Heat and Mass Transfer in Stagnation Point Flow of Maxwell Nanofluid Towards a Vertical Stretching Sheet with Effect of Induced Magnetic Field

(is paper presents a mathematical model analysis of heat and mass transfer in a two-dimensional flow of electrically conducting, thermally radiative, and chemically reactive Maxwell nanofluid towards a vertical stretching and permeable sheet embedded in a porous medium. Boundary layer approximation and suitable transformations are used to reduce the governing differential equations convenient for computation. Eventually, the transformed nonlinear differential equations along with the corresponding boundary conditions are solved in the framework of optimal homotopy analysis method. (e effects of induced magnetic field, buoyancy force, viscous dissipation, heat source, Joule heating, and convective boundary condition are analyzed in detail. (e rates of heat, mass, and momentum transfer with respect to the relevant parameters are also examined in terms of the local Nusselt number, Sherwood number, and skin friction coefficients, respectively. Among the many results of the study, it is shown that the induced magnetic field, flow velocity, and temperature profiles are increasing functions of the Maxwell parameter. (e results of the present study are also in a close agreement with previously published results under common assumptions.


Introduction
Based on the deformation of fluids in response to the applied shear stress, fluids can be classified into two categories, namely, Newtonian and non-Newtonian. Fluids such as honey, corn starch, lubricating oils, and paints are non-Newtonian fluids having unpredictable viscosity and responses to applied forces. In 1867, Maxwell developed a constitutive model that is used to describe stress relaxation of air. Now, this model has been extensively utilized to analyze the viscoelastic flow characteristics of many industrially important fluids such as liquid polymers, paints, paper pulps, shampoos, flour dough, certain engine oils, and other viscoelastic fluids. On the other hand, flows near stagnation point are commonly encountered in flows around tips of aircrafts, rockets, ships, submarines, etc. ese flows also occur in the cooling of electronic devices and nuclear reactors and in many other hydrodynamic processes. e first classical investigation of stagnation point flow of viscous fluids over a flat stationary plate was made by Hiemenz in 1911. He developed an exact solution to the Navier-Stokes equations governing the flow phenomena. e study of fluid flow, heat transfer, and mass diffusion near stagnation point has significantly contributed in understanding and solving problems in manufacturing and engineering. In order to enhance the heat transfer capabilities of traditional fluids, Choi and Eastman [1] introduced the concept of nanofluids in 1995, and they showed experimentally that embedding of nanometer-sized particles with the common base fluids such as water, oil, and ethylene glycol mixture radically increased the thermal conductivity of the fluid. Due to their wide range of applications, a number of experimental and theoretical investigations have been conducted to examine the flow properties of nanofluids in a variety of flow situations. For instance, the heat transfer phenomena of Nanofluid moving over a wedge surface are presented by Ibrahim and Tulu [2].
Recently, Zhang et al. [3] analyzed the three-dimensional flow of nanofluid among rotating circular plates. Arain et al. [4] deal with the boundary layer flow characteristics of Carreau fluid in the presence of nanoparticles. Further, Alwawi et al. [5] analyzed the effects of magnetic field and nanoparticles in boundary flow of Casson nanofluids.
A number of researchers have been interested in conducting several investigations on stagnation point flow of Maxwell nanofluids. For instance, Nagendramma et al. [6] considered a two-dimensional stagnation point flow of Maxwell nanofluid over a stretching surface in the presence of thermal radiation and viscous dissipation. Ramesh et al. [7] carried out a numerical study to analyze and discuss a stagnation point flow of a Maxwell nanofluid towards a permeable stretching sheet. Bai et al. [8] investigated a two-dimensional stagnation point flow of Maxwell nanofluid over a stretching surface by taking both the convective heat transfer and the Brownian motion into account. Haritha et al. [9] investigated the combined effect of Navier slip and convective boundary conditions on an unsteady two-dimensional flow of a Maxwell fluid over a stretching surface in the presence of magnetic field, thermal radiation, heat source, and chemical reaction. Mushtaq et al. [10] studied Buoyancy effects in stagnation point flow of Maxwell fluid utilizing non-Fourier heat flux approach. Recently, Khan et al. [11] investigated the heat and mass diffusion effects in the stagnation point flow of Maxwell nanofluid over a stretchable sheet with slip conditions and chemical reaction. Aziz and Shams [12] studied the heat transfer processes due to the flow of electrically conducting Maxwell nanofluid in the context of volumetric entropy generation.
