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 the magnetic 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.
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 [1–6]. 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 [7–15]. If the working fluid contains concentrated suspensions, then the wall slip can occur . Khaled and Vafai  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 . Yu and Ameel  imposed nonlinear slip boundary conditions on flow in rectangular microchannels. Erdogan  studied deeply the solution to the Stokes problem under nonslip conditions at the wall. Ayub and Zaman  studied the effects of suction and blowing for orthogonal flow impinging on a wall. Khan et al.  discussed the flow of Sisko fluid through a porous medium. Ariel et al.  considered the flow of elasticoviscous fluid with partial slip. Raptis et al. [22–25] studied steady and unsteady free convection and mass transfer flow through a porous medium. Penton  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  are recovered by taking magnetic field parameter and slip parameter to be zero.
2. 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 velocityat the bottom wall and mass suction velocityat the top wall; corresponds to injection andcorresponds to suction. The governing equation for this problem can be obtained as  whereis the velocity of the fluid in the-direction which is along the wall direction,is the distance alon -axis g,is the time,,,is the dynamic viscosity,is the kinematic viscosity,is the density of the fluid,is the electric conductivity of the fluid,is the applied magnetic field, andis 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 whereis the slip parameter (gives the usual no slip condition at the wall) andis the velocity at the upper wall. The problem exactly reduces to the problem of Fang  if we take,Equations (1), (2), and (3) can be made dimensionless by defining Then (1), (2), and (3) become where,, and(Reynolds number). Decomposinginto two parts, say, transient part and a steady state part, Then we have two separate problems and the steady state part will be The solution of (10) with BCs (11) and (12) can be obtained as whereand. When (no slip) and (no magnetic field), then (13) becomes Equation (14) the result number (7) of Fang . If there is no mass transfer at the walls, then, soand, and (13) becomes If there is no mass transfer at the walls and magnetic field is absent, then,, and slip parameterthen (10), (11), and (12) will collapse into Its solution is, which is conventional Couette flow. When,, and , then (13) gives, except at the bottom wall (at the bottom wallandWhen,, and , then (13) gives, except at the upper wall (at the upper wall and. The transient part problem becomes The solution can be derived by using Laplace transformation techniques . The Laplace transform pair is defined by over that range of values offor which the integrals exist. Here,is a parameter, real or complex,is the operator that transformsinto, called Laplace transform operator, andis the inverse Laplace transform operator. The solution can be shown as whereand. In inverse Laplace transform of the above equation we have simple pole atand infinite number of poles (located on the negative real axis) at,, where=and are given by The transient part velocity will be wherestands for residue andis given by We have where The residue atgives steady velocity Therefore, By using (13), we get
The transient part velocity from (24) becomes Therefore, the overall transient solution from (9) becomes To recover Fang  result number (13) we substitute,,, andinto (32), and we get Since, so (23) gives Therefore; also we have, but, so Using,, andinto (33) we obtain Consider So (36) becomes Result number (13) of Fang  is exactly followed in (38).
2.2. 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  whereis the Prandtl number,is the Eckert number, is temperature of the bottom fixed plate, is temperature of the top moving plate,is dimensionless temperature, and is the steady state velocity given by (13). The solution of (39) with BCs (35) for, is where When viscous dissipation term is negligible, then energy equation (39) is Its solution is 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 linearly dependent upon the Eckert number. The Nusselt number at the walls will be. The Nusselt number for the bottom wall, , is The Nusselt number for the upper wall,, is When, 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,, then result number (18) of Fang  can be derived.
3. Graphs and Discussion
In this part we discuss the variation of the transient part velocity, overall transient velocity, and steady state velocitywith distance from the wallfor different values of Reynolds number, magnetic field parameter, slip parameter , and time.
Figures 1 and 2 show the variation of transient part velocitywith distance from the wallfor several values of magnetic parameter, by keepingandfixed. Figure 1 shows that when there is mass suctionat the top wall, with increase in magnetic field, transient part velocity decreases in magnitude. Figure 2 shows that when there is mass injectionat the bottom wall, with increase in magnetic field, transient part velocitydecreases and will become weaker as compared to the case of suction. From Figure 3 it is observed that for suction at top wall and for fixed values of,transient part velocitydecreases in magnitude with increase in time. Figure 4 shows that for injection at bottom wall and for fixed values of,transient part velocitydecreases in magnitude with increase in time. From Figures 3 and 4 it is seen that the transient part velocity will decay with time, which is consistent with what we expected. From Figures 3 and 4 it is clear that, after a certain time, the transient part velocity will die away and velocity will become developed. Figures 5 and 6 indicate variation of overall transient velocitywithfor fixed values ofand. Figure 5 shows that, for, with increase in magnetic parameter, overall transient velocitydecreases. Figure 6 shows that, for, with increase in magnetic parameter, overall transient velocitydecreases but weaker. The overall transient velocities forat different times are depicted in Figure 7. Figure 7 shows that for mass injection at the bottom wall, overall transient velocityincreases with time.
Figures 8 and 9 illustrate the variation of steady state velocitywithfor several values of slip parameterand for fixed value of. Figureshows that, for suction at the top wall with increase in slip parameter, steady state velocityincreases. Figure 9 shows that, for injection at the bottom wall with increase in slip parameter, steady state velocitydecreases.
Now we discuss the variation of the steady state temperature distributionwith distance from the surfacefor 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 distributionwithfor several values ofand for fixed values of,, and. Figure 10 shows that for, (injection) temperatureat a point decreases with increase in. Figure 11 shows that, for, (suction) temperature with increase in increases at a point. Comparison of Figures 10 and 11 shows that, for mass suction at the bottom wall, the maximum temperature will exist in the fluid because of viscous dissipation.
Figures 12 and 13 illustrate the variation of Nusselt numbers at the bottom wall with, for fixed values of,, andand for several values of. Figure 12 shows that when there is mass suction at the bottom wall, heat transfer rateincreases with increase in. Figure 13 shows that when there is mass injection at the bottom wall, heat transfer ratedecreases with increase in. Figures 14 and 15 depict the variation of Nusselt numbers at the top wall with, for fixed values of,, andand for several values of. Figure 14 shows that when there is mass suction at the top wall, heat transfer rateincreases in magnitude with increase in. Figure 15 shows that when there is mass injection at the top wall, heat transfer rateincreases with increase in.
4. Final Remarks
In this study exact solutions for the velocity field and temperature field in the presence of magnetic field, porosity, and slip parameter are constructed. A uniform magnetic field is applied transversely to the flow. The expressions of the velocity field and temperature field for flow subjected to the slip conditions between the two parallel plates and fluid are obtained by means of Laplace transform. The so-obtained solutions, depending on the initial and the boundary conditions, are presented as sum of the steady state transient solutions. The results of Fang  are also recovered by taking ,. Graphical results for mass transfer reveal that it has significant influence on the velocity distribution, temperature distribution, and heat transfer rate at the walls. The current analysis will be useful in dealing with real engineering problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper. The authors do not have a direct financial relation with the commercial identity mentioned in the paper that might lead to a conflict of interests for any of them.
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