# Numerical analysis of unsteady MHD flow near a stagnation point of a two-dimensional porous body with heat and mass transfer, thermal radiation, and chemical reaction

- Stanford Shateyi
^{1}Email author and - Gerald Tendayi Marewo
^{2}

**2014**:218

https://doi.org/10.1186/s13661-014-0218-z

© Shateyi and Marewo; licensee Springer 2014

**Received: **18 August 2014

**Accepted: **12 September 2014

**Published: **2 October 2014

## Abstract

The problem of unsteady MHD flow near a stagnation point of a two-dimensional porous body with heat and mass transfer in the presence of thermal radiation and chemical reaction has been numerically investigated. Using a similarity transformation, the governing time-dependent boundary layer equations for the momentum, heat and mass transfer were reduced to a set of ordinary differential equations. This set of ordinary equations were then solved using the spectral local linearization method together with the successive relaxation method. The study made among others the observation that the local Sherwood number increases with increasing values of the unsteadiness parameter and the Schmidt number. The fluid temperature was found to be significantly reduced by increasing values of the Prandtl number and the thermal radiation parameter. The velocity profiles were found to be reduced by increasing values of the chemical reaction and the Schmidt number as well as by the magnetic parameter.

## Keywords

## 1 Introduction

Uniform fluid flow over bodies of various geometries has been considered by many researchers over the years due to their numerous applications in industry and engineering. Due to complexity and non-linearity of the modeling governing equations exact solutions are difficulty to obtain. To that end, many researchers have employed different analytical and numerical methods. In recent years, the study of stagnation flow has gained tremendous research interest. Stagnation flow is the fluid motion near the stagnation point. The fluid pressure, and the rates of heat and mass transfer are highest in the stagnation area. A flow can be stagnated by a solid wall or a free stagnation point or a line can exist in the interior of the fluid domain. The study of stagnation point flow was pioneered by Hiemenz in 1911 [1]. Wang [2] investigated the stagnation flow toward a shrinking sheet and found that the convective heat transfer decreases with the shrinking rate due to an increase in the boundary layer thickness. Motsa *et al.*[3] studied the Maxwell fluid for two-dimensional stagnation flow toward a shrinking sheet. Shateyi and Makinde [4] investigated the steady stagnation point flow and heat transfer of an electrically conducting incompressible viscous fluid with convective boundary conditions.

The study of heat generation or absorption in moving fluids is of great importance in problems dealing with chemical reactions and those concerned with dissociating fluids. Heat generation effects may alter the temperature distribution and consequently, the particle deposition rate in nuclear reactors, electronic chips. Chamkha and Ahmed [5] studied the problem of MHD heat and mass transfer by mixed convection in the forward stagnation region of rotating sphere in the presence of heat generation and chemical reaction effects. Bararnia *et al.*[6] investigated analytically the problem of MHD natural convectional flow of a heat generation fluid in a porous medium.

Fluid flows with chemical reaction have key importance in many processes such as drying evaporation at the surface of a water body, energy transfer in a wet cooling electric power, food processing, groves of fruit trees, *etc*. The molecular diffusion of species in the presence of a chemical reaction within or at the boundary layer always exists during several practical diffusive operations. Several researchers have studied flows with chemical reaction reactions. Pal and Talukdar [7] presented the combined effects of Joule heating and a chemical reaction on unsteady MHD mixed convection with viscous dissipation over a vertical plate in the presence of a porous medium and thermal radiation. Hayat *et al.*[8] examined the mass transfer effect on unsteady three-dimensional flow of a coupled stress fluid over a stretching surface with chemical reaction. Najib *et al.*[9] investigated the stagnation point flow and mass transfer with a chemical reaction past a stretching shrinking cylinder.

