 Open Access
 Total Downloads : 445
 Authors : Dr.P.Venkat Raman, Mr.S.Sathyendar, Mr.A.Ramesh Babu
 Paper ID : IJERTV1IS8484
 Volume & Issue : Volume 01, Issue 08 (October 2012)
 Published (First Online): 29102012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Flow through Porous Medium Between Two CoAxial Cylinders with Rotation and Linear Motion
Dr.P.Venkat Raman 
Mr.S.Sathyendar 
Mr.A.Ramesh Babu 
Director of MCA, Alluri 
Asst Professor, Department 
Ph.D Scholar, Regd. No.510CH803,Department of Chemical Engineering, National Institute of Technology, Rourkela 769008 (Odisha), India 
Institute of Management 
of Mathematics, Vaagdevi 

Sciences, 
College of Engineering, 

Warangal 506001 (A.P), 
Bollikunta, Warangal 

India 
506005 (A.P), India 
Abstract
In the present chapter, we examined the flow of a Newtonian fluid passing through complex porous medium between two coaxial cylinders which are rotating about the axis of the flow and moving parallel to the axis of the cylinders with different velocities. Flow under different situations is studied. In each case, the results are obtained when the permeability of the medium is very large and is very small. The classical Darcys effect is found very near to the axis of the cylinders and the nonDarcian effect is found near the boundaries of the flow. The effect of the permeability coefficient of the porous medium is examined in each case.
Key words and phrases: Newtonian fluid, Coaxial cylinders, Porous medium, Permeability.

Introduction
Due to applications in solid mechanics, transpiration cooling, food preservation, cosmetic Industry, blood flow and artificial dialysis, the problem of flows through porous medium has received a great attention in technological as well as in biophysical fields. The problem of flow under pressure gradient in the angular region between cylinders has important applications in hydrology. Many researchers Sharma and Gupta [1] worked on the subject. Rotating fluid flows analysis is of interest because of their applications in different branches of engineering and meteorology. Rott and Lawellen [2] have given an extensive study of rotating flows and their various applications. Raptis
and Perdikis [3] have discussed the oscillating flow in the presence of free convective flow through a porous medium. Beghel et al. [4] have examined the two dimensional unsteady free convection flow of a viscous incompressible fluid through a rotating porous medium. Pattabhi Ramacharyulu [5] examined the steady laminar flow of a viscous liquid through annulus whose walls rotate and move with constant velocities. Narasimha Charyulu and Pattabhi Ramacharyulu [6] have investigated the flow of a viscous liquid through porous region contained between two cylinders by applying the generalized Brinkmans law [7].

Formulation and solution of the problem
To investigate the problem of flow of Newtonian fluid through porous medium contained between two coaxial cylinders which are rotating slowly about the axis of the cylinders and moving slowly parallel to the axis of the cylinders. We consider the cylindrical coordinate system (r, , z) such that the zaxis lies along the length of common axis of the cylinder and r in the radial direction. The physical quantities are independent of due to symmetry of the flow and they are independent of z as the cylinders are considered to be infinite in length.
The cylinders (r = a and r = b) are assumed to rotate with angular velocities, a and b are moving with linear velocities wa,, wb parallel to the zaxis respectively.
The equation of motion of a viscous liquid through a porous medium as proposed by Brinkman is
0 P 2
k V V
together with the equation of continuity
(2.1)
(2.3)
is taken to balance the centrifugal force generated by
Div V= 0
(2.2)
the velocity component v(r), where
G P
is a
where P is the pressure,V
is the velocity field, is
constant and c is the constant of integration. z
the coefficient of viscosity of the fluid and k is the Equation (2.1) gives
permeability constant of the medium.
d 2 v 1 dv
2 1
(2.4)
The choice of the velocity
V [0, v(r), w(r)]
satisfies
2 r v 0
the equation of continuity (2.2) and the pressure
ddr2 w
r1ddrw 1
k G
2 w
P c Gz
r
r
1 [v(r)]2 dr
0
(2.5)
dr r dr k
(2.14)
a
Drag on the cylinders r = a and r = b respectively
With the boundary conditions
D 2
a
1 ka* b exp (b a) /
k w
b exp (b a) /
k w
w(r) wa ,
v(r) aa
at r a
a
k 2
1 b a
a
w(r) wb ,
v(r) bb
at r b
(2.6)
(2.15)
D 2
b 1
*
a exp (b a) /
k w
b exp (b a) /
k w
Solving equations (2.4) and (2.5) with the boundary conditions (2.6), we get
b k 2 ka 1 b
a b
(2.16)
v(r) aaT1 (r, b) bbT1 (r, a)
T1 (a, b)
a
(2.7)
2.2 When the permeability k , 1/k 0 (i.e., in the clear region)
(b2 a 2 ) ( ) a 2b2
w T (r, b) w T (r, a) ka* [T (r, b) T (r, a) T (a, b)]
v(r) b a r a b
w(r) a 0 b 0 0 0 0
b2 a 2
b2 a 2 r
(2.17)
T0 (a, b)
(bw aw ) (bw aw ) ab
(2.8) w(r) b a r a b
With
b2 a 2
b2 a 2 r
T (x, y) I
x K y
I y K x
G 2 2
2 2 log(r / a)
i i
a* G
i
k
i
k
i
k
k
i = 0, 1
(2.9)
4a r

