Analysis of the MHD Boundary Layer Flow over a Nonlinear Stretching Sheet in a Porous Medium Using Semianalytical Approaches

,e purpose of the research is to inquire the outcomes of viscous and ohmic dissipation on the MHD flow in porous media in the region of suction and injection. A flow model of nonlinear ODEs with assisting boundary conditions is tackled with the help of computational software by using various standard techniques. ,e effects of relevant parameters on the concentration, thermal, and velocity distribution are illustrated graphically; also, the skin friction coefficient and flow rates of heat and mass transfer are calculated and shown in a tabular way. An analysis of the consequences proves that the flow field is effectively appreciable by injection and suction. Comparison with the already published work is made and found to be in good agreement.


Introduction
e boundary layer (BL) of velocity relates to the existence of shear stress and velocity gradient, the temperature BL is connected with the heat transfer and temperature gradient, while the concentration BL is associated with the concentration gradient and species transfer. e porous medium where the fluid is in motion may be responsible for the formation of these three BLs. ese three layers are affected by each other. Hydromagnetics or magnetohydrodynamics is basically the study of the flow having magnetic characteristics and behavior of electrically transmitted fluids. e basic idea regarding the MHD [1,2] is that, in a moving conductive fluid, the magnetic field induces current which enhances the fluid to get polarized and changes the magnetic field itself reciprocally. Some common examples of magnetofluids are electrolytes, salt water, liquid metals, and plasma. e initiative in the field of MHD was firstly taken by Faraday in 1832 in which he proposed the concept about the interaction of the sea flow with earth as a magnetic field. Later, in 1937, Lazarus and Hartman found the exact solutions of MHD equations for the first time in the history. ey had also performed some experiments on the liquid metal flow in the MHD channel. After them, in 1942, Hannes Alfven described the class of MHD waves, later known as Alfven waves. For the achievements in the field of MHD, he was conferred an honorary Nobel prize in 1970.
Currently, MHD has a wide range of applications in geophysics, astrophysics, and agriculture sciences [3,4]. Also in the technical field, the MHD study has considerable interest because of its fruitful applications in the industrial field, for instance, MHD liquid metal flow control, cooling of nuclear reactors, high-temperature plasma, MHD power generators, and biological transportation. Since the last few decades, the study of a 2D boundary layer flow (BLF) of incompressible and viscous fluids around a shrinking/stretching sheet is worthy of attention due to its vast applications in engineering disciplines [2][3][4] [3,6] numerically. Bhargava and Chandra had studied the numerical solution of heat transfer analysis and MHD flow around a nonlinear stretching sheet fixed in a porous medium [7]. Also, Alinejad and Samarbakhsh [4] investigated viscous flow with heat dissipation on a nonlinear stretching sheet [1,8], while Zaimi et al. worked with a permeable stretching sheet with viscous effects. Later, Dessie et al. [8,9,10] amplified Zaimi's work by making a part of MHD flow with viscous dissipation effects [7]. Recently, Oderinu et al. and Bhargava et al. [7,8] solved the MHD partial differential equations and obtained the results in the form of series solutions with slip BCs in the porous medium. In many extents of engineering and industrial areas of interest, the flow through porous media plays an important role in MHD flow meters, MHD pumps, and MHD power generators [3,4]. Moreover, in polymer industries, wide range of applications are to be found in a particular flow on a stretching sheet. A little while back, Anwar et al. and Cortell et al. have studied viscous conducting fluid flows around a stretching sheet with the constant rate of stretching [6,11]. Sharma et al. [12] had inquired the influence of chemical reaction and ohmic dissipation in the permeable medium, and Pattnaik et al. [13] had studied the MHD flow with constant suction/injection in a porous medium. Prakash et al. [14] worked on the influence of the MHD nanofluid flow on the inclined stretching sheet under viscous and ohmic heating effects. Recently, viscous and ohmic dissipation on magnetohydrodynamic nanofluid flow in the porous medium had been analyzed with suction/injection effects by Jagadha et al. [14,15]. Also, various semianalytical approaches for the MHD squeezing flow had been thoroughly discussed by Ullah et al. [16], while Jabeen et al. [17] had presented a comparison of semianalytical methods for the MHD flow in the porous medium. Mabood et al. [17,18] numerically analyzed the multiple slip results on the MHD unsteady flow of mass and heat transfer around a stretching sheet. Emeka [19] reported the viscous dissipation and Soret influence on the MHD fluid flow with thermal radiation. A basic objective of the current study is to inquire the mass and heat transfer rates on a steady and viscous fluid flow around a nonlinear stretching sheet fixed in a permeable medium under the influence of viscous and ohmic dissipation, Eckert number, Soret number, and Prandtl number. e set of MHD equations is actually the amalgamation of Navier-Stokes and Maxwell equations. For this purpose, to overcome the difficulties concerning the closed-form solution of the nonlinear boundary value problem, some semianalytical methods are taken into account. Many other research studies in the literature with great interest have largely used perturbation techniques HPM, HAM, ADM, and successive approximation method. But in this work, we will mainly focus on the behavior of ADM, DTM, and VIM along with Pade approximation [6,17,18]. For the solution of nonlinear and linear ODEs and PDEs, these methods are strongly effective and reliable which can be applied directly to nonlinear equations in physics, engineering, and mathematics. Although these methods have some drawbacks, they are applicable in the small region although invalid in an unbound domain. To tackle this issue, we have considered Pade approximation [17,19].