Unsteady mixed convection flows do not necessarily possess similarity solutions in many practical applications. The unsteadiness and non-similarity in such flows may be due to the free stream velocity or due to the curvature of the body or due to the surface mass transfer or even possibly due to all these effects. Many investigators have confined their studies to either steady non-similar flows or to unsteady semi-similar flows because of the mathematical difficulties involved in obtaining non-similar solutions for such problems. Patil *et al.*[10] numerically studied the combined effects of thermal radiation and Newtonian heating on unsteady mixed convection flow along a semi-infinite vertical plate. Admon *et al.*[11] studied the behavior of unsteady free convection of a viscous and incompressible fluid in the stagnation point region of a heated three-dimensional body considering the temperature-dependent internal heat generation. Ahmad and Nazar [12] investigated the unsteady MHD mixed convection boundary layer flow of a viscoelastic fluid near the stagnation point for a vertical surface. Chamkha and Ahmed [13] investigated the effects of heat generation/absorption and chemical reaction on unsteady MHD flow heat and mass transfer near a stagnation point of three-dimensional porous body in the presence of a uniform magnetic field.

The effect of radiation is important in many non-isothermal situations. If the entire system involving the polymer extrusion process is placed in a thermally controlled environment, then radiation could become very important. Understanding radiation heat transfer in the system can lead to a desired product with a sought characteristic. Mahmoud [14] considered the effects of variable thermal radiation on the flow and heat transfer of an electrically conducting micropolar fluid over a continuously stretching surface with varying temperature in the presence of a magnetic field. Recently, Rashidi *et al.*[15] found the analytic solutions using the homotopy analysis method for velocity, temperature, and concentration distributions to study the steady magneto hydrodynamic fluid flow over a stretching sheet in the presence of buoyancy forces and thermal radiation. Hassan and Rashidi [16] presented an analytical solution for three-dimensional steady flow of a condensation film on an inclined rotating disk by the optical homotopy analysis method. Basiri Parsa *et al.*[17] applied the semi-numerical techniques known as the optimal analysis method (HAM) and the Differential Transform Method (DTM) to study the magneto-hemodynamic laminar viscous flow of a conducting physiological fluid in a semi-porous channel under a transverse magnetic field. Recently Khan *et al.*[18] studied the numerical simulation for unsteady MHD flow and heat transfer of a couple stress fluid over a rotating disk.

The present study aims to investigate the combined effects of thermal radiation, heat generation, viscous dissipation, and chemical reaction on an unsteady mixed convection flow near a stagnation point of two-dimensional porous body. The unsteadiness is induced due to the time-dependent moving plate velocity as well as by the free stream velocity. The paper seeks to compare the performance of two recently developed methods, namely the spectral local linearization method (SLLM) and the spectral relaxation method (SRM). The results generated from these two methods are also validated against the Matlab $bvp4c$ routine technique.

## 2 Mathematical formulation

## 3 Similarity analysis

We assume that the velocity varies inversely as a linear function of time making it possible to transform equations (2)-(5) into a set of self-similar equations.

where $M=\sigma {B}_{0}^{2}/\rho a$ is the magnetic field parameter, *λ* is the unsteadiness parameter, ${\lambda}_{1}=Gr/R{e}_{l}^{2}$, ${\lambda}_{2}=G{r}_{c}/R{e}_{l}^{2}$ are the buoyancy parameters, $Pr=\mu {c}_{p}/k$ is the Prandtl number, $Gr=g\beta {l}^{3}({T}_{w}-{T}_{\mathrm{\infty}})/{\nu}^{2}$ is the Grashof number, $G{r}_{c}=g\beta {l}^{3}({C}_{w}-{C}_{\mathrm{\infty}})/{\nu}^{2}$ is the modified Grashof number, $R{e}_{l}=a{l}^{2}/\nu $ is the Reynolds number, $\delta ={Q}_{0}/\rho {c}_{p}a$ is the heat generation/absorption parameter, ${E}_{c}=\nu {a}^{2}{x}^{2}/{c}_{p}(1-\lambda \tau )({T}_{w}-{T}_{\mathrm{\infty}})$, $Sc=\nu /D$ is the Schmidt number, $\gamma =\frac{{k}_{r}}{a}(1-\lambda \tau )$.

where ${f}_{w}=-{V}_{w}{((1-\lambda \tau )/a\nu )}^{\frac{1}{2}}$ is the suction/injection parameter.

