Analysis of MHD Fluids around a Linearly Stretching Sheet in Porous Media with Thermophoresis, Radiation, and Chemical Reaction

*is paper presents the comparative analysis of MHD boundary layer fluid flow around a linearly stretching surface in the presence of radiative heat flux, heat generation/absorption, thermophoresis velocity, and chemical reaction effects in a permeable surface. *e governing equations are highly nonlinear PDEs which are converted into coupled ODEs with the help of dimensionless variables and solved by using semianalytical techniques. *e numerical and graphical outcomes are observed and presented via tables and graphs. Also, the Nusselt and Sherwood numbers and skin friction coefficient are illustrated by tables. On observation of heat and mass transfer, it was noticed that Maxwell fluid dominates the other fluids such as Newtonian, Williamson, and Casson fluid due to high rate of thermal conductivity, and hence, Maxwell fluid has better tendency for heat andmass transfer than other Newtonian and non-Newtonian fluids.


Introduction
ere exist different types of fluids in nature such as water, oil, alcohol, glycerine, gasoline, polymer melts, colloidal suspension, condensed milk, ketchup, clay, paints, slurries, gels, drilling muds, toothpaste, mayonnaise, cheese, blood, and printer ink. Some of them are the fluids that obey Newton's viscosity law and are dropped in the category of Newtonian fluids because in them, the relationship of viscosity is constant with shear rate. But, some of the fluids have nonlinear viscosity interaction with shear rate, so those fluids are referred as non-Newtonian fluids. e governing non-Newtonian fluid equations are highly nonlinear and complex in nature; thus, due to the complexity, the single Navier-Stokes relation is not enough to handle all the rheological properties of non-Newtonian fluids. To overcome this deficiency, further, the non-Newtonian fluids are classified into three basic categories: (i) e differential type (ii) e rate type Other well-known rate-type fluid is Maxwell's fluid. is simplest viscoelastic fluid was firstly proposed by Maxwell in 1867. By this special type of the fluid model, relaxation time effects can be counted. e decreasing direction of thermal gradient movement of small particles is referred as thermophoresis of particles. Simply, one can say thermophoresis helps to gather the small particles on the less heated or cold surface. By Aitken's phenomenon, the molecules of the gas have greater velocity of migration from the hot region when compared with the velocity of migrated molecules from the cold region. e faster moving molecules have intensity to collide more forcefully with the particles. So, this attained velocity by the particles is called thermophoresis velocity, while due to thermal gradient, the force which is endured by the particles is called thermophoretic force which is opposite to the direction of temperature gradient.
Hayat and Qasim and Noor had investigated the influences of chemical reaction and thermophoresis along a vertical stretching sheet of MHD Maxwell fluid flow [1,2]. In 2013, Stanford also studied MHD Maxwell fluid flow by considering the effects of thermophoresis and chemical reaction and concluded that the newly developed scheme discussed in their work has an ability to solve nonlinear coupled ODEs [3]. Hayat et al. discussed the non-Newtonian fluid in the porous medium with heat source/sink properties and found the dual solutions for momentum and thermal profiles [4]. Heat and mass transfer analysis with chemically reacting MHD non-Newtonian fluids such as Casson and Maxwell was theoretically investigated by Abbasi and Shehzad [5]. e effects of three-dimensional MHD upper convected Maxwell fluid with joule heating, viscous dissipation, and thermophoresis velocity on heat and mass transfer analysis [6] were studied by Bilal et al., and they presented the comparison of Maxwell fluid with Newtonian fluid [7]. ermophysical properties on viscoelastic fluids over a vertical surface were investigated by Koriko et al. [8] with nth order of chemical reaction. Comparative analysis on behavior of Maxwell, Oldroyd-B, and Jeffrey nanofluids was discussed by Sandeep, in which they flow around a stretching surface with heat source/sink [8][9][10]. e other type of non-Newtonian fluid is Williamson fluid suggested by Williamson in 1929. e texture of the fluid model enlightened the shear thinning behavior of non-Newtonian fluids. e flow model is viscoinelastic, i.e., the fluid containing both viscous and elastic properties. Recently, many researchers considered MHD Williamson fluid [10][11][12] and analyzed the effects of thermophoresis velocity, heat source/sink, thermal radiation, and ohmic dissipation. [13][14][15][16]. Also, the comparative study of various fluids which was considered upon a permeable stretching sheet was discussed by Parmar and Jain, Kumar et al., and Raju et al. [14][15][16].
A well-known fluid having pseudo-plastic phenomenon is Casson fluid which was recommended by Casson in 1959.
is fluid has shear thinning properties with high shear viscosity and yield stress. Human blood, honey, concentrated juices, ketchup, jelly, etc., are some common examples of Casson fluid [17][18][19]. In numerous areas of biotechnology, thermal engineering, geophysics [20,21] and as well as astrophysical studies, heat and mass transfer in various non-Newtonian fluids has been fascinated by researchers and scientists because of its fruitful applications in multiple fields [21].
MHD two-dimensional and three-dimensional Casson fluid flow on a shrinking/stretching sheet [18][19][20][21] was studied by many researchers. Sumalatha and Bandari [22] analyzed the heat source/sink variations on a Casson fluid over a nonlinear stretched surface and concluded that by increasing radiation and heat source/sink parameter, the temperature of the fluid enhances.
Latter, researchers focused on the MHD Casson flow and discussed the effects of various physical parameters [23][24][25]. MHD Casson fluid with slip boundary on a moving edge was analyzed by Raju and Sandeep [26] and found dual solutions for accelerating and decelerating flow.

