A Simplified Finite Difference Method (SFDM) for EMHD Powell–Eyring Nanofluid Flow Featuring Variable Thickness Surface and Variable Fluid Characteristics

We study constant and variable fluid properties together to investigate their effect on MHD Powell–Eyring nanofluid flow with thermal radiation and heat generation over a variable thickness sheet. *e similarity variables assist in having ordinary differential equations acquired from partial differential equations (PDEs). A novel numerical procedure, the simplified finite difference method (SFDM), is developed to calculate the physical solution. *e SFDM described here is simple, efficient, and accurate. To highlight its accuracy, results of the SFDM are compared with the literature. *e results obtained from the SFDM are compared with the published results from the literature. *is gives a good agreed solution with each other. *e velocity, temperature, and concentration distributions, when drawn at the same time for constant and variable physical features, are observed to be affected against incremental values of the flow variables. Furthermore, the impact of contributing flow variables on the skin friction coefficient (drag on the wall) and local Nusselt (heat transfer rate on the wall) and Sherwood numbers (mass transfer on the wall) is illustrated by data distributed in tables. *e nondimensional skin friction coefficient experiences higher values for constant flow regimes especially in comparison with changing flow features.


Introduction
Nowadays, the use of thermal analysis in industry for non-Newtonian fluids is undergoing far reaching consequences covering the processes in biology to the mechanical devices, namely, electronics machinery. erefore, it is worth investigating to optimize the flow of heat transfer for the system. One way in which the underlying system's thermal conductivity is enhanced is to use the nanofluids. e concept of variable thickness surface is helpful in reducing the weight of structural elements.
Magnetohydrodynamics (MHD) deals with the interaction between electrically conducting fluid and magnetic field. Studying MHD flow has great value towards learning metallurgical and metal-working processes.
A nanofluid is a mixture of nanometer sized particle and base fluid. Nanofluids provide numerous uses in manufacturing, electronics, physical science, and biotechnology. e nanofluids demonstrate improved thermal conductivity due to their unique physical properties. erefore, considering nanofluids is one of the ways in enhancing thermal conductivity.
Daniel et al. [1] showed that the nanofluid flow over a variable sheet thickness is sensitive to a thermal radiation with an increase in temperature. Fang et al. [2] provided the flow results adjacent to variable thicked surface. Alsaedi et al. [4] provided analysis for the MHD bioconvective nanofluid with gyrotactic microorganisms. A similar study by Atif et al. [3] discussed micropolar nanofluid. e magnetic effect on a third-grade fluid with chemically reactive species has been put forward in [5]. In their study, Khan et al. [6] discussed stagnation point flow with variable properties and obtained solution numerically. Hayat et al. [7] studied stagnation point flow for carbon-water nanofluid. e studies lead by Yasmeen et al. [8] and Hayat et al. [10] have shown flow over a sheet. In the study of Yasmeen et al. [8], the authors assumed ferrorfluid in their study, but Hayat et al. [10] considered melting heat transfer effects. In their work, Hayat et al. [9,11] considered variable properties, but in the latter work, they discussed unsteady three dimensional convection flows. Das et al. [12] underlined analysis on second-grade MHD fluid flows. Bhatacharyya et al. [13] set up exact solution with varying wall temperature for Casson fluid. Raju et al. [14] used FEM to find the solution of unsteady MHD convection Couette flow. Farooq et al. [15] reported fluid flow adjacent to a variable thicked Riga plate. Kumar et al. [16] discussed Brownian motion, slip effects, and thermophoresis effects near a variable thicked surface. Anantha et al. [17] described bioconvective fluid flow in Carreau fluid adjacent to variable thicked surface. Kumar et al. [18] addressed MHD Cattaneo-Christov flow past a cone as well as wedge in their research. Daniel et al. [19] recorded MHD nanofluid flow with slip and convective conditions. Salahuddin et al. [20] used numerical method to present magnetic Williamson flow of the Cattaneo-Christov heat flux model with varying thickness. Reddy et al. [21] took MHD Williamson nanofluid flow with varying physical properties near a variable thickness sheet. Malik et al. [22] discussed variable viscosity and MHD flow