# On a new numerical analysis of the Hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal radiation and heat source/sink

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

**2014**:170

https://doi.org/10.1186/s13661-014-0170-y

© Shateyi and Marewo; licensee Springer. 2014

**Received: **4 March 2014

**Accepted: **26 June 2014

**Published: **26 September 2014

## Abstract

This paper employs a computational iterative approach known as the spectral local linearization method (SLLM) to analyze the Hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal radiation and heat source/sink. To demonstrate the reliability of our proposed method, we made comparison with the Matlab $bvp4c$ routine technique, and an excellent agreement was observed. The governing partial differential equations are transformed into a system of ordinary differential equations by using suitable similarity transformations. The results are obtained for velocity, temperature, skin friction and the Nusselt number.

**MSC:** 65PXX, 76-XX.

## Keywords

## 1 Introduction

Theoretical studies of magnetohydrodynamic flow and heat transfer over stretching surfaces have received great attention by virtue of their numerous applications in the fields of metallurgy and chemical engineering. Such applications include geothermal reservoirs, wire and fiber coating, food stuff processing, reactor fluidization, enhanced oil recovery, packed bed catalytic reactors and cooling of nuclear reactors. The primary aim in extrusion is to maintain the quality of the surface of the extricate. When the magnetic strength is strong, Hall currents cannot be neglected. In an ionized gas where the density is low and/or the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before suffering collisions, and a current is induced in a direction normal to both electric and magnetic fields. Due to Hall currents, the electrical conductivity of the fluid becomes anisotropic and this causes the secondary flow. The Hall effect is important when the Hall parameter, which is the ratio between the electron-cyclotron frequency and the electron-atom-collision frequency, is high. Hall currents are of great importance in many astrophysical problems, Hall accelerators, Hall sensors and flows of plasma in MHD power generators.

Examples of such studies include Sakiadis [1], [2] who did pioneering work on boundary layer flow on a continuously moving surface. Shateyi and Motsa [3] carried out a numerical analysis of the problem of magnetohydrodynamic boundary layer flow, heat and mass transfer rates on steady two-dimensional flow of an electrically conducting fluid over a stretching sheet embedded in a non-Darcy porous medium in the presence of thermal radiation and viscous dissipation. Shateyi [4] investigated thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching surface with suction and blowing. Singh *et al.*[5] investigated two-dimensional unsteady flow of a viscous incompressible fluid about a stagnation point on a permeable stretching sheet. Shateyi and Motsa [6] numerically investigated the unsteady heat, mass and fluid transfer over a horizontal stretching sheet. More recently, Shateyi and Marewo [7] studied the magnetohydrodynamic boundary layer flow with heat and mass transfer of an UCM fluid over a stretching sheet in the presence of viscous dissipation and thermal radiation. Jhankal [8] considered the problem of unsteady boundary layer and heat transfer over a stretching surface in the presence of a transverse magnetic field.

Pal and Mondal [9] presented a model to study the effects of temperature-dependent viscosity and variable conductivity on mixed convective diffusion of species over a stretching sheet. When the conducting fluid is an ionized gas, and the strength of the applied magnetic field is large, the conductivity normal to the magnetic field is reduced due to the free spiraling of electrons and ions about the magnetic lines of force before collisions occur and a current is induced in a direction normal to both electric and magnetic fields. This phenomenon is known as the Hall effect or Hall current. The conductivity of the fluid is anisotropic and the effect of Hall current cannot be neglected when the medium is rarefied or if a strong magnetic field is present. The study of MHD viscous flows with Hall current has important applications in problems of Hall accelerators and sensors as well as flight magnetohydrodynamics. Pop and Watanabe [10] presented the problem of free convection flow of a conducting fluid which is permeated by a transverse magnetic field and the Hall effect is taken into account. Abd El-Aziz [11] investigated the effect of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in the presence of a strong magnetic field.