Mathematical Formulation
We have considered the steady-state two-dimensional flow of viscous incompressible fluids. e flow is taken into account at various situations of fluid behavior caused by the linearly stretched sheet having stretching velocity u w � ax, where a is a +ive constant. Also, T w and C w are the fluid's temperature and concentration near the permeable stretched surface. With constant intensity B o , magnetic field is applied in y direction. Temperature and concentration are considered in the direction of chemical reaction, thermal radiation, and thermophoretic velocity. e governing boundary layer equations are 2 Mathematical Problems in Engineering Here, the horizontal and vertical components of velocity u and v are in direction of x-and y-axis and β, λ and, We are the Casson, Maxwell, and Williamson parameters. e radioactive heating flux in the temperature equation is calculated and simplified by the Rosseland approximation, i.e., where σ * is the Stefan-Boltzmann constant and k * is the mean absorption coefficient. e temperature variation within the flow is supposed to be small enough, so T 4 can be taken as a linear function of temperature neglecting the higher-order term while expanding T 4 in Taylor series about T ∞ which is obtained as [34] Neglecting the higher-order terms of the above said statement beyond the first degree in the series, we get Also, V T is the thermophoretic velocity, defined and expressed as V T � (]k ro /T ro ); here, T ro is the reference temperature, and k ro is the coefficient of thermophoretic velocity. e model 8-10 is solved by using semianalytical techniques such as HAM [13], VIM [39,40], and ADM [40] along with Pade approximation [39,40]. ese standard techniques are strongly effective, reliable, and convenient in the literature and can be applicable directly on linear and nonlinear equations. Although, for their closed form solution while dealing with the unbounded domain, the Pade approximation has been considered [39,40].
Introducing the similarity transformation [14], e nonlinear coupled PDEs are transformed into ODEs by the help of dimensionless variables such as

Results and Discussion
In this part of the article, the observed consequences are exhibited graphically and numerically. For the pertinent values of physical parameters Also, the momentum boundary layer thickness reduced while increasing this parameter M. is is because the fact when magnetic field is imposed in the ⊥ direction on the electrically conducting fluid originate like the drag force (also named as Lorentz's force) that acts on the opposite direction of flow which hence produces a deceleration flow and causes the depreciation in the velocity profile. But, reverse effects are to be seen in case of thermal and concentration profiles. In them, we have seen the extreme behavior of Maxwell fluid by varying M.

