International Journal of Physical Sciences

An investigation is made for unsteady magnetohydrodynamic (MHD) thin film flow of a third grade fluid down an inclined plane. The non-linear partial differential equation governing the flow and heat transfer are reduced to a system of non-linear algebraic equations using implicit finite difference approximation to obtain velocity and temperature profile. The effect of various physical parameter on both velocity and temperature profile obtained are studied through several graphs. It is noticed that the velocity and temperature profile decreases due to increase in third grade parameter and magnetic parameter. 
 
   
 
 Key word: Non-Newtonian fluid, magnetohydrodynamic (MHD) flow, third grade fluid, implicit finite difference approximation, heat transfer.


INTRODUCTION
Recently, many authors have carried out investigation on the flows of electrical conducting non-Newtonian fluid down an inclined plane.These fluids also known as Rivilin-Ericksen third grade fluid are molten plastics, emulsion, pulps, ink-jet printing; micro fluidics, hemodynamic, the flow of synovial fluid in joints and large variety of industrial products that have viscoelastic behaviour in their motion.These viscoelastic fluids phenomenons are responsible for normal stress effects, contrary to what the Newtonian viscous effect does.The normal stress effects shows fluid elasticity, which is added to the viscous effects when the fluid is in motion.Since there is no single constitute equation that can be use to analyze all the non-Newtonian fluids, various linear and non linear equations have been proposed.A third grade fluid is a subclass of non-Newtonian fluid and its governing non-linear equation has successfully studied and treated in many literatures.Hayat et al. (2008) investigate the influence of a magnetic field on the unsteady flow of an incompressible third grade electrically conducting fluid bounded by a rigid plate.They obtain the solution of the equations of conservation of mass and momentum in balance analytically using Lie symmetry analysis.Chakraborty (2001), studied the laminar convention flow of an incompressible electrically conducting second order viscoelastic stratified fluid in porous medium down an inclined channel.He was able to obtain the expression for viscous drag, the rate of heat transfer at the plate and flow flux for fluid and particle.Mohyuddin (2007), investigate an exact solution of unsteady flow of a viscous fluid due to a sudden pull with constant velocities of non-coaxial rotations between two porous infinite disks.He uses two different methods to obtain the solution at large time and the solution at small times.Nayak et al. (2012) studied the unsteady convective flow of a third grade fluid past an infinite *Corresponding author.E-mail: waheedlawal207@yahoo.com.vertical porous plate with uniform suction applied at the plate using an implicit finite difference method.They were able to show the effect of some physical parameter on the velocity field through several graphs.Gamal (2011) examined the effects of magnetic field on thin film of unsteady micropolar fluid through a porous medium.He considered these thin films for three different geometries.Aiyesimi et al. (2012) investigate the steady magnetohydrodynamic (MHD) flow of a third grade fluid down an inclined plane with ohmic heating.They analysed the effect of magnetic parameter and Brinkman number on the velocity and temperature profile obtained.
In this work a detailed investigation of the unsteady MHD thin film flow of a third grade fluid down an inclined plane in the presence of uniform magnetic field applied externally transverse to the direction of flow is presented.The heat transfer analysis is also carried out.The effect of physical parameter such as magnetic parameter, third grade parameter, gravitational parameter, Prandtl number and Eckert number on both velocity and temperature distribution are discussed and shown graphically.

BASIC EQUATION AND FORMULATION OF THE PROBLEM
The basic equations governing the unsteady MHD flow of an incompressible electrically conducting fluid are the field equation.( ) is the current density, σ is the electrical conductivity, E is the electrical field which is not considered that is ( ), denote the material derivative, p is the pressure, f is the external body force and τ is the Cauchy stress tensor which for a third grade fluid satisfies the constitution equation.

