A Note on the Unsteady Incompressible MHD Fluid Flow with Slip Conditions and Porous Walls

This work is concerned with the influence of uniform suction or injection on flow and heat transfer analysis of unsteady incompressible magnetohydrodynamic (MHD) fluid with slip conditions. The resulting unsteady problem for velocity and heat transfer is solved by means of Laplace transform.The characteristics of the transient velocity, overall transient velocity, steady state velocity and heat transfer at the walls are analyzed and discussed. Graphical results reveal that themagnetic field, slip parameter, and suction (injection) have significant influences on the velocity, and temperature distributions, which also changes the heat transfer behaviors at the two plates. The results of Fang (2004) are also recovered by keeping magnetic field and slip parameter absent.


Introduction
Navier-Stokes equations are the basic equations of fluid mechanics.Exact solutions of Navier-Stokes equations are rare due to their inherent nonlinearity.Exact solutions are important because they serve as accuracy checks for numerical solutions.Complete integration of these equations is done by computer techniques, but the accuracy of the results can be established only by comparison with exact solutions.In the literature, there are a large number of Newtonian fluid flows for which exact solutions are possible [1][2][3][4][5][6].The effects of transverse magnetic field on the flow of an electrically conducting viscous fluid have been studied extensively in view of numerous applications to astrophysical, geophysical, and engineering problems [7][8][9][10][11][12][13][14][15].If the working fluid contains concentrated suspensions, then the wall slip can occur [16].Khaled and Vafai [3] studied the effect of the slip on Stokes and Couette flows due to an oscillating wall.However, the literature lacks studies that take into account the possibility of fluid slippage at the walls.Applications of these problems appear in microchannels or nanochannels and in applications where a thin film of light oil is attached to the moving plates or when the surface is coated with special coating such as a thick monolayer of hydrophobic octadecyltrichlorosilane [17].Yu and Ameel [18] imposed nonlinear slip boundary conditions on flow in rectangular microchannels.Erdogan [6] studied deeply the solution to the Stokes problem under nonslip conditions at the wall.Ayub and Zaman [19] studied the effects of suction and blowing for orthogonal flow impinging on a wall.Khan et al. [20] discussed the flow of Sisko fluid through a porous medium.Ariel et al. [21] considered the flow of elasticoviscous fluid with partial slip.Raptis et al. [22][23][24][25] studied steady and unsteady free convection and mass transfer flow through a porous medium.Penton [26] presented the transient solution for the flow due to the oscillating plate.
In this note, the flow of an incompressible, unsteady, viscous, MHD fluid with slip conditions is considered.Unsteady means time dependent flow, and we are looking for the effects of different parameters on flow with the variation of time.Unsteady and steady velocity profiles with mass transfer will be presented and solved exactly.There is mass injection from one plate and the same amount of suction on the other plate.The steady state temperature is also solved and discussed.When the fluid motion is set up from rest, the velocity field contains transients, determined by the initial conditions which gradually disappear in time.The effect of magnetic field and time on the transient velocity and on overall transient velocity has been seen graphically for both injection and suction.The effect of slip parameter on steady state velocity for injection/suction is shown graphically.Steady state temperature profiles and heat transfer rate at the walls (Nusselt number at the walls) are also discussed for injection/suction for different Prandtl and Reynolds numbers.The results of Fang [1] are recovered by taking magnetic field parameter and slip parameter to be zero.

Theoretical Derivation
2.1.Transient Velocity Profiles.Consider an incompressible, viscous, unsteady flow problem, in which there is slip between the bottom wall and fluid and also between top wall and fluid.There are mass injection velocity V  at the bottom wall and mass suction velocity V  at the top wall; V  > 0 corresponds to injection and V  < 0 corresponds to suction.The governing equation for this problem can be obtained as [2]  (, ) where (, ) is the velocity of the fluid in the -direction which is along the wall direction,  is the distance alon axis g,  is the time, ] = /,  =  2 0 /,  is the dynamic viscosity, ] is the kinematic viscosity,  is the density of the fluid,  is the electric conductivity of the fluid,  0 is the applied magnetic field, and  is the MHD factor or parameter.For the boundary conditions we consider the existence of slip between the velocity of the fluid at the walls and speed of the walls: ( Initial condition is where  is the slip parameter ( = 0 gives the usual no slip condition at the wall) and  0 is the velocity at the upper wall.
Here,  is a parameter, real or complex,  is the operator that transforms (, ) into (, ), called Laplace transform operator, and  −1 is the inverse Laplace transform operator.The solution can be shown as where  =   /2 and  = √  2  /4 +  + .In inverse Laplace transform of the above equation we have simple pole at  = 0 and infinite number of poles (located on the negative real axis) at   = −( 2  +  2 + ), ( = 1, 2, 3, . ..),where   =  , (  = √  2  /4 +   + ) and are given by The transient part velocity will be where Re  stands for residue and   (, ) is given by We have where The residue at  = 0 gives steady velocity Therefore, By using (13), we get Re [  (, )  ] =0 =   .

