INFLUENCE OF TEMPERATURE-DEPENDENT VISCOSITY ON THE MHD COUETTE FLOW OF DUSTY FLUID WITH HEAT TRANSFER

This paper studies the effect of variable viscosity on the transient Couette flow of dusty fluid with heat transfer between parallel plates. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates. The parallel plates are assumed to be porous and subjected to a uniform suction from above and injection from below. The upper plate is moving with a uniform velocity while the lower is kept stationary. The governing nonlinear partial differential equations are solved numerically and some important effects for the variable viscosity and the uniform magnetic field on the transient flow and heat transfer of both the fluid and dust particles are indicated.


Introduction
The study of the flow of dusty fluids has important applications in the fields of fluidization, combustion, use of dust in gas cooling systems, centrifugal separation of matter from fluid, petroleum industry, purification of crude oil, electrostatic precipitation, polymer technology, and fluid droplets sprays.
The hydrodynamic flow of dusty fluids was studied by a number of authors [6-8, 13, 14].Later, the influence of the magnetic field on the flow of electrically conducting dusty fluids was studied [1,5,11,12,16].Most of these studies are based on constant physical properties.More accurate prediction for the flow and heat transfer can be achieved by taking into account the variation of these properties, especially the variation of the fluid viscosity with temperature [9].Klemp et al. [10] have studied the effect of temperaturedependent viscosity on the entrance flow in a channel in the hydrodynamic case.Attia and Kotb [4] studied the steady MHD fully developed flow and heat transfer between two parallel plates with temperature-dependent viscosity.Later, Attia [3] has extended the problem to the transient state.
In the present work, the effect of variable viscosity on the unsteady flow of an electrically conducting, viscous, incompressible dusty fluid and heat transfer between parallel nonconducting porous plates is studied.The fluid is flowing between two electrically insulating infinite plates maintained at two constants but different temperatures.An external uniform magnetic field is applied perpendicular to the plates.The upper plate is moving with a uniform velocity while the lower is kept stationary.The magnetic Reynolds number is assumed small so that the induced magnetic field is neglected.The fluid is acted upon by a constant pressure gradient and its viscosity is assumed to vary exponentially with temperature.The flow and temperature distributions of both the fluid and dust particles are governed by the coupled set of the momentum and energy equations.The Joule and viscous dissipation terms in the energy equation are taken into consideration.The governing coupled nonlinear partial differential equations are solved numerically using the finite difference approximations.The effects of the external uniform magnetic field and the temperature-dependent viscosity on the time development of both the velocity and temperature distributions are discussed.