In the aforementioned papers, and in many other early works, the effect of induced magnetic field is usually neglected by assuming an extremely small magnetic Reynolds number and/or getting the mathematical analysis of the problem simple. However, in various practical situations such as power generation, manufacturing of glass, purification of crude oil, and geophysics, the induced magnetic field effect is significant. To this end, extremely few studies are available in open literature that examine the effect of induced magnetic field in stagnation point flow of Maxwell nanofluids. Ibrahim [13] considered the effect of induced magnetic field in examining a stagnation point flow and heat transfer due to upper-convected Maxwell nanofluid over a stretching sheet. In his numerical study, the skin friction coefficient, local Nusselt number, and Sherwood number are found to be decreasing with the increase in the stretching parameter and magnetic field strength. His study also showed that the increase in the convective parameter causes a decrease in the local Sherwood number but an increase in the local Nusselt number. e review of related literature reveals that even though several investigations have been made in the area under consideration, the field remains an active area of research with a number of study gaps. us, motivated by the aforementioned early works, this study attempts to examine the influences of magnetic field, thermal radiation, heat source, viscous dissipation, Joule heating, chemical reaction, permeability, and porosity parameters on stagnation point flow of Maxwell nanofluid towards a vertical stretching sheet. Further, a relatively recent and more reliable optimal homotopy analysis method has been implemented successfully to solve the resulting model equations. e results of the present study are also in close agreement with previous works under some restricted assumptions.

Mathematical Formulation of the Flow System
In this study, it is assumed that as a steady laminar flow of Maxwell nanofluid is approaching a convectively heated vertical sheet, the flow divides into two streams keeping the stagnation point fixed at the origin. Also, a uniform magnetic field B of strength B 0 is applied normally to the vertical sheet. Magnetic Reynolds number is taken to be large enough to consider the effect of induced magnetic field. Further, an incompressible, electrically conducting, thermally radiative and chemically reactive fluid is considered, and the permeable sheet is stretching with speed U w � ax, where a > 0 is a constant. It is also assumed that the flow system is embedded in a porous medium as illustrated in Figure 1. Accordingly, the equations governing the aforementioned flow system can be extended from Ibrahim [13], Hayat and Nadeem [14] as where (u, v) and (H 1 , H 2 ) denote the components of fluid velocity vector V and induced magnetic field vector H along x and y directions, ρ f , μ m , λ 0 , σ, κ and β 0 denote density, magnetic permeability, viscoelasticity, electric conductivity, thermal conductivity, and thermal expansion of the nanofluid, respectively; the quantities g and K 0 represent the acceleration due to gravity and permeability of the porous medium, respectively; υ � (μ/ρ f ) is kinematic viscosity of the nanofluid with μ denoting coefficient of dynamic viscosity; U e and H e are the free stream velocity and induced magnetic field; T and C denote temperature and nanoparticle volume fraction; T ∞ and C ∞ are the corresponding ambient values of temperature and concentration; α � (κ/(ρC p ) f ) is thermal diffusivity, and τ � ((ρC p ) p / (ρC p ) f ) is the ratio of effective heat capacities of nanoparticle and the ordinary fluid; D B and D T are the Brownian and thermophoresis diffusion coefficients; k * and σ * are the mean absorption and the Stefan-Boltzmann constants, respectively; the coefficient Q 0 stands for heat generation (when Q 0 > 0) and heat absorption (when Q 0 < 0); the rates K r > 0 and K r < 0 denote destructive and constructive reaction rates, respectively. Taking the boundary conditions, at y � 0, and as y ⟶ ∞,  Figure 1: Sketch of the flow problem.