Shateyi and Motsa [12] numerically analyzed variable viscosity on magnetohydrodynamic fluid flow and heat transfer over an unsteady stretching surface with the Hall effect. Pal [13] studied the influence of Hall current and thermal radiation on flow and heat transfer characteristics in a viscous fluid over an unsteady stretching permeable surface. Zaman [14] considered the effects of Hall current on the flow of an incompressible, unsteady, viscous, MHD fluid with slip conditions. Lastly, Ali *et al.*[15] investigated heat and mass transfer of a steady flow of an incompressible electrically conducting fluid due to stretching plate under the influence of an applied uniform magnetic field and the effects of Hall current.

Governing equations modeling MHD flow and heat transfer over stretching surfaces are highly nonlinear, thereby exact solutions are impossible to obtain. Therefore, numerical solutions have always been developed and modified as a bid of getting more accurate and stable solutions. The current study seeks to investigate the Hall effect on MHD flow and heat transfer over an unsteady stretching permeable surface in the presence of thermal radiation and heat source/sink. We propose to numerically solve the present problem using a recently developed iterative method known as the spectral local linearization method (SLLM), Motsa [16]. The SLLM approach is based on transforming a nonlinear ordinary differential equation into an iterative scheme. The iterative scheme is then blended with the Chebyshev spectral method [17]. A similar approach to our current proposed method can be found in Motsa and Shateyi [18], [19] and Motsa *et al.*[20].

## 2 Mathematical formulation

where $b$ is a constant with $b\ge 0$.

We must remark that the particular forms of ${U}_{w}$ and ${T}_{w}(x,t)$ have been specifically chosen in order to come up with similarity transformations which then transform the governing partial differential equations into a set of nonlinear ordinary differential equations.

*et al.*[21], we introduce the following dimensionless functions $f$ and $\theta $, and the similarity variable $\eta $,

where ${\tau}_{w}=\mu {(\frac{\partial u}{\partial y})}_{y=0}$ is the wall shear stress, and ${q}_{w}=-\kappa {(\frac{\partial T}{\partial y})}_{y=0}$ is the surface heat flux, where $\mu $ and $\kappa $ are the dynamic viscosity and thermal conductivity, respectively.

## 3 Method of solution

### 3.1 Basic idea of the spectral local linearization method (SLLM)

of $m$ differential equations, where $i=1,2,\dots ,m$, and each ${H}_{i}$ is a function of $\eta \in [a,b]$. Also, ${L}_{i}$ and ${N}_{i}$ are linear and nonlinear components, respectively.

Basically, the SLLM is an iterative method for solving differential equations such as (16) which begins with an initial approximation ${\mathbf{Z}}_{0}$ of $\mathbf{Z}$. Successive application of the SLLM generates approximations ${\mathbf{Z}}_{1},{\mathbf{Z}}_{2},\dots $ , where ${\mathbf{Z}}_{r}=[{Z}_{1,r},\dots ,{Z}_{m,r}]$ for each $r=0,1,2,\dots $ . Upon linearizing nonlinear component ${N}_{i}$, differential equation (16) shall be solved numerically using the Chebyshev spectral collocation method [17].

Equation (19) shall be solved using the Chebyshev spectral collocation method. For the sake of brevity, we leave out the details of this method. We urge the interested reader to see, for example, [22].

### 3.2 Solving current problem using the SLLM

that satisfy boundary conditions (36)-(39) so that the SLLM generates subsequent approximations ${f}_{r}$, ${p}_{r}$, ${g}_{r}$, ${\theta}_{r}$ for each $r=1,2,\dots $ .

## 4 Results and discussion

The numerical results iteratively generated by the SLLM for the main parameters that have significant effects on the flow properties are presented in this section. All the SLLM results presented in this work were obtained using $N=50$ collocation points, and we are glad to highlight that convergence was achieved in as few as six iterations. We take the infinity value ${\eta}_{\mathrm{\infty}}$ to be 40. The magnetic field is taken quite strong by assigning large values of $M$ to ensure the generation of Hall currents. In order to validate our numerical method, it was compared to MATLAB routine $bvp4c$ which is an adaptive Lobatto quadrature iterative scheme.

