ION SLIP EFFECT ON UNSTEADY HARTMANN FLOW WITH HEAT TRANSFER UNDER EXPONENTIAL DECAYING PRESSURE GRADIENT

The unsteady Hartmann flow of an electrically conducting, viscous, incompressible fluid bounded by two parallel nonconducting porous plates is studied with heat transfer taking the ion slip into consideration. An external uniform magnetic field and a uniform suction and injection are applied perpendicular to the plates, while the fluid motion is subjected to an exponential decaying pressure gradient. The two plates are kept at different but constant temperatures while the Joule and viscous dissipations are included in the energy equation. The effect of the ion slip and the uniform suction and injection on both the velocity and temperature distributions is examined.


Introduction
The magnetohydrodynamic flow between two parallel plates, known as Hartmann flow, is a classical problem that has many applications in magnetohydrodynamic (MHD) power generators, MHD pumps, accelerators, aerodynamic heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil, and fluid droplets and sprays.Hartmann and Lazarus [6,7] studied the influence of a transverse uniform magnetic field on the flow of a conducting fluid between two infinite parallel, stationary, and insulated plates.Then, a lot of research work concerning the Hartmann flow has been obtained under different physical effects [1,2,5,8,10,11,[13][14][15].In most cases, the Hall and ion slip terms were ignored in applying Ohm's law as they have no marked effect for small and moderate values of the magnetic field.However, the current trend for the application of magnetohydrodynamics is towards a strong magnetic field, so that the influence of electromagnetic force is noticeable [5].Under these conditions, the Hall current and ion slip are important and they have a marked effect on the magnitude and direction of the current density and consequently on the magnetic force term.Tani [14] studied the Hall effect on the steady motion of electrically conducting and viscous fluids in channels.Soudalgekar et al. [11] and Soundalgekar and Uplekar [10] studied the effect of the Hall currents on the steady MHD Couette flow with heat transfer.The temperatures of the two plates were assumed either to be constant [11] or to vary linearly along the plates in the direction of the flow [10].Abo-El-Dahab [1] studied the effect of Hall current on the steady Hartmann flow subjected to a uniform suction and injection at the bounding plates.Later, Attia [4] extended the problem to the unsteady state with heat transfer in the presence of a constant pressure gradient, taking the Hall effect into consideration while neglecting the ion slip.
In the present study, the unsteady flow and heat transfer of an incompressible, viscous, electrically conducting fluid between two infinite nonconducting horizontal porous plates are studied with the consideration of both the Hall current and the ion slip.The fluid is acted upon by an exponential decaying pressure gradient, a uniform suction and injection, and a uniform magnetic field perpendicular to the plates.This problem is chosen due to its occurrence in many industrial engineering applications [9] and because the governing equations can be solved in closed form.This is important because these closed-form solutions can serve as known solutions for parameter effects and the calibration of numerical solutions and can be used as tools for design and understanding of flow behavior in systems involving such situations.The induced magnetic field is neglected by assuming a very small magnetic Reynolds number [5,13].The two plates are maintained at two different but constant temperatures.This configuration is a good approximation of some practical situations such as heat exchangers, flow meters, and pipes that connect system components.The cooling of these devices can be achieved by utilizing a porous surface through which a coolant, either a liquid or gas, is forced.Therefore, the results obtained here are important for the design of the wall and the cooling arrangements of these devices.The equations of motion are solved analytically using the Laplace transform method while the energy equation is solved numerically taking the Joule and the viscous dissipations into consideration.The effect of the magnetic field, the Hall current, the ion slip, and the suction and injection on both the velocity and temperature distributions is studied.

Description of the problem
The two nonconducting plates are located at the y = ±h planes and extend from x = −∞ to ∞ and z = −∞ to ∞.The lower and upper plates are kept at the two constant temperatures T 1 and T 2 , respectively, where T 2 > T 1 .The fluid flows between the two plates under the influence of an exponential decaying pressure gradient dP/dx in the x-direction which is a generalization of the case of constant pressure gradient.A uniform suction from above and injection from below with uniform velocity v o is applied at t = 0.The whole system is subjected to a uniform magnetic field B o in the positive y-direction.This is the total magnetic field acting on the fluid since the induced magnetic field is neglected.From the geometry of the problem, it is evident that ∂/∂x = ∂/∂z = 0.The existence of the Hall term gives rise to a z-component of the velocity.Thus, the velocity vector of the fluid is with the initial and boundary conditions u = w = 0 at t ≤ 0 and u = w = 0 at y = ±h for t > 0. The temperature T(y,t) at any point in the fluid satisfies both the initial and boundary conditions T = T 1 at t ≤ 0, T = T 2 at y = +h, and The fluid flow is governed by the momentum equation where ρ and μ are, respectively, the density and the coefficient of viscosity of the fluid.If the Hall and ion slip terms are retained, the current density J is given by where σ is the electric conductivity of the fluid, β is the Hall factor, and Bi is the ion slip parameter [13].This equation may be solved in J to yield where Be = σβB o is the Hall parameter [13].Thus, in terms of (2.4), the two components of (2.2) read (2.5) To find the temperature distribution inside the fluid, we use the energy equation [9] ρc ∂T ∂t where c and k are, respectively, the specific heat capacity and the thermal conductivity of the fluid.The second and third terms on the right-hand side represent the viscous and Joule dissipations, respectively.The problem is simplified by writing the equations in the nondimensional form.The characteristic length is taken to be h, and the characteristic time is ρh 2 /μ 2 while the characteristic velocity is μ/ρh.We define the following nondimensional quantities: ) The initial and boundary conditions for the velocity become and the initial and boundary conditions for the temperature are given by (2.12)

