Spectral relaxation computation of electroconductive nanofluid convection flow from a moving surface with radiative flux and magnetic induction

A theoretical model is developed for steady magnetohydrodynamic viscous flow resulting from a moving semi-infinite flat plate in an electrically conducting nanofluid. Thermal radiation and magnetic induction effects are included in addition to thermal convective boundary conditions. Buongiorno’s two-component nanoscale model is deployed, which features Brownian motion and thermophoresis effects. The governing nonlinear boundary layer equations are converted to nonlinear ordinary differential equations by using suitable similarity transformations. The transformed system of differential equations is solved numerically, employing the spectral relaxation method (SRM) via the MATLAB R2018a software. SRM is a simple iteration scheme that does not require any evaluation of derivatives, perturbation, and linearization for solving a nonlinear system of equations. Effects of embedded parameters such as sheet velocity parameter λ , magnetic field parameter β , Prandtl number Pr, magnetic Prandtl number Prm, thermal radiation parameter Rd, Lewis number Le, Brownian motion parameter Nb , and thermophoresis parameter Nt on velocity, induced magnetic field, temperature, and nanoparticle concentration profiles are investigated. The skin-friction results, local Nusselt number, and Sherwood number are also discussed for various values of governing physical parameters. To show the convergence rate against iteration, residual error analysis has also been performed. The flow is strongly decelerated, and magnetic induction is suppressed with greater magnetic body force parameter, whereas temperature is elevated due to extra work expended as heat in dragging the magnetic nanofluid. Temperatures are also boosted with increment in nanoscale thermophoresis parameter and radiative parameter, whereas they are reduced with higher wall velocity, Brownian motion, and Prandtl numbers. Both hydrodynamic and magnetic boundary layer thicknesses are reduced with greater reciprocal values of the magnetic Prandtl number Pr m . Nanoparticle (concentration) boundary layer thickness is boosted with higher values of thermophoresis and Prandtl number, whereas it is diminished with increasing wall velocity, nanoscale Brownian motion parameter, radiative parameter, and Lewis number. The simulations are relevant to electroconductive nanomaterial processing.


Notation
Roman: C f : Skin-friction coefficient C w : Nanoparticle concentration at the moving wall C ∞ : Nanoparticle concentration in the free stream D : Differentiation matrix in spectral algorithm D T : Buongiorno nanoscale thermophoresis diffusion coefficient D B : Buongiorno nanoscale Brownian diffusion coefficient f: Dimensionless stream function f : Vector function at the collocation points f : Dimensionless velocity g: Dimensionless magnetic induction stream function g : Dimensionless induced magnetic field stream function gradient F r+1 (η) : Current iteration level of dimensionless velocity G r+1 (η)): Current iteration level of dimensionless magnetic stream function H 1 , H 2 : Dimensional magnetic stream function components K: Thermal conductivity of magnetic nanofluid K S : Radiative mean absorption coefficient Le : Lewis number N: Collocation points deployed in spectral method Nu x : Local Nusselt number O : Size of the matrix in spectral algorithm Pr : Prandtl number Pr m : Magnetic Prandtl number q m : Nanoparticle mass transfer rate at the wall q r : Fourier heat flux q w : Heat transfer rate at the wall Rd : Radiation parameter (Rosseland number) Re x : Local Reynolds number Sh x : Local Sherwood number T: Magnetic nanofluid temperature T w : Temperature at the moving wall T ∞ : Temperature in the free stream U : Velocity of the free stream (at boundary layer edge) U w : Velocity of moving wall (sheet) u, v: Dimensional velocity components along the x-axis and y-axis, respectively Greek: α : Thermal diffusivity of the nanofluid α 1 : Magnetic diffusivity of nanofluid B : Magnetic body force number λ : Moving wall velocity parameter η : Dimensionless similarity variable (transverse coordinate) η ∞ : Infinity value (to edge of boundary layer domain) ρ : Magnetic nanofluid density (ρC p ) f : Heat capacity of the base fluid σ * : Stefan-Boltzmann constant θ: Dimensionless temperature function ϕ : Dimensionless nanoparticle concentration function μ: Magnetic nanofluid dynamic viscosity ν : Magnetic nanofluid kinematic viscosity μ 0 : Magnetic nanofluid permeability τ : Heat capacity of the magnetic nanoparticles. τ j : Gauss-Lobatto collocation points τ w : Shear stress at the wall ψ : Dimensional stream function : Dimensional magnetic stream function

Introduction
