Open Access

Variable Viscosity on Magnetohydrodynamic Fluid Flow and Heat Transfer over an Unsteady Stretching Surface with Hall Effect

Boundary Value Problems20102010:257568

DOI: 10.1155/2010/257568

Received: 16 July 2010

Accepted: 16 August 2010

Published: 18 August 2010

Abstract

The problem of magnetohydrodynamic flow and heat transfer of a viscous, incompressible, and electrically conducting fluid past a semi-infinite unsteady stretching sheet is analyzed numerically. The problem was studied under the effects of Hall currents, variable viscosity, and variable thermal diffusivity. Using a similarity transformation, the governing fundamental equations are approximated by a system of nonlinear ordinary differential equations. The resultant system of ordinary differential equations is then solved numerically by the successive linearization method together with the Chebyshev pseudospectral method. Details of the velocity and temperature fields as well as the local skin friction and the local Nusselt number for various values of the parameters of the problem are presented. It is noted that the axial velocity decreases with increasing the values of the unsteadiness parameter, variable viscosity parameter, or the Hartmann number, while the transverse velocity increases as the Hartmann number increases. Due to increases in thermal diffusivity parameter, temperature is found to increase.

1. Introduction

Fluid and heat flow induced by continuous stretching heated surfaces is often encountered in many industrial disciplines. Applications include extrusion process, wire and fiber coating, polymer processing, foodstuff processing, design of various heat exchangers, and chemical processing equipment, among other applications. Stretching will bring in a unidirectional orientation to the extrudate, consequently the quality of the final product considerably depends on the flow and heat transfer mechanism. To that end, the analysis of momentum and thermal transports within the fluid on a continuously stretching surface is important for gaining some fundamental understanding of such processes. Since the pioneering study by Crane [1] who presented an exact analytical solution for the steady two-dimensional flow due to a stretching surface in a quiescent fluid, many studies on stretched surfaces have been done. Dutta et al. [2] and Grubka and Bobba [3] studied the temperature field in the flow over a stretching surface subject to a uniform heat flux.

Elbashbeshy [4] considered the case of a stretching surface with variable surface heat flux. Chen and Char [5] presented an exact solution of heat transfer for a stretching surface with variable heat flux. P. S. Gupta and A. S. Gupta [6] examined the heat and mass transfer for the boundary layer flow over a stretching sheet subject to suction and blowing. Elbashbeshy and Bazid [7] studied heat and mass transfer over an unsteady stretching surface with internal heat generation.

Abd El-Aziz [8] analyzed the effect of radiation on heat and fluid flow over an unsteady stretching surface. Mukhopadyay [9] performed an analysis to investigate the effects of thermal radiation on unsteady boundary layer mixed convection heat transfer problem from a vertical porous stretching surface embedded in porous medium. Recently, Shateyi and Motsa [10] numerically investigated unsteady heat, mass, and fluid transfer over a horizontal stretching sheet.

In all the above-mentioned studies, the viscosity of the fluid was assumed to be constant. However, it is known that the fluid physical properties may change significantly with temperature changes. To accurately predict the flow behaviour, it is necessary to take into account this variation of viscosity with temperature. Recently, many researchers investigated the effects of variable properties for fluid viscosity and thermal conductivity on flow and heat transfer over a continuously moving surface.

Seddeek [11] investigated the effect of variable viscosity on hydromagnetic flow past a continuously moving porous boundary. Seddeek [12] also studied the effect of radiation and variable viscosity on an MHD free convection flow past a semi-infinite flat plate within an aligned magnetic field in the case of unsteady flow. Dandapat et al. [13] analyzed the effects of variable viscosity, variable thermal conducting, and thermocapillarity on the flow and heat transfer in a laminar liquid film on a horizontal stretching sheet.

Mukhopadhyay [14] presented solutions for unsteady boundary layer flow and heat transfer over a stretching surface with variable fluid viscosity and thermal diffusivity in presence of wall suction. The study of magnetohydrodynamic flow of an electrically conducting fluid is of considerable interest in modern metallurgical and great interest in the study of magnetohydrodynamic flow and heat transfer in any medium due to the effect of magnetic field on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. Many industrial processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid. During this process, these strips are sometimes stretched. In these cases, the properties of the final product depend to a great extent on the rate of cooling. By drawing these strips in an electrically conducting fluid subjected to magnetic field, the rate of cooling can be controlled and the final product of required characteristics can be obtained. Another important application of hydromagnetics to metallurgy lies in the purification of molten metals from nonmetallic inclusion by the application of magnetic field.

