Unsteady Couette Flow through a Porous Medium in a ... · 150 M. JANA . ET AL. tem has been obtained by Singh. et al. [9]. Guria . et al. [10] have described the unsteady Couette

Open Journal of Fluid Dynamics, 2012, 2, 149-158 http://dx.doi.org/10.4236/ojfd.2012.24016 Published Online December 2012 (http://www.SciRP.org/journal/ojfd)

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: [email protected]

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

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

Layer

1. Introduction

The flow between two parallel plates is a classical problem that has many applications in accelerators, aerody- namic heating, electrostatic precipitation, polymer tech- nology, 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-

Copyright © 2012 SciRes. OJFD

M. JANA ET AL. 150

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

uv2K . 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 mag-nitude 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

uv

2K 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 embed- ded 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 0t , the fluid as well as the plates of the channel are assumed to be at rest. At time the upper plate at 0t 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.

u0

y

, ,u v w

The equation of continuity gives 0v

y

which on inte-

gration yields 0constantv v , where for suc-tion and

0v 00 0v for the blowing at the plate.

The x-, y- and z-components of Navier-Stokes equation are

2

0

u

y22 ,

k

u u

t yv w u

(1)

0

10

pv

y k

, (2)

2

0 2

w w wv u

t y y

2 ,

kw (3)

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

The initial and boundary conditions are

0

0

0 0

0, , for 0 ,

0, , at 0 for

, 0, , at f

u w v v y h t

u w v v y t

u u w v v y h

0

0,

or t

,

0.

(4)

Introducing the non-dimensional variables

1 1 20 0

, , ,y u v t

hu w

h u u

(5)

Figure 1. Geometry of the problem.


M. JANA ET AL.


151

Equations ( Taking the Laplace transform, Equation (8) becomes 1) and (3) become 2

21 1 11 12 2

1Re 2 ,

u u uK w u

(6) 2

22 2

d d 1Re 2i 0,

dd

q qs K q

(12)

221 1 1

1 12 2

1Re 2 ,

w w wK u w

where (7)

where

0

, e sq q d .

(13) 2

2 hK

is the rotation parameter and

0Rev h

The boundary conditions for ,q s are

the Reynolds number and 2

2 02

k u

the po-

rosity param(6) and (7), we have

eter. Combing Equations

10, 0 and 1, .q s q s

s (14)

The solution of Equation (12) subject to the boundary conditions (14) is 2 1q q q2

2 2Re 2i ,K q q

(8)

where

1Re 1

21 sinh,

sinh

s aq s

s s ae ,

(15)

1 1i and i 1q u w . (9)

The initial and boundary conditions for ,q are where

22

2

Re 1=

4a

,0 0 for 0 1 andq (10) 0,

(11) 0, 0 and 0, 1 for 0.q q

2i .K

(16)

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

2 1

Re 12

22 21

sinh i 2 π 1 e, s π e

sinh i π i

nn

n

nq n

n

in

(17)

where

11 2

2 22 222 2 2 4

2

1 Re 1 Re 1π i , , 4

4 42n n K

2 .

(18)

On separating into a real and imaginary parts, we get

2

2 2 2 2

1 S S C Re 12

1 2 2 2 2 2 2 2 21

π2 2 2 2

π 1 sin πe 2

π 4

π cos 2 2 sin 2 e ,

n

n

n

C n nu

S C n

n

(19)

2

2 2 2 2

1Re 1

21 2 2 2 2 2 2 2 21

π2 2 2 2

π 1 sin πe 2

π 4

2 cos 2 π sin 2 e ,

n

n

n

C S S C n nw

S C n

n

(20)

here w

sinh cos , cosh sin ,

sinh cos , cosh sin .