Nanoscale colloidal suspensions containing solid nanoparticles and fibers are called nanofluids and were first introduced by Choi (1995). Choi et al. (2001) showed that the addition of a small number of nanoparticles to conventional heat transfer liquids such as water, toluene, oil and ethylene glycol, etc. enhances the thermal conductivity of the original fluids. In recent years, much interest has been given to the study of nanofluids owing to their greatly enhanced thermal conductivity properties (Keblinski et al., 2005). A large number of numerical and experimental studies have been carried out by numerous researchers on thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer characteristics of nanofluids (Choi et al., 2004;Liu et al., 2005;Maiga et al., 2005;Das et al., 2006Das et al., , 2007Kang et al., 2006;Tyler et al., 2006;Tiwari & Das, 2007;Abu-Nada, 2008;Oztop & Abu-Nada, 2008;Tzou, 2008;Yu et al., 2008;Choi, 2009). Nanoparticles can help to extract energy from the core of the earth, improve the efficiency of the cooling system employed in nuclear reactors, and can also help in targeted drag delivery in pharmaceutics. An exquisite collection of published works on nanofluids is given by Das et al. (2007), and comprehensive references are also provided in the review papers of Mujumdar (2007, 2008a, b). Magnetohydrodynamics (MHD) is the study of the motion of electrically conducting fluids with magnetic properties. These include both inviscid and viscous fluids. A current induced in the presence of a magnetic field in an electrically conducting fluid polarizes the liquid, and as a result, the magnetic field also changes. The fluids can be conducting electrically in various applications for the processing of materials in chemical and mechanical engineering, and as such, they will react to an applied magnetic field. Such a system may be used to monitor, e.g. the heat transfer levels on stretch sheets, and to fine-tune the final materials to industry requirements. This process takes advantage of the effect of the Lorentzian drag force that abates flows in a perpendicular direction of the magnetic field applied (transverse) (Hughes & Young, 1966). MHD boundary layer flows also arise in MHD power generator designs, liquid metal manipulation, plasma studies, cooling of nuclear reactors, induction heating, plasma cutting, etc. Jafar et al. (2011) examined MHD flow and heat transfer over stretching/shrinking sheets with an external magnetic field. The effects of thermal radiation of MHD flow, free convection flow, and viscoelastic flow were analysed, respectively, by Chamkha (1997), Raptis (1998b), Jumah et al. (2001), and Mohammadein and El-Amin (2000). Recently, MHD boundary layer flow with heat and mass transfer of nanofluids has been given a great deal of attention by the researchers owing to significant thermal enhancement properties achieved by these nano-engineered liquids. Furthermore, by combining magnetic and nanofluid material properties, one can obtain magnetic nanofluids with exceptional magnetic responsive features. Magnetic actuation offers specific capabilities as it can be controlled momentarily and spatially, and can be applied externally to the device, offering a noninvasive remote control approach. The boundary layer flow of nanofluids over a moving surface in a flowing fluid was investigated by Bachok et al. (2010). Ferdows et al. (2012) studied the steady 2D flow of an incompressible viscous and electrically conducting nanofluid generated by a stretching sheet in the vertical direction with viscous dissipation in saturated porous media. Unsteady boundary layer flows of nanofluids from a stretching sheet with thermal radiation in the presence of a magnetic field were numerically addressed by Ferdows et al. (2012). Mabood et al. (2015) and Mustafa et al. (2015) studied MHD flow and nanofluid heat transfer due to a nonlinear stretching sheet. Srinivasacharya et al. (2015) investigated the steady laminar MHD nanofluid flow over a wedge in the presence of a variable magnetic field. MHD boundary layer flow of nanofluid over a continuously stretching surface was studied by Rasheed et al. (2017). Yohannes and Shankar (2014) successfully analysed melting heat transfer in nanofluid's MHD flow over a permeable exponentially stretching sheet. The behavior of nanofluid and heat and mass transfer of free convective MHD of Ag-Kerosene oil nanofluid was elucidated by Upreti et al. (2020). The obtained results showed that the heat and mass transfer rates grew consistently as the quantity of surface mass flux shifted from injection to suction domain. Nadeem et al. (2019) analysed the characteristics of heat and mass transfer for the three-dimensional unsteady, laminar, and an incompressible flow of a micropolar nanofluid.