## 4 Methods of solution

### 4.1 Spectral local linearization method (SLLM)

#### 4.1.1 Basic idea

*L*and

*N*are the linear and non-linear components, respectively, and

*H*is a given function of

*η*. Let

**w**be the

*n*-tuple consisting of independent variable

*z*and its derivatives. If we assume that

*N*is a function of

**w**only, then equation (13) may be replaced with the linearized form

which shall be solved using the Chebyshev Spectral Collocation Method [20]. We chose this method due to its ease of implementation and relatively high rate of convergence.

#### 4.1.2 Application

but equation (10) remains unchanged.

- 1.
The infinite interval $[0,\mathrm{\infty})$ on the

*η*axis is replaced by the finite interval $[0,{L}_{1}]$, where ${L}_{1}$ is sufficiently large. - 2.
The truncated problem domain $[0,{L}_{1}]$ on the

*η*axis is mapped onto the computational domain $[-1,1]$ on the*ξ*axis. - 3.
The computational domain is partitioned using the Chebyshev collocation points ${\xi}_{0},{\xi}_{1},\dots ,{\xi}_{N}$, where $-1={\xi}_{N}<{\xi}_{N-1}<\cdots <{\xi}_{0}=1$.

For a more detailed explanation of these steps, see for example [21] and [22].

*e.g.*[22], transforms equations (20)-(22) and (23)-(25), by the transformation ${f}^{\prime}=p$ with boundary condition $f(0)={f}_{w}$, to the discrete system

which are chosen so that they satisfy boundary conditions (11) and (12). Successive application of the SLLM generates approximations ${f}_{r+1}$, ${p}_{r+1}$, ${\theta}_{r+1}$, ${\varphi}_{r+1}$ for each $r=0,1,2,\dots $ .

### 4.2 Successive relaxation method (SRM)

Just like the SLLM, the SRM also makes use of the transformation $p={f}^{\prime}$ on the governing equations (8)-(10). Hence, we begin with the transformed equations (15) and (16), and equation (10), which is invariant under this transformation.

and boundary conditions (23)-(25), respectively.

Just like with the SLLM, the following steps are done in a similar manner for the SRM:

For each linear system in equations (40)-(43), include the corresponding boundary conditions.

Choose suitable initial approximations ${f}_{0}$, ${p}_{0}$, ${\theta}_{0}$, ${\varphi}_{0}$ required by the SRM to generate subsequent approximations ${f}_{r+1}$, ${p}_{r+1}$, ${\theta}_{r+1}$, ${\varphi}_{r+1}$ for each $r=0,1,2,\dots $ .

## 5 Results and discussion

In this section we present a comprehensive numerical parametric study is conducted and the results are reported graphically and in tabular form. Numerical simulations were carried out to obtain approximate numerical values of the quantities of engineering interest. The quantities are the surface shear stress ${f}^{\u2033}(0)$, surface heat transfer ${\theta}^{\prime}(0)$, and surface mass transfer ${\varphi}^{\prime}(0)$. In both the SLLM and the SRM numerical simulations, a finite computational value of ${\eta}_{\mathrm{\infty}}=30$ was chosen in the *η* direction. This was reached through numerical experimentations. This value was found to give accurate results for all the governing physical parameters and beyond the value, the results did not change within prescribed significant accuracy. The number of collocation points used in both SLLM and SRM was $Nx=50$ in all the cases considered in this investigation. We set our tolerance level to be $\epsilon ={10}^{-8}$ which we regard to be good enough for any engineering numerical approximation.