(b

a )
log(b / a)
(2.18)
where Ii , Kiare modified Bessel functions.
2.1 When the permeability of the medium is very small (i.e., k 0, 1/k )
Equation (2.9) becomes,


SPECIAL CASES
3.1 Case (i). Flow through porous region between cylinders which are rotating with no motion parallel to the axis of the tube (i.e., wa
= wb = 0)
The velocity components are given by
1 k
v(r) aaT1 (r, b) bbT1 (r, a)
(3.1)
Ti (x, y) 2
and we get
exp (x y) /
xy
k exp (x y) / k
(2.10)
T1 (a, b)
ka* [T (r, b) T (r, a) T (a, b)] w(r) 0 0 0
(3.2)
a
b
k
v(r) 1 a3 / 2 exp (r a) /
k b3 / 2 exp (b r) /
with
, x
T0 (a, b)
y
y
x
r
(2.11)
Ti (x, y) Ii
Ki
k
Ii
k
Ki
k k
w(r) 1 a1/ 2w exp (r a) /
k b1/ 2w exp (b r) /
k
a* G
i = 0, 1
r a b
where
Ii , Ki
.
are modified Bessel functions.
ka* 1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) /
k 1
(2.12)
3.1.1 When the permeability of the medium is very small (i.e., k 0, 1/k )
Moment actig on the cylinders r = a and r = b respectively
The velocity components are given by
a
b
v(r) 1 a3 / 2 exp (r a) / k b3 / 2
exp (b r) /
k
2 3
a 2 b
1 3b r
M a 2a a
b exp(b a) /
k
(3.3)
2
k a
k a
(2.13)
w(r) ka* 1 a1/ 2 exp (r a) /
k b1/ 2 exp (b r) /
k 1
a 1
3b b2 3 r
M 2b2 exp(b a) / k
(3.4)
b b a
b
a
2
k
k
Moment acting on the cylinders r = a and r = b
respectively
G a 2 r 2 (b2 a 2 ) log(r / a)
3
a 2 b
1 3b
4
log(b / a)
2
a
M 2a 2 a
k
2 a
a b exp(b a) /
1 3b
k
k a
(3.5)
b2 3
(3.16)
3.3 Case (iii). Flow through porous region between cylinders with outer cylinder fixed
M b 2 b b a exp (b
a) / k
k
a b k 2
. (3.6)
Drag on the cylinders r = a and r = b respectively
and inner cylinder is rotating and moving
along the axis of the tube (i.e., b wb 0)
D 2
a
1 ka*
b exp (b a) /
k
In this case, the velocity components are
a
a
k 2
1
(3.7)
v(r) aaT1 (r, b)
(3.17)
D 2
b
1 ka* 1
a exp (b a) /
k
T1 (a, b)
b
k 2 b
(3.8)
w T (r, b) ka* [T (r, b) T (r, a) T (a, b)] w(r) a 0 0 0 0
T0 (a, b)
3.1.2 When the permeability k , 1/k 0 (i.e., in the clear region)
with
T (x, y) , x
y
y
x
(3.18)
The velocity components are
i Ii
Ki
k
Ii
k
Ki
k k
(b2 a 2 ) ( ) a 2b2
G
i = 0, 1
v(r) b a r a b
(3.9)
a*
b2 a
b2 a 2 r .
G .
log(r / a)
where
Ii , Ki
are modified Bessel functions
w(r)
a 2 r 2 (b2 a 2 )
4
log(b / a)
(3.10)
3.3.1 When k 0, 1/k , then
These results coincide with the results of Narasimha
The velocity components are
Charyulu and Pattabhi Ramacharyulu [24].
3.2 Case (ii). Flow through porous region
v(r) 1 a3/ 2
r a
w(r) 1 a1/ 2 w
exp (r a) /
exp (r a) /
k
k
(3.19)
between cylinders which are moving with no r. a
rotation along the axis (i.e.
The velocity components are
a b 0)