When the conducting fluid is an ionized gas and the strength of the applied magnetic field is large, the normal conductivity of the magnetic field is reduced to the free spiraling of electrons and ions about the magnetic lines force before suffering collisions and a current is induced in a normal direction to both electric and magnetic field. This phenomenon is called Hall effect. When the medium is a rare field or if a strong magnetic field is present, the effect of Hall current cannot be neglected. The study of MHD viscous flows with Hall current has important applications in problems of Hall accelerators as well as flight magnetohydrodynamics.

Mahmoud [15] investigated the influence of radiation and temperature-dependent viscosity on the problem of unsteady MHD flow and heat transfer of an electrically conducting fluid past an infinite vertical porous plate taking into account the effect of viscous dissipation. Tsai et al. [16] examined the simultaneous effects of variable viscosity, variable thermal conductivity, and Ohmic heating on the fluid flow and heat transfer past a continuously moving porous surface under the presence of magnetic field. Abo-Eldahab and Abd El-Aziz [17] presented an analysis for the effects of viscous dissipation and Joule heating on the flow of an electrically conducting and viscous incompressible fluid past a semi-infinite plate in the presence of a strong transverse magnetic field and heat generation/absorption with Hall and ion-slip effects. Abo-Eldahab et al. [18] and Salem and Abd El-Aziz [19] dealt with the effect of Hall current on a steady laminar hydromagnetic boundary layer flow of an electrically conducting and heat generating/absorbing fluid along a stretching sheet.

Pal and Mondal [20] investigated the effect of temperature-dependent viscosity on nonDarcy MHD mixed convective heat transfer past a porous medium by taking into account Ohmic dissipation and nonuniform heat source/sink. Abd El-Aziz [21] 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 magnet.

The present paper deals with variable viscosity on magnetohydrodynamic fluid and heat transfer over an unsteady stretching surface with Hall effect. Fluid viscosity is assumed to vary as an exponential function of temperature while the fluid thermal diffusivity is assumed to vary as a linear function of temperature. Using appropriate similarity transformation, the unsteady Navier-Stokes equations along with the energy equation are reduced to a set of coupled ordinary differential equations. These equations are then numerically solved by successive linearization method. The effects of different parameters on velocity and temperature fields are investigated and analyzed with the help of their graphical representations along with the energy.

2. Mathematical Formulation

We consider the unsteady flow and heat transfer of a viscous, incompressible, and electrically conducting fluid past a semi-infinite stretching sheet coinciding with the plane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq1_HTML.gif then the fluid is occupied above the sheet https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq2_HTML.gif The positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq3_HTML.gif coordinate is measured along the stretching sheet in the direction of motion, and the positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq4_HTML.gif coordinate is measured normally to the sheet in the outward direction toward the fluid. The leading edge of the stretching sheet is taken as coincident with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq5_HTML.gif axis. The continuous sheet moves in its own plane with velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq6_HTML.gif , and the temperature https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq7_HTML.gif distribution varies both along the sheet and time. A strong uniform magnetic field is applied normally to the surface causing a resistive force in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq8_HTML.gif direction. The stretching surface is maintained at a constant temperature and with significant Hall currents. The magnetic Reynolds number is assumed to be small so that the induced magnetic field can be neglected. The effect of Hall current gives rise to a force in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq9_HTML.gif direction, which induces a cross flow in that direction, and hence the flow becomes three dimensional. To simplify the problem, we assume that there is no variation of flow quantities in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq10_HTML.gif direction. This assumption is considered to be valid if the surface is of infinite extent in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq11_HTML.gif direction. Further, it is assumed that the Joule heating and viscous dissipation are neglected in this study. Finally, we assume that the fluid viscosity is to vary with temperature while other fluid properties are assumed to be constant. Using boundary layer approximations, the governing equations for unsteady laminar boundary layer flows are written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ1_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ2_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ3_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ4_HTML.gif
(2.4)
subject to the following boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ5_HTML.gif
(2.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq13_HTML.gif are the velocity components along the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq14_HTML.gif - and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq15_HTML.gif -axis, respectively, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq16_HTML.gif is the velocity component in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq17_HTML.gif direction, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq18_HTML.gif is the fluid density, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq19_HTML.gif is the coefficient of thermal expansion, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq20_HTML.gif is the kinematic viscosity, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq21_HTML.gif is the acceleration due to gravity, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq22_HTML.gif is the specific heat at constant pressure, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq23_HTML.gif is the temperature-dependent thermal conductivity.