**Comparison of the SLLM results of**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with those obtained by SRM as well as by**
$\mathit{b}\mathit{v}\mathit{p}\mathbf{4}\mathit{c}$
**for different values of the magnetic parameter**
M
**, with**
$\mathit{\lambda}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
${\mathit{\lambda}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{E}\mathit{c}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\delta}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{f}\mathit{w}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

SLLM | SRM | bvp4c | ||||||
---|---|---|---|---|---|---|---|---|

M | it | time (sec) | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | it | time (sec) | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | time (sec) | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ |

1 | 9 | 1.37 | 2.48466051 | 34 | 3.74 | 2.48466051 | 15.27 | 2.48466051 |

3 | 8 | 0.74 | 2.88909949 | 21 | 3.32 | 2.88909949 | 17.86 | 2.88909949 |

5 | 8 | 0.96 | 3.24098431 | 17 | 2.81 | 3.24098431 | 18.53 | 3.24098431 |

10 | 7 | 0.82 | 3.98172655 | 13 | 1.76 | 3.98172655 | 18.37 | 3.98172655 |

*fw*on the skin friction, the Nusselt number and the Sherwood number. Blowing fluid, with $fw<0$, into the system reduces these three physical quantities whereas sucking fluid, $fw>0$, out of the system increases the three physical quantities under consideration.

**Effect of the magnetic parameter on**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{\lambda}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
${\mathit{\lambda}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{E}\mathit{c}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\delta}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{f}\mathit{w}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

M | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

0 | 2.25937547 | 0.37888463 | 0.49382391 |

2 | 2.69541174 | 0.37785044 | 0.49091212 |

5 | 3.24098431 | 0.37663629 | 0.48843976 |

**Effect of the transpiration parameter**
${\mathit{f}}_{\mathit{w}}$
**on**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{\lambda}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
${\mathit{\lambda}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{E}\mathit{c}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\delta}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{M}\mathbf{=}\mathbf{2}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

${\mathit{f}}_{\mathit{w}}$ | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

−0.5 | 2.12463544 | 0.18753160 | 0.36279720 |

0.0 | 2.39659776 | 0.27651084 | 0.42802020 |

0.5 | 2.69541174 | 0.37785044 | 0.49091212 |

1.0 | 3.01855731 | 0.48914126 | 0.56352188 |

**The influence of the heat generation/absorption parameter**
δ
**on**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{\lambda}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
${\mathit{\lambda}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{E}\mathit{c}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{M}\mathbf{=}\mathbf{2}$
**;**
$\mathit{f}\mathit{w}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

δ | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

−0.5 | 2.67966177 | 0.56304464 | 0.48999569 |

0.0 | 2.71117974 | 0.41294180 | 0.49072523 |

0.5 | 2.74660998 | 0.13156107 | 0.49424792 |

*λ*on the shear surface stress, heat transfer on the surface and mass transfer. We consider the accelerating cases only, $\lambda >0$. Increasing the unsteadiness parameter greatly increases the skin friction but reduces both the heat and the mass transfer on the surface.

**The influence of the unsteadiness parameter**
λ
**on**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{\delta}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
${\mathit{\lambda}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{E}\mathit{c}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{M}\mathbf{=}\mathbf{2}$
**;**
$\mathit{f}\mathit{w}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

λ | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

0 | 2.61770412 | 0.44070738 | 0.52781664 |

0.5 | 2.69541174 | 0.37785044 | 0.49091212 |

1.0 | 2.77234572 | 0.30219385 | 0.45252502 |

1.5 | 2.84976599 | 0.20022768 | 0.41157441 |

**The effect of the temperature buoyancy parameter**
${\mathit{\lambda}}_{\mathbf{1}}$
**on**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{\delta}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
$\mathit{\lambda}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{E}\mathit{c}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{M}\mathbf{=}\mathbf{2}$
**;**
$\mathit{f}\mathit{w}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

${\mathit{\lambda}}_{\mathbf{1}}$ | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