ka*
1 a1/ 2
r
exp (r
a) /
k
b1/ 2
exp (b
r) /
k 1
v(r) 0
(3.11)
3.3.1 When k , 1/k 0, then
(3.20)
w T (r, b) w T (r, a) ka* [T (r, b) T (r, a) T (a, b)]
w(r) a 0 b 0 0 0 0
a 2 r b2
a
T0 (a, b)
(3.12)
v(r) 1
b2 a 2 r 2
(3.21)
with

w ar w ab2
, x y
y x
w(r) a a
Ti (x, y) Ii Ki Ii Ki
b2 a 2 (b2 a 2 )r
k k
a* G
k
k
i = 0, 1
G a 2 r 2 (b2 a 2 ) log(r / a)
where
Ii , Ki
.
are modified Bessel functions
4
log(b / a)
(3.22)
3.2.1 When k 0, 1/k , then
v(r) 0
(3.13)
3.4 Case (iv). Flow when inner cylinder is fixed and outer cylinder is rotating and
moving along the axis of the tube (i.e.,
w(r) 1 a1/ 2 w exp (r a) / k b1/ 2 w exp (b r) / k a wa 0
r a b )


ka*
1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) /
k 1
Then,

b T (r, a)
(3.23)
3.2.2 When k , 1/k 0, then
v(r) 0
(3.14)
v(r) b 1
T1 (a, b)
w T (r, a) ka* [T (r, b) T (r, a) T (a, b)]
b 0 0 0 0
w(r) (bwb awa ) r (bwa awb ) ab
(3.15)
w(r)
T0 (a, b)
(3.24)
b2 a 2
b2 a 2 r
with
T (x, y) , x
y
y
x
i Ii
Ki
k
Ii
k
Ki
k k
G a 2 r 2 (b2 a 2 ) log(r / a)
a* G
i = 0, 1
.
4
log(b / a)
(3.34)
where
Ii , Ki
are modified Bessel functions
3.6 Case (vi). Flow through porous region

When k 0, 1/k , then
The velocity components are
between cylinders which are rotating and only inner cylinder is moving (i.e., wb = 0))
b
v(r) 1
3 / 2
b exp (b r) /
k
(3.25)
v(r) aaT1 (r, b) bbT1 (r, a)
(3.35)
r T1 (a, b)
w(r) 1 b1/ 2 w exp (b r) / k
r

ka*
1
b
a1/ 2 exp (r a) /
k
b1/ 2 exp (b r) /
k
1
w T (r, b) ka* [T (r, b) T (r, a) T (a, b)] w(r) a 0 0 0 0
T0 (a, b)
r


When k , 1/k 0, then
(3.26)
with
, x
y y
x
(3.36)
b2 r a 2
Ti (x, y) Ii
Ki Ii
Ki
b2 2 a
r 2
v(r) b 1
2
(3.27)
a* G
k
k
k
k
i = 0, 1
w(r)
bwb r b2 a 2

wb a b
(b2 a 2 )r
where
Ii , Ki
.
are modified Bessel functions.
G a 2 r 2 (b2 a 2 ) log(r / a)
4
log(b / a)
3.6.1 When k 0, 1/k , then
(3.28)
v(r)
1 a3 / 2
r
a exp (r
a) /
k
b3 / 2
b exp (b
r) /
k


