Heat and mass transport of an advection-diffusion viscous fluid past a magnetized multi-physical curved stretching sheet with chemical reaction

ABSTRACT

Analysis of 2D magnetohydrodynamic flow of viscous fluid over a magnetized multi-physical curved mechanism is numerically conducted. The flow equations consist of a magnetic field transport, Maxwell’s equations, energy equation and concentration equation. Governing model is generated and establishes the boundary layer equations. Numerical computation by the Keller-Box shooting method is implemented with Jacobi’s iterative technique. The flow behaviours are anticipated against most interesting parameters. The novelty of this study focuses on the mathematical development of the flow problem with significant results. These results are applicable in manufacturing of stretchable materials.

KEYWORDS:

• Viscous fluid
• curved sheet
• advection-diffusion

1. Introduction

The studies of magnetohydrodynamic (MHD) fluids in explanation of electrically conducting fluid flow and magnetic field interactions have been extensively investigated in the recent years. The MHD is a broad field in science and engineering which deliberates the complexity and dynamic behaviours of electrically conducting fluids like electrolytes, for insurance, salt water, plasma confinement, liquid distortion into metal, and etc. The physical insight of the terrestrial magnetic fields maintained by the fluid motion in the earth core, the solar magnetic fields that generate sunspots and solar flares, and the galactic field that influences the formation of stars are of great interest to many researchers of ages. However, there are various MHD problems that concern the physiological fluids with significant applications in health sciences. These include magnetic cell separation, humans body blood circulation, treatment of arterial and hyperthermia, drug delivery, medical analysis of breathing, and blood reduction during surgeries. MHD flow generators and accelerators which are influenced by the magnetic fields are superficially useful in industry to levitate liquid metals, pumps, cooling of nuclear reactor containment vessel, treatment of fusion reactors, dispersion of metals, fusing metals in an electric furnace, stir and heat [1]. Numerous literature on MHD flows have been analysed in consideration of two-dimensional coordinates that solely caused by linear and nonlinear stretching of flat or curved sheets. Influence of fixed and variable magnetic fields over plane and curved surfaces has widely been considered. Among the earlier studies, Greenspan and Carrier [2] presented an electromagnetic field action on boundary layer of an incompressible flow over semi-infinite plate. Ashraf et al. [3] and Davies [4] considered similar flow problems with uniform and variable magnetic fields impacts under large Reynolds and magnetic Reynolds numbers, respectively. Airal and Adeel [5] analysed the stability of MHDs of Reiner-Philippoff fluid flow over magnetized stretching flat surface. Jiang et al. [6] discussed MHD fractional Burger’s fluid flow through a porous medium with heat and mass transfer influenced by a chemical reaction, whereas Pavlov [7] documented an incompressible viscous fluid in a MHD flow caused by deformation of flat surface. Veera et al. [8] examined MHD elastico-viscous fluid flow through a rotating porous medium with hall and ion slip effects. Amos et al. [9] studied heat and mass transfer of MHD Casson fluid past a free convective and dissipating sheet in the presence of variable viscosity and thermal conductivity. Variable properties of MHD heat and mass transfer in a stretching porous medium are presented by Swain et al. [10]. In addition to the above literature, more studies can be seen in refs [11–15]. Applications of heat transfer in the manufacturing industries have been of great field of interest to researchers due to growing in demand, emerging technologies, and enhancement of heat flow at high heat and impacts of magnetic field have been recorded valuable in controlling thermal boundary layer in high thermal engineering processes. For instance, non-isothermal conditions, polymer extrusion, cooling of continuous tiles, electromagnetic casting, freed and wind up rolls on conveyor belts, powered engines, centrifugal pumps, solar collector, rubber sheets and other activities involving thermally control environment. Reddy et al. [16] investigated heat transfer of MHD fluid along a porous stretching cylinder with heat absorption/generation. Zheeshan et al. [17] analysed elastic-viscous heat transfer of MHD flow in wire-coating process. Heat transfer of MHD viscous liquid in the presence of radiation and transpiration over a stagnation point is documented by Mahabaheshwar et al. [18]. Nagaraju and Garvandha [19] discussed effect of externally applied constant suction of MHD viscous fluid in a circular pipe with heat transport. Soid et al. [20] examined heat transport of MHD flow over an unsteady shrinking sheet in the presence of ohmic heating. Macha et al. [21] documented MHD flow of an unsteady Maxwell nanofluid and radiation effects over a stretching surface. Krishna et al. [22] studied hybrid flow of Casson and nanofluids influenced by a radiation and MHD over an accelerated exponentially infinite porous vertical surface. Impacts of induced magnetic field and thermal radiation are outlined in a magneto-convection flow of dissipative fluid by Kumar et al. [23]. Ali J. Chamkha [24] investigated a natural convection of MHD flow coupled with heat and mass transfer about a truncated cone in the presence of radiation. Heat sources (viscous dissipation, radiation and Joule heating) contribute more significant impacts towards heat transfer and elucidate more realistic thermal behaviours at high radiation, surface friction and fluid viscosity. Therefore, inclusion of viscous dissipation has been theoretically relevance in the studies of thermal and polymer dynamics whose operations involve high temperature. Likewise, the effect of frictional heating caused by surface induction increases dissipation rapidly, relatively and uniformly to heat flows and hence these are useful for food pasteurization, blanching, dehydration and evaporation. Abdelhafez et al. [25] considered impact of Joule heating and Maxwell nanofluid with higher order reactive time-dependent MHD viscous fluid, whereas Ali et al. [26] conducted flow of MHD thermos-convective non-isothermal nanofluid with finite element in the presence of Dufour and Soret effects. Kumar et al. [27] studied the effect of nonlinear radiation with particle-liquid suspension of Williams fluid over a stretching surface. Abbas et al. [28] analysed thermal radiation of MHD viscous fluid embedded in a vertical porous entropy generation channel. Sanni et al. [29] examined heat transfer of hydromagnetic viscoelastic fluid in the presence of radiative flux past a continuous curved stretching sheet. Sanni et al. [30] investigated flow of viscous fluid caused by a nonlinear boundary-driven curved stretching sheet with heat transfer under variable applied magnetic field. The fluid concentration contributes a remarkable effect when substantial amount of volume of fluid are utilized in the production and engineering applications. For instance, condensation process of drawing plastic and liquid film, paper production, plastic and rubber sheets, crystal growing and metal revolve. Also, the convective mass transport phenomena in the presence of first-order chemically reactive species have been of great practical importance to many branches of science. Examples include freezing effect on crops damage, wet cooling of tower due to energy transfer, cooling of turbine blades due to high aero-speed, geothermal reservoir, poly melts and engineering chemical processes. Hayat et al. [31] examined the flow of MHD fluid over a convectively thermal radiative curved surface with chemical reaction. Sanni et al. [32] carried out the treatment of nonlinear radiative and convective boundary driven flow of hydromagnetic non-Newtonian fluid influenced by viscous dissipation and chemical effects. Imran et al. [33] analysed a comprehensive study of fractional MHD viscous fluid subject to convective and generalized boundary conditions. Syahirah et al. [34] documented heat transport of mixed convective MHD hybrid nanofluid past an inclined radiative and shrinking permeable plate. Hayat et al. [35] studied the boundary layer fluid flow of nonlinear convective curved surface with mass condition. For further studies, readers are referred to [36–42]. From the above cited literature, it can be seen that advection diffusion of viscous fluid has not been studied over a curved geometry. The objective of this present study is to communicate such a novel analysis for the completeness and mathematical understanding of fluid behaviours over a curved stretching mechanism. The effort concentrates in the modelling of an advection-diffusion equation of viscous fluid using curvilinear coordinate. This analysis requires to serve as fundamental in utilization of induction equation to generate a complete MHD models for curved structures. Therefore, this work is confined to the most recent articles in the direction of this study. The governing equations of the flow problem consist of Newton’s law of motion and pre-Maxwell form of the law of electrodynamics which explain mutual interaction of fluid flow and magnetic fields. Further, it communicates the theoretical examinations <>of heat and mass transports of viscous fluid past a magnetized curved stretching sheet. Hence, the results can be applied in the manufacturing of stretchable curved sheet.

2. Flow formulation

Two-dimensional steady flow of an incompressible Navier-Stokes equation of viscous fluid over a convectively magnetized curved stretching surface is considered. The fluid transport incorporates advection-diffusion equation of $\mathit{B}(B_r,B_x,0)$ in $\mathit{r}-$ and $\mathit{x}-$ directions. The surface is magnetized and stretched with velocity $B_x(\mathit{x})$ and $u_w(\mathit{x})$, respectively. Energy and concentration equations include viscous dissipation, Joule heating and chemical reaction, respectively (refer Figure 1 for the flow geometry)

Figure 1. Physical Problem’s geometry.

For the conservation of mass and momentum equations:

(2.1) $\mathrm{\nabla }\mathbf{.}\mathbf{V}=0,$(2.1)

(2.2) $\mathit{\rho }\frac{\mathit{D}\mathbf{V}}{\mathit{Dt}}=-\mathrm{\nabla }\mathbf{p}+\mathit{\mu }{\mathrm{\nabla }}^{\mathbf{2}}\mathbf{V}+(\mathit{J}\times \mathbf{B}),$(2.2)

in which $\mathit{\rho }$ defines the density of fluid, $\mathit{\mu }$ is the dynamic viscosity, $\mathbf{V}=\mathit{V}(\mathit{v},\mathit{u},0)$ is the field velocity, $\mathit{J}\times \mathit{B}$ generates the Lorentz force as $\mathit{J}$ denotes the electric current density, and $\mathbf{B}=(B_r,B_x,0)$ is the magnetic induction vector.