Analytical solution of the equations of motion
Equations (2.8) and (2.9) are the two equations of motion which, if solved, give the two components of the velocity field as functions of space and time.Multiplying (2.9) by i and adding to (2.8), we obtain with the initial and boundary conditions where V = u + iw.Equations (3.1) and (3.2) can be solved using the method of Laplace transform (LT) [12] to obtain V as functions of y and t.The real part of V represents the x-component of the velocity while the imaginary part represents the z-component.
Taking LT of (3.1) and (3.2), we have where V (y,s) = L(V (y,t)), −F(s) is the LT of the pressure gradient, K(s) = A + s, and A = Ha 2 ((1 + BiBe) − iBe)/((1 + BiBe) 2 + Be 2 ).The solution of (3.3) with y as an independent variable is given as V (y,s) = F(s) K 1 + exp(Sy/2) sinh(S/2)sinh(qy) sinh(q) − cosh(S/2)cosh(qy) cosh(q) , ( where q 2 = S 2 /4 + K. Using the complex inversion formula and the residue theorem [12], the inverse transform of V (y,s) is determined as where C and α are two parameters characterizing the form of the pressure gradient and α = 0 corresponds to the case of constant pressure gradient, (3.7) The expression for the complex velocity V is to be evaluated for different values of the parameters Ha, Be, Bi, and S. The velocity components u and w are, respectively, the real and imaginary parts of V .

Numerical solution of the energy equation
The exact solution of the equations of motion, given by (3.5), determines the velocity field for different values of the parameters Ha, Be, Bi, and S. The values of the velocity components, when substituted in the right-hand side of the inhomogeneous energy equation (2.10), make it too difficult to solve analytically.The energy equation is to be solved numerically with the initial and boundary conditions given by (2.12) using finite differences [3].The Crank-Nicolson implicit method is applied.The finite difference equations are written at the mid-point of the computational cell and the different terms are replaced by their second-order central difference approximations in the y-direction.The diffusion term is replaced by the average of the central differences at two successive time levels.The viscous and Joule dissipation terms are evaluated using the velocity components and their derivatives in the y-direction which are obtained from the exact solution.Finally, the block tridiagonal system is solved using Thomas' algorithm.All calculations have been carried out for C = −5, α = 1, Pr = 1, and Ec = 0.2.It is observed that the velocity component u reaches the steady state faster than w which, in turn, reaches the steady state faster then T. This is expected, since u is the source of w, while both u and w act as sources for the temperature.Figure 5.2 shows the time evolution of u and w at the centre of the channel y = 0 for various values of the Hall parameter Be and the ion slip parameter Bi.In this figure, Ha = 3 and S = 0.It is clear from Figure 5.2(a) that increasing the parameter Be or Bi increases u.This is because the effective conductivity (σ/{(1 + BiBe) 2 + Be 2 }) decreases with increasing Be or Bi which reduces the magnetic damping force on u.In Figure 5.2(b), the velocity component w increases with increasing Be, since w is a result of the Hall effect.On the other hand, increasing the ion slip parameter Bi decreases w for all values of Be as a result of decreasing the source term of w (BeHa 2 u/{(1 + BiBe) 2 + Be 2 }) and increasing its damping term (Ha 2 w/{(1 + BiBe) 2 + Be 2 }).The influence of the ion slip on w becomes more pronounced for higher values of Be.