Case (v). Flow when inner and outer cylinders are rotating and only outer cylinder is moving along the axis of the tube (i.e. wa = 0)
w(r) 1 a1/ 2 w
r a
exp (r a) /
k
(3.37)
v(r) aaT1 (r, b) bbT1 (r, a)
T (a, b)
(3.29)

ka*
1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) /
k 1
1
w T (r, a) ka* [T (r, b) T (r, a) T (a, b)] w(r) b 0 0 0 0
3.5.2 When k , 1/k 0, then
(3.38)
T (a, b)
(b2 a 2 ) ( ) a 2b2
0 v(r) b a r a b
(3.39)
(3.30)
b2 a 2
b2 a 2 r
with

wa ar w ab
2
a
,
w(r) b2 a 2 (b2 a 2 )r
T (x, y) I x K y I y K x
i i k i k
i k i k
G 2 2
2 2 log(r / a)
a* G
.
i = 0, 1
4a r

(b

a ) log(b / a)
(3.40)
where
Ii , Ki
are modified Bessel functions
3.7 Case (vii). Flow through porous region

When k 0, 1/k , then
between cylinders which are moving with
b
vr) 1 a3 / 2
r a
exp (r a) /
k b3 / 2
exp (b r) /
k
same velocity and rotating along the axis of the tube (i.e. wa = wb = w)
b
w(r) 1 b1/ 2 w
exp (b r) /
k
(3.31)
v(r) aaT1 (r, b) bbT1 (r, a)
T1 (a, b)
(3.41)
r
w[T (r, b) T (r, a)] ka* [T (r, b) T (r, a) T (a, b)]

ka*


1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) /
k 1
w(r) 0 0 0 0 0
T0 (a, b)

When k , 1/k 0, then
(3.32)
with
(3.42)
2 2 2 2
T (x, y) , x
y
y
x
v(r) (b b a a ) r (a b ) a b
i Ii
Ki
k
Ii
k
Ki
k k
b2 a 2
bw r
w a 2b
b2 a 2 r
(3.33)
a* G
i = 0, 1
w(r) b b .
i i
b2 a 2
(b2 a 2 )r
where I , Kare modified Bessel functions

When k 0, 1/k , then
a 0 b 0 0 0 0
b
v(r) 1 a3 / 2
r a
exp (r a) /
k b3 / 2
exp (b r) / k
w(r)
w T (r, b) w T (r, a) ka* [T (r, b) T (r, a) T (a, b)] T0 (a, b)
(3.43)
(3.54)
w(r) w a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) / k
with
T (x, y) , x
y
y
x
* 1
1/ 2
1/ 2
i Ii
Ki
k
Ii
k
Ki
k k

ka
a exp (r a) /
r
k b
exp (b r) /
k 1
a* G
i = 0, 1


When k , 1/k 0, then
(3.44)
where
Ii , Ki
.
are modified Bessel functions.
(b2 a 2 ) ( ) a 2b2
v(r) b a r a b
b2 a 2
b2 a 2 r
(3.45)
3.9.1 When k 0, 1/k , then
w(r) (r ab)w
G a 2 r 2 (b2 a 2 ) log(r / a)
v(r) 1 a3/ 2
exp (r a) /
k
b a
4
log(b / a)
r a
(3.55)
(3.46)
w(r) 1 a1/ 2 w exp (r a) / k b1/ 2 w exp (b r) /
k
r a b

Case (viii). Flow when inner and outer cylinders are moving and only outer cylinder

ka*
1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) /
k 1
is rotating along the axis of the tube (i.e.
a 0)

When k , 1/k 0, then
(3.56)

b T (r, a)

(3.47)
a 2 r b2
a
v(r) b 1
T1 (a, b)
v(r)
.
1
b2 a 2 r 2
(3.57)
w T (r, b) w T (r, a) ka* [T (r, b) T (r, a) T (a, b)]
a 0 b 0 0 0 0
w(r) (bwb awa ) r (bwa awb ) ab
w(r)
a 0 b 0 0 0 0
T0 (a, b)
b2 a 2
b2 a 2 r
(3.48)
G a 2 r 2 (b2 a 2 ) log(r / a)
with
T (x, y) , x
y
y
x
4
log(b / a)
(3.58)
i Ii
Ki
k
Ii
k
Ki
k k
a* G
.
i = 0, 1
3.10 Case (x). Flow when the cylinders are rotating with same angular velocities and
where
Ii , Ki
are modified Bessel functions
moving parallel to the axis of the tube (i.e.