Steady State Temperature. It is too difficult to exactly solve the transient energy equation with viscous dissipation
for this problem by using (, ).We solve the steady state energy equation, which in dimensionless form is given by [1] where   = ]/ is the Prandtl number,   =  2 0 /  ( 1 −  2 ) is the Eckert number,  1 is temperature of the bottom fixed plate,  2 is temperature of the top moving plate,  is dimensionless temperature ( = ( −  1 )/( 2 −  1 )), and   is the steady state velocity given by (13).The solution of (39) with BCs (35) for   ̸ = 1, is where When viscous dissipation term is negligible, then energy equation ( 39) is Its solution is The Nusselt number for the upper wall,  = 1, is +  2 ( −  1 )  2(− 1 ) −  3  2 }          . (45) When   = 1, then the energy equation is The solution of ( 46) is where The Nusselt number in this case is It is worth recalling in the vicinity of (40) that when  = 0,  = 0, then result number (18) of Fang [1] can be derived.

Graphs and Discussion
In this part we discuss the variation of the transient part velocity   , overall transient velocity , and steady state velocity   with distance from the wall  for different values of Reynolds number   , magnetic field parameter , slip parameter , and time .Figures 1 and 2 show the variation of transient part velocity   with distance from the wall  for several values of magnetic parameter , by keeping  and  fixed.Figure 1 shows that when there is mass suction   < 0 at the top wall, with increase in magnetic field, transient part velocity decreases in magnitude.Figure 2 shows that when there is mass injection   > 0 at the bottom wall, with increase in magnetic field, transient part velocity   decreases and will become weaker as compared to the case of suction.From Figure 3   with increase in magnetic parameter , overall transient velocity  decreases.Figure 6 shows that, for   = 5 > 0, with increase in magnetic parameter , overall transient velocity  decreases but weaker.The overall transient velocities for   = 5 at different times are depicted in Figure 7.
Figure 7 shows that for mass injection at the bottom wall, overall transient velocity  increases with time.Figures 8 and 9 illustrate the variation of steady state velocity   with  for several values of slip parameter  and for fixed value of . Figure 8 shows that, for suction at the top wall with increase in slip parameter , steady state velocity   increases.Figure 9 shows that, for injection at the bottom wall with increase in slip parameter , steady state velocity   decreases.Now we discuss the variation of the steady state temperature distribution  with distance from the surface  for   when there is mass suction at the bottom wall, heat transfer rate   1 increases with increase in   .Figure 13 shows that when there is mass injection at the bottom wall, heat transfer rate   1 decreases with increase in   .Figures 14 and 15 depict the variation of Nusselt numbers at the top wall with   , for fixed values of   , , and  and for several values of   .Figure 14 shows that when there is mass suction at the top wall, heat transfer rate   2 increases in magnitude with increase in   .Figure 15 shows that when there is mass injection at the top wall, heat transfer rate   2 increases with increase in   .

Final Remarks
In this study exact solutions for the velocity field and temperature field in the presence of magnetic field, porosity,
linearly dependent upon the Eckert number.The Nusselt number at the walls will be   = |/|  .The Nusselt number for the bottom wall,  = 0, is  (     − 1) [1 +      { 1 ( 2(+ 1 ) − 1) So the first term in (40) is the solution of energy equation when viscous dissipation term is negligible and the second term in (40) is the temperature profile from viscous dissipation and MHD.It is found from (40) that the temperature profile is it is observed that for suction at top wall and for fixed values of ,  transient part velocity   decreases in magnitude with increase in time.Figure4shows that for injection at bottom wall and for fixed values of ,  transient part velocity   decreases in magnitude with increase in time.From Figures3 and 4it is seen that the transient part velocity will decay with time, which is consistent with what we expected.From Figures3 and 4it is clear that, after a certain time, the transient part velocity will die away and velocity will become developed.Figures5 and 6indicate variation of overall transient velocity  with  for fixed values of  and .
different values of five dimensionless parameters: Reynolds number   , Prandtl number   , Eckert number   , slip parameter , and magnetic parameter .Variation of the Nusselt number at walls for different Prandtl numbers is also discussed.Figures 10 and 11 elucidate the variation of steady state temperature distribution  with  for several values of   and for fixed values of   , , and .