Free convection flow of a nanofluid in the presence of a magnetic field from a vertical semi-infinite flat plate was investigated by Hamad et al. (2011). The impact of thermophoresis and Brownian motion in heat transfer enhancement in natural convection of nanofluids was examined by Haddad et al. (2012). Daniel et al. (2018) studied the combined effects of thermal stratification, applied magnetic field, and thermal radiation on a boundary layer flow of electrically conducting nanofluid over a nonlinear stretching sheet. The radiative heat transfer is found sensitive to an increase in the fluid temperature and thicker thermal boundary layer thickness. Das et al. (2016) studied the impact of thermophoresis and thermal radiation on heat and mass transfer for the second-grade fluid flow passing through a semi-infinite stretching sheet with convective surface heat flux and explored that the fluid velocity and temperature in the boundary layer region rise significantly for increasing the values of thermal radiation parameter.
The above studies did not consider magnetic induction effects, which become significant at higher magnetic Reynolds numbers, e.g. MHD nanomaterials processing Aly & Sayed, 2017). Additionally, in high-temperature nanomaterial processing, thermal radiation is significant. Mathematical models of magnetic nanoliquid fabrication processes therefore require radiative heat flux to be also incorporated, as elaborated in several recent studies Uddin et al., 2015;Madkour, 2019). In this work, thus, a new model is presented for MHD induction convection flow of an electrically conducting nanofluid from a moving semi-infinite stretching surface with appreciable thermal radiative heat transfer. This constitutes the novelty of the present simulations, which for the first time use a spectral relaxation method (SRM) and combine properly electromagnetic induction, moving sheet velocity effects, nonlinear radiative flux (for high-temperature material processing), and heat and mass transfer via the Buongiorno model. The Rosseland diffusion flux model is deployed for optically thick magnetic nanofluids (Khan et al., 2013;. Using suitable similarity transformations, the momentum and energy equations are converted into dimensionless coupled ordinary differential equations and solved with a numerical technique known as the SRM (Bég et al., , 2016Haroun et al., 2015). Validation with the earlier study of Shateyi and Prakash (2014) is included. The effects of embedded parameters, i.e. moving sheet velocity parameter λ, magnetic field parameter β, Prandtl number Pr, magnetic Prandtl number Prm, thermal radiation parameter Rd, Lewis number Le, Brownian motion parameter Nb and thermophoresis parameter Nt on velocity, induced magnetic field, temperature, nanoparticle concentration, skin friction, local Nusselt number, and Sherwood number profiles, are visualized. A residual error analysis is included for the convergence rate against iteration. The simulations are relevant to multiphysical magnetic nanoliquid material processing (Ali et al., 2018).

Governing Equations for Magnetic Induction Nanofluid Flow
Consider the 2D, steady, viscous MHD thermal convection boundary layer flow of an electrically conducting nanofluid from a moving semi-infinite surface (sheet). The physical model is shown in Fig. 1. A uniform magnetic field of force H 0 is applied to the surface in the normal direction. In contrast, the normal component of the induced magnetic field H 2 disappears when it hits the wall and the parallel component H 1 approaches the value of H 0 . It is also assumed that the velocity of the free stream is U, and that of the plate (sheet) is U w = λU, where λ is a velocity parameter. Maxwell displacement current, Hall current, viscous and Ohmic dissipation, and also chemical reaction effects are neglected. At the moving surface, the temperature and the nanoparticle concentration take constant values T w and C w , respectively, while the free stream values are prescribed as T ∞ and C ∞ , respectively. Under the boundary layer approximations, using the Buongiorno two-component nanoscale model (Buongiorno, 2006) [as elaborated in detail by Bég (2018)] and the Rosseland diffusion flux model for radiative transfer Ferdows et al., 2014;Uddin et al., 2015), the conservation equations can be written as follows: where u and v are the velocity components along the xaxis and the y-axis, respectively, T is the nanofluid temperature, ρ and ν are the nanofluid density and kinematic viscosity coefficient, respectively, μ 0 is the magnetic permeability, α 1 is the magnetic diffusivity, α = K (ρCp) f is the thermal diffusivity of the fluid, K is the thermal conductivity of magnetic nanofluid, (ρC p ) f is the heat capacity of the base fluid, q r is the heat flux, D B is the Brownian diffusion coefficient, D T is the thermophoresis , and (ρC p ) p is the heat capacity of the nanoparticles.