When k 0, 1/k , then
a b )
v(r) 1 b3/ 2 exp (b r) /
r b
k
(3.49)
v(r) [aT1 (r, b) bT1 (r, a)]
T (a, b)
(3.59)
a
w(r) 1 a1/ 2 w exp (r a) /
k b1/ 2 w exp (b r) /
k
1
w T (r, b) w T (r, a) ka* [T (r, b) T (r, a) T (a, b)]
b
r w(r) a 0 b 0 0 0 0

ka*
1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) /
k 1
T0 (a, b)
(3.60)


When k , 1/k 0, then
(3.50)
with
T (x, y) , x
y
y
x
b2 r
a 2
i Ii
Ki
k
Ii
k
Ki
k k
2 2
v(r) b 1
b a
r 2
(3.51)
* G
i = 0, 1
(bw
a .
w(r)
b awa ) r (bwa awb ) ab
b2 a 2
b2 a 2 r
where
Ii , Ki
are modified Bessel functions.
G a 2 r 2 (b2 a 2 ) log(r / a)
3.10.1 When k 0, 1/k , then
4 log(b / a)
(3.52)
v(r) a3 / 2 exp (r a) /
r
k b3 / 2 exp (b r) /
k
3.9 Case (ix). Flow through porous region between cylinders which are moving and only
w(r) 1 a1/ 2 w exp (r a) /
(3.61)
k b1/ 2 w exp (b r) / k
inner cylinder is rotating (i.e. b
0)
r a b
v(r) aaT1 (r, b)
(3.53)

ka*
1 a1/ 2 exp (r a) /
r
k b1/ 2 exp (b r) / k 1
T1 (a, b)
(3.62)
3.10.2 When k , 1/k 0, then
w( y) ka* k cosh( y)
v(r) r
(bw aw ) (bw aw ) ab
(3.63)
cosh(h / k )
1
(3.72)
w(r) b a r a b
b2 a 2
b2 a 2 r
3.12.1 When the permeability of the medium is
G a 2 r 2 (b2 a 2 ) log(r / a)
very small (i.e., k 0, 1/k )
4
log(b / a)
(3.64)
Then,
3.11 Case (xi). Flow when the pressure gradient is absent (i.e. G = 0)
w( y) ka* exp (h y) /
k 1
(3.73)
We take the pressure gradient to be absent, then G =
0. In this case, the velocity components are given by
3.12.2 When the permeability k , 1/k 0 (i.e., in the clear region)
v(r) aaT1 (r, b) bbT1 (r, a)
a* p y 2.
T1 (a, b)
w(r) waT0 (r, b) wbT0 (r, a)]
(3.65)
(3.66)
w( y)
2 h
1
2
(3.74)
with
T0 (a, b)

Results and Discussions
T (x, y) , x
y
y
x
i Ii
Ki
k
Ii
k
Ki
k k
4.1 Radial velocity
a* G
.
i = 0, 1
When there is a rotation as well as linear motion and
where
Ii , Ki
are modified Bessel functions.
in the case when there is a rotation and there is no linear motion, the velocity profiles for different

When k 0, 1/k , then
permeabilities show parabolic nature. The increase in
b
v(r) 1 a3 / 2
r a
exp (r a) /
k b3 / 2
exp (b r) /
k
the permeability makes the velocity to increase [Figure 1(a) and Figure 2(a)]. When there is no
w(r) 1 a1/ 2 w exp (r a) /
(3.67)
k b1/ 2 w exp (b r) / k
rotation but only linear motion results the disappearance of parabolic profiles [Figure 3].
r a b
(3.68)
From Figure 4(a) and Figure 5(a), it is observed that the velocity profiles are reversed. The velocity at the

When k , 1/k 0, then rotating and moving cylinder is higher than the
(b2 a 2 ) ( ) a 2b2
velocity of the fluid at the cylinder which is having
v(r) b a r a b
b2 a 2
. (bw aw )
b2 a 2 r
(bw aw ) ab
(3.69)
no rotation and no linear motion. It is also observed that increasing the permeability increases the
w(r) b a r a b
(3.70)
velocity at different points. From Figure 6(a) and
b2 a 2
b2 a 2 r
Figure 7(a), the radial velocities profiles are not affected whether the outer cylinder or inner cylinder