for Casson fluid by using the Keller-Box method. e work on variable thickness and radiation for Casson fluid is presented in [23]. Hydromagnetic distribution in nanofluid flow with variable viscosity and radially stretching sheet has been discussed in [24]. In [25], they reported a solution of MHD flow of nanofluid with variable thickness. Turkyilmazoglu [26] mentioned effects of radiation with varied viscosity on a time-dependent MHD porous flow. Prasad et al. [27] noticed MHD fluid flow solution over a stretching surface with thermophysical varying features. Vajravelu et al. [28] discussed variable fluid flow characteristics over a vertical surface. Das [29], Mukhopadhyay [30], and Rahman and Eltayeb [31] reported the effect of variable properties in fluid flows. Hayat et al. [32] discussed melting heat transfer for Powell-Eyring fluid. Jalil and Asghar [33] also used Powell-Eyring fluid for their analysis and found a solution by the Lie group method. Fluid flow with numerical and series solutions over an exponentially stretchable surface with the Powell-Eyring model has been discussed in [34]. In [35], the flow of Powell-Eyring non-Newtonian fluid has been discussed. Mustafa et al. [36] discussed MHD boundary layer nanofluid for second-grade fluid. Motsumi et al. [37] discussed thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate. In their work, Andersson and Aarseth [39] revisited Sakiadis flow with variable fluid properties. For nanofluid and other relevant areas of research, it would be worth reading these references [40][41][42][43][44].
Constant fluid properties of Newtonian and non-Newtonian fluids have been found in most of the above studies. Variable fluid property analysis is uncommon with variable sheet thickness. e current work is an extension of the work presented in Irfan et al. [45] where the Newtonian nanofluid is considered. To the best of the authors' knowledge, no research has been done on the Powell-Eyring nanofluid with constant and variable fluid properties together. To fill this gap, under the influence of different kinematics and dynamics, we examine constant and variable fluid properties to find critical difference between these when the non-Newtonian Powell-Eyring fluid is taken into account. Furthermore, we introduce a new numerical procedure, SFDM, to find the solution of the ordinary differential equations (ODEs).
e novelty of the current work lies in addressing the nanofluid of Powell-Eyring along with constant and variable fluid properties. A novel computational technique, the SFDM, has been tested for the Powell-Eyring nanofluids in addition to theoretical modelling of fluid flow. e SFDM is simple, accurate, and easy to implement in MATLAB. e layout of the paper is as follows. Section 2 is devoted to obtaining mathematical equations of the physical system. Section 3 deals with both constant and variable fluid features. Section 4 offers overview of a numerical process. Section 5 describes the results and the discussion. e findings of the paper are summed up in Section 6.

Theoretical Model
Consider a magnetohydrodynamic (MHD) two-dimensional steady laminar flow of Powell-Eyring nanofluid over a nonlinear uneven stretching sheet emerging from the narrow slit with variable fluid characteristics. Assume that varying magnetic field is directed perpendicular to the flow motion and is defined by is the variable permeability. e surface has a nonlinear stretching velocity U w � a 0 (x 1 + b) n . Furthermore, the thickness of the sheet is varied by the relation y 1 � A(x 1 + b) 1− n/2 , in which A is a very small constant. Fluid flow configuration is illustrated in Figure 1. e assumption of small magnetic Reynolds number leads to disregarding the induced magnetic field.
After incorporating aforementioned fluid flow assumptions in the following equations of motion, we get [45] where (u 1 , v 1 ), μ 1 , ρ 1 , and C p are the velocity components, the dynamic viscosity, the density, and the specific heat capacity, respectively. Moreover, E * 1 (x 1 ), B 1 (x 1 ), T 1 , and C 1 are the electric field, the magnetic field, the temperature of the fluid, and nanoparticle volume fraction, respectively. Also, T w is the wall temperature, and the ambient temperature is denoted by T ∞ . e parameters D B and D T are characterized as the Brownian diffusion and thermophoretic diffusion coefficients, respectively. In τ � (ρ 1 c) p /(ρ 1 c) f , (ρ 1 c) p is the effective heat capacity of the nanoparticle and (ρ 1 c) f is the heat capacity of the fluid. q r represents radiation due to heat flow, and Q 1 (x 1 ) is heat generation/absorption parameter. e boundary conditions needed to solve (1)-(4) are given by U w = a 0 (x 1 + b) n , Tw