Consider the set of Maxwell’s equation and Ohm’s law in a steady mode, the reduced form [1]

(2.3) $\mathrm{\nabla }\times \mathbf{E}=0,\mathrm{\nabla }\mathbf{.}\mathbf{B}=0,\mathrm{\nabla }\times \mathbf{B}={\mu }_s\mathit{J},$(2.3)

where

(2.4) $\mathit{J}=\mathit{\sigma }(\mathbf{E}+\mathbf{V}\times \mathbf{B}).$(2.4)

Here, $\mathit{\sigma }$ represents fluid electric conductivity, $\mathit{E}$ symbolizes an applied electric field, and ${\mu }_s$ is the permeability of free space. Utilizing Eqs. (2.3) and (2.4), Eq. (2.2) becomes

(2.5) $\mathit{\rho }\frac{\mathit{D}\mathbf{V}}{\mathit{Dt}}=-\mathrm{\nabla }\mathbf{p}+\mathit{\mu }{\mathrm{\nabla }}^{\mathbf{2}}\mathbf{V}+(\mathbf{B}\mathbf{.}\mathbf{V})\frac{\mathbf{B}}{{\mu }_s}-\mathrm{\nabla }\left(\frac{{\mathbf{B}}^{\mathbf{2}}}{2{\mu }_s}\right).$(2.5)

Introduce the pressure difference in the form

(2.6) $\mathrm{\nabla }\mathbf{p}{ }^{\mathrm{\prime }}=-\mathrm{\nabla }\mathbf{p}-\frac{1}{2}\mathrm{\nabla }\left(\frac{{\mathbf{B}}^{\mathbf{2}}}{2{\mu }_s}\right),$(2.6)

and utilizing Eq. (2.6), the momentum Eq. (2.5) reduce to

(2.7) $\mathit{\rho }\frac{\mathit{D}\mathbf{V}}{\mathit{Dt}}=-\mathrm{\nabla }\mathbf{p}{ }^{\mathrm{\prime }}+\mathit{\mu }{\mathrm{\nabla }}^{\mathbf{2}}\mathbf{V}+(\mathbf{B}\mathbf{.}\mathbf{V})\frac{\mathbf{B}}{{\mu }_s}.$(2.7)

Employing curvilinear coordinate operator- $\mathrm{\nabla }=({\hat{e}}_{\mathit{r}}{\mathrm{\partial }}_r,\frac{{\hat{e}}_{\mathit{x}}}{1+\mathit{kr}}{\mathrm{\partial }}_x,0)$ and curvature- $\mathit{k}=\frac{1}{R}$. Equations (2.1)-(2.7) yield

(2.8) ${\mathrm{\partial }}_r\mathit{v}+\frac{{\mathrm{\partial }}_x\mathit{u}}{1+\mathit{kr}}=-\frac{\mathit{kv}}{1+\mathit{kr}}$(2.8)

(2.9) $\mathit{v}{\mathrm{\partial }}_r\mathit{v}+\frac{u{\mathrm{\partial }}_x\mathit{v}-\mathit{k}{u}^2}{1+\mathit{kr}}=-\frac{1}{\rho }{\mathrm{\partial }}_r\mathit{p}+\frac{\mu }{\rho }\left({\mathrm{\partial }}_r^2\mathit{v}+\frac{{\mathrm{\partial }}_x^2\mathit{v}-k^2\mathit{v}-2\mathit{k}{\mathrm{\partial }}_x\mathit{u}}{{(1+\mathit{kr})}^2}+\frac{k{\mathrm{\partial }}_r\mathit{v}}{1+\mathit{kr}}\right)+\frac{1}{\rho {\mu }_s}\left(B_r{\mathrm{\partial }}_r{B}_r+\frac{B_x{\mathrm{\partial }}_x{B}_r}{1+\mathit{kr}}-\frac{k{B}_x^2}{1+\mathit{kr}}\right)$(2.9)

(2.10) $\mathit{v}{\mathrm{\partial }}_r\mathit{u}+\frac{u{\mathrm{\partial }}_x\mathit{u}+\mathit{kuv}}{1+\mathit{kr}}=-\frac{1}{\mathit{\rho }(1+\mathit{kr})}{\mathrm{\partial }}_x\mathit{p}+\frac{\mu }{\rho }\left({\mathrm{\partial }}_r^2\mathit{u}+\frac{{\mathrm{\partial }}_x^2\mathit{u}-k^2\mathit{u}+2\mathit{k}{\mathrm{\partial }}_x\mathit{v}}{{(1+\mathit{kr})}^2}\right),\text{break}+\frac{1}{\rho {\mu }_s}\left(B_r{\mathrm{\partial }}_r{B}_x+\frac{B_x{\mathrm{\partial }}_x{B}_x}{1+\mathit{kr}}+\frac{k{B}_r{B}_x+\mathit{k}{\mathrm{\partial }}_r\mathit{u}}{1+\mathit{kr}}\right).$(2.10)

Thus, advection-diffusion equations in curvilinear coordinate yield

(2.11) ${\mathrm{\partial }}_r{B}_r+\frac{{\mathrm{\partial }}_x{B}_x}{1+\mathit{kr}}=-\frac{k{B}_r}{1+\mathit{kr}}$(2.11)

(2.12) $B_r{\mathrm{\partial }}_r\mathit{v}-\mathit{v}{\mathrm{\partial }}_r{B}_r-\frac{B_x{\mathrm{\partial }}_x\mathit{v}+\mathit{u}{\mathrm{\partial }}_x{B}_r}{1+\mathit{kr}}=-\mathit{\lambda }\left({\mathrm{\partial }}_r^2{B}_r+\frac{{\mathrm{\partial }}_x^2{B}_r-k^2{B}_r-2\mathit{k}{\mathrm{\partial }}_x{B}_x}{{(1+\mathit{kr})}^2}\right)-\mathit{\lambda }\left(-\frac{k{\mathrm{\partial }}_r{B}_r}{1+\mathit{kr}}\right),$(2.12)

(2.13) $B_r{\mathrm{\partial }}_r\mathit{u}-\mathit{v}{\mathrm{\partial }}_r{B}_x+\frac{B_x{\mathrm{\partial }}_x\mathit{u}-\mathit{u}{\mathrm{\partial }}_x{B}_x}{1+\mathit{kr}}=-\mathit{\lambda }\left({\mathrm{\partial }}_r^2{B}_x+\frac{{\mathrm{\partial }}_x^2{B}_x-k^2{B}_x+2\mathit{k}{\mathrm{\partial }}_x{B}_r}{{(1+\mathit{kr})}^2}\right)-\mathit{\lambda }\left(-\frac{k{\mathrm{\partial }}_r{B}_x}{1+\mathit{kr}}-\frac{ku{B}_r+\mathit{kv}{B}_x}{\mathit{\lambda }(1+\mathit{kr})}\right).$(2.13)

where $\mathit{\lambda }=(\mathit{\mu \sigma }{)}^{-1}$ indicates the magnetic diffusivity. It is worth mentioning that the governing equations are presented first time.

3. Boundary-Layer Analysis

With appropriate scaling, we define the non-dimensional variables as follows:

(3.1) $\bar{x}=\frac{x}{l},\bar{r}=\frac{r}{U_{\delta }},\bar{k}=\mathit{k\delta },\bar{P}=\frac{P}{\rho U_{\mathrm{\infty }}^2},\bar{u}=\frac{u}{U_{\mathrm{\infty }}},\bar{v}=\frac{\mathit{vl}}{U_{\mathrm{\infty }}\mathit{\delta }},$(3.1)

(3.2) $B_r=\frac{B_0\mathit{l}}{\mathit{\delta }}\sqrt{\frac{\mu }{\rho U_{\mathrm{\infty }}\mathit{l}}}{\bar{B}}_{\mathit{r}},B_x=B_0\sqrt{\frac{\mu }{\rho U_{\mathrm{\infty }}\mathit{l}}}{\bar{B}}_{\mathit{x}},$(3.2)

Likewise, it is worth noting that Eq. (3.2) is presented first time. Substituting Eqs. (3.1) and (3.2) in Eqs. (2.8)-(2.13), one can get the following

(3.3) ${\mathrm{\partial }}_r\mathit{P}=\frac{k{u}^2}{1+\mathit{kr}}+\mathrm{\Gamma }\frac{k{B}_x^2}{1+\mathit{kr}}$(3.3)

(3.4) $\mathit{v}{\mathrm{\partial }}_r\mathit{u}+\frac{u{\mathrm{\partial }}_x\mathit{u}+\mathit{kuv}}{1+\mathit{kr}}=-\frac{{\mathrm{\partial }}_x\mathit{P}}{1+\mathit{kr}}+\mathit{\nu }\left({\mathrm{\partial }}_r^2+\frac{\nu k{\mathrm{\partial }}_r\mathit{u}}{1+\mathit{kr}}\right)\text{break}+\mathrm{\Gamma }\left(B_r{\mathrm{\partial }}_r{B}_x+\frac{k{B}_r{B}_x}{1+\mathit{kr}}\right)-\frac{\nu k^2\mathit{u}}{{(1+\mathit{kr})}^2}$(3.4)

(3.5) $B_r{\mathrm{\partial }}_r\mathit{u}-\mathit{v}{\mathrm{\partial }}_r{B}_x+\frac{B_x{\mathrm{\partial }}_x\mathit{u}+\mathit{kv}{B}_x}{1+\mathit{kr}}=\mathit{\beta }\left({\mathrm{\partial }}_r^2{B}_x+\frac{k{\mathrm{\partial }}_r{B}_x}{1+\mathit{kr}}-\frac{k^2{B}_x}{{(1+\mathit{kr})}^2}\right)+\frac{u{\mathrm{\partial }}_x{B}_x+\mathit{ku}{B}_r}{1+\mathit{kr}},$(3.5)

where $\mathrm{\Gamma }=\frac{B_0}{\sqrt{\mathit{\rho }{\mu }_s}}$ is the magnetic constant, $\mathit{Re}=\frac{\rho u_w}{\mathit{\mu }}$ is the Reynolds number, and $\mathit{\beta }=\frac{\lambda }{\nu }$ denotes magnetic diffusivity of the fluid. Note that both Eqs. (2.8) and (2.11) are identically satisfied.

