2.1 INTRODUCTION
Flow between rotating porous boundaries is of practical as well as theoretical interest. The
problems of gaseous diffusion, boundary cooling and lubrication of porous bearings are a few
examples. These have applications in aerospace, chemical, civil, environmental, mechanical and
bio-mechanical engineering.
The unsteady squeezing flow of a viscous incompressible fluid between two parallel disks
moving normal to their own planes occurs in hydrodynamical machines, particularly in turbomachinery. These squeezing flows are useful in polymer processing, compression and injection
moulding. Lubrication equipment is also modeled by squeezing flows. Earlier studies of
squeezing flows involved the solution of Reynolds equation. The study involving full NavierStokes equations is more useful in the analysis of porous thrust bearing and squeeze films
involving high velocities. The problem of unsteady squeezing of a viscous incompressible fluid
between two parallel disks moving normal to their own surfaces is basic unsteady flow.
The unsteady motion of a viscous, incompressible and electrically conducting fluid
squeezed between two parallel disks have been analyzed by Bhattacharya et al (1996a). The
lower disk was set in rotation with an arbitrary time dependent angular velocity (t )   (t ) ,
while upper disk approaches the lower disk with a time-dependent velocity. They analyzed the
effect of Reynolds number and magnetic parameter on velocity field, load and torque. They
found that the load on the upper disk increased significantly with increase in magnetic parameter
and it also increased with decrease in gap between the disks. Torque on the lower disk increased
with increase in magnetic parameter and angular velocity of this disk. The problem of a squeeze
film between two rotating disks, one with a porous facing, was studied by Hai (1971). He
derived the governing equations according to lubrication theory. It was found that the fluid in the
film region satisfied the modified Reynolds equation and the flow in the porous region satisfied
Poisson's equation. The problem was solved analytically using Fourier expansions. He obtained
the solution for load capacity and pressure distribution. The film-thickness and time relation was
taken in integral form. Wang (1976) obtained similarity solution, when a viscous fluid is
squeezed between two parallel disks spaced a distance  1  t / T apart. Rukmani and Usha
(1994) have studied squeezing flow between two disks of varying width. The solution is obtained
22
as a power series in a single non-dimensional parameter (squeeze number). They studied the
cases
(i) constant velocity squeezing, (ii) constant squeezing force and (iii) constant power squeezing.
The gap width h( ) is obtained when the top disk moves with constant velocity, constant force
or constant power. Jackson (1963) examined the squeezed film of liquid between two parallel
surfaces. Approximate expressions of velocity were obtained by an approximate iterative
solution of the continuity and momentum equations. The radial pressure distribution in a
squeezed film was found to be due partly to the action of viscosity and partly to inertia effects.
The latter causes the relationship between the reaction on the surfaces and their relative velocity
to be non-linear. This effect was significant for conditions where the Reynolds number based
upon the distance between the surfaces and their relative velocity is greater than unity. The
results obtained should be of interest in connection with the study of the performance of
transiently loaded bearings in reciprocating engines, and a possible application in the field of
chemical engineering might arise in connection with the phenomenon of adhesion. Usha and
Vimala (2002) used energy integral approach to study the behavior of curved squeeze film taking
into consideration the inertial effects. Ishizawa (1966) has shown that when the angular
velocities of the disks were time-dependent, the Navier-Stokes equations describing the flow
between two disks could also be reduced to a pair of coupled non-linear ordinary differential
equations.
Bhattacharya et al (1996b) examined the unsteady flow and heat transfer of a viscous
fluid confined between two parallel disks. The disks were allowed to rotate with different time
dependent angular velocities, and the upper disk approaches the lower disk with a constant
speed. They solved the governing partial differential equations numerically by a fourth-order
accurate compact finite difference scheme. They have analyzed the effect of the rate of
squeezing and disk angular velocities on the normal forces and torques on the rotating surfaces.
Elshekh et al (1996) examined the flow of an electrically conducting viscous fluid film squeezed
between two rotating disks which at time t were spaced a distance D (1   t )
1/ 2
apart. The
combined effects of vertical motion of the upper disk, the rotational motion of the two disks, the
magnetic forces on the velocity profiles, the load capacity and the torques that the fluid exerts on
the disks were studied. Their results showed that the rotational motion of the lower disk and the
axial component of the magnetic force have opposite effects on the load capacity and the torque
23
exerted on the lower disk. They observed that the torque exerted on the two disks reaches its
maximum absolute value when the two disks rotate in opposite directions with the same angular
velocity.
The validity of the no-slip condition at the surface of a saturated porous material is a
matter of reasonable doubt. When saturated, the surface pores are filled with fluid to the level of
their respective perimeters, so that a smooth boundary of the shape of the nominal surface
results. When the surface is so smoothened out, no microscopic irregularities are left to cause
the no-slip on macroscopic scale. The existence of the slip velocity is connected with the
presence of a thin layer of streamwise moving fluid just beneath the surface of the porous
material. The fluid in this layer is pulled along by the flow external to the porous material. The
