Analysis of Thermal Boundary Layer Flow over a Vertical Plate with Electrical Conductivity and Convective Surface Boundary Conditions

This paper analyses the thermal boundary layer flow over a vertical plate with electrical conductivity and convective surface boundary conditions. Transforming the governing nonlinear partial differential equations into a set of coupled non-linear ordinary differential equations by using the usual similarity transformation, the resulting coupled nonlinear ordinary differential equations are solved numerically by RungeKutta fourth order method with shooting technique. The behaviour and properties of thermo physical parameters in the fluid flow on the structure of the velocity and temperature fields are presented graphically and discussed.


INTRODUCTION
The study of heat transfer is an integral part of natural convection flow and a class of boundary layer theory. The quantity of heat transferred is highly dependent on the fluid motion within the boundary layer.
Convective heat transfer studies are very important in processes involving high temperature such as gas turbines, nuclear plants, thermal energy storage, etc. The solution for the laminar boundary layer problem on a horizontal flat plate was obtained by Heinrich Blasius [1] and since then it has been a subject of current research. Aziz [2] investigated a similarity solution for laminar thermal boundary layer over a flat-plate with a convective surface boundary condition. Bataller [3] presented a numerical solution for the combined effects of thermal radiation and convective surface heat transfer on the laminar boundary layer about a flat-plate in a uniform stream of fluid (Blasius flow) and about a moving plate in a quiescent ambient fluid. Cortell [4] in his work presented a numerical solution of the Classical Blasius Flat-Plate Problem using a Runge-Kutta algorithm for higher order initial value problem. Cortell [5] investigated a similarity solution for flow and heat transfer of a quiescent fluid over a nonlinearly stretching surface. Gabriella [6] used an iterative transformation method for the solution of boundary layer problem of a non-Newtonian power law fluid flow along a moving plate surface. There was a drag coefficient dependent on the velocity ratio and on the power law exponent.
He [7] worked on a simple perturbation approach to blasius equation. In his paper, he coupled the iteration method with the perturbation method to solve the well-known Blasius equation. Makinde and Olanrewaju [8] conducted a study on the effects of buoyancy force on thermal boundary layer over a vertical plate with convective surface boundary conditions. In Olanrewaju and etal [9], it was assumed that the lower surface of the plate is in contact with the hot fluid while a stream of cold fluid flows steadily over the upper surface with a heat source that decayed exponentially. Makinde [10] studied analysis of non-newtonian reactive flow in a cylindrical pipe. Makinde and Sibanda [11] conducted a study on magneto hydrodynamic mixed convective flow and heat and mass transfer past a vertical plate in a porous medium with constant wall suction. Shrama and Gurminder [12] looked at the effect of temperature dependent electrical conductivity on steady natural convection flow of a viscous incompressible low Prandtl (Pr<<1) electrically conducting fluid along an isothermal vertical non-conducting plate in the presence of transverse magnetic field and exponentially decaying heat generation. The study of an incompressible viscous and electrically conducting fluid in the presence of a uniform transverse magnetic field was investigated by Watunade and pop [13].
This paper is an extension of [8] with buoyancy force, convective surface boundary condition and electrical conductivity parameters. The numerical solutions of the resulting momentum and the thermal similarity equations are reported for representative values of thermo physical parameters characterizing the fluid convective process.

