Analytical Simulation for Magnetohydrodynamic Maxwell Fluid Flow Past an Exponentially Stretching Surface with First-Order Velocity Slip Condition

The study of fluid flow upon an exponentially stretching surface has significant importance due to its applications in technological phenomena at the industrial level. These applications include condensing process of fluid film, heat exchanger processes, extrusion of plastic sheet in aerodynamics, cooling process of metal sheet, and growth of crystals, etc. Keeping in view all these applications, in this paper, we have discussed the magnetohydrodynamic flow of Maxwell fluid past an exponentially stretching sheet. The stretching surface is considered to be slippery by imposing the velocity slip condition. The magnetic field impact is taken into consideration. Furthermore, heat radiation, Joule heating, Brownian motion, and thermophoresis are also considered. The modeled system is reduced to ordinary differential equations with the help of similarity variables. For the analytical solution, we have used the homotopy analysis method. Furthermore, HAM is compared with the shooting method and found to be in great agreement. The squared residual error of the fluid flow problem at 15th order of approximations for Newtonian and non-Newtonian cases has been investigated. It is found that the fluid flow problem converges quickly for the case of non-Newtonian fluid as compared to Newtonian fluid. In addition, the velocity profile increases while the thermal and concentration profiles reduce with greater values of Darcy number. The thermal profile is the increasing function of the Brownian motion parameter and Eckert number whereas the concentration profile is the reducing function of the Brownian motion parameter and Eckert number. With the augmentation in Darcy number, the permeability strength of porous medium increases which concludes the increasing conduct of thermal and mass transportation.


