Unsteady Unidirectional MHD Flow of Voigt Fluids Moving between Two Parallel Surfaces for Variable Volume Flow Rates

The velocity profile and pressure gradient of an unsteady state unidirectional MHD flow of Voigt fluids moving between two parallel surfaces under magnetic field effects are solved by the Laplace transform method. The flow motion between parallel surfaces is induced by a prescribed inlet volume flow rate that varies with time. Four cases of different inlet volume flow rates are considered in this study including 1 constant acceleration piston motion, 2 suddenly started flow, 3 linear acceleration piston motion, and 4 oscillatory piston motion. The solution for each case is elaborately derived, and the results of associated velocity profile and pressure gradients are presented in analytical forms.


Introduction
Magnetohydrodynamics MHD is an academic discipline, which studies the dynamic behaviors of the interaction between magnetic fields and electrically conducting fluids.Examples of such fluids are numerous including plasmas, liquid metals, and salt water or electrolytes.The MHD flow is encountered in a variety of applications such as MHD power generators, MHD pumps, MHD accelerators, and MHD flowmeters, and it can also be expanded into various industrial uses.
During the past decades, a great deal of papers in literatures used a combination of Navier-Stokes equations and Maxwell's equations to describe the MHD flow of the Newtonian and electrically conducting fluid.Sayed-Ahmed and Attia 1 examined the effect of the Hall term and the variable viscosity on the velocity and temperature fields of the MHD flow.Attia 2 studied the unsteady Couette flow and heat transfer of a dusty conducting rate of shear strain, and μ is the viscosity coefficient.Here G, μ are the material properties and are assumed to be constant.When G 0, 2.2 reduces to that of Newtonian fluid.
The problem of the unsteady flow of incompressible Voigt fluid between the parallel surfaces is considered.The dynamic equation is In the above equation, T denotes the total stress tensor, ρ the fluid density, V the velocity vector, b the body force field, and ∇ the divergence operator.The continuity equation is Using the Cartesian coordinate system x, y, z , the x-axis is taken as the centerline direction between these two parallel surfaces, y is the coordinate normal to the plate, z is the coordinate normal to x and y, respectively, and the velocity field is assumed in the form where u is the velocity in the x-coordinate direction and i is the unit vector in the x-coordinate direction.This effectively assumes that the flow is fully developed at all points in time.

Methodology of Solution
Since the governing equation with boundary conditions and initial condition are known, the problem is well posed.In general, it is not an easy question to solve this kind of equation by the method of separation of variables and eigenfunctions expansion.In this paper, the Laplace transform method is used to reduce the two variables into a single variable.This procedure greatly reduces the difficulties of treating these partial differential 9 and integral equations 11 .
The governing equation of motion in x-direction and the strain function are ∂p ∂x τ yx e G/μ t dt .

3.2
As these two surfaces are 2h apart, the boundary conditions are u h, t 0, ∂u 0, t ∂y 0.

3.3
The initial condition is related to the inlet volume flow rate by Using the inverse transform formula, the pressure gradient distribution can also be obtained.

Illustration of Examples
Hereafter, we will solve the cases proposed by Das and Arakeri 10 with the Voigt fluid to understand the different flow characteristics between these two fluids under the same condition.
For the first case, the piston velocity u p t moves with a constant acceleration and for the second one, the piston starts suddenly from rest and then maintains this velocity.These two solutions are used to assess the trapezoidal motion of the piston, namely, the piston has three stages: constant acceleration of piston starting from rest, a period of constant velocity, and a constant deceleration of the piston to a stop.Finally, the oscillatory piston motion is also considered.

Constant Acceleration Piston Motion
The piston motion of constant acceleration can be described by the following equation:

4.3
From the above expression, the integration is determined using complex variable theory, as discussed by Arpaci 38 .It is easily observed that s 0 is a pole of order 2.
Therefore, the residue at s 0 is Res 0

4.4
The other singular points are the roots of following transcendental equation: Setting m iα, we have where R y α n h cos α n h − cos α n y and Q, Q s 1n , Q s 2n are defined in 4.9 .The first term on the right-hand side of 4.10 the steady state velocity and the second term, the transient response of the flow to an abrupt change either in the boundary conditions, body forces, pressure gradient, or other external driving force.
Equation 3.19 is used to determine the pressure gradient in this flow field and follows the same procedure for solving velocity profile Res 0

4.11
Therefore, the pressure gradient is 4.12 Q, Q s 1n , Q s 2n are defined in 4.9 .

Suddenly Started Flow
For a suddenly started flow between the parallel surfaces, u p 0, for t ≤ 0, U p , for t > 0, 4.13 where U p is the constant velocity.
In which case, the velocity profile is where R y α n h cos α n h − cos α n y , Q, Q s 1n , Q s 2n are defined in 4.9 , and the pressure gradient is 4.15

Linear Acceleration Piston Motion
The piston motion of linear acceleration can be described by the following equation: where a p is the constant acceleration, U p is the final velocity after acceleration, and t 0 is the time period of acceleration.
In which case, the velocity profile is where R y α n h cos α n h − cos α n y , Q, Q s 1n , Q s 2n are defined in 4.9 , and the pressure gradient is 4.18

Oscillatory Piston Motion
The oscillating piston motion starting from rest is considered.The piston motion is described as u p 0, for t ≤ 0, U 0 sin ωt , for t > 0.

4.19
Taking the Laplace transform of 4.

Conclusions
In this paper, the analytical solutions of unsteady unidirectional MHD flow of Voigt fluids under magnetic field effects for different piston motion that provide different volume flow rates are derived and solved by Laplace transform technique.The results are presented in analytical forms.
The boundary conditions 3.7 and 3.8 are used to solve the two arbitrary coefficients C 1 and C 2 .Substituting C 1 and C 2 into 3.10 gives h −h u y, t dy u p t 2h Q t , 3.4where u p t is the given average inlet velocity and Q t is the given inlet volume flow rate.The above governing equation, boundary conditions, and initial condition are prescribed and can be solved by the following calculation of Laplace transform.Differentiating 3.2 with respect to time and taking Laplace transform, then we have Adding Res 0 , Res s 1n , and Res s 2n , a complete solution for constant acceleration case is obtained as n , n 1, 2, 3, ..., ∞, are zeros of 4.6 , then Substituting 4.20 into 3.17 to find the velocity profile, the poles are simple poles at s ±iω and the roots of αh cos αh − sin αh 0. The solution to the velocity profile is cos α n h−cos α n y , Q, Q s 1n , Q s 2n are defined in 4.9 , Ω y, s is defined by 3.16 , and the pressure gradient is obtained as