C

S C

S

(21)

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 0v for the blowing at the plates) and Re > 0 (corres ing to 0 0v

Solutions for Small Time

pond

Following Carslaw and Jaegasolution of (12) subject to the bobtained by Laplace transform technique in the following form

M. JANA ET AL. 152

2

21 12i Re 1 1

,nK

nq

222

0 0

2 2

22 1 2 1

e e 2i 4

erfc erfc2 2

Reerfc erfc

8 2 2

k n

n n

n n

c dj j

c dcj dj

K

(22)

where ,2 1 , 2 1c m d m

1erfc erfc ,n n

xj x j d

0erfc erfc d , erfc .x

j x j x x erfc

The solution (22) can be written as

1

2

0

i 4 ,

0,2,4,6, ,

nn

rk

K T

r

(2where

22

12i Re 12

2

1, e e 2

K

q

3)

0

21 1

erfc erfc2 2

Reerfc erfc ,

4 2 2

0,2,4,6,

r rr

k

T

r r

c dj j

c dcj dj

r

(24)

On separating into a real and imaginary parts, we get the velocity distributions for the primary and the secon-da

ry flow as

21

Re 1 2 221 eu , cos 2 , sin 2 e ,P K Q K

(25)

21

Re 1 2 221 e , cos 2 , sin 2w Q K P K e ,

(26)

where

240 22 4

43

66 2

222

2 42

236

64

1 1, 4 4 4

1 124

4, 2 4 4

68 4

P T T K

KT

KQ K T T

KK T

4T

(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 1 and 1 against

uw in Figures 2-6 for several values of

the rotation parameter 2K , Reynolds number Re, porosity parameter and time . It is seen from Figure 2 that the primary velocity 1 decreases and the magnitude of the secondary velocity 1 increases with an increase in rotation number

uw

2K . Figure 3 reveals that both the primary velocity 1 and the magnitude of the secondary velocity 1 decreases with increase in Rey- nolds number Re. It is observed from Figure 4 that both the primary velocity 1 and the magnitude of the secondary velocity 1 increase with an increase in porosity parameter

uw

uw

. 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.


M. JANA ET AL. 153

Fi .

gure 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 me has an accelerating influence on the flow field. Thus we can control the velocity field by introducing porous m m in a rotating system. It is observed from Figure 5 at both the primary velocity and the magnitude the secondary velocity eases with an n times

dium

ediu th

of increase i

1u incr 1w

. For small values of time, we have dr e velocity omponents on using the solution give quations nd the general s n 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

awn th

olutio c

n by E 1u

(25) and

d (21w

6) a an

0 due to the primary and the secondary flows n by are give

22 2

0

π i2 21

222 2

1

i

i 2 π 1 ee

sinh i π i

x y

nnRe

n

q

n

n

(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 0 as


M. JANA ET AL. 154

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

nt t

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

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


M. JANA ET AL. 155

2 2 2 2

222 2 2 2 2 21

π2 2 2 2

ecosh 2 cos 2 π 4

π cos 2 2 sin 2 e

xn

n

n

n

2 21Re 2 sinh cos cosh sin 2 π 1

nn

(29)

2 2 2 2

2 21Re

222 2 2 2 2 21

π2 2 2 2

2 sinh cos cosh sin 2 π 1e

cosh 2 cos 2 π 4

2 cos 2 π sin 2 e

n

yn

n

n

n

n

(30)

The numerical values of the non-dimensional shear

stresses x and y at the plate 0 number

are presented in Figures 8- 0 ag t Reynolds Re for several values o

1f

ains2K , and . It is om Figure 8 that

both the ue he shea s seen fr

r stresseabsolute val of t x and y increase with increase in rotation parameter 2K while

the shear stresses the absolute value of x increases and that of the she ar stress y decrRe. Rey

eases with an increase in nolds number Re

uanforces dom

l forcesgood

d q

e i

a and

is the ratio tifies the relative i

given fl lo

domireement with

show t

of i

w R

h

nm

ynoldnant at high Rey-

the p

ertial forcesto visc portanceof these t y of for The visc r nant at e n

are nolds g yspoint of 10value

um-

ical

ous fw

ous bers while the inertia

num view.

oro t

fo

bers

ces anpes

rces a

. It is a Figures 9

ow conditions. s

hat the absolute

of the shear stress x increases while the absolute value of the shear stress y increases with a rin either porosi

n inc ease ty parameter or time .

For small times, the non-dimensional shear stresses ud e to the primary and secondary flows at the stationary

plate 0 are given by

2

2

0

12i

22 12

0

i

1e 2i 4 ,

0,1,2,3, ,

x y

nKn

rn

q

K Y

r

(31)

where

2 12 1

0

2 2

22

1 2 1erfc

2

1 2 12 1 Re erfc

8 2

Re 2 1erfc .