Following Elbashbeshy and Bazid [22], we assume that the stretching velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq24_HTML.gif is to be of the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ6_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq26_HTML.gif are positive constants with dimension reciprocal time. Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq27_HTML.gif is the initial stretching rate, whereas the effective stretching rate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq28_HTML.gif is increasing with time. In the context of polymer extrusion, the material properties and in particular the elasticity of the extruded sheet vary with time even though the sheet is being pulled by a constant force. With unsteady stretching, however, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq29_HTML.gif becomes the representative time scale of the resulting unsteady boundary layer problem.

The surface temperature https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq30_HTML.gif of the stretching sheet varies with the distance https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq31_HTML.gif along the sheet and time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq32_HTML.gif in the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ7_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq33_HTML.gif is a (positive or negative; heating or cooling) reference temperature.

The governing differential equations (2.1)–(2.4) together with the boundary conditions (2.5) are nondimensionalized and reduced to a system of ordinary differential equations using the following dimensionless variables:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ8_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq34_HTML.gif is the physical stream function which automatically assures mass conservation (2.1) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq35_HTML.gif is constant.

We assume the fluid viscosity to vary as an exponential function of temperature in the nondimensional form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq36_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq37_HTML.gif is the constant value of the coefficient of viscosity far away from the sheet, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq38_HTML.gif is the variable viscosity parameter. The variation of thermal diffusivity with the dimensionless temperature is written as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq39_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq40_HTML.gif is a parameter which depends on the nature of the fluid, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq41_HTML.gif is the value of thermal diffusivity at the temperature https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq42_HTML.gif

Upon substituting the above transformations into (2.1)–(2.4) we obtain the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ9_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ10_HTML.gif
(2.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ11_HTML.gif
(2.11)
where the primes denote differentiation with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq43_HTML.gif , and the boundary conditions are reduced to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ12_HTML.gif
(2.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ13_HTML.gif
(2.13)

The governing nondimensional equations (2.9)–(2.11) along with the boundary conditions (2.12)-(2.13) are solved using a numerical perturbation method referred to as the method of successive linearisation.

3. Successive Linearisation Method (SLM)

The SLM algorithm starts with the assumption that the independent variables https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq44_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq45_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq46_HTML.gif can be expressed as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ14_HTML.gif
(3.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq49_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq50_HTML.gif are unknown functions and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq52_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq53_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq54_HTML.gif ) are approximations which are obtained by recursively solving the linear part of the equation system that results from substituting (3.1) in the governing equations (2.9)–(2.13). The main assumption of the SLM is that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq56_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq57_HTML.gif become increasingly smaller when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq58_HTML.gif becomes large, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ15_HTML.gif
(3.2)
Thus, starting from the initial guesses https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq59_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq60_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq61_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ16_HTML.gif
(3.3)
which are chosen to satisfy the boundary conditions (2.12) and (2.13), the subsequent solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq62_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq63_HTML.gif are obtained by successively solving the linearised form of equations which are obtained by substituting (3.1) in the governing equations, considering only the linear terms. In view of the assumption (3.2), the exponential term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq64_HTML.gif can be approximated as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ17_HTML.gif
(3.4)
Thus, using (3.4), the linearised equations to be solved are given as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ18_HTML.gif
(3.5)
subject to the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ19_HTML.gif
(3.6)

where the coefficient parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq65_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq66_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq67_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq68_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq70_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq71_HTML.gif are defined in the appendix.

