Unsteady Couette Flow through a Porous Medium in a Rotating System

An investigation has been made on an unsteady Couette flow of a viscous incompressible fluid through a porous medium in a rotating system. The solution of the governing equations has been obtained by the use of Laplace transform technique. It is found that the primary velocity decreases and the magnitude of the secondary velocity increases with an increase in rotation parameter. The fluid velocity components are decelerated by an increase of Reynolds number. An increase in porosity parameter leads to increase the primary velocity and the magnitude of the secondary velocity. It is also found that the solution for small time converges more rapidly than the general solution. The asymptotic behavior of the solution is analyzed for small as well as large values of rotation parameter and Reynolds number. It is observed that a thin boundary layer is formed near the moving plate of the channel and the thicknesses of the boundary layer increases with an increase in porosity parameter.


Introduction
The flow between two parallel plates is a classical problem that has many applications in accelerators, aerodynamic heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil, fluid droplets and sprays.Such a flow model is of great interest, not only for its theoretical significance, but also for its wide applications to geophysics and engineering.A lot of research work concerning the flow between two parallel plates studied in a rotating system have appeared, for example, Batchelor [1], Ganapathy [3], Gupta [4] and Mazumder [5].The flows through porous medium are very much prevalent in nature and therefore, the study of such flows has become of principal interest in many scientific and engineering applications.This type of flows has shown their great importance in petroleum engineering to study the movements of natural gas, oil and water through the oil reservoirs; in chemical engineering for the filtration and water purification processes.Further, to study the underground water resources and seepage of water in river beds one need the knowledge of the fluid flow through porous medium.Therefore, there are number of practical uses of the fluid flow through porous media.Rotation has an immense importance in various phenomena such as in cosmical fluid dynamics, meteor-ology, geophysical fluid dynamics, gaseous and nuclear reactors and many engineering applications, that is why, the study of Couette flow through porous medium in a rotating system enhances an interest to the researchers due to its applications in the aforesaid area.Such a study has a greater importance in the design of turbines and turbo mechanics, in estimating the flight path of rotating wheels and spin-stabilized missiles.A large number of investigations has been made on the flow through a porous medium in a rotating system.In general, most of solutions for unsteady flows of viscous fluids are in a series form.These series may be rapidly convergent for large values of the time but slowly convergent for small values of the time or vice versa.Sometimes, it can be difficult to obtain the solution for small values of the time but it can be easy to obtain it for large values of the time and the opposite can also be true.Vidyanidhi and Nigam [6] studied the channel flow between rotating parallel plates under constant pressure gradient.Jana and Dutta [7] studied the steady Couette flow of a viscous incompressible fluid between two infinite parallel plates, one stationary and the other moving with uniform velocity, in a rotating frame of reference.Singh and Sharma [8] have presented the three dimensional Couette flow through porous media.A periodic solution of oscillatory Couette flow through a porous medium in rotating sys-tem has been obtained by Singh et al. [9].Guria et al. [10] have described the unsteady Couette flow in a rotating system.Das et al. [10] have studied the unsteady Couette flow with an oscillatory velocity of one of the plates in a rotating system.The unsteady MHD Couette flow in a rotating system has been investigated by Das et al. [12].Attia [13] has studied the effect of porosity on unsteady Couette flow with heat transfer in the presence of uniform suction and injection.Israel-Cookey et al. [14] have presented the MHD oscillatory Couette flow of a radiating viscous fluid in a porous medium with periodic wall temperature.The unsteady hydromagnetic Couette flow through a porous medium in a rotating system have been presented Prasad and Kumar [15].Das et al. [16] have studied the Couette flow through porous medium in a rotating system.
In the present paper, we have studied the unsteady Couette flow between two infinite horizontal parallel plates in a porous medium in a rotating system when one of the plate moving with uniform velocity and the other one held at rest.The fluid and plates are in a state of rigid body rotation with uniform angular velocity  .The solutions for the velocity distributions as well as shear stresses have been obtained for small time as well as for large time by the Laplace transform technique.It is found that the primary velocity 1 decreases and the magnitude of the secondary velocity 1 increases with an increase in rotation parameter u v 2 K .The primary velocity 1 and the magnitude of the secondary velocity 1 decrease with an increase in Reynolds number Re.An increase in porosity parameter u v  leads to increase in the values of both the primary velocity 1 and the magnitude of the secondary velocity 1 .It is also found that the solution for small time converges more rapidly than the general solution.For the steady state solution, the asymptotic behavior of the solution is analysed for small as well as large values of rotation parameter u v 2 K and Reynolds number Re.It is observed that a thin boundary layer is formed near the stationary plate and the thicknesses of the boundary layer increases with an increase in porosity parameter  .