The temperature and concentration equations are as follows [32,35]

(3.6) $\mathit{v}{\mathrm{\partial }}_r\mathit{T}+\frac{u{\mathrm{\partial }}_x\mathit{T}}{1+\mathit{kr}}=\frac{1}{\rho C_p}\left[K_{\ast }\left({\mathrm{\partial }}_r^2\mathit{T}+\frac{k{\mathrm{\partial }}_r\mathit{T}}{1+\mathit{kr}}\right)+\mathit{\mu }{\left({\mathrm{\partial }}_r\mathit{u}-\frac{\mathit{ku}}{1+\mathit{kr}}\right)}^2\right]+\frac{1}{\rho C_p}\left(\frac{k^2\mathit{\sigma }{B}_0^2{u}^2}{{(1+\mathit{kr})}^2}-{\mathrm{\partial }}_r{q}_o\right)$(3.6)

(3.7) $\mathit{v}{\mathrm{\partial }}_r\mathit{C}+\frac{u{\mathrm{\partial }}_x\mathit{C}}{1+\mathit{kr}}=D_{\ast }\left({\mathrm{\partial }}_r^2\mathit{C}+\frac{k{\mathrm{\partial }}_r\mathit{C}}{1+\mathit{kr}}\right)+k_o(\mathit{C}-C_{\mathrm{\infty }}),$(3.7)

in which $C_p$ is the fluid-specific heat capacity, $\mathit{T}$ is the temperature, $K_{\ast }$ is the thermal conductivity, $q_o=\left(\frac{-4{\sigma }^{\ast }}{3k^{\ast }}\right){\mathrm{\partial }}_r{T}^4$ is the heat flux subject to Rosseland approximation such that ${\sigma }^{\ast }$ and $k^{\ast }$ denote Stefan-Boltzmann constant and mean spectral absorption coefficient, respectively. $\mathit{C}$ represents the fluid concentration, $D_{\ast }$ is the chemical diffusion coefficient, and $k_{\ast }$ is the first-order chemical reaction rate.

Expressing the temperature variation $T^4$ about $T_{\mathrm{\infty }}$ using Taylor’s series, one can get

(3.8) $T^4\approx 4T_{\mathrm{\infty }}^3\mathit{T}-3T_{\mathrm{\infty }}^4.$(3.8)

Hence, incorporating Eq. (3.8) in Eq. (3.6), we have

(3.9) $\mathit{v}{\mathrm{\partial }}_r\mathit{T}+\frac{u{\mathrm{\partial }}_x\mathit{T}}{1+\mathit{kr}}=\frac{1}{\rho C_p}\left[K_{\ast }\left(1+\frac{16{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{3k^{\ast }{K}_{\ast }}\right){\mathrm{\partial }}_r^2\mathit{T}+\frac{k{\mathrm{\partial }}_r\mathit{T}}{1+\mathit{kr}}\right]+\frac{1}{\rho C_p}\left[\mathit{\mu }{\left({\mathrm{\partial }}_r\mathit{u}-\frac{\mathit{ku}}{1+\mathit{kr}}\right)}^2+\frac{k^2\mathit{\sigma }{B}_0^2{u}^2}{{(1+\mathit{kr})}^2}\right].$(3.9)

Subject to relevant boundary conditions of the form:

$\mathit{u}{|}_{\mathit{r}=0}=U_0\mathit{x},\mathit{v}{|}_{\mathit{r}=0}=0,$

(3.10) $\mathit{u}{|}_{\mathit{r}\to \mathrm{\infty }}=0,{\mathrm{\partial }}_r\mathit{u}{|}_{\mathit{r}\to \mathrm{\infty }}=0,$(3.10)

$B_x{|}_{\mathit{r}=0}=\mathit{B}(\mathit{x}),B_r{|}_{\mathit{r}=0}=0,$

(3.11) $B_x{|}_{\mathit{r}\to \mathrm{\infty }}=0,$(3.11)

(3.12) $K_{\ast }{\mathrm{\partial }}_r\mathit{T}{|}_{\mathit{r}=0}=h_0(\mathit{T}-T_0),\mathit{T}{|}_{\mathit{r}\to \mathrm{\infty }}\to T_{\mathrm{\infty }},$(3.12)

(3.13) $D_{\ast }{\mathrm{\partial }}_r\mathit{C}{|}_{\mathit{r}=0}=D_0(\mathit{C}-C_w),\mathit{C}{|}_{\mathit{r}\to \mathrm{\infty }}\to C_{\mathrm{\infty }},$(3.13)

such that $U_0(1/\mathit{t})$ is the stretching constant. $T_0$, $T_{\mathrm{\infty }}$, $h_0$ and $D_0$ are the surface temperature, ambient temperature, convective heat transport and convective mass transport, respectively.

4. Similarity analysis

Consider the surface velocity and magnetic component of the form $\mathit{u}(\mathit{x})=U_0\mathit{xg}{ }^{\mathrm{\prime }}(\mathit{\xi })$ and $B_x(\mathit{x})=B_0\mathit{xh}{ }^{\mathrm{\prime }}(\mathit{\xi })$, while the temperature and concentration are given as $\mathit{\theta }=\frac{T-T_{\mathrm{\infty }}}{T_0-T_{\mathrm{\infty }}}$ and $\mathit{\psi }=\frac{C-C_{\mathrm{\infty }}}{C_0-C_{\mathrm{\infty }}}$, respectively.

In relation to $\mathit{V}(\mathit{v},\mathit{u},0)$ and $\mathit{B}(B_r,B_x,0)$, we define

$\mathit{u}(\mathit{x})={\mathrm{\partial }}_r[\mathit{R}(1+\mathit{kr})\mathit{\psi }]$

(4.1) $B_x(\mathit{x})={\mathrm{\partial }}_r[\mathit{R}(1+\mathit{kr})\mathit{A}],$(4.1)

where $\mathit{\psi }$ and $\mathit{A}$ imply the distinguish stream function and analogous flux function.

By invoking

$\mathit{v}=-\frac{1}{k}{\mathrm{\partial }}_x\mathit{\psi },$

(4.2) $B_r(\mathit{r})=-\frac{1}{k}{\mathrm{\partial }}_x\mathit{A},$(4.2)

one can integrate Eq. (4.1) and get

$\mathit{\psi }=\frac{k}{1+\mathit{kr}}\sqrt{\frac{\nu }{U_0}}{U}_0\mathit{xg}{ }^{\mathrm{\prime }}(\mathit{\xi }),$

(4.3) $\mathit{A}=\frac{k}{1+\mathit{kr}}\sqrt{\frac{\nu }{U_0}}{B}_0\mathit{xh}{ }^{\mathrm{\prime }}(\mathit{\xi }),$(4.3)

such that Eq. (4.2) after incorporating Eq. (4.3) becomes

$\mathit{v}=-\frac{U_0}{1+\mathit{kr}}\sqrt{\frac{\nu }{U_0}}\mathit{g}(\mathit{\xi }),$

(4.4) $B_r(\mathit{r})=-\frac{B_0}{1+\mathit{kr}}\sqrt{\frac{\nu }{U_0}}\mathit{h}(\mathit{\xi }),$(4.4)

Note that non-dimensional independent variables include

(4.5) $\mathit{R}=\mathit{k}\sqrt{\frac{\nu }{U_0}},\mathit{\xi }=\mathit{r}\sqrt{\frac{U_0}{\mathit{\nu }}}.$(4.5)

Therefore, similarity quantities in view of temperature and concentration fields follow

$\mathit{T}=(T_0-T_{\mathrm{\infty }})\mathit{\theta }(\mathit{\xi })+T_{\mathrm{\infty }},$

(4.6) $\mathit{C}=(C_0-C_{\mathrm{\infty }})\mathit{\psi }(\mathit{\xi })+C_{\mathrm{\infty }}.$(4.6)

Using Eqs. (4.4)-(4.6), Eqs. (2.8)-(2.10), (3.3)-(3.5), (3.9) and (3.7) reduce to