Introduction
The effect of magnetohydrodynamic (MHD) fluid flow is a physical property that describes the movement of an electrically conducted fluid with the impact of externally applied magnetic effects. The salt water, electrolytes, and plasma, etc. are MHD fluids. MHD fluid flow has plentiful industrial applications, for instance, drug targeting and cooling of reactors, etc. MHD is based upon the induction of electric current by applied magnetic field through a conducted moving fluid. The idea of MHD was first floated by Swedish electrical engineer Alfven [1], a developer of Alfven's waves. Many investigators have used the concept of MHD in various investigations. Turkyilmazoglu [2] has inspected the thermal characteristics for MHD fluid flow upon stretching a rotary disk by using Joule heating and viscous dissipation in the thermal equations. In this study, it has been noticed that an augmentation in magnetic effects has opposed the flow of fluid and has supported the thermal flow. Kumar et al. [3] investigated the influences of thermal radiations upon MHD fluid flow over an exponential stretched sheet. A comparison has been accumulated in this work with a fine agreement between the claimed and established results. Khan et al. [4,5] have established incredible results for MHD fluid flows and heat transfers by employing different flow conditions. The authors have proven that the augmented values of the magnetic field have enhanced the transmission of heat while they have decreased the flow behavior.
The fluid that exhibits viscoelastic characteristics, which is fluid with the behavior of viscosity as well as elasticity, is termed as Maxwell fluid. The Maxwell fluid model was proposed first by James Clerk Maxwell [6]. Due to its importance, many researchers have used Maxwell fluid in the investigation of mass and heat transfer. Islam et al. [7] inspected the impact of thermal radiations on Maxwell nanoparticle flow upon a stretching cylinder. Semi-numerical technique has been used in the study and has established that due to enhancement in Maxwell parameter, the stress relaxation phenomenon has been augmented, due to which the flow of fluid has declined. Shehzad et al. [8] have discussed the forced convection Maxwell fluid flow with the impact of thermophoresis particle movements past a spinning disk. Khan et al. [9] have analyzed the MHD Maxwell fluid flowing on the stretched surface. In this examination, the heat and mass transmission have been investigated by using the impacts of variations in thermal conductivity and the Cattaneo-Christove effects. It has been noticed in this work that, with an augmentation in Deborah number, more resistance has been offered to the fluid flow particles that have decreased the flow and concentration of fluid and have enhanced the thermal characteristics. Ahmad et al. [10] have inspected the unsteady Maxwell fluid flow past a time-dependent stretching sheet and have established that augmented values of Brownian motion have jumped up the thermal profile and have jumped down the flow and concentration of fluid. Farooq et al. [11] inspected the MHD and Maxwell fluid flow over the exponential stretched surface by using the famous Boungiorno model.
The study of fluid flow upon the exponentially stretching surface is of significant importance due to its applications in technological phenomena at the industrial level. These applications comprise condensing process of fluid film, heat exchanger processes, extrusion of plastic sheet in aerodynamics, cooling process of metal sheet, and growth of crystals, etc. After the revolutionary work of Sakiadis [12,13], many investigators have carried out different examinations for fluid flow upon exponentially stretching surfaces with a prime focus upon the mass and thermal flow. Nadeem et al. [14] have discussed the influences of slip effects over nanoparticle flow over a stretching sheet. The authors of this investigation have observed that with augmentation of stretching parameter, the microorganism and Nusselt numbers have enhanced while the skin fraction has declined with this increase. Lund et al. [15] examined the MHD fluid flow upon an exponentially stretching surface with the suction condition. Waini et al. [16] have inspected the mixed convective flow of hybrid nanofluid over an exponentially stretched medium and have established that the solid nanoparticles have decreased the flow profiles and have increased the thermal transmission. Patil et al. [17] inspected the influences of roughness upon mixed convection nanoparticle flow over an exponentially stretched surface. It has been observed in this work that the thermal flow rate of the wall is considerably controlled by the addition of nanoparticles. The topic can be further studied in the Refs. [18][19][20][21][22][23].
A surface that contains void spaces is termed a porous surface such as rubber, some rocks, woods, and sponges, etc., which are available in nature. Normally, a porous surface is described by the porosity of that surface. Some porous surfaces are manmade, such as ceramics and cements, etc. There are various applications of porous media in applied sciences and engineering, for instance, rock mechanics, geomechanics, petroleum geology, geophysics, and biophysics, etc. Because of its importance, many investigations have been conducted by different researchers. Alazmi and Vafai [24] have analyzed the thermal and fluid flow upon a porous medium in the combination of the fluid layer. Fatunmbi et al. [25] have discussed micropolar MHD fluid flow past a porous medium using slip conditions. It has been observed in this work that augmentation in stretching parameter has adverse effects upon the flow and thermal profiles. Krishna et al. [26] have inspected the time-dependent MHD flow of micropolar fluid upon permeable surface using the influence of ion slip and Hall current. In this investigation, the effects of Sherwood and Nusselt numbers have been investigated numerically with the influence of different physical parameters. Hayat et al. [27] have inspected the time-dependent nanoparticle fluid flow upon a permeable surface using different flow conditions and characteristics. In this investigation, Darcy-Forchheimer and Boungiorno's models have also been incorporated by the authors in the flow problem. It has been noticed in this study that augmented values of Darcy number and inertia parameters have enhanced the thermal profiles and have decreased the flow profiles. These same effects have also been observed for augmentation in Brownian effects. Asjad et al. [28] inspected the MHD fluid flow in a permeable medium with the help of fractional order derivatives. The authors have used the transform method to determine the exact solution. Khan et al. [29] have inspected the generation of entropy for thermos-solutal convection problems over porous medium. In this study, a newer mass Bejan number has been described for the first time using two salt concentrations. Jafar et al. [30] discussed MHD nanoparticle flow over a permeable and stretching surface with thermal radiation. The authors have used the Keller Box technique for the approximate solution of the problem.
In light of the above literature review, we are interested to investigate the Maxwell fluid flow through an exponentially stretching surface with first-order velocity slip condition using the porous medium. The homotopy analysis method is employed in order to investigate the proposed model analytically. Furthermore, heat radiation, Joule heating, Brownian motion, and thermophoresis are also considered.