The radiative heat flux q r is evaluated by using Rosseland diffusion approximation (Hossain & Takhar, 1996) and following Raptis (1998a) among other researchers, to be where σ * and K s are the Stephan-Boltzman constant and the Rosseland mean absorption coefficient, respectively. The temperature differences within the flow are sufficiently small under the assumption that T 4 may be expressed as a linear combination of the temperature as shown in Chamakha (1997). Expanding T 4 in a Taylor series about T ∞ and neglecting the higher order terms, we obtain Using (7) and (8) in equation (5), we obtain The prescribed boundary conditions at the wall (plate) and in the free stream for equations (1)-(6) are defined as follows: At y = 0: At y → ∞:

Similarity Transformations
We now introduce the following similarity transformations, to normalize the boundary layer equations: Here, ψ is the stream function, which is defined as u = ∂ψ ∂y and v = − ∂ψ ∂x , and is the magnetic stream function, defined by H 1 = ∂ ∂y and H 2 = − ∂ ∂x . Furthermore, η is the dimensionless similarity variable (transverse coordinate), f and g are dimensionless stream and magnetic stream functions, respectively, θ is the dimensionless temperature function, and ϕ is the dimensionless nanoparticle concentration. Mass conservation (1) and magnetic field continuity (2) are satisfied identically. Substituting the similarity variables into equations (3)-(6) gives the following similarity ordinary nonlinear differential momentum, magnetic, thermal, and species (nanoparticle) boundary layer equations: The dimensionless boundary conditions for the tenth-order problem now become Here, primes denote differentiation with respect to η, f is the dimensionless velocity, g is the dimensionless induced magnetic field stream function gradient, Pr = ν α is the Prandtl number, Pr m = ν α1 is the magnetic Prandtl number, β = μ0H0 2 ρU 2 is the magnetic body force number, Rd = 4σ * T∞ 3 KKs is the radiation pa- is the thermophoresis parameter, and Le = ν DB is the Lewis number. The physical quantities of interest in magnetic material processing are the skin-friction coefficient C f , the local Nusselt number Nu x , and the local Sherwood number Sh x . These parameters, respectively, characterize the surface drag, wall heat transfer rate, and wall nanoparticle mass transfer rate. The skin-friction coefficient is defined as Here, τ w is the shear stress at the surface of the wall (sheet), which is given by Using equation (18) in (17), we obtain the dimensionless skinfriction coefficient (surface drag) as The Nusselt number is defined as where the heat transfer rate at the surface is given by Using equation (21) in (20), the dimensionless wall heat transfer rate is obtained as The nanoparticle mass transfer rate at the surface is given by The Sherwood number is defined as From equations (23) and (24), the dimensionless nanoparticle mass transfer rate becomes Here, Re x = Ux ν is the local Reynolds number. According to Bachok et al. (2010), Re x −1/2 Nu x and Re x −1/2 Sh x are referred to as the reduced Nusselt number and reduced Sherwood number, which are represented by −θ (0) and −ϕ (0), respectively.