Mathematical Problems in Engineering
where U w , T w , C w denote the surface velocity, temperature, and nanoparticle concentration, respectively; V w is the mass transmission at the surface of the stretching sheet; V 0 is the constant value of velocity; k and h f are thermal diffusivity and convective heat transfer coefficients, respectively; H 0 is the initial induced magnetic field strength; a and b are positive constants denoting velocity rate of the stretching sheet and the fluid in the free stream, respectively.
In order to further simplify the mathematical analysis, we use the following transformations: where η is a dimensionless similarity variable; f ′ (η), g(η), θ(η) and φ(η) denote the dimensionless functions for velocity, induced magnetic field, temperature, and concentration profiles. Introducing the stream function ψ(x, y) with the property u � (zψ/zy) and v � − (zψ/zx), the continuity equation for velocity in equation (1) is identically satisfied. Also, from the definition of H 1 and H 2 in equation (9), the continuity equation for induced magnetic field equation (2) was satisfied identically. Computing the required partial derivatives and substituting the values into the governing equations (3)-(6), the following system of nonlinear differential equations can be obtained: where the prime indicates differentiation with respect to η; M � (μ m H 2 0 /ρυ 2 ) and H 2 a � (σB 2 0 /aρ f )(u − U e ) are parameters representing the induced and the external magnetic field strengths, respectively; λ � aλ 0 ≥ 0 is the Deborah number representing the Maxwell viscoelastic parameter; K p � (υx/U w K 0 ) is the porosity parameter; δ � (b/a) is a velocity ratio representing the sheet stretching rate, with δ ≥ 1 and δ < 1, respectively, implying that the fluid free stream velocity is greater than the sheet velocity and vice is a gravity dependent parameter for assisting (when r > 0) and opposing (when r < 0) flows; ) is the buoyancy ratio; ε � (1/μ m συ) > 0 is the magnetic diffusivity number; P r � (υ/α) and S c � (υ/D B ) are the Prandtl number and the Schmidt number, respectively; are the nanofluid parameters that denote Brownian motion, which represent mass transfer and thermophoresis effects, which represent heat transfer characteristics, respectively; Q � (xQ 0 /(ρC p ) f U w ) is the heat generation (for Q > 0) or absorption (for Q < 0) parameter; and c � (K r x/U w ) is the chemical reaction parameter, with c > 0 and c < 0 indicating the destructive and generative chemical reaction rates, respectively. Also employing the transformation in equation (9), the boundary conditions in equations (7) and (8) can be reduced as follows: where the parameter S � (V 0 / �� aυ √ ) denotes injection and suction for S > 0 and S < 0, respectively; is the Biot number representing the convective parameter.
From a practical point of view, we exploit the behavior of Skin friction C f , local Nusselt number Nu x , and Sherwood number Sh x in the region of stagnation point defined, respectively, as where is the mass flux at the stretching surface.

Method of Solution
e homotopy analysis method (HAM) is one of the relatively recent mathematical methods showing greater efficiency over the past few years. e method is also known for its flexibility and high accuracy to give analytic approximations to the solutions of many important problems in science and engineering. Details of the method can be found in Liao [15,16]. In order to implement the homotopy analysis method, we first choose the following auxiliary linear operators: satisfying the conditions where C i (i � 1 − 10) are integral constants to be determined by the boundary conditions. Next, based on the linear operators and the boundary conditions, we take the following initial approximations: e nonzero auxiliary functions can also be selected as Finally, the parameter values δ � ε � 1.5, P r � S c � 1, 1 are used throughout this study unless otherwise stated. In order to do the required computations, we employ a powerful computational software, namely, the BVPh2.0, which is a HAM-based Mathematica package, developed by Zhao and Liao [17]. Now, before presenting the results of the study, we give the following convergence analysis of the method. Since convergence of the HAM solutions depends strongly on the Mathematical Problems in Engineering proper choice of the so-called convergence-control parameters, we determine the admissible values of Z f , Z g , Z θ and Z φ as follows. According to Liao [15], one of the mechanisms to identify the range of such values is to sketch the Z-curves as shown in Figure 2. is result illustrates that the Z-curves are nearly horizontal in the intervals: So, any value in these intervals may be taken to get convergent solutions to our flow problem.
