RADIATION AND CHEMICAL REACTION EFFECTS ON UNSTEADY VISCOELASTIC FLUID FLOW THROUGH POROUS MEDIUM Frontiers in Heat and Mass Transfer

In this paper, we attempt to analyze the effects of radiation and chemical reactions on laminar boundary layer flow of an electrically conducting unsteady viscoelastic incompressible fluid flow along a vertical semi-infinite plate passing through porous channel. The coupled nonlinear partial differential equations for energy, momentum, and mass diffusions are solved analytically with the use of perturbation technique. Moreover, the analytical expressions for fluid velocity, temperature, and concentration distributions are obtained and interpreted using the software MATLAB. The enhancement of velocity is prominent with the growth of G r , G m , N r and Kr but a reverse pattern is observed with the rise of P r . The physical values of skin friction coefficient, Nusselt number and Sherwood number are estimated remembering their importance in engineering and technology fields. The graphical augmentation are instituted in possible cases for various viscoelastic parameters in combination of other dimensionless flow parameters. The simulations presented here relate to metal extraction in chemical and polymer processes .


INTRODUCTION
In the past few decades, the application of boundary layer flow for non-Newtonian fluids has been continuously increasing, as it plays an important role in different fields like engineering, processing, metal extraction, aerospace, polymer processing, and contemporary technology. Non-Newtonian fluid flow can also be practically applicable in bioengineering, for instance, there are various viscoelastic fluid models to describe the blood circulation process through animal arteries, then also to understand the mechanism of dialysis of blood through artificial kidneys one can study viscoelastic pulsatile flow.
There are many hydrometallurgical industries and transport affairs which are governed by the combined effect of buoyancy forces due to mass diffusion and thermal energy in presence of radiation and chemical reaction effects. Further, these processes occur in the combustion system, solar collectors, nuclear reactor safety and chemical engineering, etc.
MHD flow through an exponentially extending sheet with a heat sink: chemical reaction and radiation impact has been first studied by (Khalili et al., 2017). Many researchers have studied various problems related to Walters B / liquid model. To mention a few, the simultaneous investigation of heat and mass transfer, as well as magnetohydrodynamic (MHD) flow through a vertically oscillating plate contained in a porous medium, was investigated using Walters Liquid Model-B / given by (Abro and Khan, 2018). The constitutive equations of some non-Newtonian fluid flows, with short memories, have been studied by (Walters, 1964) and (Bread and Walters, 1964). On an unstable MHD heat and mass transfer flow across an accelerated inclined vertical plate, finite element Soret Dufour impacts have been investigated by † Corresponding author. Email: bamdeb.dey@dbuniversity.ac.in Bejawada et al. (2021).
Again, heat and mass transmission effects on unsteady MHD free convection flow past an infinite vertical plate, a numerical solution has been studied by (Kumari et al., 2021). Chemical reaction and hall effect in heat and mass transfer of MHD Casson fluid under radiation over an exponentially permeable stretching sheet was analyzed by (Ganesh et al., 2021). In the presence of thermal radiation and chemical reaction with suction, MHD boundary layer flow of nanofluid and heat transfer across a porous exponentially stretched sheet have been suitably illustrated by (Reddy et al., 2017) in presence of first-order chemical reaction and Joule heating. Also, the effects of thermal diffusion and chemical reaction on MHD flow on a vertical surface with a generation of heat has been discussed by (Bhavana et al., 2018). The hydromagnetic oscillatory slip flow effects on viscoelastic fluid have been studied by (Dey and Choudhury, 2019). The effects of Chemical Reaction and Suction/Blowing MHD Boundary Layer Flow of Nanofluid and Heat Transfer Over a Nonlinear Stretching sheet was illustrated by  for nanofluids. Analysis of heat transfer and mass transfer results on MHD viscoelastic fluid flow on a stretching-sheet in presence of a chemical reaction was done by (Nayak et al., 2016).  have discussed the effects of thermal diffusion and radiation on MHD dusty viscoelastic fluids between two parallel moving plates. (Jonnadula, 2015) have examined the influence of chemical reactions and thermal radiation on MHD viscous fluid flow. The unsteady viscous fluid flow in presence of chemical reaction and radiation has been analyzed by (Mishra et al., 2014). Chemical reaction over a curved surface with nonlinear thermal radiation and slip condition has been analyzed for nanofluids by (Ramzan et al., 2020). (Sivaraj and Kumar, 2013) have found the analytical solution for chemically reacting dusty viscoelastic fluid in presence of MHD. They show that convective boundary conditions and thermal radiation have a more significant effects on the dusty fluid in their regular porous channel. The combined effects of Soret-Dufour on unsteady MHD flow of dusty fluid flow over inclined porous

