IMPACT OF CHEMICAL REACTION ON MASS TRANSFER AND MELTING HEAT TRANSFER OF POROUS MHD CASSON FLUID FLOW

Melting and chemical reaction impacts on heat and mass transfer of MHD Casson fluid flow over a porous stretching surface is examined numerically in this article. The governing partial differential equations are converted by using adequate transformations and the resulting ordinary differential equations are solved numerically using finite difference scheme along with Thomas algorithm. The graphical illustrations are presented for velocity, temperature and concentration distributions. Also Skin friction, Nusselt number and Sherwood number are elucidated for chosen values of various parameters. To validate the numerical method employed, the present results are compared with the existing literature and found to be in good agreement.


INTRODUCTION
The model developed by Newtonian fluids cannot explain all the various flow characteristics in this modern technology. For this reason, theory of non-Newtonian fluid flow has become significant. As there are numerous applications of these studies such as casting, production of paper, extrusion of polymers. Casson fluid is one among such non-Newtonian liquids which has attracted many scholars due to its applications in drillings techniques, bio-engineering operations and many more. Some important Casson fluids are jelly, tomato sauce, human blood and soon.
Research work on various properties of heat transfer in Newtonian and non-Newtonian liquids can be found in the literature [1,2,3]. In consideration of Viscous dissipation, effect of heat flux on non-Newtonian fluid over a porous moving surface was investigated by Kishan and Shashidar [4]. Hayat and Qasim [5] analyzed the impact of thermophoresis on Maxwell fluid flow. Mixed convection with the effects of various parameters on MHD flow is examined by Narayan [6].
The model of the Casson fluid flow was initially represented by Casson [7] to predict the performance of pigment-oil suspensions. The flow of Casson fluid with radiation effects over stretching surface in porous medium was discussed by Pramanik [8]. The reactions of heterogenous/homogeneous in the flow of Casson fluid was explained by Khan et al. [9]. The impact of Soret and Dufour on the flow of Casson fluid was examined by Hayat et al. [10]. The unsteady MHD flow of Casson liquid past a stretching plate considering various parameters was discussed by Aziz and Afify [11]. Eldabe and Salwa [12] investigated the flow of Casson fluid between rotating cylinders. Khalid et al [13] has presented his studies on convective motion of Casson fluid past a oscillating vertical plate. Heat transfer of Casson liquid past a porous rotating plate was analysed by Raju and Sandeep [14]. Mukhopadhyay [15] constructed Unsteady flow of Casson liquid past a stretching surface with suction/injection. Whereas the Unsteady heat transfer flow of Casson liquid past a moving plate was discussed by Mustafa et al. [16].
In MHD, mass transfer plays significant role because of its significance in many scientific approaches such as convective transfer of molecules and atoms. Mass transfer of MHD Casson liquid past a stretching surface was investigated by Bhattacharyya [17]. Influence of suction on 151 POROUS MHD CASSON FLUID FLOW MHD heat mass transfer past a shrinking surface is investigated by Muhamin and Khamis [18].
Melting heat transfer process in non-Newtonain fluids has played significant role because of its applications in thermal engineering such as solidification, recovery of geothermal energy, magma and many more. Melting effects on MHD Williamson nanofluid flow was investigated by Krishnamurthy et al [19]. Prasannakumara et al [20] analysed the phenomenon of melting on MHD Dusty fluid flow past a stretching surface considering the effects of heat absorption/generation and radiation. Impact of melting on MHD flow past a moving plate with radiation was observed by Kalidas [21]. Sarkar [22] shown the influence of melting MHD Casson nanofluid wedge flow. MHD Carreau fluid flow over melting surface is examined by Ramdev et al. [23].
Heat transfer in the presence of radiation has large number of applications in various industrial processes such as Satellites, missiles, space vehicles and soon. Heat and mass transfer with radiation over a moving plate in porous medium was addressed by Makinde [24]. Heat transfer characteristics on the flow of Casson nanofluid over a stretching surface was examined by Ibrahim et al [25].
The past studies reveals that few works exist in the literature presenting the influence of mass transfer on MHD Casson fluid flows. Recently, Fazle and Kalidas [26] presented the melting effects on the MHD flow of Casson fluid over stretching surface. So our present investigation is to fill this gap. The theme of the present study is to extend the effort of Fazle and Kalidas [26].
The present work is to concentrate on heat and mass transfer of MHD Casson fluid flows past a stretching surface. In this study, fluid is considered in porous medium, also melting effects and thermal radiation are included. The equations governing the flow are solved numerically and the impact of prominent parameters is analysed graphically. Figure 1 depicts the mathematical model of a 2D steady flow of a MHD Casson fluid past a stretching sheet. Here we consider that the sheet is horizontal and linear in the existence of a porous medium also the fluid is melting at a steady rate into a constant property. Cartesian model is preferred such that x-axis is along the surface and y-direction is perpendicular to the sheet and flow is confined to y > 0. The fluid is assumed to be conducting electrically and subjected to applied magnetic field in normal direction. Let uw(x) = cx be the stretching sheet velocity and the external flow velocity be ue(x) = ax, where c > 0 and a > 0 be the constants. The melting and free stream temperatures of the liquid be taken as Tm and T∞ accordingly. The constituent equations determining the Casson liquid is represented by:

FORMULATION OF MODEL
Where π = eijeij, is the deformation rate, πc is a critical value of the non-Newtonian fluid, Py is fluid yield stress, eij is the (i, j) th component of the deformation rate, μB is the plastic dynamic viscosity.
Under the above conditions, the equations governing the flow are [26]: Subject to the limit conditions, And Here K is permeability, u, v are the velocity components, k is thermal conductivity, λ is latent heat α is Thermal diffusivity, ν is kinematic viscosity, ρ is the density of the fluid, β is Casson fluid parameter, cs is heat capacity, cp is the specific heat at constant pressure, qr is radiative heat flux, D is mass diffusion and k1 is rate of reaction.
Using the approximation of Rosseland [27], we have Where σ * is Stefan-Boltzmann constant, and k * the absorption coefficient respectively Using equation (8), equation (4) can be expressed as To transform the equations governing the flow, the following similarity variables are hosted: Where ψ is the stream function, using which the velocity components are defined as Now by means of equation (11), the governing equations are formulated as Subject to the boundary conditions: Where,

NUMERICAL PROCEDURE
The formulated governing equations (12), (13) and (14)  The procedure employed to obtain the numerical solution for the mathematical model is presented as follows: ➢ Initially, momentum equation is linearized by employing Quasi Linearization Method [28] ➢ By means of Implicit finite difference scheme, the system of linear Ordinary differential equations is converted to form difference equations ➢ These algebraic equations are then expressed in matrix-vector form ➢ Thomas Algorithm is applied to solve this linear system.       The impact of magnetic parameter and permeability parameter on velocity distribution is revealed in figure 1. It is clear that the amplitude of the velocity reduces with the escalation in the magnetic parameter. And hence lessening the thickness of velocity boundary layer. It is due to the fact that, when magnetic force is applied normal to the flow, a resistive type of force is generated and by accelerating the magnetic effect, the drag force rises and which decelerates the velocity of the flow. And the outcome of permeability is to reduce the velocity which is shown is figure 1(b). These outcomes are seen when the stretching parameter ε = 1.5, whereas a reverse trend is observed when ε=0.5. Figure  In Figure 3, the temperature profiles are plotted, for a variety of values of M and Ω keeping other parameters as fixed. It is evident from the figures that, the temperature of the fluid amplifies with the rise in magnetic parameter as well as permeability parameter for both the cases of ε = 0.5 and ε =1.5. This is justified due to the fact the Lorentz force helps to enhance the fluid temperature.
The impact of β on temperature distribution is illustrated in figure 4(a). It is evident from the graph that the fluid temperature acts as a growing function of β for both cases ε = 1.5 and ε = 0.5.
Correspondingly, the impact of Prandtl number is to enhance the temperature of the Casson fluid for both the cases which is illustrated in figure 4(b).  Figure 5(a) explains the impact of radiation on the fluid temperature on the melting surface. It is illustrated graphically, that the temperature reduces with the increase in radiation parameter. It is due to the reason that radiation enhances the rate of heat transfer on melting surface. And the influence of Melting parameter Me on temperature is sketched in figure 5(b). The temperature of the Casson fluid decreases monotonically from Tm at the surface to T∞ at infinity. For both the cases of stretching parameter, temperature falls down at the boundary layer. The impact of Schmidt number on concentration profiles is displayed in figure 6(a). Graphical representation shows that concentration profile is increasing function of Sc. Figure 6(b) demonstrates the concentration profiles with the variation of chemical reaction parameter. It is evident from the graph that an increase in the chemical reaction causes reduction in concentration field.

CONCLUSIONS:
The development of Casson fluid flow with heat and mass transfer over porous melting stretching surface in the presence of chemical reaction is investigated numerically. The converted flow equations are solved numerically using Finite difference scheme with Thomas algorithm.
The influence of various pertinent parameters on velocity, temperature and concentration fields is elucidated graphically. The significant results are listed as follows: ➢ The impact of magnetic parameter, permeability, Prandtl number and Casson fluid parameter is to decline velocity, accordingly velocity boundary layer thickness reduces.
➢ Thermal radiation and melting effects tends to depreciate the temperature distribution. ➢ Schmidt number helps to boost the concentration of the liquid.
➢ Enhancement in chemical reaction parameter reduces the concentration.
➢ Impact of melting and stretching parameter is to reduce Skin friction.
➢ Nusselt number enhances with melting and falls with stretching parameter.
➢ Sherwood number increases with chemical reaction and decreases with stretching parameter.