Mathematical Formulation
We consider the steady 2D viscous incompressible flow of the electrically transmitted fluid around a nonlinear stretching sheet fixed in a porous medium. e applied magnetic field acts in the perpendicular direction to the flow with uniform strength B 0 . e magnetic field is assumed insignificant as the magnetic Reynolds number is considered small enough. e physical configuration of the flow regime is presented in Figure 1.
e basic two-dimensional BL equations subject to the above consideration are zu zx where u and v are the velocity components, ] is the coefficient of kinematic viscosity, ρ is the fluid density, B is the strength of the induced magnetic field, T is the temperature, k/(ρC p ) is the thermal diffusivity, q r is the radiative heat flux, C is the concentration of the fluid, D m are the mass diffusion coefficients, and T m and k T are the mean temperature and the ratio of thermal diffusion. Also, U w � cx n (n > 1) is the stretching velocity of the sheet, and B(x) � B 0 (x)x (n− 1)/2 and k p (x) � k p′ x 1− n are the variable magnetic field and variable permeability, respectively. Governing equations associated with the BCs are as follows: where v w � v 0 x (n− 1)/2 is the blowing/suction velocity. For the case of suction, v w < 0, and for the case of blowing, v w > 0. Also, T 0 and C 0 are positive concentration and temperature references. e radiative heat flux term in the temperature equation is simplified by the Rosseland approximation, i.e., q r � − (4σ * /3k * )(zT 4 /zy), where σ * is the Stefan-Boltzmann constant and k * is the mean absorption coefficient. Temperature variation in the flow is supposed to be small enough, so T 4 can be considered a linear function of temperature; neglecting the higher order term while expanding T 4 in Taylòr series about T ∞ is obtained as ∞ as taken by the authors in [12,14,19]. e continuity equation is satisfied by choosing the stream function Ψ which is Introducing similarity transformation [2,13,20], Also, the transformed BCs are where dimensionless parameters in (5)-(7) such as magnetic field, permeability parameters, Prandtl, Eckert, thermal radiation, Schmidt, and Soret numbers are, respectively:

Results and Discussion
e set of nonlinear ODEs (2)-(6) with described BCs is solved with the symbolic computational software MAPLE by using various standard techniques.
e comparison with previous studies is discussed in Tables 1 and 2 [2,8]. e skin friction coefficient f ″ (0), the rate of heat transfer θ ′ (0), and the mass transfer rate ϕ ′ (0) are tabulated in Tables 3-5 for various values of relevant parameters. Figures 2(a), 2(e), and 2(k) depict the behavior of the magnetic field on the velocity profile. It is described in an electrically conducting fluid that the magnetic field presence produces Lorentz force which acts reversely in the direction of the flow if the magnetic field is enforced in the perpendicular direction which slows down the fluid velocity. Figures 2(b), 2(f ), and 2(l) describe the behavior of the permeability parameter kp. It is observed that as we increase the value of kp, the velocity profile increases. Physically, whenever in the porous medium, the holes are large enough and then the resistance of the porosity medium

Mathematical Problems in Engineering
can be neglected. So the velocity profile increases because the presence of the permeable surface increases the resistance to the fluid. In Figures 2(c), 2(j), and 2(o), suction and injection parameters are obtained on velocity, thermal, and concentration profiles. It is observed that, on a stretching sheet, suction on the BL slows down the backflow, whereas injection increases the reverse flow strength. It is observed in Figure 2(g) that thermal BL thickness reduces when the Prandtl number (Pr) increases. Pr is the ratio of momentum diffusivity to thermal diffusion. So when Pr gets smaller, the thermal BL thickness becomes larger as compared to the velocity BL thickness. Also, thermal BL thickness reduces when Pr increases. Figure 2(d) shows the variation in the velocity profile. It is clearly seen that when we increase the value of n, the velocity profile decreases. Figure 2(h) shows the heat transfer dissipation factor or Eckert number (Ec) indicating the interrelationship between the fluid flow's kinetic energy and enthalpy differences of the BL. It expresses the conversion of kinetic energy into internal energy. So, it is observed that the thermal BL thickness increases when viscous dissipation increases. In Figure 2(i), the effects of the thermal radiation parameter R have been observed. It is noticed in the graphical representation that, with the increment of the value of R, the thermal BL thickness increases which is due to the fact that the radiation in the thermal BL increases when the value of R increases. So the temperature gets higher, which causes the increment of the temperature profile.

Conclusion
e steady flow of the viscous incompressible Newtonian fluid having a magnetic field with porous influence is analyzed under the viscous dissipation and ohmic dissipation effects by using similarity transformation and assisting BCs.
e system of coupled nonlinear ODEs is tackled with the help of computational software (MAPLE) by using various standard techniques. ese techniques are considered effective and convenient in the literature. e findings of this study are presented via tables and graphs, which are in good agreement by physical behavior already discussed in the literature. e following points are to be noticed in the whole discussion:

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