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org 2 medium has been investigated by (Pandya et al., 2017). (Choudhury and Dey, 2019) have studied the unsteady thermal radiation effects on MHD connective ship flow of viscoelastic fluid. Furthermore, (Srinivasacharya and Reddy, 2016) have successfully illustrated mixed convection heat and mass transport over a vertical plate in a power-law fluidsaturated saturated porous medium: chemical reaction and radiation impacts. The survey of literature pacifies that, flow of viscoelastic fluid through porous medium for various outlook have been demonstrated by many researchers and the details engrossed readers are referred to the references.
In most of the above studies, chemical reaction and radiation effects on fluid flow are discussed for both Newtonian and non-Newtonian fluids. But in many practical situations, it has been seen that models based on natural non-Newtonian fluid flow are extensively applicable in many engineering and industrial applications. In presence of chemical reaction or thermal radiation, some models based on viscoelastic fluid flows through porous medium are found in the works of many researchers. The combined effects of chemical reaction and thermal radiation on unsteady viscoelastic fluid flow through porous media is considered here. In the present study, the inclusion of porous media is important since it greatly prevents heat loss in heat transport processes and also accelerates the process of heating/cooling. Also, the fluids may be chemically reactive so for the mass transport equation chemical reaction term is vital.

Formulation of the problem
We consider an incompressible, unsteady, laminar, electrically conducting, boundary layer flow of a viscoelastic fluid through a vertical semi-infinite plate via a porous medium in presence of radiation and chemical reactions. Further, the time-dependent suction is also considered. The physical configuration of the problem is shown in figure-1 Energy of mass diffusion: For radiation, the Rosseland approximation can be taken as Using Taylor's series T 4 can be expanded linearly about the free-stream temperature T∞ and neglecting all higher order teams we get 4 ≅ 4 ∞ 3 − 3 ∞ 4 (6) Using equations (5) and (6), the energy equation reduces to * + * * = 2 * 2 + 16 * ∞ 3 3 2 * 2 (7) From equation (1), it can be assumed that suction is only dependent on time and can be assume as follows: * = − 0 (1 + * * ), where ≪ 1 (8) the negative sign, specifies that the suction is towards the plate. To get dimensionless partial differential equation, we introduce the following dimensionless parameters Making use of the above dimensionless parameters in equations (2), (7) and (4), the governing equations reduce to The boundary conditions are = 0: = 1, = 1 + , = 1 + → ∞: → 0 , → 0, → 0 } (12)

Results and Discussion
The goal of our present analysis is to highlight the effects of chemical reaction and radiation on unsteady viscoelastic fluid flow through a porous medium. The analytical outcomes are discussed in the previous section. In the present section, the simulation of divergent flow parameters has been analyzed using MATLAB code. The consequences of physical parameters, such as viscoelastic parameter k1, Grashof number Gr, modified/solutal Grashof number Gm, Nr the radiation parameter, Schmidt number Sc, chemical reaction parameter k, porosity parameter K on fluid flow variables like velocity, skin friction, concentration, and temperature have been analyzed using simulation and   The Grashof number has a favourable effect on velocity since it minimises frictional power. As a result, the Grashof number may be raised to lessen the effect of friction. Again, ejection of mass occurs due to the fluid motion relation with mass transfer. The free convective parameter for mass transfer is represented by Gm. The positive non-zero value of Gm implies that the concentration at the boundary surface is more than the free stream concentration. Figures 2 and 4 indicates that with the increased value of Gm, fluid viscosity enhances and which amplifies the velocity furthermore. From figures 2 and 5, it is observed that with the grow of thermal radiation parameter (Nr) there is a rising trend in the fluid velocity in both Newtonian and viscoelastic fluid which is due to the internal heat generation. Effect of chemical reaction parameter (Kr) is shown in figures 2 and 6. It has been noticed that with the enhancement value of Kr, a considerable amount of growth in velocity is observed.    Figure: 9 Effect of shearing stress against Gm The Nusselt and Sherwood number which characterizes the rate of heat and mass transfer respectively are not notably affected by the changed values of viscoelastic parameters k1. Also, from the expressions (39) and (40) it is clear that the influence of viscoelastic parameter on temperature and concentration profile has no significance, although they are Figure: 10 Effect of shearing stress against M considerably affected by Newtonian fluid. Since our prime focus is to find the effects of viscoelasticity on the Sherwood number, and the Nusselt number, and these are not affected by the parameter k1, so instead of graphical representation, we are expressing their values for different parameters in tabular form ( Table-1). To ensure the validity of our findings, we compared them to previously published work for viscous fluid (Mishra et al., 2014) and found significant.

CONCLUSIONS
The key aspects of this study are highlighted here. The final ordinary differential equations are facilitated with some flow parameters and almost every parameter plays an effective role in the fluid flow region. The conclusion of this study is as follows • The fluid velocity enhances sharply near the plate then decreases gradually and approaches to zero after attaining the peak.