Mathematical Problems in Engineering
With the following transformations [1,15]: equation (1) is satisfied through ψ. In the above, ] 0 � μ 0 /ρ 1 is the ambient kinematic viscosity. By using above transformations, equations (2)-(4) resulted into e transformed boundary conditions are written as Assuming e transformed boundary conditions on the new domain are written as e quantities appearing above are grouped into Mathematical Problems in Engineering where α is the wall thickness parameter, b is a positive constant, a 0 is a rate of stretching sheet, Ec is the Eckert number, ϵ is the thermal conductivity parameter, M is a magnetic parameter, k 1 is the thermal conductivity of the fluid, s is a heat source parameter, σ is the electrical conductivity, σ * is the Stefan-Boltzmann constant, Pr 0 is the ambient Prandtl number, Le is the Lewis number, Nb is the Brownian parameter, Nt is the thermophoresis parameter, λ is a free stream parameter, Rd is a thermal radiation parameter, Kp is a permeability parameter, k * is the mean absorption coefficient, Gr is the Grashof number, and B 0 is applied magnetic field.

Analysis on Fluid Properties
First consider constant thermophysical properties of liquids followed by the variable physical properties.

Case A: Constant Fluid
Features. In such scenario, equations (10)-(12) reduce into the following: Illustration of the skin friction coefficient in Table 1 reveals an outstanding alignment of the SFDM (discussed below) with Daniel et al. [1] and Fang et al. [2].

Case B: Variable Fluid Features.
e variation in viscosity for water due to change in temperature is illustrated in Table 2 (see White [38]). e viscosity decreases by a factor of 6. However, less change is noted in density. is motivates us to study the variable fluid properties. erefore, viscosity and thermal conductivity vary accordingly with temperature [39].
where c is a fluid property. Hereafter, the subscript "r" denotes reference value. When T o ≈ T r , we obtain where is the fluid viscosity parameter. Substituting equation (19) into (10), we get Similarly, the variable thermal conductivity is articulated by varying temperature as [21] Introducing equation (21) into (11), one obtains e skin friction coefficient C f is determined in the following manner:  Table 2: Water as a function of temperature (White [38]).  (23) in which τ w is the wall shear stress defined as Using equation (24) in (23), the skin friction coefficient in dimensionless form is defined in the following section.

e Skin Friction Coefficients (Cases A and B).
e skin friction coefficients for Cases A and B are written as Likewise, we define the local Nusselt and the Sherwood numbers in the following manner.

e Local Sherwood Number
where Re x is the local Reynolds number.

Simplified Finite Difference Method (SFDM)
e simplified finite difference method (SFDM) has been introduced in [45]. is scheme is motivated from the work by Na [46]. e algorithmic steps involved in the SFDM are as follows: (1) Reduction of higher-order ODE into a system of first-and second-order ODEs. (2) Linearization of nonlinear ODE through the use of Taylor series. (3) Use of finite differences to discretize the linear second-order ODE. (4) Finally, the obtained algebraic system is solved efficiently by LU decomposition. (5) Repeating the above procedure will produce solutions in θ and ϕ.
e summary of the involved steps in the SFDM is also shown in Figure 2. e results are computed for N � 1000 grid points in the η direction. However, the number of grid points was varied in some calculations to achieve better accuracy. e iterative procedure has been done with tolerance of machine epsilon in MATLAB. Assuming f ′ � F in equation (15), we may write en, this expression is written as and we replace dF/dη by forward difference approximation: e coefficients of second-order ODE are 6 Mathematical Problems in Engineering After manipulating equations (28)-(33), the linear algebraic system in F is written as [45] (34) where the coefficients are defined by