−0.5 | 2.23773235 | 0.36058187 | 0.52781664 |

0.0 | 2.46917284 | 0.36975021 | 0.49091212 |

0.5 | 2.69541174 | 0.37785044 | 0.45252502 |

**The influence of the Eckert number**
Ec
**on**
${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**,**
$\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$
**with**
$\mathit{\delta}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{R}\mathbf{=}\mathbf{1}$
**;**
$\mathit{P}\mathit{r}\mathbf{=}\mathbf{0.71}$
**;**
$\mathit{\lambda}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
**;**
${\mathit{\lambda}}_{\mathbf{2}}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{\gamma}\mathbf{=}\mathbf{0.1}$
**;**
$\mathit{M}\mathbf{=}\mathbf{2}$
**;**
$\mathit{f}\mathit{w}\mathbf{=}\mathbf{0.5}$
**;**
$\mathit{S}\mathit{c}\mathbf{=}\mathbf{0.22}$

Ec | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ | $\mathbf{-}{\mathit{\varphi}}^{\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ |
---|---|---|---|

0 | 2.69467396 | 0.40027022 | 0.49088572 |

0.1 | 2.70208324 | 0.17491583 | 0.49115055 |

0.5 | 2.70956330 | 0.05303825 | 0.49173820 |

*i.e.*, the buoyancy effect) increases, the convection cooling effect increases and hence the fluid flow accelerates. Therefore both the temperature and the concentration reduce.

*R*means a decrease in the Rosseland radiation absorptivity

*k*1. Thus the divergence of the radiative heat flux decreases as ${k}_{1}$ increases the rate of radiative heat transferred from the fluid and consequently the fluid temperature decreases.

*λ*1 in the concentration profiles. Increasing the values of this parameter enhances the solutal boundary layer, thereby increasing the concentration distribution within the fluid flow.

## 6 Conclusion

- 1.
Increasing the values of the magnetic field parameter resulted in increases in the skin-friction coefficients, whereas the Nusselt number and the Sherwood number and the velocity profiles decrease with increasing values of the magnetic parameter.

- 2.
The skin-friction coefficient, the Nusselt number, and the Sherwood number increase with fluid injection.

- 3.
The skin-friction coefficients also increase with increasing values of the heat source, the unsteadiness parameter, and the buoyancy parameters as well as the Eckert number.

- 4.
The presence of a heat source has significant effects on the Nusselt number as well as on the temperature distribution in the fluid flow.

- 5.
The thermal and solutal boundary layer thicknesses increase with increasing values of the unsteadiness parameter and Eckert number but decrease with increasing values of the buoyancy parameters.

## Greek letters

*β* thermal expansion coefficient

${\beta}_{c}$ compositional expansion coefficient

*η* similarity variable

*σ* electrical conductivity

*λ* unsteadiness parameter

${\lambda}_{1}$, ${\lambda}_{2}$ buoyancy parameters

*μ* coefficient of viscosity

*ν* kinematic viscosity

*θ* dimensionless temperature

*ϕ* dimensionless concentration

## Nomenclature

*a*, *b* velocity gradient parameters at the boundary layer edge in the *x*- and *y*-directions

${B}_{O}$ magnetic induction

${C}_{f}$ skin-friction coefficient

${c}_{p}$ heat capacity at constant pressure

*Ec* Eckert number

*D* mass diffusion

*g* gravitational acceleration

*f* dimensionless stream function

${k}_{c}$ chemical reaction parameter

*k* thermal conductivity coefficient

*M* Hartmann number

*Pr* Prandtl number

*R* thermal radiation parameter

${Q}_{0}$ heat generation/absorption coefficient

$R{e}_{l}$ Reynolds number

*Sc* Schmidt number

*T* temperature

*t* dimensional time

${u}_{e}$ dimensional free stream velocity component in *x*-direction

$(u,v)$ velocity components

$(x,y)$ transverse and normal directions

## Declarations

### Acknowledgements

The authors wish to acknowledge financial support from the University of Venda and NRF. The authors are very grateful to the reviewers for their constructive suggestions.

## Authors’ Affiliations

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