Description of the problem
The dusty fluid is assumed to be flowing between two infinite horizontal plates located at the y = ±h planes.The dusty particles are assumed to be uniformly distributed throughout the fluid.The two plates are assumed to be electrically nonconducting and kept at two constant temperatures, T 1 for the lower plate and T 2 for the upper plate with T 2 > T 1 .The upper plate is moving with a uniform velocity U o while the lower is kept stationary.A constant pressure gradient is applied in the x-direction and the parallel plates are assumed to be porous and subjected to a uniform suction from above and injection from below.Thus the y-component of the velocity is constant and denoted by v o .A uniform magnetic field B o is applied in the positive y-direction.By assuming a very small magnetic Reynolds number the induced magnetic field is neglected [17].The fluid motion starts from rest at t = 0, and the no-slip condition at the plates implies that the fluid and dust particles velocities have neither a znor an x-component at y = ±h.The initial temperatures of the fluid and dust particles are assumed to be equal to T 1 and the fluid viscosity is assumed to vary exponentially with temperature.Since the plates are infinite in the xand z-directions, the physical variables are invariant in these directions.The flow of the fluid is governed by the Navier-Stokes equation [17] where ρ is the density of clean fluid, μ is the viscosity of clean fluid, u is the velocity of fluid, u p is the velocity of dust particles, σ is the electric conductivity, p is the pressure acting on the fluid, N is the number of dust particles per unit volume, and K is a constant.The first three terms in the right-hand side are, respectively, the pressure gradient, viscosity, and Lorentz force terms.The last term represents the force term due to the relative motion between fluid and dust particles.It is assumed that the Reynolds number of the relative velocity is small.In such a case the force between dust and fluid is proportional to the relative velocity [14].The motion of the dust particles is governed by Newton's second law [14] where m p is the average mass of dust particles.The initial and boundary conditions on the velocity fields are, respectively, given by For t > 0, the no-slip condition at the plates implies that (2.4) Heat transfer takes place from the upper hot plate towards the lower cold plate by conduction through the fluid.Also, there is a heat generation due to both the Joule and viscous dissipations.The dust particles gain heat energy from the fluid by conduction through their spherical surface.Two energy equations are required which describe the temperature distributions for both the fluid and dust particles and are, respectively, given by [15] ) where T is the temperature of the fluid, T p is the temperature of the particles, c is the specific heat capacity of the fluid at constant pressure, C s is the specific heat capacity of the particles, k is the thermal conductivity of the fluid, γ T is the temperature relaxation time (= 3Prγ p C s /2c), γ p is the velocity relaxation time (= 2ρ s D 2 /9μ), ρ s is the material density of dust particles (= 3ρ p /4πD 3 N), and D is the average radius of dust particles.The last three terms in the right-hand side of (2.5) represent the viscous dissipation, the Joule dissipation, and the heat conduction between the fluid and dust particles.The initial and boundary conditions on the temperature fields are given as (2.7) The viscosity of the fluid is assumed to depend on temperature and is defined as μ = μ o f (T).For practical reasons which are shown to be suitable for most kinds of fluids [2,10], the viscosity is assumed to vary exponentially with temperature.The function f (T) takes the form [2,10], f (T) = e −b(T−T1) , where the parameter b has the dimension of [T] −1 and such that at T = T 1 , f (T 1 ) = 1 and then μ = μ o .This means that μ o is the viscosity coefficient at T = T 1 .The parameter a 1 may take positive values for liquids such as water, benzene, or crude oil.In some gases like air, helium, or methane a 1 it may be negative, that is, the coefficient viscosity increases with temperature [2,10].
The temperature variations within a convective flow give rise to variations in the properties of the fluid, in the density and viscosity, for example.An analysis including the full effects of these is so complicated that some approximations become essential.The equations are commonly used in a form known as the Boussinesq approximation.In the Boussinesq approximation, variations of all fluid properties other than the density are ignored completely.Variations of the density are ignored except insofar as they give rise to gravitational force [18].Therefore, a buoyancy force term may be included in the Navier-Stokes equation which equals −αρΔT, where α is the coefficient of expansion of the fluid.Such a buoyancy term may be neglected on the basis of either ΔT small, that is, T 2 − T 1 is small, or small α which is a reasonable approximation for liquids and perfect gases [18].
The problem is simplified by writing the equations in the nondimensional form.The characteristic length is taken to be h and the characteristic velocity is U o .We define the following nondimensional quantities: ) , a is the viscosity variation parameter, ) is the Eckert number, L o = ρh 2 /μ o γ T is the temperature relaxation time parameter.In terms of the above nondimensional variables and parameters (2.1) to (2.7) take the form (the hats are dropped for convenience) ) ) (2.14) Equations (2.9), (2.10), (2.12), and (2.13) represent a system of coupled and nonlinear partial differential equations which are solved numerically under the initial and boundary conditions (2.11) and (2.14) using the finite difference approximations.A linearization technique is first applied to replace the nonlinear terms at a linear stage, with the corrections incorporated in subsequent iterative steps until convergence is reached.Then the Crank-Nicolson implicit method is used at two successive time levels [2].An iterative scheme is used to solve the linearized system of difference equations.The solution at a certain time step is chosen as an initial guess for next time step and the iterations are continued till convergence, within a prescribed accuracy.Finally, the resulting block tridiagonal system is solved using the generalized Thomas algorithm [2].Finite difference equations relating the variables are obtained by writing the equations at the midpoint of the computational cell and then replacing the different terms by their second-order central difference approximations in the y-direction.The diffusion terms are replaced by the average of the central differences at two successive time levels.The computational domain is divided into meshes each of dimension Δt and Δy in time and space, respectively.We define the variables v = ∂u/∂y and H = ∂θ/∂y to reduce the second-order differential equations (2.9) and (2.12) to first-order differential equations which are (2.15) The variables with bars are given initial guesses from the previous time steps and an iterative scheme is used at every time to solve the linearized system of difference equations.Computations have been made for R = 0.5, G = 0.8, λ = 5, Pr = 1, Ec = 0.2, and L o = 0.7.Grid-independence studies show that the computational domain 0 < t < ∞ and −1 < y < 1 can be divided into intervals with step sizes Δt = 0.0001 and Δy = 0.005 for time and space, respectively.Smaller step sizes do not show any significant change in the results.Convergence of the scheme is assumed when all of the unknowns u, v, u p , T, H, and T p for the last two approximations differ from unity by less than 10 −6 for all values of y in −1 < y < 1 at every time step.Less than 7 approximations are required to satisfy these convergence criteria for all ranges of the parameters studied here.