Computational Solution with SRM
The SRM is an iterative algorithm proposed for nonlinear systems of differential equations in which some of the unknown functions have exponentially decaying profiles. The method has been identified to be very efficient and convenient in solving boundary-layer problems in which at least one of the profiles such as velocity, temperature, or concentration decays exponentially. The SRM approach is derived from importing and linearizing the Gauss-Seidel concept of decoupling a system of equations by a simple rearrangement of the order in which they are described and sequentially resolved. The resulting SRM iterative scheme is then implemented using the Chebyshev Spectral Collocation process wherein such polynomials provide good accuracy and fast convergence in the computations. This approach has been recently introduced by Motsa (2014). SRM has been applied by Motsa and Makukula (2013) to solve the swirling Von Karman disk flow of a Reiner-Rivlin fluid with Joule heating, viscous dissipation, and wall transpiration. It has also been implemented in thermal ignition flows of rheological rocket gel propellants by Awad et al. (2014) and in micropolar convective wall plumes in geothermic by Haroun et al. (2015). The SRM algorithm, when applied to the transformed equations (9)-(12), may be summarized as follows: I. Reduce the order of the velocity and induced magnetic equations by introducing the transformation f (η) = F(η) and g (η) = G(η) and express the original equations in terms of F(η) and G(η). II. The iteration scheme is developed by assuming that only linear terms in F(η) and G(η) are to be evaluated at the current iteration level [denoted by F r+1 (η) and G r+1 (η)], and all other terms (linear and nonlinear) in f(η) and g(η) are assumed to be known from the previous iteration [denoted by f r (η) and g r (η)]. Also, besides nonlinear terms in F(η) and G(η) are also calculated at the previous iteration. III. The process is repeated for the other governing variables using the updated solutions of the variables determined in the previous equation.
The strategy stated above for decoupling a linear algebraic system of equations is analogous to the Gauss-Seidel relaxation method. Applying this algorithm leads to a series of linear differential equations with variable coefficients that are solved here using Chebyshev spectral collocation methods (Awad et al., 2014;Haroun et al., 2015). Due to their noticeable high precision and relative efficiency, spectral methods are used here in discretizing and for the subsequent solution of variable coefficient linear differential equations with smooth solutions over a simple domain. To apply the SRM to the nonlinear ordinary differential equations, we define f (η) = F(η) and g (η) = G(η) and then orchestrate the problem as the following set of equations: The associated boundary conditions become In the context of the SRM algorithm, we obtain the following iteration scheme: Subject to the boundary conditions: To solve equations (33)-(38), we discretize them using the Chebyshev spectral method. For brevity, we omit the details of the Spectral method and refer interested readers to the standard monographs of Canuto et al. (1988) and Trefethen (2000). To apply the spectral method, the domain is transformed to the interval [−1, 1] on which the governing equation is defined. The transformation η = L(τ + 1)/2 is used to map the interval [0, L] to [−1, 1], where L is chosen to be large enough to approximate the conditions at infinity numerically. In spectral collocation methods, a differentiation matrix D of size (N + 1) × (N + 1) is introduced, which is used to approximate the derivatives of the unknown variables at the collocation points as the matrix vector where p is the order of derivatives. We choose the Gauss-Lobatto collocation points to define the nodes in [−1, 1] as follows: Applying the Chebyshev pseudo-spectral method on equations (33) to (38), we obtain Here, In equations (39)-(44), the size of the matrix O is (N + 1) × 1, diag [] is a diagonal matrix, all of size (N + 1) × (N + 1), whereN is the number of grid points, f, g, F, G, θ, and ϕ are the values of f, g, F, G, θ, and ϕ, respectively, when evaluated at the grid points, and the subscript r denotes the iteration number. Equations (39)-(44) constitute the SRM scheme and can be solved using the spectral collocation method starting from the following initial conditions, which are chosen to satisfy the boundary conditions: CPU times are of the order of several minutes, and the solutions converge quickly.

Validation of SRM
The nonlinear boundary value problem is defined by equations (12)-(15) subject to the boundary conditions (16a) to (16b) have been solved numerically using the SRM. The numerical method is programmed using the MATLAB R2018a software. The SRM results are obtained here using N = 50 collocation points, and the infinity value η ∞ is taken as 15.