Once each solution for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq72_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq73_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq74_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq75_HTML.gif ) has been found from iteratively solving (3.5), the approximate solutions for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq77_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq78_HTML.gif are obtained as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ20_HTML.gif
(3.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq79_HTML.gif is the order of SLM approximation. Since the coefficient parameters and the right-hand side of (3.5), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq80_HTML.gif , are known (from previous iterations), the equation system (3.5) can easily be solved using any numerical methods such as finite differences, finite elements, Runge-Kutta-based shooting methods, or collocation methods. In this work, (3.5) are solved using the Chebyshev spectral collocation method. This method is based on approximating the unknown functions by the Chebyshev interpolating polynomials in such a way that they are collocated at the Gauss-Lobatto points defined as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ21_HTML.gif
(3.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq81_HTML.gif is the number of collocation points used (see e.g. [2325]). In order to implement the method, the physical region https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq82_HTML.gif is transformed into the region https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq83_HTML.gif using the domain truncation technique in which the problem is solved on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq84_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq85_HTML.gif . This leads to the following mapping:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ22_HTML.gif
(3.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq86_HTML.gif is the scaling parameter used to invoke the boundary condition at infinity. The unknown functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq88_HTML.gif are approximated at the collocation points by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ23_HTML.gif
(3.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq89_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq90_HTML.gif th Chebyshev polynomial defined as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ24_HTML.gif
(3.11)
The derivatives of the variables at the collocation points are represented as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ25_HTML.gif
(3.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq91_HTML.gif is the order of differentiation and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq92_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq93_HTML.gif being the Chebyshev spectral differentiation matrix (see e.g., [23, 25]). Substituting (3.9)–(3.12) in (3.5) leads to the matrix equation given as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ26_HTML.gif
(3.13)
in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq94_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq95_HTML.gif square matrix and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq97_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq98_HTML.gif column vectors defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ27_HTML.gif
(3.14)
in which
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ28_HTML.gif
(3.15)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq99_HTML.gif are defined in the appendix. After modifying the matrix system (3.13) to incorporate boundary conditions, the solution is obtained as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ29_HTML.gif
(3.16)

4. Results and Discussion

In this section, we give the SLM results for the six main parameters affecting the flow. We remark that all the SLM results presented in this paper were obtained using https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq100_HTML.gif collocation points. For validation, the SLM results were compared to those by Matlab routine https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq101_HTML.gif and excellent agreement between the results is obtained giving the much needed confidence in using the successive linearization method. Tables 13 give a comparison of the SLM results for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq103_HTML.gif at different orders of approximation against the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq104_HTML.gif In Table 1, we observe that full convergence of the SLM is achieved by as early as the third order, substantiating the claim that SLM is a very powerful technique. We observe in this table that the variable viscosity parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq105_HTML.gif significantly affects the skin friction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq106_HTML.gif The skin friction increases as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq107_HTML.gif increases. We observe also in this table that the local Nusselt number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq108_HTML.gif decreases as the fluid variable viscosity parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq109_HTML.gif increases. The lower part of Table 1 depicts the effects of variable diffusivity parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq110_HTML.gif on the local skin friction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq111_HTML.gif and the local Nusselt number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq112_HTML.gif It can be observed that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq113_HTML.gif does not have significant effect on the skin friction but very significant effects on the local Nusselt number. As https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq114_HTML.gif increases, the skin friction slightly decreases but the local Nusselt number is greatly reduced.
Table 1

Comparison between the present successive linearisation method (SLM) results and the bvp4c numerical results for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq116_HTML.gif for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq118_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq119_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq120_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq121_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq122_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq123_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq124_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq125_HTML.gif

0.1

1.554880

1.554902

1.554902

1.554902

1.270615

1.270618

1.270618

1.270618

0.2

1.654744

1.654780

1.654780

1.654780

1.262638

1.262637

1.262637

1.262637

0.3

1.759494

1.759550

1.759550

1.759550

1.254515

1.254506

1.254506

1.254506

0.4

1.869278

1.869358

1.869358

1.869358

1.246255

1.246233

1.246233

1.246233

0.5

1.984248

1.984356

1.984356

1.984356

1.237868

1.237829

1.237829

1.237829

0.6

2.104569

2.104702

2.104702

2.104702

1.229367

1.229305

1.229305

1.229305

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq126_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq127_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq128_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq129_HTML.gif