( ) ( )
where pI is the isotropic stress due to constraint Aiyesimi et al. 947 incompressibility, µ is the dynamics viscosity, ( ) ( ) are the material constants, ⊥ indicates the matrix transpose, ( ) are the first three Rivlin-Ericksen tensors.In the absence of modified pressure gradient, Equations (1) to (3) along with Equations (4) and (5) yield   ( ) with relevant boundary and initial condition ( ) The energy equation for the thermodynamically compatible third grade fluid with viscous dissipation, work done due to deformation and Joule heating is given as: ( ) with boundary and initial condition ( ) and k are respectively, specific heat capacity and thermal conductivity of the fluid.T is the temperature and δ T is the temperature of the ambient fluid.Equations ( 7) and ( 11 with the boundary and initial condition ( ) For energy equation, we have With the boundary and initial condition ( )

NUMERICAL RESULTS AND DISCUSSION
The momentum Equation ( 15) and energy Equation ( 19) represent a couple system of non-linear partial differential equations, which are solved numerically under the boundary and initial conditions ( 16) to ( 18) and ( 20) to (22) using finite difference approximation.Figures 1 to 3 .It is found in the Figure 4 that increase in parameter Μ decreases u and its steady state time for all values of β .This is due to the magnetic damping force on u which is increases with increase in Μ .Also the rate of transport is considerably reduced with increase in magnetic field parameter.It shows that the transverse magnetic field opposes the transverse phenomena.
From this figure, we noticed that the parameter β has a small effect on the steady state time of u and this effect is clearly shown when the value of Μ become smaller and smaller.Figure 5 illustrate the effect of magnetic parameter M on T depends on time.Temperature profile T slightly decreases as magnetic parameter M increases largely at all times.And as the values of β increases the temperature profile approach to its steady state.The reason for this is that for small times, an increase in Μ increase the Joule dissipation which is proportional to Μ .For large time, increasing in Μ decreases u largely and in turn decrease the Joule and viscous dissipation. .Figure 6 shows that increasing in value of gravitational parameter K increases u for all values of β .This is because increase in value of K correspond to increasing the angle of inclination which shows that by increasing the angle of inclination of inclined plane, the velocity increases.Figure 7 shows increase in temperature profile T as a result of increase in the value of gravitational parameter K for all values of β .

Conclusion
The unsteady MHD thin film flow of a third grade fluid down an inclined plane with heat transfer is studied.The governing equations were developed and solved numerically using finite difference method.A systematic study on the effects of third grade parameter and other physical parameters controlling the flow and heat transfer characteristics is examined.The important findings of the graphical analysis of the results of the problem are as follows: (i) The effect of third grade parameter Μ and magnetic parameter is seen to decrease the velocity and temperature of the fluid at a small time but make them reach the steady state at large time; (ii) The effect of gravitational parameter is also seen to increase the velocity of fluid thereby increasing the Joule and viscous dissipation which in turn increases the temperature of the fluid; (iii) The Prandtl number and Eckert number decreases and increases respectively the temperature of the fluid thereby decrease its thickness.

B
ρ is the density of the fluid, v is the fluid velocity B is the magnetic induction.So that and b are applied and induced magnetic field respectively) and ) are simplified together with boundary and initial condition ( ) ( ) in the nondimensional form.We define the following non-dimensional quantities:

.
Figures 4 and 5 show the time evolution of velocity profile u and temperature profile T at

Figures
Figures 6 and 7 show the time evolution of u and T at 5 .0 = η

Figure
Figure 11 reveal that the increase of Prandtl number r P results in the decreasing of temperature profile at a certain time ( 5 .0 = t ) as the fluid is flowing down in an incline plane.The reason is that the thermal boundary layer thickness will decrease with increase in the values of Prandtl number r P which also results to slow rate of thermal diffusion.The effect of increasing the values of Eckert number c E is to increase the temperature profileT in an incline plane as shown in Figure12.This behaviour of temperature enhancement can be seen as heat energy is stored in the fluid due to frictional heating.

Figure 8 .Figure 9 .
Figure 8.Time development of Temperature profile T at

Figure 12 .
Figure 12.Effect of Eckert number ( c E ) on Temperature profile T at