(4.7) $\frac{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}(\mathit{\xi })}{\mathit{k}+\mathit{\xi }}=\mathit{P}{ }^{\mathrm{\prime }}(\mathit{\xi })-{\beta }_0(\mathit{h}{ }^{\mathrm{\prime }}(\mathit{\xi }){)}^2,$(4.7)

(4.8) $\frac{k}{\mathit{k}+\mathit{\xi }}\left(2\mathit{P}-\mathit{gg}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}-\frac{\mathit{gg}{\mathit{ }}^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}+{(\mathit{g}{\mathit{ }}^{\mathrm{\prime }})}^2\right)=\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\frac{1}{\mathit{k}+\mathit{\xi }}\left(\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}-\frac{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}\right),\text{break}-\frac{\mathit{kN}}{{\mathit{R}}_{\mathit{em}}(\mathit{k}+\mathit{\xi })}\left(\mathit{hh}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\frac{\mathit{hh}{\mathit{ }}^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}\right),$(4.8)

(4.9) $\frac{k}{\mathit{k}+\mathit{\xi }}\left(\mathit{gh}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}-\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}\mathit{h}-\mathit{g}{ }^{\mathrm{\prime }}\mathit{h}{ }^{\mathrm{\prime }}+\frac{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}\mathit{h}}{\mathit{k}+\mathit{\xi }}\right)=\frac{1}{\mathit{pm}}\left(\mathit{h}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\frac{\mathit{h}{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}-\frac{\mathit{h}{\mathit{ }}^{\mathrm{\prime }}}{{(\mathit{k}+\mathit{\xi })}^2}\right),$(4.9)

(4.10) $(1+\mathit{Rd})\mathit{\theta }{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\frac{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}+\mathit{Br}{\left(\mathit{g}{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}-\frac{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}\right)}^2=-\frac{\mathit{kPrg\theta }{\mathit{ }}^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}-\frac{k^2\mathit{NBr}{(\mathit{g}{\mathit{ }}^{\mathrm{\prime }})}^2}{{(\mathit{k}+\mathit{\xi })}^2},$(4.10)

(4.11) $\mathit{\psi }{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\frac{1}{\mathit{k}+\mathit{\xi }}\mathit{\psi }+\mathit{Sc}\frac{k}{\mathit{k}+\mathit{\xi }}\mathit{g\psi }{ }^{\mathrm{\prime }}=\mathit{wPe\psi },$(4.11)

where ${\beta }_0=\frac{\mathrm{\Gamma }{B}_0}{U_0^2}$ denotes the magnetic constant, $\mathit{N}=\frac{\sigma B_0^2\mathit{l}}{\mathit{\rho u}}$ is the Stuart number (interaction parameter), $R_{\mathit{em}}={\mu }_s\mathit{\sigma ul}$ s the magnetic Reynolds number, $\mathit{pm}=\frac{\nu }{\rho }$ is magnetic Prandtl number, $\mathit{Rd}=1+\frac{16{\sigma }^{\ast }{T}_{\mathrm{\infty }}^3}{3k^{\ast }{K}_{\ast }}$ is the radiation parameter, $\mathit{Ec}=\frac{u^2}{C_p(T_0-T_{\mathrm{\infty }})}$ is the Eckert number, $\mathit{Pr}=\frac{\mu C_p}{K_{\ast }}$ is the Prandtl number, $\mathit{Br}=\mathit{Pr}\ast \mathit{Ec}$ is the Brinkman number, $\mathit{Sc}=\frac{\nu }{D_0}$ is the Schmidt number, $\mathit{w}=\frac{k_{\ast }\mathit{\mu }}{u^2}$ is the rate of reaction, and $\mathit{Pe}=\mathit{Re}\ast \mathit{Sc}$ is the Peclet number.

After using Eq. (4.7) in Eq. (4.8), we obtain

(4.12) $g^{\mathit{iv}}+\frac{2\mathit{g}{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\mathit{kgg}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}-\mathit{kg}{ }^{\mathrm{\prime }}\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}}{\mathit{k}+\mathit{\xi }}-\frac{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}-\mathit{kgg}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}}{{(\mathit{k}+\mathit{\xi })}^2}+\frac{\mathit{g}{\mathit{ }}^{\mathrm{\prime }}-\mathit{kgg}{ }^{\mathrm{\prime }}}{{(\mathit{k}+\mathit{\xi })}^3},\text{break}-\frac{\mathit{k}{(\mathit{g}{\mathit{ }}^{\mathrm{\prime }})}^2+2\mathit{kg}{ }^{\mathrm{\prime }}+2\mathit{k\beta }{(\mathit{h}{\mathit{ }}^{\mathrm{\prime }})}^2}{{(\mathit{k}+\mathit{\xi })}^2}+\frac{\mathit{Nkhh}{\mathit{ }}^{\mathrm{\prime }}}{R_{\mathit{em}}{(\mathit{k}+\mathit{\xi })}^3},\text{break}=\frac{\mathit{N}(\mathit{khh}{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\mathit{kh}{ }^{\mathrm{\prime }}\mathit{h}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }})}{R_{\mathit{em}}(\mathit{k}+\mathit{\xi })}+\frac{\mathit{N}(\mathit{khh}{\mathit{ }}^{\mathrm{\prime }}{ }^{\mathrm{\prime }}+\mathit{k}{(\mathit{h}{\mathit{ }}^{\mathrm{\prime }})}^2)}{R_{\mathit{em}}{(\mathit{k}+\mathit{\xi })}^2},$(4.12)

subject to

(4.13) $\mathit{g}{|}_{\mathit{\xi }=0}=0,\mathit{g}{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }=0}=1,\text{breakg}{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }\to \mathrm{\infty }}=0,\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }\to \mathrm{\infty }}=0,$(4.13)

(4.14) $\mathit{h}{|}_{\mathit{\xi }=0}=0,\mathit{h}{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }=0}=1,\mathit{g}{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }\to \mathrm{\infty }}=0,$(4.14)

(4.15) $\mathit{\theta }{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }=0}=\mathit{Bt}(\mathit{\theta }(0)-1),\mathit{\theta }{|}_{\mathit{\xi }\to \mathrm{\infty }}=0,$(4.15)

(4.16) $\mathit{\psi }{ }^{\mathrm{\prime }}{|}_{\mathit{\xi }=0}=\mathit{Bc}(\mathit{\psi }(0)-1),\mathit{\psi }{|}_{\mathit{\xi }\to \mathrm{\infty }}=0,$(4.16)

where $\mathit{Bt}=\frac{h_0}{K^{\ast }}\sqrt{\frac{\mu }{\rho U_0}}$ and $\mathit{Bc}=\frac{D_0}{D_w}\sqrt{\frac{\mu }{\rho U_0}}$ are thermal and concentration Biot numbers, respectively. The physical quantities are indispensable which provide scientific values in view of engineering and industrial applications. These are skin friction coefficient- $C_{\mathit{skf}}$, magnetic flux coefficient- $C_{\mathit{mf}}$, rate of heat transfer- $\mathit{Nu}$ (Nusselt number), and rate of mass transfer- $\mathit{Sh}$ (Schmidt number) are computed as follows:

$C_{\mathit{skf}}=\frac{{\mathrm{\partial }}_r\mathit{u}{|}_{\mathit{r}=0}}{0.5\mathit{\rho }{U}_0^2{x}^2},$

$C_{\mathit{mf}}=\frac{{\mathrm{\partial }}_r{B}_x{|}_{\mathit{r}=0}}{\sqrt{{\mu }_s\mathit{\sigma }{U}_0(B_0\mathit{x})}},$

$\mathit{Nu}=\frac{x{q}_0}{K_{\ast }(\mathit{T}-T_0)},$

(4.17) $\mathit{Sh}=\frac{x{H}_x}{K_{\ast }(\mathit{C}-C_0)},$(4.17)

such that

(4.18) $q_0=-K_{\ast }{\mathrm{\partial }}_r\mathit{T}{|}_{\mathit{r}=0},H_x=-D_0{\mathrm{\partial }}_r\mathit{C}{|}_{\mathit{r}=0}.$(4.18)

Using Eq. (4.18) in Eq. (4.17), one can get

$-0.5\mathit{R}{e}^{0.5}{C}_{\mathit{skf}}=\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(0)$

$\frac{R_{em}^{0.5}}{\mathit{R}{e}^{0.5}}{C}_{\mathit{mf}}=\mathit{h}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(0)$

$\mathit{R}{e}^{-0.5}\mathit{Nu}=-\mathit{\theta }{ }^{\mathrm{\prime }}(0),$

(4.19) $\mathit{R}{e}^{-0.5}\mathit{Sh}=-\mathit{\psi }{ }^{\mathrm{\prime }}(0).$(4.19)

The values of $\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(0)$, $\mathit{h}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(0)$ and $-\mathit{\theta }{ }^{\mathrm{\prime }}(0)$ are plotted while $-\mathit{\psi }{ }^{\mathrm{\prime }}(0)$ are presented in tabular form.