Results and discussions
The exponential dependence of the viscosity on temperature results in decomposing the viscous force term in the momentum equation into two terms.The variations of these resulting terms with the viscosity variation parameter a and their relative magnitudes have an important effect on the flow and temperature fields in the absence or presence of the applied uniform magnetic field.
Figures 3.1 and 3.2 indicate the variations of the velocities u and u p at the center of the channel (y = 0) with time for different values of the viscosity variation parameter a and for Ha = 0 and S = 0.The figures show that increasing a increases the velocity and the time required to approach the steady-state.The effect of the parameter a on the steadystate time is more pronounced for positive values of a than for negative values.Notice that u reaches the steady state faster than u p .This is because the fluid velocity is the source for the dust particles velocity.Figure 3.1 shows also that the influence of a on u p is negligible for some time and then increases as the time develops.for T than for T p , as T p always follows T. It is noticed that the steady-state values of T p coincide with the corresponding steady-state values of T, and the time required for T p to reach the steady state, which depends on a, is longer than that for T.
The application of the uniform magnetic field adds one resistive term to the momentum equation and the Joule dissipation term to the energy equation.Figures 3.5 and 3.6 present the influence of the viscosity variation parameter a on the evolution of both the velocities u and u p at the center of the channel, respectively, for Ha = 1 and S = 0.The magnetic field results in a reduction in the velocities and the steady-state time for all values of a due to its damping effect.
Figures 3.7 and 3.8 present the influence of the viscosity variation parameter a on the evolution of the temperatures T and T p at the center of the channel, respectively, for Ha = 1 and S = 0. Increasing the magnetic field increases the temperatures for all positive values of a except for very small time.This is because the magnetic field has a resistive effect which becomes more pronounced as time develops especially with the case of negative a which has the same resistive effect.their steady-state times as a result of pumping the fluid from the slower lower half region to the center of the channel.The influence of suction on u and u p is more pronounced for higher values of the parameter a.
Figures 3.11 and 3.12 present the influence of the viscosity variation parameter a on the evolution of the temperatures T and T p at the center of the channel, respectively, for   Ha = 0 and S = 1.It is shown that increasing suction velocity decreases both T and T p and their steady-state times.This results from pumping the fluid from colder lower half region to the center of the channel.The effect of suction on T and T p is more apparent for higher values of a.    3.14 present the influence of the viscosity variation parameter a on the steady-state profile of the velocities u and u p , respectively, for Ha = 0.5 and S = 0.5.It is clear that increasing a increases u and u p for all values of y due to the increase in viscosity.It is clear also that the steady-state velocity attains more than three times the wall velocity due to the effect of the applied pressure gradient.Figures 3.15   present the influence of the viscosity variation parameter a on the steady-state profile of the temperatures T and T p , respectively, for Ha = 0.5 and S = 0.5.Increasing a increases both T and T p as a result of increasing the velocities and their gradients which increase the viscous and Joule dissipations.Also, it is shown that the temperatures exceed unity for some locations (i.e., the temperature of the upper plate) due to the heating effect of the dissipations.

Conclusions
In this paper the effect of a temperature-dependent viscosity, suction and injection velocity, and an external uniform magnetic field on the unsteady flow and temperature distributions of an electrically conducting viscous incompressible dusty fluid between two parallel porous plates has been studied.The viscosity was assumed to vary exponentially with temperature and the Joule and viscous dissipations were taken into consideration.The most interesting result was the cross-over of the temperature curves due to the variation of the parameter a and the influence of the magnetic field in the suppression of such cross-over.On the other hand, changing the magnetic field results in the appearance of cross-over in the temperature curves for a given negative value of a. Also, changing the viscosity variation parameter a leads to asymmetric velocity profiles about the central plane of the channel (y = 0) which is similar to the effect of variable percolation perpendicular to the plates.

Figures 3. 13
Figures 3.13and 3.14 present the influence of the viscosity variation parameter a on the steady-state profile of the velocities u and u p , respectively, for Ha = 0.5 and S = 0.5.It is clear that increasing a increases u and u p for all values of y due to the increase in viscosity.It is clear also that the steady-state velocity attains more than three times the wall velocity due to the effect of the applied pressure gradient.Figures 3.15 and 3.16 Figures 3.13and 3.14 present the influence of the viscosity variation parameter a on the steady-state profile of the velocities u and u p , respectively, for Ha = 0.5 and S = 0.5.It is clear that increasing a increases u and u p for all values of y due to the increase in viscosity.It is clear also that the steady-state velocity attains more than three times the wall velocity due to the effect of the applied pressure gradient.Figures 3.15 and 3.16 Figures 3.13and 3.14 present the influence of the viscosity variation parameter a on the steady-state profile of the velocities u and u p , respectively, for Ha = 0.5 and S = 0.5.It is clear that increasing a increases u and u p for all values of y due to the increase in viscosity.It is clear also that the steady-state velocity attains more than three times the wall velocity due to the effect of the applied pressure gradient.Figures 3.15 and 3.16