Comparison of the SLLM results of $-{f}^{\u2033}(0)$, ${g}^{\prime}(0)$, $-{\theta}^{\prime}(0)$ with those obtained by $bvp4c$ for different values of the unsteadiness parameter

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

bvp4c | SLLM | bvp4c | SLLM | bvp4c | SLLM | |

1 | 2.06334096 | 2.06334096 | 0.17551632 | 0.17551632 | 0.95973533 | 0.95973533 |

2 | 2.27277685 | 2.27277685 | 0.15185421 | 0.15185421 | 1.30758931 | 1.30758931 |

3 | 2.46649701 | 2.46649701 | 0.13459762 | 0.13459762 | 1.54422525 | 1.54422525 |

Comparison of the SLLM results of $-{f}^{\u2033}(0)$, ${g}^{\prime}(0)$, $-{\theta}^{\prime}(0)$ with those obtained by $bvp4c$ for different values of the magnetic parameter

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

bvp4c | SLLM | bvp4c | SLLM | bvp4c | SLLM | |

1 | 2.06334096 | 2.06334096 | 0.17551632 | 0.17551632 | 0.51730365 | 0.51730365 |

3 | 2.40059801 | 2.40059801 | 0.41757839 | 0.41757839 | 0.46087659 | 0.46087659 |

5 | 2.69187810 | 2.69187810 | 0.59380491 | 0.59380491 | 0.43117279 | 0.43117279 |

Comparison of the SLLM results of $-{f}^{\u2033}(0)$, ${g}^{\prime}(0)$, $-{\theta}^{\prime}(0)$ with those obtained by $bvp4c$ for different values of the Hall parameter

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

bvp4c | SLLM | bvp4c | SLLM | bvp4c | SLLM | |

0.1 | 2.20591638 | 2.20591638 | 0.03127850 | 0.03127850 | 0.49780303 | 0.49780303 |

0.5 | 2.15366831 | 2.15366831 | 0.13112526 | 0.13112526 | 0.50406091 | 0.50406091 |

1.0 | 2.06334096 | 2.06334096 | 0.17551632 | 0.17551632 | 0.51730365 | 0.51730365 |

## 5 Conclusion

The present work analyzed MHD unsteady flow and heat transfer of an electrically conducting fluid over a stretching sheet in the presence of thermal radiation and the Hall effect. Much attention was given to trying to investigate how the velocity field, skin friction, temperature distribution and heat transfer are influenced by the parameters of importance in this study. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations by using suitable similarity variables. The resultant system of nonlinear ordinary differential equations is solved numerically by a recently developed technique known as the spectral local linearization method. The accuracy of the SLLM is validated against the MATLAB in-built $bvp4c$ routine for solving boundary value problems. The following conclusions were drawn in our investigation.

An excellent agreement was observed between our results and those obtained using the $bvp4c$ routine technique giving confidence to our present results.

The unsteadiness parameter $A$ has significant effects on the velocity components and temperature profiles. The maximum axial velocity, transverse velocity and temperature profiles are attained when the flow is steady ($A=0$).

Increasing the values of the magnetic field strength decreases the momentum boundary layer thickness while increasing the thermal boundary layer thickness.

The velocity components are enhanced as the Hall parameter increases.

The fluid temperature increases with increasing values of thermal radiation as well as a heat source.

The heat transfer rate and the skin friction coefficient in the $x$-direction are increased while the skin friction in the *z*-direction decreases as the unsteadiness parameter increases.

The skin friction coefficients are enhanced while the heat transfer rate is depressed by increasing the values of magnetic strengths.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd.**Open Access** This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.