Case (xii). Flow between two infinite parallel plates
Flow through the porous region bounded by two infinite parallel plates y = Â± h can be derived by taking a ~ b = 2h, as the distance between the two plates with respect to the Cartesian coordinate system (x, y, z). The velocity components are given by
is moving are at rest. From Figure 9(a) and Figure 10 (a), when the cylinders are in linear motion and one of the cylinders in rotation, the velocity profiles are reversed such that the velocity will be higher at the rotation cylinder and lower at the nonrotating cylinder.
4.2 Linear velocity
v(r) 0
ka* [T (r, b) T (r, a) T (a, b)] w(r) 0 0 0
(3.70)
Figure 2(b) shows even in the absence of linear motion of the cylinders, the rotation of the cylinder results the fluid flow along the axis.
In the absence of rotation of the cylinder the linear
T1 (a, b)
ka* b sinh(r a) a sinh(r b)
velocity profiles shows different flow profiles as
r sinh(a b)
If r = b + h + y, then we get
1
(3.71)
shown in the figure.3. From Figure 4(b) and Figure 5(b), the velocity profiles are in reverse direction when one of the cylinders is rotating and moving while other is fixed.
k=0.1
k=0.2
k=0.3
In general, it is observed that from equations (2.11)
and (2.12) when 1 = r a, 2 = b – r, becomes very 5
large, then
and
v(r) 0
w(r) ka*= a constant.
4.5
4
These equations show that as the distance from the axis to the wall increases, the velocity w(r) = a constant. The classical Darcy effect is felt in a core very near to the axis of the cylinder. v(r) = 0 shows that there exist a thin layer between the cylinders far away from the boundaries of the cylinder, where the velocity v(r) is zero.
______ v(r) ______
3.5
3
2.5
2
1.5
k=0.1
k=0.2
k=0.3
1
5
4.5
0.5
0
0 0.5 1 1.5 2 2.5
_______ r _______
4
3.5
Figure 2(a). Velocity profiles v(r) of the fluid for different values of k
(a 1, b 2, a 2, b 3, wa wb 0)
______ v(r) ______
3
2.5
k=0.1
k=0.2
k=0.3
0.6
2
1.5
0.5
w(r)
1
0.4
0.5
0
0 0.5 1 1.5 2 2.5
_______ r _______
0.3
0.2
Figure 1(a). Velocity profiles v(r) of the fluid for different values of k
k=0.1 k=0.2
k=0.3
(a 1, b 2, a 2, b 3, wa 2, wb 3)
3
0.1
0
0 0.5 1 1.5 2 2.5
r
Figure 2(b): Velocity profiles w(r) of the fluid for
w (r)
2 different values of k
(a 1, b 2, a 2, b 3, wa wb 0)
1
k=0.1
k=0.2
k=0.3
0.5
1
1.5
2
2.
3
0
0 0.5 1 1.5 2 2.5 2
w(r)
1
1
2
0
3 0 5
r 1
Figure 1(b). Velocity profiles w(r) of the fluid for different values of k
(a 1, b 2, a 2, b 3, wa 2, wb 3)
2
3 r
Figure 3. Velocity profiles of the fluid for different values of k
(a 1, b 2, a 0, b 0, wa 2, wb 3)
k=0.1 k=0.2
k=0.3
2.5
k=0.1
k=0.2 k=0.3
1.5
1
0.5
0
0
0.5
1 1.5
2
2.5
r
0
0.2 0 0.5 1 1.5 2 2.5
w (r)
2 0.4
0.6
v(r)
0.8
1
1.2
1.4
1.6
1.8
2
r
Figure 4(a). Velocity profiles of the fluid for different values of k
k=0.1 k=0.2
k=0.3
(a 1, b 2, a 2, b 0, wa 2, wb 0)
Figure 5(b). Velocity profiles of the fluid for different values of k
(a 1, b 2, a 0, b 3, wa 0, wb 2)
2.5
w(r)
2
1.5
5
k=0.1
k=0.2
k=0.3
4.5
4
3.5
______ v(r) ______
3