5. Numerical solution

Initialize the first-order system of Equations (36)–(38)

(5.1) $\left(\begin{array}{c}\mathit{h}{ }^{\mathrm{\prime }}\\ {\mathit{h}{\mathit{ }}^{\mathrm{\prime }}}_1\\ {\mathit{h}{\mathit{ }}^{\mathrm{\prime }}}_2\end{array}\right)=\left(\begin{array}{c}{h}_1\\ h_2\\ \frac{-h_2+\mathit{pm}(\mathit{kg}{h}_2-\mathit{h}{g}_2-h_1{g}_1)}{\mathit{k}+\mathit{\xi }}+\frac{h_1-\mathit{kpmh}{g}_1}{{(\mathit{k}+\mathit{\xi })}^2}\end{array}\right),$(5.1)

(5.2) $\left(\begin{array}{c}\mathit{\theta }{ }^{\mathrm{\prime }}\\ {\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}_1\end{array}\right)=\left(\begin{array}{c}{\theta }_1\\ \frac{-1}{1+\mathit{Rd}}\left[\frac{{\theta }_1+\mathit{kPrg}{\theta }_1}{\mathit{k}+\mathit{\xi }}+\frac{NBr{k}^2{(g_1)}^2}{{(\mathit{k}+\mathit{\xi })}^2}\right]\end{array}\right),$(5.2)

(5.3) $\left(\begin{array}{c}\mathit{\psi }{ }^{\mathrm{\prime }}\\ {\mathit{\psi }{\mathit{ }}^{\mathrm{\prime }}}_1\end{array}\right)=\left(\begin{array}{c}{\psi }_1\\ \frac{-\mathit{\psi }}{\mathit{k}+\mathit{\xi }}-\frac{k\mathrm{Scg}{\psi }_1}{\mathit{k}+\mathit{\xi }}+\mathit{wPe\psi }\end{array}\right),$(5.3)

and

(5.4) $\left(\begin{array}{c}\mathit{g}{ }^{\mathrm{\prime }}\\ {\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}_1\\ {\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}_2\\ {\mathit{g}{\mathit{ }}^{\mathrm{\prime }}}_3\end{array}\right)=\left(\begin{array}{c}{g}_1\\ g_2\\ g_3\\ -\frac{2g_3-g_2-2\mathit{k}{g}_1+\mathit{kg}{g}_2}{\mathit{k}+\mathit{\xi }}\\ -\frac{k{g}_1^2-g_2-2\mathit{k}{g}_1+\mathit{kg}{g}_2-2\mathit{k\beta }{h}_1^2}{{(\mathit{k}+\mathit{\xi })}^2}\\ -\frac{g_1-\mathit{kg}{g}_1}{{(\mathit{k}+\mathit{\xi })}^3}+\frac{\mathit{N}(\mathit{kh}{\mathit{h}{\mathit{ }}^{\mathrm{\prime }}}_2+\mathit{k}{h}_1{h}_2)}{R_{\mathit{em}}(\mathit{k}+\mathit{\xi })}\\ -\frac{N(k{h}_1{h}_2+\mathit{k}{h}_1^2)}{R_{\mathit{em}}{(\mathit{k}+\mathit{\xi })}^2}\\ +\frac{N(kh{h}_1)}{R_{\mathit{em}}{(\mathit{k}+\mathit{\xi })}^3}\end{array}\right).$(5.4)

By adopting standard Keller-Box shooting technique with Jacobi iterative technique [39]

Let

$\mathit{m}{ }^{\mathrm{\prime }}=m_1,\mathit{m}{{\mathit{ }}^{\mathrm{\prime }}}_1=m_2,\mathit{m}{{\mathit{ }}^{\mathrm{\prime }}}_2=m_3,\dots ,\mathit{m}{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{n}-1}=H_1(\mathit{\xi },m_1,m_2,m_3,\dots ,m_{\mathit{k}-1}),$

$\frac{m_i^{n+1}-m_{i-1}^n}{\mathrm{\Delta h}}=(m_1{)}_{i-1/2}^n,$

$\frac{(m_1{)}_i^{n+1}-(m_1{)}_{i-1}^n}{\mathrm{\Delta h}}=(m_2{)}_{i-1/2}^n,$

$\dots$

(5.5) $\frac{({\mathit{m}}_{\mathit{n}-1}{)}_i^{n+1}-(m_{\mathit{n}-1}{)}_{i-1}^n}{\mathrm{\Delta h}}=H_2({\xi }_{i-1/2}^n,m_{i-1/2}^n,(m_1{)}_{i-1/2}^n\dots ,(m_{\mathit{k}-1}{)}_{i-1/2}^n)$(5.5)

Now, implementing Eq. (5.5) explicitly into Eqs. (5.1)-(5.4) subject to Eqs. (4.13)-(4.16),

the magnetic field equations become:

$h_i^{n+1}-h_{i-1}^n+0.5\mathrm{\Delta h}((h_1{)}_i^n+(h_1{)}_{i-1}^n)=0,$

$h_0^{n+1}=0$

$(h_1{)}_i^{n+1}-(h_1{)}_{i-1}^n+0.5\mathrm{\Delta h}((h_2{)}_i^n+(h_2{)}_{i-1}^n)=0,$

$(h_1{)}_0^{n+1}=1$

$(h_2{)}_i^{n+1}-(h_2{)}_{i-1}^n=\frac{-\mathrm{\Delta h}}{4(\mathit{k}+\mathit{\xi })}\left((h_2{)}_i^n+(h_2{)}_{i-1}^n+\frac{(h_1{)}_i^n-(h_1{)}_{i-1}^n}{\mathit{k}+\mathit{\xi }}\right)\text{break}-\frac{\Delta h\mathrm{kpm}{A}_1}{4(\mathit{k}+\mathit{\xi })}$

(5.6) $(h_2{)}_0^{n+1}=c_1,$(5.6)

-the temperature field:

${\theta }_i^{n+1}-{\theta }_{i-1}^n+0.5\mathrm{\Delta h}(({\theta }_1{)}_i^n+({\theta }_1{)}_{i-1}^n)=0,$

${\theta }_0^{n+1}=c_2$

$({\theta }_1{)}_i^{n+1}-({\theta }_1{)}_{i-1}^n=\frac{-\mathrm{\Delta h}}{1+\mathit{Rd}}\left(\frac{kPr(g_i^n-g_{i-1}^n)(({\theta }_1{)}_i^n+({\theta }_1{)}_{i-1}^n)}{4(\mathit{k}+\mathit{\xi })}\right)\text{break}-\frac{\mathrm{\Delta }\mathit{h}}{1+\mathit{Rd}}\left(\frac{({\theta }_1{)}_i^n+({\theta }_1{)}_{i-1}^n}{4(\mathit{k}+\mathit{\xi })}\right)\text{break}-\frac{\mathrm{\Delta }\mathit{h}}{1+\mathit{Rd}}\left(\frac{k^2\mathit{BrN}{((g_1{)}_i^n-(g_1{)}_{i-1}^n)}^2}{4(\mathit{k}+\mathit{\xi }{)}^2}\right)\text{break}-\frac{\Delta hBr{A}_2}{4(1+\mathit{Rd})}$

(5.7) $({\theta }_1{)}_0^{n+1}=\mathit{Bt}({\theta }_0^{n+1}-1),$(5.7)

-the concentration field:

${\psi }_i^{n+1}-{\psi }_{i-1}^n+0.5\mathrm{\Delta h}(({\psi }_1{)}_i^n+({\psi }_1{)}_{i-1}^n)=0,$

${\psi }_0^{n+1}=c_3$

$({\psi }_1{)}_i^{n+1}-({\psi }_1{)}_{i-1}^n=-\mathrm{\Delta h}\left(\frac{({\psi }_1{)}_i^n+({\psi }_1{)}_{i-1}^n}{4(\mathit{k}+\mathit{\xi })}\right)$

$-\mathrm{\Delta h}\left(\frac{kSc(g_i^n+g_{i-1}^n)({\psi }_1{)}_i^n+({\psi }_1{)}_{i-1}^n}{4(\mathit{k}+\mathit{\xi })}\right)$

$-0.5\Delta \mathit{hwPe}({\psi }_i^{n+1}-{\psi }_{i-1}^n)$

(5.8) $({\psi }_1{)}_0^{n+1}=\mathit{Bc}({\psi }_0^{n+1}-1),$(5.8)

-and velocity field:

$g_i^{n+1}-g_{i-1}^n+0.5\mathrm{\Delta h}((g_1{)}_i^n+(g_1{)}_{i-1}^n)=0,$

$g_0^{n+1}=0$

$(g_1{)}_i^{n+1}-(g_1{)}_{i-1}^n+0.5\mathrm{\Delta h}((g_2{)}_i^n+(g_2{)}_{i-1}^n)=0,$

$(g_1{)}_0^{n+1}=1$

$(g_2{)}_i^{n+1}-(g_1{)}_{i-1}^n+0.5\mathrm{\Delta h}((g_2{)}_i^n+(g_2{)}_{i-1}^n)=0,$

$(g_2{)}_0^{n+1}=c_4$

$(g_3{)}_i^{n+1}-(g_3{)}_{i-1}^n=-\mathrm{\Delta h}\left(\frac{A_3}{4(\mathit{k}+\mathit{\xi })}+\frac{A_4}{4(\mathit{k}+\mathit{\xi }{)}^2}-\frac{A_5}{4(\mathit{k}+\mathit{\xi }{)}^3}+\frac{kN{A}_6}{R_{\mathit{em}}(\mathit{k}+\mathit{\xi })}\right),$

(5.9) $(g_3{)}_0^{n+1}=c_5.$(5.9)

In these equations, $c_j$ for $\mathit{j}=1(1)5$ is initial guesses and $A_1,\dots ,A_6$ are given in the APPENDIX. The initial guesses are handled using MATLAB solver. The correctness of the these guesses is checked against end point ${\xi }_{\mathit{max}}$. Otherwise, the values are updated and iterations continue with step size, $\mathrm{\Delta h}={10}^{-2}$ until solutions are found which satisfy the end point initially at infinity and subject to the stopping criterion given as

(5.10) $|\parallel W^{\mathit{n}+1}{\parallel }_2-\parallel W^n{\parallel }_2|<\in .$(5.10)

Here, $\in$ remains the fixed tolerance value less than ${10}^{-8}$ to ascertain the numerical results. The algorithm is implemented through ode45 built-in MATLAB command.