0.1

1.554880

1.554902

1.554902

1.554902

1.270615

1.270618

1.270618

1.270618

0.2

1.554140

1.554159

1.554159

1.554159

1.196543

1.196541

1.196541

1.196541

0.3

1.553464

1.553482

1.553482

1.553482

1.132811

1.132803

1.132803

1.132803

0.4

1.552845

1.552861

1.552861

1.552861

1.077289

1.077278

1.077278

1.077278

0.5

1.552274

1.552289

1.552289

1.552289

1.028406

1.028392

1.028392

1.028392

0.6

1.551747

1.551761

1.551761

1.551761

0.984976

0.984958

0.984958

0.984958

Table 2

Comparison between the present successive linearisation method (SLM) results and the bvp4c numerical results for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq131_HTML.gif for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq132_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq133_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq134_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq135_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq136_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq137_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq138_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq139_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq140_HTML.gif

0.1

1.346973

1.346977

1.346977

1.346977

1.298217

1.298219

1.298219

1.298219

1.0

1.554880

1.554902

1.554902

1.554902

1.270615

1.270618

1.270618

1.270618

2.0

2.094695

2.094728

2.094728

2.094728

1.205903

1.205873

1.205873

1.205873

3.0

2.780752

2.780758

2.780758

2.780758

1.142601

1.142533

1.142533

1.142533

4.0

3.524973

3.524963

3.524963

3.524963

1.092001

1.091925

1.091925

1.091925

5.0

4.296202

4.296187

4.296187

4.296187

1.052905

1.052838

1.052838

1.052838

6.0

5.081869

5.081855

5.081855

5.081855

1.022458

1.022404

1.022404

1.022404

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq141_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq142_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq143_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq144_HTML.gif

0.1

1.711146

1.711172

1.711172

1.711172

1.254049

1.254052

1.254052

1.254052

1.0

1.554880

1.554902

1.554902

1.554902

1.270615

1.270618

1.270618

1.270618

2.0

1.438664

1.438677

1.438677

1.438677

1.285089

1.285092

1.285092

1.285092

3.0

1.394031

1.394040

1.394040

1.394040

1.291251

1.291254

1.291254

1.291254

4.0

1.374422

1.374429

1.374429

1.374429

1.294079

1.294082

1.294082

1.294082

5.0

1.364411

1.364417

1.364417

1.364417

1.295553

1.295556

1.295556

1.295556

6.0

1.358689

1.358695

1.358695

1.358695

1.296405

1.296408

1.296408

1.296408

Table 3

Comparison between the present successive linearisation method (SLM) results and the bvp4c numerical results for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq146_HTML.gif for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq148_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq149_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq150_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq151_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq152_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq153_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq154_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq155_HTML.gif

0.1

1.356050

1.356062

1.356062

1.356062

0.982230

0.981936

0.981936

0.981936

0.5

1.471732

1.471752

1.471752

1.471752

1.161685

1.161666

1.161666

1.161666

1.0

1.608655

1.608677

1.608677

1.608677

1.336560

1.336569

1.336569

1.336569

1.5

1.737464

1.737489

1.737489

1.737489

1.485834

1.485849

1.485849

1.485849

2.5

1.973469

1.973500

1.973500

1.973500

1.741847

1.741866

1.741866

1.741866

3.0

2.082331

2.082365

2.082365

2.082365

1.855701

1.855719

1.855719

1.855719

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq156_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq157_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq158_HTML.gif

2nd ord.

3rd ord.

4th ord.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq159_HTML.gif

0.1

1.542490

1.542494

1.542494

1.542494

0.405270

0.405254

0.405254

0.405254

0.5

1.551892

1.551905

1.551905

1.551905

1.032273

1.032252

1.032252

1.032252

1.0

1.557830

1.557862

1.557862

1.557862

1.529016

1.529053

1.529053

1.529053

1.5

1.561763

1.561812

1.561812

1.561812

1.916129

1.916220

1.916220

1.916220

2.5

1.567062

1.567138

1.567138

1.567138

2.535077

2.535240

2.535240

2.535240

3.0

1.569013

1.569099

1.569099

1.569099

2.798249

2.798436

2.798436

2.798436

From Table 2 (upper part), it is observed that the Hartmann number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq160_HTML.gif tends to greatly increase the local skin friction at the unsteady stretching surface. This is because the increase in the magnetic strength leads to a thinner boundary layer, thereby causing an increase in the velocity gradient at the wall. We also observe that the local Nusselt number decreases as the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq161_HTML.gif increase. We observe in the lower part of Table 2 that the local skin friction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq162_HTML.gif is reduced as the Hall parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq163_HTML.gif increases, but the Nusselt number increases as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq164_HTML.gif increases.