existence of a slip velocity has been observed experimentally by Beavers and Joseph (1967).
The analysis of the squeezed film between porous rectangular disks was extended by Hai
(1972) to include the effect of velocity slip at the fluid and porous material interface. He
presented the modified equations for calculating the pressure, the load-carrying capacity; and the
film thickness and time relation. He observed that the existence of slip velocity will further
reduce the load-carrying capacity and the response time of the squeeze film. Prakash and Vij
(1976) analyzed the squeeze film between two rotating annular disks, one with a porous facing.
They have included the effect of velocity slip at the porous surface through the Beavers-Joseph
slip model. The problem was solved analytically using the separation of variables method. They
observed that slip velocity has reduced the load capacity and the response time of the squeeze
film.
Rajvanshi (1981) investigated the effect of slip velocity in the axial current induced pinch
effect on the squeeze-film behavior for porous annular disks on the assumptions of
hydromagnetic lubrication to the Navier-Stokes equations. He studied the effect of slip velocity
and an axial current induced pinch on the load capacity and film thickness-time. Laun et al
(1999) investigated analytical solutions for squeeze flow with partial wall slip. The results for the
squeeze force as a function of the squeezing speed reduce to the Stefan and Scott equations in the
no-slip limit. They noted that the slip velocity at the plate increases linearly with the radius upto
the rim.
Unsteady flow between two rotating disks with heat transfer was studied by Ibrahim
(1991). It was found that the temperature of the fluid increased with the decrease of Prandtl
24
number and the heat flux parameter individually. It was established that the rotation of the two
disks has a very small effect on the temperature of the fluid and the heat transfer process, while
the rapid normal motion of the upper disk has a dominant effect on the temperature of the fluid
and the heat transfer process. Singh and Rajvanshi (1989) have analyzed the flow between two
rotating disks one being naturally permeable. Laminar flow between a fixed impermeable disk
and a porous rotating disk have analyzed by Kavenuke et al (2009).
Rashidi et al (2008) investigated the flow of a viscous incompressible fluid between two
parallel disks due to the normal motion of the disks. The disks were placed at a distance
z  l (1   t )1 / 2 apart, where l is the position of the disks at time t  0 and  a constant of
1
dimension [T ] which characterizes unsteadiness. The unsteady Navier-Stokes equations were
reduced to a nonlinear fourth order differential equation by using similarity solutions. The
governing equations were solved analytically by means of Homotopy analysis method (HAM).
They have verified the validity of the HAM numerically by fourth-order Runge-Kutta.
Hamza and MacDonald (1984) considered the case where two parallel disks in an
unsteady rotation were subjected to a velocity component in a direction perpendicular to their
planes. They obtained a similarity solution of the governing equations and results were compared
with those obtained numerically.
Bhatt and Hamza (1996) have analyzed similarity solutions for the squeeze film flow
between two rotating naturally permeable disks. The disk permeabilities were taken as
k i (1   t  ) at any time t  , i  1, 2 for the lower and upper disks respectively. The effects of
rotation and permeabilities have been studied on velocity profiles, the load capacity, and the
torques on the disks. They observed that if the disks have same permeability, then there exists a
critical value of permeability above and below which the behaviour of various physical
quantities changes. Kumari et al (1995) examined MHD flow between two parallel infinite disks
such that upper disk performed normal oscillation towards the lower disk. The governing NavierStokes equations were reduced to a set of ordinary differential equations. These have been solved
by using shooting method. The effects of squeeze Reynolds numbers, Hartmann number and
rotation of the disk on the flow pattern, load and torque have been studied in detail.
25
Rosemarie (2007) examined the temperature and heat transfer profiles of MHD flow
squeezed between two permeable parallel rotating disks in presence of a magnetic field. She has
investigated the cases when (i) each disk was set at a constant temperature
and (ii) when upper disk was maintained at a constant uniform temperature and the lower disk
was subjected to heat flux.
This chapter has been divided into two parts.
In Part-I the squeezing flow of viscous incompressible fluid in a highly permeable medium
between two parallel, permeable rotating disks has been investigated. The effect of rotation and
permeability has been studied on the velocity profiles. The slip condition at the surface of the
permeable disks has been assumed. The permeability of the disks enters through slip condition.
The disk permeability can be taken to be the
same or it can be set to be different for each disk.
In Part-II the squeezing flow between two parallel rotating disks has been analyzed. Both
the disks are made up of a porous material with an impermeable surface at the bottom. The
region between two disks is divided into two parts. The upper part designated as region-I, is
filled with viscous incompressible fluid, while the lower part is region-II, and is highly
permeable porous medium. The flow in the region-I is governed by Navier-Stokes equations,
while in region-II Brinkman equations are valid. The effect of permeability of the disks enters
through the slip conditions imposed on the disks.
PART-I: SQUEEZING FLOW BETWEEN TWO PARALLEL
ROTATING NATURALLY PERMEABLE DISKS.
2.2 FORMULATION OF THE PROBLEM
26
We consider a thin film of a highly permeable medium saturated with Newtonian
fluid squeezed between two parallel circular disks with different permeability. The disks are
allowed to rotate in their own planes about the z  -axis with different angular velocities. The two