More systematically, the optimal values for the convergence-control parameters are determined by reducing the squared residual error: Upon using the BVPh2.0 Mathematica package, it has been found that Further, the respective residual errors and the required values of derivatives are presented against the orders of HAM approximations in Table 1.
It is clear from Table 1 and Figure 3 that increasing the order of HAM approximation decreases the squared residual errors and leads the series to converge. e sketch in Figure 3 clearly indicated that the total error is decaying extremely fast for increasing orders of the first few iterations.
is guarantees that the method is convergent for the selected optimal values of the convergence parameters. To further ensure the validity of our work, we made a comparison of our results with respect to some other previous reports in the absence of extended physical effects as presented in Table 2. Table 2 verifies that, for the selected values of the parameters, the values of the physical quantities of practical interest determined in this study are in excellent agreement with those of the results reported by Hayat and Nadeem [14].

Results and Discussions
In this section, we present the most significant results of the study in graphical or tabular forms followed by brief discussions. e impacts of various thermophysical parameters on the dimensionless velocity f ′ (η), induced magnetic field g(η), temperature θ(η), and concentration φ(η) profiles in the stagnation point region are presented as follows: e effects of external magnetic field in the flow field profiles can be expressed in terms of the Hartman number Ha as presented in Figure 4.
Here, it is worth mentioning that the term (σB 2 0 /ρ)(u − U e ) can be viewed as the combination of the imposed pressure term ((σB 2 0 /ρ)U e ) and the Lorentz force ((σB 2 0 /ρ)u). Unlike free convection, for forced convection regime, the external flow velocity U e is greater than the velocity u of the fluid in the boundary layer region. at is, the imposed pressure overcomes the Lorentz force. is corresponds to an assisting force in the direction of main flow.
us, the increase in external magnetic field advances the flow velocity and the induced magnetic field. However, the opposite behavior is observed for temperature profile. e influence of stretching parameter δ on the dimensionless profiles in the region of stagnation point has been investigated and illustrated graphically in Figures 5(a) and 5(b). It is revealed that as the parameter δ increases, the velocity profile also increases for both δ < 1 and δ ≥ 1 but with inverted boundary layer structures. is is due to the fact that as the velocity ratio parameter increases, the velocity of the straining motion near the stretching surface also increases from the smooth intersection of the external fluid and the sheet. Also, the induced magnetic field profile was found to be increasing for δ < 1 and decreasing for δ ≥ 1. On the other hand, temperature and concentration profiles have remained decreasing as we move away from the stretching sheet. e influences of buoyancy ratio parameter N r on dimensionless velocity, induced magnetic field, temperature, and concentration profiles for both assistive and resistive flows are analyzed and given in Figures 6(a) and 6(b).
For resistive flow r < 0, it can be observed from Figure 6(a) that both the velocity and induced magnetic field profiles are increasing functions of N r . On the other hand, temperature and concentration profiles slowly decrease with N r . e opposite behavior of the profiles can be noticed for the assistive flow r > 0 as shown in Figure 6(b). is is true because, in the assistive flow, the buoyancy force acts against the flow field.      [14] Present study [14] Present study [14] Present study As pointed out in Figures 7(a) and 7(b), the velocity profile is increasing function of λ. Physically, at higher Deborah numbers, elasticity of the material increases, while viscosity decreases, which enhances the fluid velocity. Similarly, the induced magnetic field also increases with the increase in the values of λ. On the other hand, the temperature gradually decreases, and no significant fluctuation of concentration is observed with increase in the values of λ. e impact of the convective parameter B i has been studied and illustrated in Figure 8. It can be noticed that the velocity, temperature, and concentration profiles can be lowered by increasing the Biot number.
It can be deduced from Figure 8 that the influence of B i is more prominent on temperature profile near the vertical sheet. It is shown that the temperature profile is declining for increasing values of B i . e reciprocal of magnetic Prandtl number ε is the ratio of the magnetic diffusivity to viscous diffusivity. Its impact in the flow field profiles has been examined and shown in Figures 9(a) and 9(b).