Table 3 depicts the effects of the unsteadiness parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq165_HTML.gif , (upper part) the Prandtl number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq166_HTML.gif (lower part) on the local skin friction, and the local Nusselt number. We observe that both of these flow properties are greatly affected by the unsteadiness parameter. They both increase as the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq167_HTML.gif increase. We also observe in this table that the Prandtl number has little effects on the skin friction but significant effects on the local Nusselt number. The local skin friction slightly increases as the values of the Prandtl number increase, while the Nusselt number is greatly increased as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq168_HTML.gif increases.

Figures 112 have been plotted to clearly depict the influence of various physical parameters on the velocity and temperature distributions. In Figure 1, we have the effects of varying the variable viscosity parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq169_HTML.gif on the axial velocity. It is clearly seen that as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq170_HTML.gif increases the boundary layer thickness decreases and the velocity distributions become shallow. Physically, this is because a given larger fluid https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq171_HTML.gif implies higher temperature difference between the surface and the ambient fluid.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig1_HTML.jpg
Figure 1

The variation axial velocity distributions with increasing values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq172_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq176_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq177_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig2_HTML.jpg
Figure 2

The variation axial velocity distributions with increasing values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq178_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq182_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq183_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig3_HTML.jpg
Figure 3

The variation axial velocity distributions with increasing values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq184_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq186_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq187_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq188_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq189_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig4_HTML.jpg
Figure 4

The variation axial velocity distributions with increasing values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq190_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq191_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq192_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq194_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq195_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig5_HTML.jpg
Figure 5

Transverse velocity profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq196_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq197_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq198_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq199_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq200_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq201_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig6_HTML.jpg
Figure 6

Transverse velocity profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq202_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq203_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq204_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq205_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq206_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq207_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig7_HTML.jpg
Figure 7

Transverse velocity profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq208_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq211_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq212_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq213_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig8_HTML.jpg
Figure 8

Transverse velocity profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq214_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq215_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq216_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq217_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq218_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq219_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig9_HTML.jpg
Figure 9

Temperature profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq220_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq221_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq223_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq224_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq225_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig10_HTML.jpg
Figure 10

Temperature profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq226_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq228_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq229_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq230_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq231_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig11_HTML.jpg
Figure 11

Temperature profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq232_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq233_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq236_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq237_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Fig12_HTML.jpg
Figure 12

Temperature profiles for various values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq238_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq239_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq240_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq242_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq243_HTML.gif

The effects of the unsteadiness parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq244_HTML.gif on the axial velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq245_HTML.gif are presented in Figure 2. It can be seen in this figure that when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq246_HTML.gif values are increased, the boundary layer thickness is reduced and this inhibits the development of transition of laminar to turbulent flow. The effect of the magnetic strength parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq247_HTML.gif on the axial velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq248_HTML.gif is shown in Figure 3. It is noticed that an increase in the magnetic parameter leads to a decrease in the velocity. This is due to the fact that the application of the transverse magnetic field to an electrically conducting fluid gives rise to a resistive type of force known as the Lorentz force. This force has a tendency to slow the motion of the fluid in the axial direction.

Figure 4 shows typical profiles for the fluid velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq249_HTML.gif for different values of the Hall parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq250_HTML.gif We observe that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq251_HTML.gif increases with increasing values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq252_HTML.gif as the effective conducting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq253_HTML.gif decreases with increasing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq254_HTML.gif which reduces the magnetic damping force on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq255_HTML.gif , and the reduction in the magnetic damping force is coupled with the fact that magnetic field has a propelling effect on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq256_HTML.gif

Figure 5 shows the effect of the variable viscosity parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq257_HTML.gif on the transverse velocity distribution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq258_HTML.gif As shown, the velocity is decreasing with increasing the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq259_HTML.gif In addition, the curves show that for a particular value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq260_HTML.gif the transverse velocity increases rapidly to a peak value near the wall and then decays to the relevant free stream velocity (zero). The effect of the unsteadiness parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq261_HTML.gif on the transverse velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq262_HTML.gif is presented in Figure 6. From this figure, it is seen that the effect of increasing the unsteadiness parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq263_HTML.gif is to decrease the transverse velocity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq264_HTML.gif greatly near the plate.