Mathematical Problems in Engineering
In matrix-vector format, it is where is a tridiagonal matrix. e column vectors F and s are 4.1. omas Algorithm. e omas algorithm [47] is implemented in MATLAB to compute the solution F. LU factorization is chosen for the matrix factorization of A 1 , i.e., where It is clear that L 1 and U 1 are the lower and unit upper triangular matrices, respectively. Variables (Λ j , ζ j ), j � 1, 2, . . . , N − 1, are related by After defining these relations, equation (36) becomes We have e unknown element z is computed from

Mathematical Problems in Engineering
We get e above equation (45) is a solution of equation (28). We discretize the relation f ′ � F as follows: which gives a required solution of equation (15). Repeating the above process, one can easily get θ and ϕ appearing in equations (16) and (17). However, all the details for these variables have been omitted for brevity. e reader is referred to [45] for further details on the SFDM. Since the SFDM results for both constant and variable fluid properties have been discussed in detail in [45], in the current work, only Table 2 is produced from the SFDM to compare accuracy with the literature. However, we obtain the skin friction coefficient and the local Nusselt numbers for Case A and Case B (Tables 3 and 4) by using MATLAB built-in solver bvp4c cf. [48].

Results and Discussion
e comparison of skin friction coefficient for Case A and Case B is given in Table 3. e skin friction coefficient increases for higher values of magnetic parameter M, power index parameter n, variable thickness parameter α, power index parameter n, and porosity parameter Kp while it decreases with the increase of parameter λ, Grashof number Gr, and electric field parameter E. e same phenomenon is observed for the skin friction coefficient in both Cases A and B. e skin friction coefficient decreases for larger values of viscosity parameter θ r in Case B. Table 4  thermophoretic parameter Nt, radiation parameter Rd, electric field parameter E, and heat source parameter s. e local Nusselt number for Case B decreases for larger values of small thermal conductivity parameter ϵ. In these tables, we can see the difference in values for the skin friction coefficient and the local Nusselt numbers. e skin friction coefficient and the local Nusselt numbers for the results of constant fluid properties have higher values when compared with the results obtained for variable fluid properties. Figure 3(a) describes the role of wall thickness parameter α in the fluid's velocity. Note that the velocity profile diminishes significantly for both Cases A and B with an increase in the values of wall thickness parameter α. It is observed that higher values of variable thickness parameter α correspond to deformation due to stretching wall, and hence the velocity profile decreases. Figure 3(b) is plotted to perceive the effect of electric field parameter E on velocity distribution. It is shown that the increase in electric field parameter E enhances the velocity of nanofluid because the electrical force introduces accelerating body force which acts in the direction of the electrical force. e Lorentz force accelerates the boundary layer flow. is force also increases the thickness of the momentum boundary layer. Hence, it results in a reduction in skin friction coefficient. Figure 3(c) presents the effect of magnetic parameter M on velocity profile. From Figure 3(c), we conclude that a rise in magnetic parameter M causes reduction in the velocity of fluid. An increase in magnetic parameter M generates a resistive force called Lorentz force in the fluid layers. is resistive force causes the reduction in velocity profile. Figure 3(d) is constructed to know the effect of power index n on velocity distribution. It is analyzed that for higher values of power index n, the velocity profile remains unchanged. Figure 4(a) reflects the influence θ r has on the velocity. e larger values of viscosity parameter θ r intensify the skin friction, which causes the reduction in the velocity profile. Figure 4 Grashof number Gr. Figure 4(d) highlights the effect of porosity parameter Kp on velocity profiles. It is seen that velocity profile decreases with the increase in Kp. Figure 5(a) presents the effect of λ on velocity profile. It is analyzed that there is no significant change in velocity profile for different