1 2.5
2
0.5
1.5
0
0 0.5 1 1.5 2 2.5
r
1
0.5
0
0 0.5 1 1.5 2 2.5
_______ r _______
Figure 4(b). Velocity profiles of the fluid for different values of k
Figure 6(a). Velocity profiles v(r) of the fluid for different values of k
k=0.1
k=0.2
k=0.3
(a 1, b 2, a 2, b 0, wa 2, wb 0)
4.5
(a 1, b 2, a 2, b 3, wa 0, wb 3)
k=0.1
k=0.2
k=0.3
0
4
v(r)
3.5
3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5
r
0.2 0 0.5 1 1.5 2 2.5
w (r)
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
r
Figure 5(a). Velocity profiles of the fluid for different values of k
(a 1, b 2, a 0, b 3, wa 0, wb 2)
Figure 6(b). Velocity profiles of the fluid for different values of k
(a 1, b 2, a 2, b 3, wa 0, wb 2)
k=0.1
k=0.2
k=0.3
2.5
2
k=0.1
k=0.2
k=0.3
1.5
1
0.5
0
0.5 0
0.5
1 1.5
2
2.5
1
1.5
2
r
5
4.5
4
3.5
______ v(r) _____
w(r)
3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5
_______ r _______
Figure 7(a). Velocity profiles v(r) of the fluid for different values of k
k=0.1 k=0.2
k=0.3
(a 1, b 2, a 2, b 3, wa 2, wb 0)
2.
5
w (r)
2
1.
5
1
0.
5
Figure 8(b). Velocity profiles of the fluid for different values of k
k=0.1
k=0.2
k=0.3
(a 1, b 2, a 2, b 3, wa wb 2)
4.5
4
v(r)
3.5
3
2.5
2
1.5
1
0.5
0
0
0 0.5 1 1.5 2 2.5
r
Figure 7(b). Velocity profiles of the fluid for different values of k
(a 1, b 2, a 2, b 3, wa 2, wb 0)
0 0.5 1 1.5 2 2.5
r
Figure 9(a). Velocity profiles of the fluid for different values of k
k=0.1 k=0.2
k=0.3
(a 1, b 2, a 0, b 3, wa 2, wb 3)
3
k=0.1
k=0.2
k=0.3
w(r)
5 2
4.5
4
_____ v(r) ______
3.5
3
2.5
1
0
0 0.5 1 1.5 2 2.5
1
2
2
1.5
1 3
0.5 r
0
0 0.5 1 1.5 2 2.5
_______ r _______
Figure 8(a). Velocity profiles v(r) of the fluid for different values of k
(a 1, b 2, a 2, b 3, wa wb 2)
Figure 9(b). Velocity profiles w(r) of the fluid for different values of k
(a 1, b 2, a 0, b 3, wa 2, wb 3)
k=0.1 k=0.2
k=0.3
4
3
k=0.1
k=0.2
k=0.3
2
1
0
1 0
0.5
1 1.5
2
2.5
2
3
4
5
r
2.5
2
1.5
v(r)
w(r)
1
0.5
0
0 0.5 1 1.5 2 2.5
r
Figure 10(a). Velocity profiles of the fluid for different values of k
k=0.1
k=0.2
k=0.3
(a 1, b 2, a 2, b 0, wa 2, wb 3)
2.5
2
w(r)
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5
r
Figure 10(b). Velocity profiles of the fluid for different values of k
k=0.1 k=0.2
k=0.3
(a 1, b 2, a 2, b 0, wa 2, wb 3)
3.5
v(r)
3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5
r
Figure 11(a). Velocity profiles of the fluid for different values of k
(a 1, b 2, a b 2, wa 3, wb 4).
Figure 11(b). Velocity profiles of the fluid for different values of k
(a 1, b 2, a b 2, wa 3, wb 4)


Conclusions
In radial velocity the increase in permeability makes the velocity to increase. Even in the absence of linear motion of cylinders, the rotation of the cylinder results the fluid along the axis and as the distance from the axis to the wall increases, the velocity becomes constant. The nonDarcian effect is observed near the boundaries and the Darcian effect is observed in the porous medium away from the boundaries.

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