Figure 7 depicts the effects of the magnetic strength https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq265_HTML.gif on the transverse velocity. We observe that close to the sheet surface an increase in the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq266_HTML.gif leads to an increase in the values of the transverse velocity with shifting the maximum values toward the plate while for most of the parts of the boundary layer at the fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq267_HTML.gif position, the transverse velocity decreases along with decreases in the boundary layer thickness as the magnetic field increases.

Figure 8 is obtained by fixing the values of all the parameters and by allowing the Hall parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq268_HTML.gif to vary. Increasing the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq269_HTML.gif from 0 to 1.5 causes the transverse flow in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq270_HTML.gif -direction to increase. However, for values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq271_HTML.gif greater than 1.5, the transverse flow decreases as these values increase as can be clearly seen on Figure 8. This is due to the fact that for larger values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq272_HTML.gif the term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq273_HTML.gif is very small, and hence the resistive effect of the magnetic field is diminished.

Figures 9 and 10 are aimed to shed light on the effects of variable viscosity and variable thermal diffusivity parameters https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq275_HTML.gif on the temperature. The distribution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq276_HTML.gif increases as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq277_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq278_HTML.gif increase as shown in Figure 9 and Figure 10, respectively. This is due to the thickening of the thermal boundary layer as a result of increasing thermal diffusivity. Figure 11 depicts the effect of the unsteadiness parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq279_HTML.gif on the temperature profiles. It can be observed that the temperature profiles decrease with the increase of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq280_HTML.gif In general, it is noted that the effect of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq281_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq282_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq283_HTML.gif is more notable than that on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq284_HTML.gif

Figure 12 presents typical profiles for the fluid temperature https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq285_HTML.gif for different values of Hartmann number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq286_HTML.gif Increases in the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq287_HTML.gif have a tendency to slow the motion of the fluid and make it warmer as it moves along the unsteady stretching sheet causing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq288_HTML.gif to increase as shown in this figure.

5. Conclusion

The problem of unsteady magnetohydrodynamic flow and heat transfer of a viscous, incompressible, and electrically conducting fluid past a semi-infinite stretching sheet was investigated. The governing continuum equations that comprised the balance laws of mass, linear momentum, and energy were modified to include the Hartmann and Hall effects of magnetohydrodynamics, and variable viscosity of the fluid was solved numerically using the successive linearization method together with the Chebyshev collocation method. Graphical results for the velocity and temperature were presented and discussed for various physical parametric values. The effects of the main physical parameters of the problem on the skin friction and the local Nusselt number were shown in Tabular form. It was found that the skin coefficient https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq289_HTML.gif is increased as the variable viscosity parameter, Hartmann number, unsteadiness parameter, or the Prandtl number is increased. It was found, however, to decrease as the thermal diffusivity parameter or the Hall parameter increases. The local Nusselt number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq290_HTML.gif was found to be decreasing as the values of the variable viscosity parameter, thermal diffusivity parameter, or Hartmann number increase and to be increasing with increasing the values of the Hall parameter, unsteadiness parameter, or the Prandtl number.

It is hoped that, with the help of our present model, the physics of flow over stretching sheet may be utilized as the basis of many scientific and engineering applications and experimental work.

Appendix

A. Definition of Coefficient Parameters

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ30_HTML.gif
(A.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ31_HTML.gif
(A.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ32_HTML.gif
(A.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_Equ33_HTML.gif
(A.4)

In the above definitions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq291_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq292_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq293_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq294_HTML.gif ) are diagonal matrices of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F257568/MediaObjects/13661_2010_Article_909_IEq295_HTML.gif .

Authors’ Affiliations

(1)
School of Mathematical and Natural Sciences, University of Venda
(2)
Department of Mathematics, University of Swaziland

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Copyright

© S. Shateyi and S. S. Motsa. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.