6. Error analysis

Theorem: Keller-Box shooting method with Jacobi iterative technique attain a maximum error and bounded if suppose the functions $m_1=m_1(\mathit{h},h_1,h_2,\mathit{g},g_1,g_2)$, $m_2=m_2(\mathit{\theta },{\theta }_1,\mathit{g},g_1)$, $m_3=m_3(\mathit{\psi },{\psi }_1,\mathit{g})$ and $m_4=m_4(\mathit{g},g_1,g_2,g_3,\mathit{h},h_1,h_2)$ are differentiable such that linearize Eq. (5.1) is diagonally dominant.

Proof:

Discretize the linearize system, Eq. (5.1) becomes

$h_i^{n+1}-h_{i-1}^n+\mathrm{\Delta h}(h_1{)}_{i-1/2}^n=0$

$(h_1{)}_i^{n+1}-(h_1{)}_{i-1}^n+\mathrm{\Delta h}(h_2{)}_{i-1/2}^n=0$

(6.1) $(h_2{)}_i^{n+1}-(h_2{)}_{i-1}^n+\mathrm{\Delta h}(m_1{)}_{i-1/2}^n=0,$(6.1)

subject to exact scheme

$h_i^E-h_{i-1}^E+\mathrm{\Delta h}(h_1{)}_{i-1/2}^E=0$

$(h_1{)}_i^E-(h_1{)}_{i-1}^E+\mathrm{\Delta h}(h_2{)}_{i-1/2}^E=0$

(6.2) $(h_2{)}_i^E-(h_2{)}_{i-1}^E+\mathrm{\Delta h}(m_1{)}_{i-1/2}^E=0,$(6.2)

such that at any grid point, errors of the solution are as follows:

$(e_1{)}_i^n=h_1^n-h_1^E,$

$(e_2{)}_i^n=(h_1{)}_i^n-(h_1{)}_i^E,$

$(e_3{)}_i^n=(h_2{)}_i^n-(h_2{)}_i^E,$

$(e_4{)}_i^n=g_1^n-g_1^E,$

$(e_4{)}_i^n=(g_1{)}_i^n-(g_1{)}_i^E,$

(6.3) $(e_4{)}_i^n=(g_1{)}_i^n-(g_1{)}_i^E.$(6.3)

By virtue of Mean Value Theorem, one can write

(6.4) $m_1(h_i^n,(h_1{)}_i^n,(h_2{)}_i^n,g_i^n,(g_1{)}_i^n,(g_2{)}_i^n)-m_1(h_i^E,(h_1{)}_i^E,(h_2{)}_i^E,g_i^E,(g_1{)}_i^E,(g_2{)}_i^E)=({\bar{e}}_1{)}_i^n.\mathrm{\nabla }{m}_1(c_1,c_2,c_3,c_4),$(6.4)

where

$c_1=h_i^n+{\epsilon }_1(e_2{)}_i^n,$

$c_2=(h_1{)}_i^n+{\epsilon }_2(e_2{)}_i^n,$

$c_3=(h_2{)}_i^n+{\epsilon }_3(e_3{)}_i^n,$

$c_4=g_i^n+{\epsilon }_4(e_4{)}_i^n,$

$c_5=(g_1{)}_i^n+{\epsilon }_5(e_5{)}_i^n,$

(6.5) $c_6=(g_2{)}_i^n+{\epsilon }_6(e_6{)}_i^n,$(6.5)

$c_i\mathit{\epsilon }[0,1]$ for $\mathit{i}=1(1)6$, and $({\bar{e}}_1{)}_i^n=[(e_1{)}_i^n,(e_2{)}_i^n,(e_3{)}_i^n,(e_4{)}_i^n,(e_5{)}_i^n,(e_6{)}_i^n].$

And for convergence region, we define

$(e_1{)}^{\mathit{n}+1}=(e_1{)}_{i-1}^n+\mathrm{\Delta h}(e_1{)}_{i-1/2}^n,$

$(e_2{)}_i^{n+1}=(e_2{)}_{i-1}^n+\mathrm{\Delta h}(e_2{)}_{i-1/2}^n,$

(6.6) $(e_3{)}_i^{n+1}=(e_3{)}_{i-1}^n+\mathrm{\Delta h}({\hat{e}}_1{)}_{i-1/2}^n\mathrm{\nabla }{m}_1,$(6.6)

from Eq. (6.6), the following inequalities are deduced

$|(e_1{)}_i^{n+1}|\le |(e_1{)}_{i-1}^n|+\mathrm{\Delta }\mathit{h}|(e_2{)}_{i-1/2}^n|,$

$|(e_2{)}_i^{n+1}|\le |(e_2{)}_{i-1}^n|+\mathrm{\Delta h}|(e_3{)}_{i-1/2}^n|,$

(6.7) $|(e_3{)}_i^{n+1}|\le |(e_3{)}_{i-1}^n|+\mathrm{\Delta h}|({\hat{e}}_1{)}_{i-1/2}^n.\mathrm{\nabla }{m}_1|,$(6.7)

such that $\mathrm{\nabla }{m}_1=[\bar{m}11,\bar{m}12,\bar{m}13,\bar{m}14,\bar{m}15,\bar{m}16]$.

Thus, Equation (6.7) can be expressed as follows:

$|(e_3{)}_i^{n+1}|\le |(e_3{)}_{i-1}^n|+\mathrm{\Delta }\mathit{h}|{\mathrm{\Sigma }}_{j=1}^6(e_j{)}_{i-1/2}^n{\hat{m}}_1^{\mathit{j}}|,$

(6.8) $\le |(e_3{)}_{i-1}^n|+\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|(e_j{)}_{i-1/2}^n{\bar{m}}_1^{\mathit{j}}|,$(6.8)

in which the maximum error is estimated as follows:

$(e_1{)}_i^n=\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}|(e_1{)}_i^n|,$

$(e_2{)}_i^n=\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}|(e_2{)}_i^n|,$

$(e_3{)}_i^n=\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}|(e_3{)}_i^n|,$

$(e_4{)}_i^n=\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}|(e_4{)}_i^n|,$

$(e_5{)}_i^n=\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}|(e_5{)}_i^n|,$

(6.9) $(e_6{)}_i^n=\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}|(e_6{)}_i^n|,$(6.9)

where $\mathit{N}$ represents the number of nodes and $(\bar{e}{)}^n=\mathit{max}[\mathit{ma}{x}_{\mathit{i}=1(1)\mathit{N}}(e_1=1(1)\mathit{N}{)}_i^n]$.

Equations (6.7) and (6.8) can be written in the form

$e_1^{n+1}\le e_1^n+\mathrm{\Delta h}{e}_2^n+{\bar{H}}_1\mathit{O}(\mathrm{\Delta h}{)}^2,$

$e_2^{n+1}\le e_2^n+\mathrm{\Delta h}{e}_3^n+{\bar{H}}_2\mathit{O}(\mathrm{\Delta h}{)}^2,$

(6.10) ${\bar{e}}^{\mathit{n}+1}\le (1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|){\bar{e}}^{\mathit{n}}+{\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2.$(6.10)

Evaluating $\mathit{n}=0,1$, and $\mathit{n}$ in the last expression of Eq. (6.10) yield

${\bar{e}}^1\le (1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|){\bar{e}}^0+{\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2,$

${\bar{e}}^2\le (1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|{)}^2{\bar{e}}^0+[1+(1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|)]{\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2,$

${\bar{e}}^{\mathit{n}}\le (1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|{)}^n{\bar{e}}^0+[1+(1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|)+\dots ,$

(6.11) $+{(1+6\mathrm{\Delta }h{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|)}^{\mathit{n}-1}]{\bar{H}}_3\mathit{O}{(\mathrm{\Delta h})}^2,$(6.11)

and sum of the nth term yields

${\bar{e}}^{\mathit{n}}\le (1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|{)}^n{\bar{e}}^0+\left(\frac{{[1+6\mathrm{\Delta }h{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|]}^{\mathit{n}}}{6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|}\right){\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2,$

(6.12) $\le (1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|{)}^n{\bar{e}}^0+\mathit{EXP}(6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|){\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2.$(6.12)

Using Eq. (6.12) in Eq. (6.10), one can get

(6.13) $e_1^n\le (1+\mathrm{\Delta h})[(1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|{)}^n{\bar{e}}^0,\text{break}+\mathit{EXP}(6(\mathit{n}-1)\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|){\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2],\text{break}+{\bar{H}}_1\mathit{O}(\mathrm{\Delta h}{)}^2,$(6.13)

(6.14) $\begin{array}{rl}& e_2^n\le (1+\mathrm{\Delta h})[(1+6\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|{)}^n{\bar{e}}^0,\\ & +\mathit{EXP}(6(\mathit{n}-1)\mathrm{\Delta h}{\mathrm{\Sigma }}_{j=1}^6|{\bar{m}}_1^{\mathit{j}}|){\bar{H}}_3\mathit{O}(\mathrm{\Delta h}{)}^2],\\ & +{\bar{H}}_2\mathit{O}(\mathrm{\Delta h}{)}^2.\end{array}$(6.14)

Hence, equations (6.13) and (6.14) give the maximum error bounds for Eq. (5.1). It is worth noting that error bounds for Eqs. (5.2)-(5.4) can be shown in the similar way [38]. See the APPENDIX for A_i where i=1(1)3.