values of λ in both Cases A and B. e thermal boundary layer thickness decreases for Case B when compared with the associated results obtained for Case A. Figure 5(b) demonstrates the effect of variable thickness parameter α on temperature profile. It is observed that an enhancement in variable thickness parameter α reduces the temperature profile. e behavior of electric field parameter E on temperature profile is portrayed in Figure 5(c). It is noted that enhancement in electric field parameter E decreases the thermal boundary layer thickness. Figure 5(d) is presented to describe the influence of small thermal conductivity parameter ϵ on temperature profile. It is examined that increment in small thermal conductivity ϵ causes enhancement in temperature profile. Figure 6(a) is presented to examine the effect of Prandtl number Pr 0 on temperature profile. It is noticed that increasing parameter Pr 0 reduces thermal conductivity of fluid which decreases temperature profile. Figure 6(b) illustrates the effect of radiation parameter Rd on temperature profile. It is seen that small increment in radiation parameter Rd intensifies the temperature profile because an increment in radiation parameter Rd provides more heat to fluid that causes enhancement in the thermal boundary layer thickness. Figure 6(c) illustrates the effect of heat source parameter s on temperature profile. It is observed that by increasing heat source parameter s, the temperature of the fluid increases, and hence the thermal boundary layer increases. Figure 6(d) is plotted to perceive the effect of power index parameter n on temperature profile. It is noted that increasing power index parameter n increases the temperature profile. Figures 7(a) and 7(b) show the effect of Brownian motion parameter Nb on the temperature and concentration profiles, respectively. It is noticed that for higher values of Nb, there is no significant increase or decrease in temperature profile, but by increasing Nb, the concentration boundary layer thickness increases. Figures 7(c) and 7(d) describe the effect of Nt on temperature and concentration profile, respectively. It is found that an increment in thermophoretic parameter Nt enhances both temperature and concentration profiles. Figure 8 depicts the influence of Le on concentration profile. We have seen that for larger values of Lewis number Le, the concentration boundary layer thickness is decreased. e reason is that increasing Lewis number Le reduces the  concentration rate diffusivity, and hence the concentration profile decreases.

Conclusions
e present study gives the results of two-dimensional EMHD flow of Powell-Eyring nanofluid over a nonlinear stretching sheet in a porous medium with variable thickness and variable liquid properties. e numerical solutions are obtained with the help of SFDM for constant fluid properties. However, bvp4c is implemented for variable fluid properties.
e key findings of the present study are as follows: (i) Momentum boundary layer thickness is higher for electric field E, parameter N, and Grashof number Gr, whereas it is going down for magnetic parameter M, porosity parameter K p , variable thickness parameter α, and fluid viscosity parameter θ r (for variable properties only). (ii) Momentum boundary layer remains almost constant when we increase the power index parameter n and fluid property parameter λ. (iii) ermal boundary layer thickness increases with the increase of radiation parameter Rd, small thermal conductivity parameter ϵ, heat source parameter s, power index n, and thermophoretic parameter Nt. (iv) ermal boundary layer thickness decreases for variable thickness parameter α, electric field parameter E, and Prandtl number Pr 0 , whereas there is a constant effect in thermal boundary layer when increasing Brownian motion parameter Nb. (v) Concentration boundary layer thickness decreases with the increase of Nt, whereas it increases with the increase of Nb and Le. (vi) Difference of Cases A and B is seen with the help of graphs. Graphs show that boundary layer thickness of Case B is different when compared to Case A. Furthermore, the skin friction coefficient is higher for constant fluid properties when seen in comparison with variable properties. e same is observed for the local Nusselt number. (vii) Lastly, the accuracy of the SFDM has been observed when comparison is drawn with the literature.