In this study, the following default parameter values are adopted for computations: Pr All data are realistic and apply to the case studied, i.e. magnetic nanofluid stretching sheet flow for coating. All graphs therefore conform to these values unless stated explicitly otherwise. To verify the accuracy of the SRM, the results are compared with the earlier solutions of Shateyi and Prakash (2014) and shown in Table 1 for skin friction, −f (0) with different values of sheet velocity parameter, λ. Excellent agreement is achieved testifying the accuracy of the present SRM code.

Results and Discussion
Extensive computations have been conducted in MATLAB, and SRM solutions are shown in Tables 2-4 and Figs 2-18. Table 2 represents the effects of Prandtl number Pr, radiation parameter Rd, moving surface (sheet velocity) parameter λ, Brownian motion parameter Nb and thermophoresis parameter Nt on the skin-friction coefficient C f , the reduced Nusselt number Nu, and the reduced Sherwood number Sh. It is observed that the rate of heat transfers at the surface increases with more significant Prandtl number (lower thermal conductivity of magnetic nanofluid reduces temperatures in the boundary layer and encourages heat transfer to the wall). In contrast, the rate of mass transfer is diminished as the Prandtl number increases (nanoparticle concentration is increased in the boundary layer regime, which decreases the rate of nanoparticle mass transfer to the wall). Increasing the Lewis number Le, and the radiation parameter Rd, increases the reduced Sherwood number at the border (more significant Lewis number implies higher thermal diffusivity relative to nanoparticle mass diffusivity, which boosts nanoparticle concentrations in the fluid regime and suppresses transport to the wall). In contrast, it reduces the heat transfer rate at the border (higher radiative flux elevates temperatures in the boundary layer, which diminishes the transfer of heat to the wall). It is also evident that increasing moving surface parameter value λ decreases the skin friction at the wall plate (since this serves to inhibit boundary layer growth and destroys momentum leading to deceleration), whereas it manifests in an elevation in reduced Nusselt and Sherwood numbers, i.e. significantly enhancing heat and nanoparticle diffusion to the wall. An increase in the Brownian motion parameter     Nb and thermophoresis parameter Nt has no substantial influence on the skin-friction coefficient. The rate of heat and mass transfer at the wall, however, increases as the Brownian motion parameter increases (which corresponds to smaller nanoparticles and encourages thermal and species diffusion to the boundary). Both the Nusselt number and the Sherwood number are reduced as the values of the thermophoresis parameter increase  since temperatures and nanoparticle concentration magnitudes in the boundary layer are elevated, and this mitigates transfer to the wall, as noted by many other researchers, e.g. Hamad et al. (2011) and Haddad et al. (2012). Table 3 depicts the heat transfer rate for various values of Prandtl number Pr with other flow parameters constrained. With negative wall velocity, the reduced Nusselt number is reduced, whereas with positive wall velocity, it is consistently elevated at all Prandtl number values.               Table 4 depicts the mass transfer rate (reduced Sherwood number) for different Lewis number Le values with other parameters constant. Reduced Sherwood number is lowered with increasing Lewis number, for negative wall velocity, whereas it is enormously increased with more significate Lewis number for the positive wall velocity case. Table 5 demonstrates the influence of various magnetic force parameters, β, with different sheet velocity parameters, λ, on skin-friction coefficient C f . It is noticed that the skin-friction coefficient reduced very rapidly due to increase in the values of these parameters.