Mathematical Formulation and Its Solution
Consider the unsteady flow of a viscous incompressible fluid between two infinite parallel porous plates embedded in a porous medium.The plates are separated by a distance h.The fluid and channel rotate in unison about an axis normal to the planes of the plates with a uniform angular velocity  .Choose a Cartesian co-ordinate system with x-axis along the lower stationary plate in the direction of the flow, the y-axis is normal to the plates and the z-axis perpendicular to xy-plane (see Figure 1).Flow within the channel is induced due to the motion of the upper plate at y h  parallel to itself in x-direction with a uniform velocity 0 .Initially, at time u 0 t  , the fluid as well as the plates of the channel are assumed to be at rest.At time the upper plate at 0 t  y h  starts moving with uniform velocity 0 along x-direction in its own plane while the lower plate at is kept fixed.The velocity components are relative to a frame of reference rotating with the fluid.Since plates of the channel are infinitely long along x and y directions, all physical quantities will be functions of z and t only.
where  and  are respectively the fluid density, the kinematic viscosity and k the permeability of the porous medium.
The initial and boundary conditions are Introducing the non-dimensional variables Equations ( Taking the Laplace transform, Equation ( 8) becomes 1) and (3) become where (7) where is the rotation parameter and The boundary conditions for  , q s   are the Reynolds number and the porosity param ( 6) and ( 7), we have The solution of Equation ( 12) subject to the boundary conditions ( 14) is where The initial and boundary conditions for   , q   are where ,0 0 for 0 1 and     0, 0 and 0, 1 for 0.
The inverse Laplace's transform of Equation ( 15) is where On separating into a real and imaginary parts, we get The solution given by Equations ( 19) and (20) exists fo for the suction at the plates).r [17], for small time, the oundary conditions ( 14) is r both Re < 0 (corresponding to 0 0  v  for the blowing at the plates) and Re > 0 (corres ing to 0 0 v 

Solutions for Small Time
pond Following Carslaw and Jaega solution of (12) subject to the b obtained by Laplace transform technique in the following form The solution (22) can be written as On separating into a real and imaginary parts, we get the velocity distributions for the primary and the seconda   ry flow as Equations ( 25) and (26) describe the fluid velocities for small time.

Results and Discussion
To study the effects of rotation, Reynolds number and porosity parameter on the velocity distributions we have presented the non-dimensional velocity components 1 and 1 against   19) and (20) in Figures 6 and 7.It is seen that the solution for small time given by Equations ( 25) and ( 26) converges more rapidly than the general solution given by ( 19) and (20).Hence, we conclude that for small times, the numerical values of the velocity components can be evaluated from Equations ( 25) and On separating into a real and imaginary parts, we get the shear stress components due to the primary and secondary flows at the stationary plate 0 The where On separating into a real and imaginary parts, we get the shear stress components due to the primary and second s as where For small time, the numerical values of the shear stress components calculated from Equations (29), (30), (33) and (34) are given in Tables 1 and 2 Case 1): When where It is evident from Equations (37) and (38) that there exists a single-deck boundary layer of thickness of order near the moving plate of the channel where  is given by (39).The thickness of this bound- ary layer decreases with ter an increase in porosity parame- since  decreases with increase in  .
Case 2): When 2 1 K  and In this case, the velocity distribut ned from the Equations (36) as

Conclusion
The unsteady Couette flow of a viscous incompressible fluid through a porous medium in a rotating system has been investigated.It is found that the primary velocity decreases and the magnitude of the secondary velocity increases with an increase in rotation parameter.The fluid velocity components decrease with a increase in the porosity of the meecondary velocities increase.That is, the porosity of the medium has an accel- the blowing at the plate.The x-, y-and z-components of Navier-Stokes equation are 2 0

Figure 1 .
Figure 1.Geometry of the problem.

K 2 K.
, Reynolds number Re, porosity parameter  and time  .It is seen from Figure2that the primary velocity 1 decreases and the magnitude of the secondary velocity 1 increases with an increase in rotation number u w Figure3reveals that both the primary velocity 1 and the magnitude of the secondary velocity 1 decreases with increase in Reynolds number Re.It is observed from Figure4that both the primary velocity 1 and the magnitude of the secondary velocity 1 increase with an increase in porosity parameter u w u w  .The presence of porous medium produces a resisting force in the flow field.So, the resistance in the

Figure 7 .
Figure 6.σ = 0.1.Velocity u 1 for general solution and solution for small time when K 2 = 2, Re = 2 and of the shear stress x  increases while the absolute value of the shear stress y  increases with a r in either porosi n inc ease ty parameter  or time  .For small times, the non-dimensional shear stresses u d e to the primary and secondary flows at the stationary plate
numerical values of the non-dimensional shear stresses x