4 2

rr

m

r

r

mY j

mm j

mj

1

(32)

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

21

Re 2 22e 0, cos 2 0, sin 2 ex C K D K ,

(33)

21

Re 2 22e 0, cos 2 0, sin 2 ey D K C K ,

(34)

where

241 12 4

4

3

6 2

222

1 32

236

4

1 10, 4 4 4

1 124

40, 2 4 4

68 4 .

C Y Y K Y

K

KD K Y Y

KK Y

(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

3

5Y

5

. 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 Equation(34) instead of Equations (29) and (30).

We shall now discuss the asymptotic behavior of the (25) and (26) for small and large values of

s (33) and

solutions 2K dy and Re, when the motion is in steady state. In the stea

state , Equation (17) becomes

sinh i, .

sinh iq

(36)

Case 1): When 2 1K and . When Re is large and

Re 12K

s bour th

is of small order of magnitude, the flow become ndary layer type. For the boundary layer flow nea e upper plate 1 , introducing the boundary layer coordinate 1 , we obtain the velocity distributions from (36) as

Re

21 e cosu

,

(37) ary flow


M. JANA ET AL. 156

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.

Figure 10. Shear stresses τx and τy for different τ when K2 = 2 and σ = 0.1.


M. JANA ET AL. 157

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

General solution Solution for small times Re\

0.04 0.05 0.06 0.04 0.05 0.06

1 0.00217 0.00359 0.00443 0.03107 0.02736 0.01407

2 0.00129 0.00212 0.00261 0.10398 0.09122 0.04630

3 0.00075 0.00123 0.00150 0.26266 0.22920 0.11421

4 0.00043 0.00070 0.00085 0.59325 0.51417 0.25021

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

General solution Solution for small times Re\

0.04 0.05 0.06 0.04 0.05 0.06

1 0.00579 0.01111 0.01502 0.01037 0.02393 0.02489

2 0.07952

3 0.00199 0.00378 0.00507 0.08480 0.19675 0.19852

4 0.00114 0.00214 0.00285 0.18683 0.43526 0.44202

0.00342 0.00654 0.00882 0.03423 0.07917

Re

21 e sinw

,

(38)

where 2

2 2

Re 4 41 ,

2 RRe.

e

K

(39)

It is evident from Equations (37) and (38) that there exists a single-deck boundary layer of thickness of order

1Re

2O

near the moving plate of the channel

where is given by (39). The thickness of this boundary layer decreases withter

an increase in porosity parame- since decreases with increase in .

Case 2): When 2 1K and In this case, the velocity distribut ned

from the Equations (36) as

1Re . ions are obtai

Re

21 e cos ,u

(40)

Re

21 e sin

,w

(41

here

)

w

21 Re 1.

(42) 2 2, 144

KK

Equations (40) and (41) sho at there e a single-decker boundary layer of thick of orde

w th xists ness r

1Re

2O

adjacent ovin of th

channel where

to the m g plate e

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 a increase in

the porosity of the me- econdary velocities in-

crease. That is, the porosity of the medium has an accelerating influenc urn, it can control he veloc ield by ucing po medium in a

. It is also ound th lution for all tim erges general solu- n. Fo state, ptoti ior of the so-

tion i ed fo as well as large values of n ter and lds num t is observed

at a t ndary form the moving plate of the channel and the thicknesses of the layer in-

parameter.

REFERENCES G. K. Batchelor, “An Introduction to Dynamics,”

Cambridge Univer , Cam 67.

] G. S. Seth, R. N. Jana and M. K. Maiti, “Unsteady Hy-dromagnetic Couette Flow in a Rotating System,” Inter-

nReynolds number. An increase indium both the primary and the s

e on the flow field. In tt ity f introd rousrotating systemsm

fmore rapidly

at the sothan the e conv

tio r steady the asym c behavlurotatio

s analyzparame

r small Reyno ber. I

th hin bou layer is ed near

creases with an increase in porosity

[1] Fluidbridge, 19sity Press

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M. JANA ET AL. 158

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[8] d . Sh K. D. Singh an rma, “T R a hree Dimensional Couette

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Unsteady Couette Flow through a Porous Medium in a ... · 150 M. JANA . ET AL. tem has been obtained by Singh. et al. [9]. Guria . et al. [10] have described the unsteady Couette