LAPLACE TRANSFORM SOLUTION OF UNSTEADY MHD JEFFRY FLUID FLOW PAST VERTICALLY INCLINED PORUS PLATE

The behavior of unsteady MHD flow of Jeffrey fluid over an inclined porous plate was analyzed in the present article. The governing partial differential equations of the flow phenomena were solved by using powerful mathematical tool Laplace transforms. The variations of velocity, temperature of the flow with respect to dissimilar physical parameters are analyzed through graphs. The parameters of engineering interest are skin friction and Nusselt number. For better understanding of the problem, variations of skin friction and Nusselt number with respect to critical parameters are tabulated.


INTRODUCTION
Non -Newtonian fluids got good attention by several researchers because of their significant industrial and technical applications. A good number of people have done work on MHD flows. Vijaya N et al.(2018) thermo physical properties of a Casson fluid through an oscillating vertical wall embedded through porous medium. Vedavathi N et al.(2017) investigated the radiation effects, of MHD free convective chemically reacting nano fluid over a semi infinite vertical plate. An analytical investigation has been done by Dharmaiah G et al.(2018), for an unsteady, two-dimensional, laminar, boundary layer flow of a viscous incompressible electrically conducting fluid along a semi-infinite vertical permeable moving plate in the presence of Diffusion-thermo and radiation absorption effects. Heat transfer analysis on a stagnation point boundary layer flow of a Jeffrey fluid over a stretching surface was done by Ehtsham A et al.(2019). Dhanalakshmi M et al.(2019)

Fig. 1 Geometry of the problem
The plate is taken along x -direction and y is taken normal to the plate. The plate is considered of infinite length along x direction and so all the physical parameters will not depend on x . The velocity components along x and y direction are taken as u and v  

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org 2 respectively. T w  and T  are the temperatures at wall and free stream temperatures. A uniform magnetic field of strength B 0 is applied normal to the plate. The transverse magnetic field and magnetic Reynolds number are assumed to be very small so that the produced magnetic field is negligible. An unsteady MHD free convective, incompressible, electrically conducting fluid flow over an inclined permeable plate was considered. The equations narrating the flow are given as follows. v 0 v is cons y ( t a n t ) The associated boundary conditions of the flow are The non dimensional variables and parameters are The transformed boundary conditions are

Laplace Transform Solution
Laplace transform technique was employed to get exact solution of the problem. Applying Laplace transform for the equations (7) and (8) and the boundary conditions (9) and (10) becomes u y s 0 y s 0 as y ( , ) , ( , )      (14) Solving the equations (11) and (12) along with the boundary conditions we obtain the expressions for the solutions of (11) and (12) Applying inverse Laplace transform for equations (15) and (16)

SKIN FRICTION AND RATE OF HEAT TRANSFER
The physical parameters of engineering interest are skin friction and Nusselt number. These parameters are useful to narrate the fluid flow and exchange of temperature near to the plate. The non dimensional skin friction and Nusselt number are u sf Nu y y 0

RESULTS AND DISCUSSIONS
The equations narrating the flow are resolved by employing Laplace transform method, to get exact analytical solutions for velocity and temperature. The influence of various physical parameters on fluid flow were analyzed graphically. The variations in skin friction, Nusselt number with respect to different physical parameters were illustrated through tables. The range of the variable y was taken in between 0 and 6. It is observed that the profiles of velocity and temperature fields reflecting the boundary conditions of the problem. Figure -2 displays the velocity profiles for different values of inclined angle. It reflects the rise in angle of inclination slow down the velocity of the fluid, because of buoyancy force due to gravity. Figure -3 shows that increase of Jeffry fluid parameter increases fluid velocity. Figure -4 depicts increase of thermal grashof number enhances fluid velocity. This is due to the fact increment of grashof number increases the buoyancy forces that lead to rise in liquid velocity. Figure -5 shows the impact of permeability parameter on fluid flow. This is because porous medium opposes the fluid flow. Lorentz forces resist the fluid flow which was accompanied with magnetic parameter. Therefore increase of magnetic parameter declines the fluid velocity. This was depicted in figure -6. Figure -7 shows the impact of Prandtl number on velocity profiles. Increase of Prandtl number results in decrease in thermal conductivity of the fluid, and hence temperature of the fluid decreases.

Conclusions
Analysis of unsteady MHD Jeffry fluid flow was done in the present article. Laplace transform technique was employed for the governing equations of the flow. The governing equations are first transformed into a set of normalized equations and then solved analytically applying Laplace transform technique to obtain the general solution. The expressions for velocity and temperature were obtained exponential and error functions. For better understanding of the problem the outcome of the phenomenon was illustrated graphically to explain the behavior of the fluid flow under the influence of different physical parameters. The velocity profiles are starting at origin and converging to zero far away from the plate. Whereas temperature profiles starting at 1 at origin and far away from the plate coinciding with the axis. This shows that the obtained solution is satisfying the boundary conditions of the present problem. Change in skin friction and Nusselt numbers with respect to different critical parameters was shown with tables. During the analysis the following conclusions were drawn. The present work can be extended by incorporating Newtonian heating for the temperature equation.