Figures 2-5 display the effects of moving wall velocity parameter, λ, on velocity, induced magnetic field, temperature, and nanoparticle concentration profiles. It is seen that increasing the velocity parameter λ leads to an increase of speed (Fig. 2), i.e. flow acceleration and induced magnetic field (Fig. 3) magnitudes. Hydrodynamic boundary layer thickness is therefore decreased, whereas magnetic boundary layer thickness is enhanced. Asymptotically smooth profiles are achieved in the free stream, verifying that an adequately large infinity boundary condition is prescribed in the MATLAB SRM code. Temperature (Fig. 4) and nanoparticle concentration (Fig. 5) are both Table 5: Skin-friction coefficient C f with Pr = 0.7, Prm = 0.5, λ = 0.1, Nt = 0.1, Nb = 0.3, Rd = 0.5 , and Le = 5.  Figures 6 and 7 visualize the effects of magnetic force number, β, on the velocity and induced magnetic field profiles. This parameter features in the magnetic induction term in the dimensionless momentum boundary layer equation (9), −βgg , which also couples the momentum equation (9) to the magnetic induction equation (10). It is an opposing body force, and increasing values of β will therefore inhibit momentum and magnetic induction in the regime. This will manifest as a decrease in velocity and induced magnetic field magnitudes and a concomitant elevation in momentum boundary layer thickness and depletion in magnetic boundary layer thickness, as noted earlier by Hughes and Young (1966), among other researchers. Figures 8 and 9 depict the influence of Brownian motion parameter Nb on temperature and nanoparticle concentration profiles. Temperature and nanoparticle concentration noticeably decrease with the increase of Nb. Conversely, in Figs 10 and 11, increasing thermophoresis parameter Nt significantly elevates temperature and concentration profiles. Thermophoretic body force encourages heat and nanoparticle mass diffusion with thermal gradient. Thermal boundary layer and nanoparticle concentration boundary layer thickness are therefore both accentuated with greater thermophoretic body force. These trends concur with many other investigations, including Bachok et al. (2010), Ferdows et al. (2012), and Mabood et al. (2015, although these studies neglected magnetic induction effects. The nanoscale effects, i.e. Brownian motion and thermophoresis, do not exert any significant influence on either nanofluid velocity or induced magnetic field profile, and therefore, these graphs are omitted for brevity. Figures 12 and 13 illustrate the variation of temperature and nanoparticle concentration distributions in response to a change in Prandtl number Pr. Increasing the Prandtl number Pr induces a substantial decrease in nanofluid temperature and suppresses thermal boundary layer thickness. The nanoparticle concentration profiles exhibit a monotonic response to Prandtl number, i.e. as Pr increases, nanoparticle diffusion is encouraged, and the nanoparticle concentration boundary layer thickness is also increasing. Figures 14 and 15 visualize the impact of magnetic Prandtl number Prm on the velocity and induced magnetic field profiles. Both velocity and induced magnetic field profiles of the nanofluid increase with elevation in Prm. This parameter equals ν α1 and is invoked because of magnetic induction effects and determines the ratio of magnetic diffusivity to viscous diffusivity. It Figure 19: Influences of magnetic force parameter, β, and sheet velocity parameter, λ, on skin-friction coefficient C f . features exclusively magnetic induction conservation equation (10). Thus, the magnetic Reynolds number is high enough for the flow field to cause magnetic field distortion. The magnetic field is considered to be undistorted by the flow for a case where this parameter is very weak (as compared with unity). However, the induction effects of large values of magnetic Reynolds number are significant and necessitate a separate conservation equation, as considered in the present model. As Prm is elevated, both momentum and magnetic induction are assisted in the regime. Momentum and magnetic boundary layer thicknesses will therefore be, respectively, increased and decreased. It is also apparent that the effect of Prm is more pronounced on magnetic induction g (η) compared to f (η). Figures 16 and 17 depict the influence of the radiation parameter Rd on nanoparticle volume fraction (concentration) and the temperature distribution, respectively, occuring in the augmented thermal diffusion term in the energy conservation equation (11), viz. ( 1+Rd Pr )θ . Rd = 16σ * T∞ 3 3KKs and expresses the relative contribution of thermal radiation heat transfer to thermal conduction heat transfer. When Rd = 1, both modes contribute equally. For Rd < 1, conduction dominates and vice versa for Rd > 1. An increase in Rd leads to a decrease of the nanoparticle volume fraction (Fig. 16) since greater energization of the nanofluid with radiative flux inhibits nanoparticle diffusion; however, the converse response is induced on temperature magnitudes (Fig. 17) that are strongly elevated with a greater radiative contribution. Nanoparticle concentration boundary layer thickness is therefore decreased, whereas thermal boundary layer thickness is enhanced with greater radiative parameter (which concurs with the earlier decrease in reduced Nusselt number at the wall). The computations agree with several other investigations, including Aly and Sayed (2017) and Uddin et al. (2015). The inclusion of radiative heat transfer in magnetic nanomaterials fabrication therefore produces more accurate temperature and nanoparticle concentration predictions than neglecting this mode of heat transfer (which leads to underprediction in temperatures and overprediction in nanoparticle volume fractions).