7. Results and graphical analysis

In this section, we discuss the numerical results of implementing Keller-Box shooting method together with Jacobi iterative technique using MATLAB ode45 built-in command. The graphs of velocity- $\mathit{g}{ }^{\mathrm{\prime }}(\mathit{\xi })$ and magnetic field- $\mathit{h}{ }^{\mathrm{\prime }}(\mathit{\xi })$ are plotted and analysed for the most emerging problem’s parameters. In this investigation, we characterized these parameters as Stuart number, $\mathit{N}$ (magnetic interaction), magnetic Reynolds number $R_{\mathit{em}}$, magnetic Prandtl number pm and curvature parameter $\mathit{k}$. For completeness of the flow situation, the effects of Prandtl number $\mathit{Pr}$, Brinkman number $\mathit{Br}$, Schmidt number $\mathit{Sc}$, Peclet number $\mathit{Pe}$ and rate of reaction $\mathit{w}$ are also given against temperature $\mathit{\theta }(\mathit{\xi })$ and concentration $\mathit{\psi }(\mathit{\xi })$ profiles, respectively. Noting that some specific parameters are kept constant- $\mathit{Pr}=4.0$, $\mathit{Br}=0.4$, $\mathit{Rd}=0.4$, $\mathit{Sc}=0.2$, $\mathit{Pe}=0.4$, $\mathit{pm}=0.5$, $\mathit{BT}=0.4$, $\mathit{BD}=0.4$, and $\mathit{w}=0.5$ except otherwise stated to establish the present solutions within the range of $0<\mathit{\xi }<10$. Figure 2 depicts the effects of $\mathit{k}$, $\mathit{N}$, and $\mathit{R}{e}_m$ on the stream function $\mathit{g}(\mathit{\xi })$.

Figure 2. Effects of $\mathit{k}$, $\mathit{N}$, and $\mathit{R}{e}_m$ on stream function.

In this graph, the stream function grows for increasing any of $\mathit{k}$ and $\mathit{N}$. Figure 2(a) confirms the existing result [29,30] of curvature $\mathit{k}$. Meaning the flow trajectories can be enhanced not only by increasing curvature parameter but also magnetic interaction parameter, see Figure 2(b). On the other hand, Figure 2(c) illuminates a reduction in the flow trajectories by increasing magnetic Reynolds number. Figures 3(a,b) portray the impacts of increasing $\mathit{k}$, and $\mathit{N}$, on velocity profiles, $\mathit{g}{ }^{\mathrm{\prime }}(\mathit{\xi })$. In these graphs, the flow velocity is optimized by increasing any of the parameters. Figure 3(a) again confirms an established result of the velocity field over a curved mechanism. From the application point of view, the flow velocity can be accelerated by controlling the surface curvature.

Figure 3. Effects of $\mathit{k}$, $\mathit{N}$, and $\mathit{R}{e}_m$ on velocity field.

Figure 3(b) reveals that increasing Stuart number, $\mathit{N}$, enhances the flow velocity of the fluid. The reason being that the greater the electromagnetic force to inertial force indicates a dominant electromagnetic force over a weak inertial force resulting in the enhancement flow speed. Figure 3(c) is plotted to show the impact of magnetic Reynold number $\mathit{R}{e}_m$, on velocity profile, $\mathit{g}{ }^{\mathrm{\prime }}(\mathit{\xi })$. The graph explicates a decreasing velocity field for slight increase in magnetic Reynolds number. Since magnetic Reynolds number measures the relative magnetic field induction through the motion of conducting fluid to magnetic diffusion, we can infer that the relative comparison with induction and diffusion is insignificant for large rise in $\mathit{R}{e}_m$ results in diminishing velocity of the fluid and thus shrinks boundary layer thickness. In fact, large magnetic Reynolds number generates a resisting force on the fluid flow due to occurrence of the back reaction from the curvedness of magnetic field lines as they swept along the fluid. Effects of curvature parameter and magnetic Prandtl number on magnetic field are presented in Figure 4. Figure 4(a) shows the behaviour of $\mathit{k}$ against magnetic field.

Figure 4. Effects of $\mathit{k}$ and $\mathit{pm}$ on magnetic field.

That is increasing $\mathit{k}$ enhances the fluid velocity and expands the magnetic boundary layer region of the flow. This result is given for the first time in literature. This means that the curvature is important in enlarging magnetic field. Figure 4(b) explains the influence of magnetic Prandtl number on magnetic field. Increase in $\mathit{pm}$ diminishes the magnetic profile due to lowering magnetic diffusivity of the fluid and consequently lessen the strength of magnetic field. Figure 5 is plotted to explain the effects of Stuart number (magnetic interaction parameter), $\mathit{N}$ together with the usual parameters $\mathit{Br}$, $\mathit{Rd}$, $\mathit{Bt}$, $\mathit{k}$, and $\mathit{Pr}$ on temperature profile, $\mathit{\theta }(\mathit{\xi })$ at constant $\mathit{Pe}=0.2$, $\mathit{Sc}=0.2$, $\mathit{w}=0.5$, ${\beta }_0=0.4$ and $\mathit{R}{e}_m=0.4$. Figure 5(a) depicts the behaviour of Stuart number against the temperature profile. It can be seen that there is an increase in the temperature region for large $\mathit{N}$. The increase occurs at earlier stage of the thermal distribution and reduces along the flow. In other words, greater electromagnetic force creates large Lorentz force which add more heat from the surface to the fluid. This means that the thermal region can be regulated by means of controlling the size of Stuart number and therefore serves as important parameter in thermal engineering. Figures 5(b-e) agree with the existing interpretations of the concerned parameters.

Figure 5. Effects of $\mathit{N}$, $\mathit{Br}$, $\mathit{Rd}$, $\mathit{Bt}$, $\mathit{k}$, and $\mathit{Pr}$ on temperature field.

That is, Figure 5(b) illustrates an increasing thermal profile for large Brinkman number, $\mathit{Br}$. This observation occurs due to significant viscous heating effect which produces more heat from the fluid to the surface. For radiation parameter, Figure 5(c) illuminates the usual impact of $\mathit{Rd}$ against thermal field. A substantial increase in temperature region is established by enhancing the parameter. Habitually, radiation is used as a supplementary heat generation. Influence of thermal Biot number, $\mathit{Bt}$, against thermal distribution, $\mathit{\theta }(\mathit{\xi })$ is portrayed in Figure 5(d). Increasing $\mathit{Bt}$ enhances the heat flow from the surface to the fluid. It plays an additional means of expanding temperature profile and quantifies the reduction in thermal conductivity. In practical sense, this observation can be substantially useful in thermal industry for manufacturing stretchable materials. Figure 5(f) maintains an existing result of the curvature against temperature field. It is well established that increasing curvature, $\mathit{k}$, reduces the thermal boundary layer and consequently less the thermal distribution. Further, while heat flow can be regulated with aid of curvature parameter, more heat flows rapidly over curved surface. Figure 5(e) reveals the consequence of increasing $\mathit{Pr}$ resulting in lessen temperature distribution as well as thermal boundary layer.

It is an established fact that confirms the decrease in fluid thermal conductivity. Figure 6 is plotted to quantify the behaviours of $\mathit{w}$, $\mathit{Pe}$, $\mathit{Sc}$, $\mathit{k}$, and $\mathit{Bc}$ parameters against concentration profile. We noticed that increasing rate of reaction $\mathit{w}$, Peclet number $\mathit{Pe}$, Schmidt number $\mathit{Sc}$ and curvature parameter $\mathit{k}$, reduced the concentration of the fluid as shown in Figure 6(a-d). To preserve repetition, more explanation can be found in refs [29,32,40], whereas we oblige to say that decreasing concentration profile as a consequence of large $\mathit{Sc}$ is due to mass diffusion of the fluid, see Figure 6(c). Figure 6(e) implies that the concentration profile can be boosted by increasing concentration Biot number, $\mathit{Bc}$. This further accentuate the observation made in Figure 6(c). From application point of view, values of the physical quantities like the skin friction coefficient ${g^{\prime }}^{\prime }\left(0\right)$, magnetic flux, $\mathit{h}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(0)$ and rate of heat transfer, $-\mathit{\theta }{ }^{\mathrm{\prime }}(0)$ are presented in Figure 7. Figure 7(a,b) presents the skin friction coefficient for the magnetic Reynolds number, $R_{\mathit{em}}$ and Stuart number $\mathit{N}$ (magnetic interaction parameter).