As shown in Figures 9(a) and 9(b), the increase in the reciprocal of magnetic Prandtl number ε causes the increase in momentum and induced magnetic field profiles near the  Mathematical Problems in Engineering stretching surface. is is evident from the relation that the reciprocal of magnetic Prandtl number is inversely proportional to momentum diffusivity, which is low near the stretching sheet.
Similarly, the effect of porosity in the flow problem has been investigated and presented as in Figures 10(a) and  10(b).
It can be observed in Figures 10(a) and 10(b) that the velocity and induced magnetic field profiles increase as porosity of the medium increases in the flow field. e influence of radiation R d in the region of stagnation point flow has been determined as depicted in Figure 11(a) and 11(b). e results in Figures 11(a) and 11(b) show that the increment in thermal radiation leads to the growth of thermal boundary layer. Physically, this is due to the fact that larger thermal radiation develops higher heat flux at the surface. It is also noted that as thermal radiation increases, the velocity and induced magnetic field profiles also increase, but the nanoparticle concentration profile reduces in the stagnation point region.
Chemical reactions occur in many industrial and engineering activities such as hydrolysis, electroplating, and combustion processes.
According to the results in Figures 12(a) and 12(b), as the destructive chemical reaction (c > 0) increases, the induced magnetic field profile has been increased near the stretching sheet, and the concentration profile decreases in the region of stagnation point. On the other hand, the concentration profile increases with the constructive chemical reaction (c < 0) causing the raise of nanoparticle concentration at the sheet. is holds as constructive chemical reactions (c < 0) result in consumption, while a destructive chemical reaction (c > 0) causes production of the chemical species.
In many situations, heat is generated within a body by a chemical, electrical, or nuclear process. e heat source parameter Q has been used to analyze the effects of heat generation and heat absorption as shown in  13(d) show that the temperature of the nanofluid increases with heat generation and decreases with heat absorption. Here, the effect of heat generation yields thermal energy; thereby, the temperature rises, while heat absorption causes the removal of thermal energy in which the temperature profile declines.
Viscous dissipation is the conversion of kinetic energy into thermal energy due to the work done against the viscous fluid. e effects of viscous dissipation can be demonstrated by using the Eckert number Ec.
It is shown in Figure 14(a) that the increases in the dissipation parameter cause the increase in temperature profile. Physically, this is true because the increase in the dissipation effect implies a rise in the frictional heating, which leads to storing of more heat energy in the flow system. Also, Figures 14(b) and 14(c) display that the velocity and induced magnetic field profiles are found to increase with the increase in the dissipation parameter.
is study also examines the permeability effects in terms of the suction/injection parameter S as depicted in Figures 15(a) and 15(b).
It can be seen from Figures 15(a) and 15(b) that both the velocity and induced magnetic field profiles are increasing functions of the injection parameter. On the other hand, the temperature and concentration profiles both decrease with the increase in the injection parameter. e impacts of some parameters on the rates of momentum, heat, and mass transfer near the stretching sheet are examined and expressed in terms of the skin friction, local Nusselt number, and Sherwood number. To this end, the values of the corresponding boundary derivatives f″(0), − θ ′ (0), and − φ ′ (0) are computed and provided in Table 3. Table 3 shows that the values of skin friction will grow if any values of δ, E c , ε, K p and R d are increased, or any values of N r , Q − and c − are decreased. e local Nusselt number can be increased by increasing N r , c and Q − or decreasing E c , ε and K p . Also, the local Sherwood number can be

Concluding Remarks
In this study, the heat and mass transfer phenomena in MHD stagnation point flow of Maxwell nanofluid towards a convectively heated, vertical, and stretching sheet embedded in a porous medium have been analyzed in the presence of induced magnetic field and various thermophysical parameters. e homotopy analysis method (HAM) has been applied to obtain series solutions for the steady twodimensional laminar flow near the stagnation point. Convergence of the method has been verified. Further, the results obtained in this study are also found in excellent agreement with previous study works under some restricted assumptions. e influences of pertinent parameters on velocity, induced magnetic field, temperature, and concentration profiles are examined as summarized below: (i) e flow velocity can be accelerated by increasing rates of stretching, external magnetic field, Maxwell parameter, convective parameter, reciprocal of