Results and discussion
Table 5.1 shows the time evolution of T at the centre of the channel y = 0 for various values of the Hall parameter Be and the ion slip parameter Bi and for Ha = 3 and S = 0. Table 5.1 indicates that increasing Be or Bi decreases T at small times and increases it at large times.This can be attributed to the fact that, for small times, u and w are small and an increase in Be or Bi decreases the Joule dissipation which is also proportional to (1/{(1 + BiBe) 2 + Be 2 }).For large times, increasing Be increases both u and w and, in turn, increases the Joule and viscous dissipations.Also, for large times, increasing Bi, although it decreases w, increases the velocity u of the main flow and consequently increases the viscous and Joule dissipations.
Figure 5.3 shows the time evolution of u and w at the centre of the channel y = 0 for various values of the Hartmann number Ha and the ion slip parameter Bi.In this figure, Be = 3 and S = 0. Figure 5.3(a) indicates that the effect of Bi on u depends on Ha.For small values of Ha, increasing Bi slightly decreases u as a result of increasing the damping force on u which is proportional to Bi. Increasing Bi more increases the effective conductivity and, in turn, decreases the damping force on u which increases u.On the other hand, for larger values of Ha, u becomes small, and increasing Bi always decreases the effective conductivity and therefore increases u.It is also clear that the effect of Bi on u becomes more apparent for higher values of Ha. Figure 5.3(b) ensures that increasing the ion slip parameter Bi decreases w for all values of Ha and that its effect is more apparent for higher values of Ha.
Table 5.2 shows the time evolution of T at the centre of the channel y = 0 for various values of the Hartmann number Ha and the ion slip parameter Bi and Be = 3 and S = 0. Table 5.2 indicates that the parameter Bi has a more pronounced effect on T for higher values of the magnetic field.It is clear that increasing Bi increases T as a result of increasing the viscous and Joule dissipations.But increasing Bi more decreases T for small t and increases it as time develops.
Figure 5.4 presents the time evolution of u and w at the centre of the channel y = 0 for various values of the suction parameter S and the ion slip parameter Bi.In this figure

Figure 5 .
Figure 5.1 shows the profiles of the velocity components u and w and temperature T for various values of time t.The figure is plotted for Ha = 3, Be = 3, Bi = 3, and S = 1.As shown in Figures 5.1(a) and 5.1(b), the profiles of u and w are asymmetric about the plane y = 0 because of the suction.It is observed that the velocity component u reaches the steady state faster than w which, in turn, reaches the steady state faster then T. This is expected, since u is the source of w, while both u and w act as sources for the temperature.Figure5.2shows the time evolution of u and w at the centre of the channel y = 0 for various values of the Hall parameter Be and the ion slip parameter Bi.In this figure, Ha = 3 and S = 0.It is clear from Figure5.2(a) that increasing the parameter Be or Bi increases u.This is because the effective conductivity (σ/{(1 + BiBe) 2 + Be 2 }) decreases with increasing Be or Bi which reduces the magnetic damping force on u.In Figure5.2(b), the velocity component w increases with increasing Be, since w is a result of the Hall effect.On the other hand, increasing the ion slip parameter Bi decreases w for all values of Be as a result of decreasing the source term of w (BeHa 2 u/{(1 + BiBe) 2 + Be 2 }) and increasing its damping term (Ha 2 w/{(1 + BiBe) 2 + Be 2 }).The influence of the ion slip on w becomes more pronounced for higher values of Be.Table5.1 shows the time evolution of T at the centre of the channel y = 0 for various values of the Hall parameter Be and the ion slip parameter Bi and for Ha = 3 and S = 0. Table5.1 indicates that increasing Be or Bi decreases T at small times and increases it at large times.This can be attributed to the fact that, for small times, u and w are small and an increase in Be or Bi decreases the Joule dissipation which is also proportional to (1/{(1 + BiBe) 2 + Be 2 }).For large times, increasing Be increases both u and w and, in turn, increases the Joule and viscous dissipations.Also, for large times, increasing Bi, although it decreases w, increases the velocity u of the main flow and consequently increases the viscous and Joule dissipations.Figure5.3shows the time evolution of u and w at the centre of the channel y = 0 for various values of the Hartmann number Ha and the ion slip parameter Bi.In this figure, Be = 3 and S = 0. Figure5.3(a)indicates that the effect of Bi on u depends on Ha.For small values of Ha, increasing Bi slightly decreases u as a result of increasing the damping force on u which is proportional to Bi. Increasing Bi more increases the effective conductivity and, in turn, decreases the damping force on u which increases u.On the other hand, for larger values of Ha, u becomes small, and increasing Bi always decreases the effective conductivity and therefore increases u.It is also clear that the effect of Bi on u becomes more apparent for higher values of Ha.Figure5.3(b)ensures that increasing the ion slip parameter Bi decreases w for all values of Ha and that its effect is more apparent for higher values of Ha.Table5.2shows the time evolution of T at the centre of the channel y = 0 for various values of the Hartmann number Ha and the ion slip parameter Bi and Be = 3 and S = 0. Table5.2indicates that the parameter Bi has a more pronounced effect on T for higher