Figure 6. Effects of $\mathit{w}$, $\mathit{Pe}$, $\mathit{Sc}$, and $\mathit{k}$ on concentration field.

Figure 7. Effects of $R_{\mathit{em}}$, $\mathit{N}$, $\mathit{pm}$, $R_{\mathit{em}}$, $R_{\mathit{em}}$, and $\mathit{N}$ on temperature field.

Figures 7(c,d) illuminate the variation of magnetic flux for $\mathit{pm}$ and $\mathit{R}{e}_m$, respectively. Figure 7(e,f) explains the variation of the Nusselt number for $\mathit{R}{e}_m$ and $\mathit{N}$. Table 1 depicts an excellent agreement between the existing papers and present results. Finally, the rate of heat transfer, $-\mathit{\theta }(0)$ and mass transfer, $-\mathit{\psi }{ }^{\mathrm{\prime }}(0)$, is presented in Table 2.uu

Table 1. Comparison of skin friction coefficient, $-\mathit{g}{ }^{\mathrm{\prime }}{ }^{\mathrm{\prime }}(0)$ as $\mathit{k}\to \mathrm{\infty }$ withliteraturee.

k	Sanni et al. [36]	Rosca & Pop [37]	Abass et al. [41]	Present
5	1.15076	1.1576	1.1576	0.900601
10	1.07172	1.0735	1.0734	0.947206
20	1.03501	1.0356	1.0355	0.977767
50	1.01380	1.0141	1.0140	0.998138
100	1.00687	1.0070	1.0070	1.005227

Table 2. Present heat$$& mass transfer rate, Nusselt and Sherwood numbers, respectively, for various K, SC, W, pe and $\mathit{Bc}$.

$\mathit{k}$	$\mathit{Sc}$	$\mathit{w}$	$\mathit{Pe}$	$\mathit{Bc}$	$\mathit{pm}$	$R_{\mathit{em}}$	$\mathit{N}$	$-\mathit{\theta }(0)$	$-\mathit{\psi }(0)$
0.4	0.2	0.5	0.2	0.4	0.2	0.4	1.0	2.478675	0.299793
-	0.3	-	-	-	-	-	-	-	0.300905
-	0.4	-	-	-	-	-	-	-	0.301995
0.6	0.2	-	-	-	0.4	-	-	1.624554	0.283780
-	-	0.7	-	-	0.6	-	-	1.583242	0.288351
-	-	0.9	-	-	0.8	-	-	1.541263	0.292206
0.8	-	0.5	-	-	0.2	-	-	1.242138	0.273295
-	-	-	0.4	-	-	0.6	-	1.245144	0.284592
-	-	-	0.6	-	-	0.8	-	1.246669	0.292493
1.0	-	-	0.2	-	-	0.4	-	0.985323	0.265890
-	-	-	-	0.6	-	-	2	0.972904	0.341574
-	-	-	-	0.8	-	-	3	0.961316	0.403220

We observe that rate of heat and mass transfer is enhanced for different values of characterizing parameters at constant $\mathit{Bt}=0.4$, $\mathit{Bc}=0.1$, $\mathit{Rd}=0.4$, ${\beta }_0=4$ and $\mathit{Pr}=2$.

8. Concluding remark

This study analysed $2\mathit{D}$ heat and mass transports of MHD advection-diffusion flow of viscous fluid under desirable multi-physical flow conditions. Solutions of the developed nonlinear boundary value problem are presented using Keller-Box shooting method with Jacobi iterative technique. Consequences of viscous dissipation, radiation, Joule heating and first-order chemical reaction in the energy and concentration equation are provided, respectively. Results of the velocity field, $\mathit{g}(\mathit{\xi })$, magnetic field, $\mathit{h}(\mathit{\xi })$, temperature field, $\mathit{\theta }(\mathit{\xi })$, and concentration field, $\mathit{\psi }(\mathit{\xi })$ are summarized as follows:

1. the velocity field is optimized either by increasing curvature, $\mathit{k}$, or magnetic interaction number, $\mathit{N}$ and synchronously enhanced the flow trajectories.

2. opposite effect is recorded by slight increase in magnetic Reynolds number. This means that the flow speed can be controlled with the help $\mathit{R}{e}_m$.

3. increasing curvature, $\mathit{k}$, enhances the magnetic field and expands its magnetic boundary layer region, whereas magnetic field is decreased for large $\mathit{pm}$ and consequently shrinks its boundary layer.

4. the temperature profile, $(\mathit{\xi })$, is increased for large magnetic interaction number, $\mathit{N}$. However, existing results for increasing $\mathit{Br}$, $\mathit{Rd}$, $\mathit{k}$ and $\mathit{Pr}$ against temperature region, $\mathit{\theta }(\mathit{\xi })$ are established without.

5. increasing thermal Biot number, $\mathit{Bt}$, and concentration Biot number, $\mathit{Bc}$, enlarge the temperature and concentration field, respectively. Likewise, standing reports of increasing $\mathit{w}$, $\mathit{Pe}$, $\mathit{Sc}$ and $\mathit{k}$ against $\mathit{\psi }(\mathit{\xi })$ are equally obtained.

6. The results of the drag force, magnetic flux, rate of heat and mass transfer are calculated and plotted graphically.

Nomenclature

$\mathit{\rho }$	=	Density of the fluid
$\mathrm{\nabla }$	=	Curvilinear operator
$\mathbf{B}$	=	Magnetic flux
$B_x,B_r$	=	Magnetic flux in $\mathit{r}$- and $\mathit{x}$- direction
$\mathit{u},\mathit{v}$	=	velocity field in $\mathit{r}$- and $\mathit{x}$- direction
$\mathit{p}$	=	Pressure
$\mathit{\mu }$	=	Dynamic viscosity
$\mathit{E}$	=	Electric Current Density
$\mathit{\sigma }$	=	Permeability of the free space
$\mathit{\lambda }$	=	Magnetic diffusivity
$\mathit{\delta }$	=	Boundary layer thickness
$\mathrm{\Gamma }$	=	Magnetic constant
$\mathit{\beta }$	=	Magnetic diffusivity
$K_{\ast }$	=	Thermal conductivity
$q_{\ast }$	=	Heat flux
${\sigma }^{\ast }$	=	Stefan-Boltzmann constant
$k^{\ast }$	=	Spectral absorption coefficient
$\mathit{p}$	=	Pressure
$\mathit{k}$	=	Dimensionless Curvature
$\mathit{R}$	=	Radius of Curvature
$\mathit{\nu }$	=	Kinematic viscosity
$U_{inf}$	=	Free stream velocity
$\mathit{l}$	=	Length of the surface
${\mathbf{B}}_o$	=	Dimensionless magnetic flux
$\mathit{Re}$	=	Reynolds number
$T_o$	=	Surface temperature
$k_o$	=	First order reaction rate
$C_p$	=	Fluid specific capacity
$h_o$	=	Convective heat transport
$D_o$	=	Convective mass transport
$\mathit{\theta }$	=	Dimensionless temperature field
$\mathit{\psi }$	=	Dimensionless concentration field
$\mathit{h}$	=	Dimensionless magnetic field
$\mathit{g}$	=	Dimensionless velocity field
$\mathit{A}$	=	Analogous flux density
$\mathit{R}{e}_m$	=	Magnetic Reynolds number
$\mathit{pm}$	=	Magnetic Prandtl number

Disclosure statement

No potential conflict of interest was reported by the author(s).

Appendix

(1) $A_1=(g_i^n+g_{i-1}^n)((h_2{)}_i^n+(h_2{)}_{i-1}^n)-(h_i^n+h_{i-1}^n)((g_2{)}_i^n+(g_2{)}_{i-1}^n)\text{break}-((h_1{)}_i^n+(h_1{)}_{i-1}^n)((g_1{)}_i^n+(g_1{)}_{i-1}^n)+\frac{(h_i^n+h_{i-1}^n)((g_1{)}_i^n+(g_1{)}_{i-1}^n)}{\mathit{k}+\mathit{\xi }},$(1)

(2) $A_2={\left(((g_2{)}_i^n+(g_2{)}_{i-1}^n)-\frac{((g_1{)}_i^n+(g_1{)}_{i-1}^n)}{\mathit{k}+\mathit{\xi }}\right)}^2,$(2)

(3) $A_3=2((g_3{)}_i^n+(g_3{)}_{i-1}^n)+\mathit{k}(g_i^n+g_{i-1}^n)((g_3{)}_i^n+(g_3{)}_{i-1}^n)-((g_1{)}_i^n+(g_1{)}_{i-1}^n)((g_2{)}_i^n+(g_2{)}_{i-1}^n),$(3)

(4) $A_4=((g_2{)}_i^n+(g_2{)}_{i-1}^n)-\mathit{k}(g_i^n+g_{i-1}^n)((g_2{)}_i^n+(g_2{)}_{i-1}^n)\text{break}-\mathit{k}((g_1{)}_i^n+(g_1{)}_{i-1}^n{)}^2+2\mathit{k}((g_1{)}_i^n+(g_1{)}_{i-1}^n)+2\mathit{k\beta }((h_1{)}_i^n+(h_1{)}_{i-1}^n{)}^2,$(4)