It is obvious from Figures 3(a)-3(c) the behavior of porosity parameter
Kp. It is observed that the velocity boundary layer decreases, and Casson fluid affects the most in this case. Generally, the porosity factor is responsible to soak the major amount of fluid from the boundary layer, so the velocity flow field decreases, but by increasing Kp amount, the thermal and concentration profiles increase, and Maxwell fluid is highly influenced than rest of the fluids under consideration. It is seen that all the profiles are decreasing, and Casson and Maxwell fluids in this situation are influenced highly by other Newtonian and non-Newtonian fluids. In fact, this trend of depreciating is pointing the physical situation of suction effect which is that it delays the separation and transition and helps to maintain the steady flow near the sheet. is observation indicates in Figure 3(a) that Casson fluid is more affected by increasing the values of s, while in Figures 4(b) and 4(c), it is seen that Maxwell fluid is extremely influenced by s. e role of Eckert number on temperature profile θ(η) is observed in Figure 5(a). Physically, by upper-going values of Eckert number which tend to increase the thermal profile, the frictional heat generation is quite obvious as the fluid moves faster along the surface which causes the particles' growth of temperature. On comparison and analysis made by the graph, it can be observed that Maxwell fluid is more influenced than the other fluids.
It is evident from Figure 5(b) that the concentration profile is rapidly compressed by the destructive distillation parameter, i.e., chemical reaction. Because this appears with many instabilities, it results in high molecular motion and uplifts the transportation process and hence retards the fluid concentration. Also, it is interesting to note that Casson fluid shows more influenced in comparison with other fluids. Behavior of Prandtl number is illustrated by Figure 6(a) with various fluid conditions. It is noticed that increasing Prandtl number causes decrement in the thermal layer, which is by the fact when thermal diffusivity decreases, the fluid temperature decreases because Prandtl's number is inversely proportional to thermal diffusivity, so it will lower the heat of the fluid for growing Pr. In Figure 6(b), it is clearly seen that Casson fluid is highly influenced as compared with other resting fluids.
By Figure 6(b), effects of thermomigration are successfully presented. From this figure, we can conclude that, by increasing the value of Soret number Sr, the concentration species rises in upward direction. Scientifically, transfer the mass flux from the lower to the maximum concentration region moved by the temperature gradient. Although all the Newtonian as well as non-Newtonian fluids' profiles are showing increasing behavior, Maxwell fluid is observed more influenced by other fluids.
It is noticed from Figure 7(a) that the radiation parameter R enlarges the temperature profile when the values of R increase. When the radiation effects exceed, the heat energy generates towards the flow which causes to elevate the heat, and hence, thermal profile proceeds. Maxwell fluid significantly affects much. Figure 7(b) indicates the influence of Schmidt number. As the values increase, the concentration profile decreases due to the inverse relation between Sc and diffusion coefficient. e reason is that the mass transfer rate gets slower as we increase the diffusion coefficient.        In Figures 8(a) and 8(b), A * and B * are plotted to know the nature of nonuniform positive heat source/sink parameter on temperature profile θ(η) which exhibits the increasing behavior of the temperature profile while increasing the value of A * and B * . It is noticed that the Maxwell fluid is highly influenced by A * , while Casson fluid is more affected by the increased value of B * . e positive values of A * and B * are responsible factors of heat generators, which release the heat energy to fluid flow, and this then causes to uplift the temperature profile. Figure 9 incorporates the effects of τ, i.e., thermophoretic parameter. It is observed that, by increasing the value of τ, the concentration species retards everywhere in the flow domain.
e effects of magnetic field, porosity, chemical reaction, radiation, heat source/sink parameters, and thermophoresis velocity and Schmidt, Soret, Eckert, and Prandtl numbers are described in Tables 1-4 for detailed intrinsic characteristics of Newtonian and non-Newtonian fluids. From Tables 1-4, we can conclude that increasing the magnetic field and porosity parameter leads to the enhancement of skin friction but reduces heat and mass flux rate, while slower down the rate of wall shear stress and heat and mass fluxes for suction/injection parameter. By rising values of skin friction − f′′(0), lessen the heat flux rate − θ(0) as more heat is absorbed; similarly, mass flux rate − ϕ(0) is also expected to reduce. Local Nusselt number was also seen to be decreased for larger values of Ec, R, A * , and B * but increases for rising values of Pr number, so it can be used as a cooling agent. e numerical outputs in the tables show the significant increment for mass flow rate as chemical reaction, thermophoresis, and Schmidt number increase. Obviously, these parameters have direct impact on the concentration profile. In these tables, it is evident that, by increasing the values of Ec, A * , B * , R, and Sr, the heat and mass flow rate decreases.

Conclusion
We have thoroughly investigated the behavior of velocity, temperature, and concentration profiles. e heat and mass transfer analysis has been performed theoretically via graphical and numerical approaches.
is research describes the chemically reacting MHD flow in the porous medium for Newtonian and non-Newtonian fluids, namely, Maxwell, Williamson, and Casson fluids, over a stretching sheet under the effects of radiation, heat source, Soret, Schmidt, and Prandtl numbers. e semianalytical solutions have been obtained by using analytical methods such as ADM, HAM, and VIM. To tackle the unbounded domain, Pade approximation has been encountered. Some useful outcomes of the current study are listed as follows: (i) Magnetic and porosity parameters (M and Kp) have the tendency to increase the heat and mass transfer rate, and Maxwell fluid is much appreciated but depreciates the velocity distributed, and here, Casson fluids dominate the other fluids when comparison is performed (ii) Mass transfer rate continuously decreases for increasing trend of Schmidt number and chemical reaction parameter, and Casson fluid is intensively affected (iii) ermal and concentration profiles for Maxwell fluid are highly influenced as compared with other Newtonian and non-Newtonian fluids (iv) Ec, Sr, R, and A * enhance the thermal and concentration profiles, and Maxwell fluid has a dominating trend over other fluids (v) Maxwell fluid has better heat and mass transfer tendency while compared with Newtonian, Williamson, and Casson fluids (vi) e thermophoretic parameter τ is constantly working as an agent to increase the mass transfer rate and hence causes the reduction in concentration boundary layer thickness

Data Availability
e data used to support the findings of this study are included within the article.

Additional Points
(i) e comparison of MHD fluid flow in the porous medium over a stretching surface is studied; (ii) it is observed that Maxwell fluid dominates the other fluids when comparison is performed for heat and mass transfer; (iii) skin friction coefficient and heat and mass fluxes are obtained; (iv) the impact of suction, radiation, thermophoresis velocity, and internal heat source/sink is observed for various fluid categories; and (v) the effects of Prandtl number, Schmidt number, and chemical reaction reduce the thermal and concentration profiles.