Problem Formulation
Consider a laminar, two-dimensional, and incompressible MHD Maxwell fluid flow past an exponentially stretching sheet with first-order velocity slip condition. The flow of Maxwell fluid is submerged in a porous medium. The stretching velocity of the surface is considered as U w = U 0 exp(x/L) along x− direction, whereas y− direction is perpendicular to the stretching sheet. Furthermore, the heat radiation, Joule heating, Brownian motion, and thermophoresis phenomena are also considered.
In view of the above assumptions, the leading equations take the form: ∂u ∂x where u and v are the velocity components, ρ is the fluid density, B 0 is the strength of magnetic field, λ is the relaxation time, q r = −4σ * /3k * ∂T 4 /∂y is the radiative heat flux, K is the permeability, c p is the specific heat, α = k/ρc p is the thermal diffusivity, D B , D T are the Brownian and thermophoresis coefficients, U 0 is reference velocity, T is the temperature, T 0 , T ∞ are the reference and ambient temperatures, C is the concentration, C 0 is the reference concentration, and C ∞ is the ambient concentration.
The boundary conditions are defined as: Here, µ is the viscosity coefficient, L 1 is the slip length, T w = T ∞ + T 0 exp(x/2L) and C w = C ∞ + C 0 exp(x/2L) are the varying temperature and concentration.
The similarity variables are defined as: where L is the reference length.
Using the above similarity transformations, Equation (1) is obvious and Equations (2)-(5) are transformed as: In the above system of equations, β = λU w /L is the Maxwell parameter, Prandtl number is Pr = µc p /k, M = 2LσB 2 0 /ρU w is the magnetic parameter, Nt = τD T (T w − T ∞ )/νT ∞ and Nb = τD B (C w − C ∞ )/ν are thermophoresis and Brownian parameters, the Lewis number is Le = ν/D B , Da = KU w /2νL is the Darcy's number, Ec = U 2 w /c p T w is the Eckert number, S = QL/U w c p is heat source/sink parameter, and thermal radiation parameter is Rd = 16σ * T 3 ∞ /3kk * . The quantities of engineering interest are described as: In view of Equation (6), Equation (11) is reduced as: where Re x = U w x/ν describes the local Reynolds number.

HAM Solution
In order to solve Equations (7)-(9) along with the respective boundary conditions defined in Equation (10), homotopy analysis method is applied. The initial guesses and linear operators are defined as: with the properties: are the arbitrary constant.

