﻿Unsteady Couette Flow through a Porous Medium in a Rotating System

Open Journal of Fluid Dynamics
Vol.2 No.4(2012), Article ID:25720,10 pages DOI:10.4236/ojfd.2012.24016

Unsteady Couette Flow through a Porous Medium in a Rotating System

Maitree Jana1, Sanatan Das2, Rabindra Nath Jana1

1Department of Applied Mathematics, Vidyasagar University, Midnapore, India

2Department of Mathematics, University of Gour Banga, English Bazar, India

Email: jana261171@yahoo.co.in

Received October 19, 2012; revised November 23, 2012; accepted December 5, 2012

Keywords: Couette Flow; Rotation Parameter; Reynolds Number; Porous Medium; Rotating System and Boundary Layer

ABSTRACT

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.

1. Introduction

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 decreases and the magnitude of the secondary velocity increases with an increase in rotation parameter. The primary velocity and the magnitude of the secondary velocity decrease with an increase in Reynolds number Re. An increase in porosity parameter leads to increase in the values of both 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. For the steady state solution, the asymptotic behavior of the solution is analysed for small as well as large values of rotation parameter 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.

2. 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 parallel to itself in x-direction with a uniform velocity. Initially, at time, the fluid as well as the plates of the channel are assumed to be at rest. At time the upper plate at starts moving with uniform velocity 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.

The equation of continuity gives which on integration yields, where for suction and for the blowing at the plate.

The x-, yand z-components of Navier-Stokes equation are

(1)

(2)

(3)

where and are respectively the fluid density, the kinematic viscosity and the permeability of the porous medium.

The initial and boundary conditions are

(4)

Introducing the non-dimensional variables

(5)

Figure 1. Geometry of the problem.

Equations (1) and (3) become

(6)

(7)

where is the rotation parameter and the Reynolds number and the porosity parameter.

Combing Equations (6) and (7), we have

(8)

where

(9)

The initial and boundary conditions for are

(10)

(11)

Taking the Laplace transform, Equation (8) becomes

(12)

where

(13)

The boundary conditions for are

(14)

The solution of Equation (12) subject to the boundary conditions (14) is

(15)

where

(16)

The inverse Laplace’s transform of Equation (15) is

(17)

where

(18)

On separating into a real and imaginary parts, we get

(19)

(20)

where

(21)

The solution given by Equations (19) and (20) exists for both Re < 0 (corresponding to for the blowing at the plates) and Re > 0 (corresponding to for the suction at the plates).

Solutions for Small Time

Following Carslaw and Jaegar [17], for small time, the solution of (12) subject to the boundary conditions (14) is obtained by Laplace transform technique in the following form

(22)

where

The solution (22) can be written as

(23)

where

(24)

On separating into a real and imaginary parts, we get the velocity distributions for the primary and the secondary flow as

(25)

(26)

where

(27)

Equations (25) and (26) describe the fluid velocities for small time.

3. 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 and against in Figures 2-6 for several values of the rotation parameter, Reynolds number Re, porosity parameter and time. It is seen from Figure 2 that the primary velocity decreases and the magnitude of the secondary velocity increases with an increase in rotation number. Figure 3 reveals that both the primary velocity and the magnitude of the secondary velocity decreases with increase in Reynolds number Re. It is observed from Figure 4 that both the primary velocity and the magnitude of the secondary velocity increase with an increase in porosity parameter. The presence of porous medium produces a resisting force in the flow field. So, the resistance in the

Figure 2. Velocities u1 and w1 for different K2 when Re = 2, σ = 0.1 and τ = 0.2.

Figure 3. Velocities u1 and w1 for different Re only when K2 = 2, σ = 0.1 and τ = 0.2.

Figure 4. Velocities u1 and w1 for different σ when K2 = 2, Re = 2 and τ = 0.2.

flow field decreases as the porosity parameter increases. This indicates that porosity of the medium has an accelerating influence on the flow field. Thus we can control the velocity field by introducing porous medium in a rotating system. It is observed from Figure 5 that both the primary velocity and the magnitude of the secondary velocity increases with an increase in times. For small values of time, we have drawn the velocity components and on using the solution given by Equations (25) and (26) and the general solution given by Equations (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 (26) instead of Equations (19) and (20).

The non-dimensional shear stresses at the stationary plate due to the primary and the secondary flows are given by

(28)

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 as

Figure 5. Velocities u1 and w1 for different time τ when K2 = 2, Re = 2 and σ = 0.1.

Figure 6. Velocity u1 for general solution and solution for small time when K2 = 2, Re = 2 and σ = 0.1.

Figure 7. Velocity w1 for general solution and solution for small time when K2 = 2, Re = 2 and σ = 0.1.

(29)

(30)

The numerical values of the non-dimensional shear stresses and at the plate are presented in Figures 8-10 against Reynolds number Re for several values of, and. It is seen from Figure 8 that both the absolute value of the shear stresses and increase with increase in rotation parameter while the absolute value of the shear stresses increases and that of the shear stress decreases with an increase in Re. Reynolds number is the ratio of inertial forces to viscous forces and quantifies the relative importance of these two types of forces for given flow conditions. The viscous forces are dominant at low Reynolds numbers while the inertial forces are dominant at high Reynolds numbers. It is a good agreement with the physical point of view. Figures 9 and 10 show that the absolute value of the shear stress increases while the absolute value of the shear stress increases with an increase in either porosity parameter or time.

For small times, the non-dimensional shear stresses due to the primary and secondary flows at the stationary plate are given by

(31)

where

(32)

On separating into a real and imaginary parts, we get the shear stress components due to the primary and secondary flows as

(33)

(34)

where

(35)

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 for several values of Re and. It is observed that for small time the shear stresses calculated from Equations (33) and (34) give better result than that calculated from Equations (29) and (30). Hence, we conclude for small times shear stress components should be evaluated from Equations (33) and (34) instead of Equations (29) and (30).

We shall now discuss the asymptotic behavior of the solutions (25) and (26) for small and large values of and Re, when the motion is in steady state. In the steady state, Equation (17) becomes

(36)

Case 1): When and.

When Re is large and is of small order of magnitude, the flow becomes boundary layer type. For the boundary layer flow near the upper plate, introducing the boundary layer coordinate, we obtain the velocity distributions from (36) as

(37)

Figure 8. Shear stresses τx and τy for different K2 when σ = 0.1 and τ = 0.2.

Figure 9. Shear stresses τx and τy for different σ when K2 = 2 and τ = 0.2.

. Shear stresses τx and τy for different τ when K2 = 2 and σ = 0.1.

Table 1. Shear stress 10τx due to primary flow when K2 = 2 and σ = 0.1.

Table 2. Shear stress −102τy due to secondary flow when K2 = 2 and σ = 0.1.

(38)

where

(39)

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 boundary layer decreases with an increase in porosity parameter since decreases with increase in.

Case 2): When and.

In this case, the velocity distributions are obtained from the Equations (36) as

(40)

(41)

where

(42)

Equations (40) and (41) show that there exists a singledecker boundary layer of thickness of order

adjacent to the moving plate of the channel where is given by (42). The thicknesses of the layer decreases with an increase in Reynolds number Re while it increases with increase in porosity parameter.

4. 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 an increase in Reynolds number. An increase in the porosity of the medium both the primary and the secondary velocities increase. That is, the porosity of the medium has an accelerating influence on the flow field. In turn, it can control the velocity field by introducing porous medium in a rotating system. It is also found that the solution for small time converges more rapidly than the general solution. For steady state, 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 layer increases with an increase in porosity parameter.

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