Finally, Fig. 18 illustrates the variation of nanoparticle concentration profile with the Lewis number Le. Le = ν DB and defines the ratio of momentum diffusion rate to nanoparticle species diffusion rate. When Le = 1, these rates are equal, and both momentum and nanoparticle concentration boundary layer thicknesses are similar. However, for Le > 1 (as considered here), momentum diffusivity exceeds species  diffusivity of the nanoparticles, which inhibits nanoparticle diffusion and results in a depletion in nanoparticle concentration boundary layer thickness. The latter may therefore be successfully controlled in nanoliquid material processing via careful selection of nanoparticles based on appropriate diffusivities. Figures 19-21 demonstrate the graphs of the skin-friction coefficient C f , the reduced Nusselt number Nu, and the reduced Sherwood number Sh for different values of magnetic parameter, Prandtl number, and Lewis number, respectively, with velocity parameter. The values of the skin-friction coefficient C f decrease as the magnetic force parameter enhances and with the increasing values of the Prandtl number, the reduced Nusselt number Nu increases very rapidly. The reduced Sherwood number Sh gets enhanced due to increase in the values of Lewis number.

Conclusions
A mathematical study has been presented for steady 2D MHD thermal convection boundary layer flow of electroconductive nanofluids over a moving surface in a uniform free stream with magnetic induction and radiative heat transfer effects. The Buongiorno model has been deployed, which incorporates the results of Brownian diffusion and thermophoresis. The partial differential equations governing the flow and heat transfer were transformed into coupled nonlinear ordinary differential equations using appropriate similarity transformations. The resulting equations were then solved by an accurate iterative method known as the SRM. The SRM algorithm is straightforward to implement since it can be constructed directly from the governing equations in their original form. Validation of the MATLAB SRM code with published results is included. The variations of the skin-friction coefficient, the heat transfer rate, and the mass transfer rate, as well as the velocity, induced magnetic field, temperature, and the nanoparticle volume fraction profiles, are illustrated in tabular and graphical forms for various values of the physical parameters. From this study, some fundamental results are summarized as follows: 1. Velocity and induced magnetic field profiles decrease with an increase in magnetic body force parameters. 2. Thermal boundary layer thickness is enhanced with increasing magnetic body force parameter, thermophoresis parameter, and radiative parameter. In contrast, they are diminished with more incredible wall velocity, Brownian motion, and Prandtl numbers. 3. The induced magnetic field stream function is affected the most by reciprocating the magnetic Prandtl number Pr m compared with the skin-friction and heat transfer coefficients. 4. Nanoparticle concentration improves with higher values of thermophoresis and Prandtl number, whereas it is suppressed with increasing moving wall, Brownian motion, radiative, and Lewis numbers.
This study has shown that SRM is a versatile and powerful numerical technique for nonlinear electromagnetic induction nanofluid materials processing flows. However, attention has been confined to Newtonian nanofluids. Future studies will explore rheological nanofluid behavior, including viscoelastic (Rana et al., 2017) and viscoplastic (Thumma et al., 2020) models, and will be communicated imminently. This study may be generalized also to consider electrical field effects (Bég et al., 2013), and this is also presently under consideration. Additionally, several other physicochemical products arise in magnetic nanomaterial synthesis, including geometrical shape effects of nanoparticles , Hiemenz stagnation and slip flows (Nadeem et al., 2020), Hall current effects (Bhatti et al., 2019), and bioconvection effects in which nanoparticles are combined with micro-organism doping to achieve dual benefits (Bhatti et al., 2020). Furthermore, alternative nanoparticles, e.g. molybdenum disulfide (Sowmya et al., 2019) and triple diffusion (Archana et al., 2018) (double nanoparticle species), may also be considered, and efforts in this direction are also being explored. All these areas constitute interesting directions for the refinement of the present simulations with the SRM numerical approach.

Funding
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.