MATERIALS AND METHODS
Consider a two-dimensional steady incompressible fluid flow coupled with heat transfer by convection over a vertical plate. A stream of cold fluid at temperature ∞ moving over the right surface of the plate with a uniform velocity ∞ while the left surface of the plate is heated by convection from a hot fluid at temperature , which provides a heat transfer coefficient ℎ (see Fig. 1). The x-axis is taken along the plate and y-axis is normal to the plate. Magnetic field of intensity is applied in the ydirection. It is assumed that the external field is zero. Incorporating the Boussinesq's approximation within the boundary layer, the governing equations of continuity, momentum and energy equations according to [8] are respectively given as: where u and v are the x(along the plate) and the y(normal to the plate) components of the velocity respectively; g is the acceleration due to gravity; x, y are the Cartesian coordinates, is the Magnetic field intensity, is the coefficient of thermal expansion, is the density of the fluid, is the Kinematic viscosity, is the coefficient of thermal conductivity, T is the temperature of the fluid, * is the electrical conductivity and it is variable with temperature as given below * = (4) is the electrical conductivity parameter. All prime symbols denotes differentiation with respect to  The velocity boundary conditions can be expressed as: The boundary conditions at the plate surface and far into the cold fluid may be written as:

Fig. 1. Flow Configuration and coordinate system
Introducing the stream function ψ(x,y) such that Thus, the continuity equation (1) is satisfied with u and v of equations (11). Using (11), equations (2) and (3) are transformed into a set of coupled non-linear ordinary differential equation as The boundary conditions (5), (6), (7) and (8) reduced to It is assumed that equations (12) and (13) have a similarity solution where the parameters and are defined as constants.
Solving the governing boundary layer equations (12) and (13) with the boundary conditions (14) and (15) numerically using Runge-Kutta fourth order method along with shooting technique and implemented on maple 17. The step size of 0.001 is used to obtain the numerical solution correct to four decimal places as the criterion of the convergence.

RESULTS AND DISCUSSION
Numerical calculations have been carried out for different values of the thermo physical parameters controlling the fluid dynamics in the flow region.  Table 2 and Figs. 6, 8 and 10, it is observed that the skin-friction and the rate of heat transfer at the plate surface increases with an increase in local Grashof number Gr, electrical conductivity parameter and convective surface heat transfer parameter Bi . It is also observed that for values of Gr > 0 as in Fig. 7 there is decrease in the temperature profile which corresponds to the cooling problem. The cooling problem is often encountered in engineering applications; for example, in the cooling of electronic components and nuclear reactors.
However, in Fig. 2 and Fig. 4, an increase in the Prandtl number Pr and magnetic field parameter M decreases the skin-friction but increases the rate of heat transfer at the plate surface. This is attributed to the fact that as the prandtl number decrease, the thermal boundary layer thickness increases, causing reduction in the temperature gradient . ꞌ(0) at the surface of the plate.
In Fig. 3 the temperature gradient reduces at the surface because low prandtl number has high thermal conductivity, causing the fluid to attain higher temperature thereby reducing the heat flux at the surface. Moreover, for such low prandtl number, the velocity boundary layer is inside the thermal boundary layer and its thickness reduces as Prandtl number decreases and so the fluid motion is confined in more and more thinner layer near the surface and thereby experiencing drag increase(skin-friction) by the fluid. In other words there is more straining motion inside velocity boundary layer resulting in the increase of skinfriction coefficient. It is also observed from Table  2 that increase in magnetic field intensity, the skin-friction coefficient decreases the rate of heat transfer near the surface; hence the surface experiences reduction in drag.

CONCLUSION
From the numerical solutions and graphical representations, increasing the Prandtl number and the Grashof number tend to reduce the thermal boundary layer thickness. Fluid temperature increases with increase in magnetic field intensity and decreases with increase in electrical conductivity parameter. Fluid velocity increases with increase in electrical conductivity parameter while it decreases with increase in magnetic field intensity.
Thermal boundary layer thickness increases with an increase in biot numbers i B and decreases with an increase in Grashof Gr and Prandtl Pr numbers. Thus, convective surface heat transfer enhances thermal diffusion while an increase in the prandtl number slows down the rate of thermal diffusion within the boundary layer. Fluid temperature increases due to increase in magnetic field intensity while it decreases due increase or decrease in electrical conductivity parameter.

ACKNOWLEDGEMENT
We appreciate the comments of the reviewers in improving the quality of the paper.