Results and Discussion
The purpose of this section is to examine the impact of embedding factors on different profiles of  Figure 4 shows the effect of Deborah number on velocity profile. The greater Deborah number reduces the velocity profile. The maximum influence of Deborah number is perceived at the surface of the fluid flow; however, this impact reduces gradually, and the properties of the non-Newtonian fluid shift to Newtonian fluid. Thus, as we increase the Deborah number, the velocity profile reduces. Figures 5-7 depict the impact of magnetic field on velocity, thermal, and concentration profiles, respectively. The velocity profile reduces while the thermal and concentration profiles augment with a greater magnetic field parameter. The greater magnetic field parameter yields to the Lorentz force which always opposes the flow of fluid. Thus, a reducing impact is observed here. Furthermore, the temperature and mass profiles are the augmenting functions of the magnetic field. Figures 8 and 9 portray the impact of thermophoretic parameters on thermal and concentration flow profiles. Both the thermal and concentration profiles are increasing functions of thermophoresis parameter. This behavior is because the greater thermophoresis parameter moves the nanoparticles of fluid from hot to colder region which results in increasing the temperature of the fluid flow. Thus, an increasing impact of thermophoresis parameter on thermal profile is depicted here. A similar impact of thermophoresis parameter on concentration profile is observed. Figures 10 and 11 show the influence of Brownian motion on thermal and concentration profiles, respectively. Brownian motion is responsible for augmentation in thermal profile and a reduction in concentration profile. The greater Brownian motion parameter heats up the concentration boundary layer thickness, which moves the fluid nanoparticles from the exponentially extending surface to motion less fluid. Therefore, the concentration profile reduces with greater Brownian motion parameter. Figures 12 and 13 signify the outcome of Eckert number on thermal and concentration profiles. The thermal profile is the increasing function, while the concentration profile is the reducing function of the Eckert number. Eckert number is mostly utilized in nuclear engineering systems for compressible flow at high speeds. The increasing Eckert number heightens the internal energy which consequently increases the temperature of the fluid. A very slight diminishing influence of Eckert number on concentration profile is depicted. Figure 14 shows the impact of Lewis number on concentration profile. The greater Lewis number reduces the concentration profile. The greater Brownian motion diffusion coefficient is responsible for the reduction in concentration of the fluid flow because the Brownian motion diffusion coefficient is inversely related to Lewis number. Thus, the reducing impact in concentration profile is depicted. Figures 15-17 Table 1 shows the comparison of the current analysis with previous studies. Here, we have found great agreement of our current study with previously published results. Table 2 shows the squared residual errors along with CPU time for velocity, temperature, and concentration profiles. Tables 3 and 4 show the impacts of different embedded factors on thermal and mass transfer rates. It is observed that the radiation parameter augments the values of Nu x and Sh x . It is obvious that the higher thermal radiation escalates the heat transfer rate while an opposite impact is observed against mass transfer rate. The heat transfer rate is a declining function of Nb and Nt. As a result, both Brownian motion and thermophoresis processes in the current work tend to push the nanoparticles away from the stretching surface. Moreover, increasing Brownian motion or thermophoresis parameters accelerates the nanoparticles' diffusion into the boundary layer, lowering the heat transfer rate. Moreover, a decreasing impact of Nb on mass transfer rate is observed while the thermophoresis parameter has increasing impact on mass transfer rate. It is observed that the higher Darcy number augments both these rates. Actually, the greater values of Da designates the porousness of porous media. Therefore, as we increase the Darcy number, the permeability strength of porous media increases, which concludes the increasing conduct in transportation rates of mass and heat. It is perceived that the higher magnetic parameter jumps down Nu x and jumps up Sh x . Physically, the addition of magnetic effects to the fluid flow creates the resistive force at the surface of the fluid flow which raises the skin friction, but at the same time, the heat transfer rate reduces. In addition, it is found that Sh x augmenting function of magnetic parameter is at the surface of the fluid while after some time the mass transfer rate is declined. It has been noticed that the higher Eckert number reduces the heat and mass transfer rates. Actually, the greater Eckert number increases the thermal and nanoparticle boundary layer thicknesses, which consequently reduce the thermal and mass transportation rates. Tables 5-7 show the comparison of HAM and numerical method for velocity, temperature, and concentration profiles, respectively. These tables are associated with Figures 15-17. x function of magnetic parameter is at the surface of the fluid while after some time the mass transfer rate is declined. It has been noticed that the higher Eckert number reduces the heat and mass transfer rates. Actually, the greater Eckert number increases the thermal and nanoparticle boundary layer thicknesses, which consequently reduce the thermal and mass transportation rates. Tables 5-7 show the comparison of HAM and numerical method for velocity, temperature, and concentration profiles, respectively. These tables are associated with Figures 15-17.

Conclusions
This work describes MHD Maxwell fluid flow through an exponentially stretching sheet. The fluid's particles are immersed by a permeable surface. The flow has been analyzed with viscous dissipation, thermal radiation, Joule heating, Brownian motion, and thermophoresis phenomena. HAM has been employed for the solution of leading equations of the proposed model. The impact of numerous variables upon the flow system has been described theoretically. After an in-depth investigation of this work, the forthcoming points have been observed:

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The flow profile increases while the thermal and concentration profiles reduce with greater Darcy number.

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The greater Deborah number augments the velocity profile.

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The augmentation in magnetic parameter reduces the velocity profile, while the temperature and concentration profile increases with the increasing magnetic parameter. • Both the thermal and concentration profiles are the increasing functions of the thermophoresis parameter.

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The thermal profile is the increasing function of the Brownian motion parameter and Eckert number, whereas the concentration profile is the reducing function of the Brownian motion parameter and Eckert number.

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With the augmentation in Darcy number, the permeability strength of porous media increases, which concludes the increasing conduct of thermal and mass transportation.

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In this investigation, the higher Eckert number reduces the heat and mass transfer rates. • Growth in magnetic parameter reduces the heat transfer rate while it increases the mass transfer rate.