disks are separated by a distance h (t ) at any time t  . The upper disk is set in motion along the
z  - axis with velocity dz * / dt * towards the lower disk which remains at a fixed position z   0.
The governing equations for flow through porous medium as suggested by Brinkman (1947) are
u

r

u
u

u 
r 
u
r



 w
w
z


u 
z 
0

v 2
r
(2.2.1)

  2u 
1 u  u   2 u    





 r 2 r  r   r 2  z 2   k u
 r 
0


1 p

 1 v  v   2 v   2 v    
v 
u v
 v
    2  2  2   v

w



r 
z 
r
r
r
z  k 0
 r r
27
(2.2.2)
(2.2.3)
u
w
r 
 w
w

z 
  2 w
1 w  2 w   




 2  
w
 r 2
 z 
r  r 
z  k 0

1 p
(2.2.4)
where u  , v  , w  are the velocity components in the direction of r  ,   and z  . p is the
pressure, k 0 is the permeability of the medium,  is the kinematic viscosity and  is the density
of the fluid. Owing to symmetry



  0.
With a view to non-dimensionalize the physical quantities, we introduce the following
characteristic length, time and angular velocity H , 
1
and  respectively. The lower and the
upper disks are assumed to rotate with angular velocities 1 (1   t  ) 1 and  2 (1   t  ) 1
respectively. The permeabilities of lower disk, upper disk and porous medium are taken in the



form k1 (1   t ), k 2 (1   t ) and k (1   t ) respectively.
The boundary conditions on the disks are assumed in the form
u
*
z
*
u
*
z
*

 1u *
1 r * 
*

,

v 
* 
* 
z *
1


t
k1 

v
,
k1*

  2u *
k
*
2
where  1 and
,
1 
*
v
*
z
*

*
w  0 on z *  0
2  *
2r* 
v 
,
* 
* 
1


t
k2 

w 
*
dh
dt
*
on z  h(t )
*
*
(2.2.5)
 2 represents slip parameters for lower and upper disk respectively.
Following Wang (1976), we introduce the following non-dimensional quantities
 r  f ( y) 
r   g ( y)
  H f ( y)

u ( y) 
, v ( y) 
, w ( y) 


2(1   t )
(1   t )
1 t

(2.2.6)
where
y
z

H 1 t
(2.2.7)

With a view to maintain simplicity, we replace r   r ,
z   z and t   t and
prime denotes derivative with respect to y.
Using equations (2.2.6) and (2.2.7), governing equations (2.2.2) – (2.2.4) take the form
r 2
1 p

 r 4(1   t ) 2
2
 f 
H f 
2
2 2




2
f
f

(
f
)

N
g

 s

k Re s 
 Re
28
(2.2.8)
g 
Re s

H 2g
k Re s
 2 f  g  2 f g  0
(2.2.9)
2
3
f 
1 p   H  f  2 f f 


 s 
 y 2(1   t )  Re
H
 H k 
(2.2.10)
 H 2
2
where Re (Squeeze Reynolds number) 
, and N 
.
2

s
The factor ‘2’ has been introduced for notational convenience. Equation (2.2.1) is satisfied
identically.
Equation (2.2.10) implies that
2 p
0
yr
(2.2.11)
Equation (2.2.8) gives
 r
 p

yr 4(1   t ) 2
2
2
2
 f iv
H f  
2
 s  2 f f   2 N gg  

k Re s 
 Re
(2.2.12)
Equation (2.2.11) and (2.2.12) imply that
f
iv


 Re  2 f f   2 N g g  
s
2
H f 
k
2
(2.2.13)
Modified boundary conditions are
f (0)  1 f (0), g (0)  1 ( g (0)  1),
f (0)  0
f (1)  2 f (1), g (1)  2 ( g (1)  s),
where i   i H / k i
f (1)  1 / 2
(2.2.14)
i  1, 2 . Ratio of the rotational velocities of the disks is defined as
s   2 / 1 .
Governing equations are solved using regular perturbation technique. The approximate
solutions for the coupled non-linear equations is obtained for Re  o (1) .
s
f and g are expanded in the following forms
( f , g )  ( f 0 , g 0 )  Re s ( f1 , g1 )  O(Re s ) 2 ,
Equations (2.2.8), (2.2.12) and (2.2.15) give
29
(2.2.15)
f 0iv 
H2 
f0
k
(2.2.16)
H f 1
 2 N g 0 g 0 
 2 f 0 f 0
k
2
f
iv
1
2
g 0 
g1
(2.2.17)
H2
g0
k
(2.2.18)
H2
g1  2 f 0 g 0  2 f 0 g 0  0
k
(2.2.19)
Corresponding boundary conditions reduce to
f 0 (0)  0,
f 1 (0)  0,
f 0 (0)  1 f 0(0),
1
f 0 (1)  ,
2
f 1 (1)  0,
f 0 (1)   2 f 0(1),
g 0 (0)  1{g 0 (0)  1},
f 1(0)  1 f 1(0),
f 1(1)  2 f 1(1),
g1 (0)  1 g1 (0),
g 0 (1)  2 {g 0 (1)  s},
g1 (1)  2 g1 (1),
(2.2.20)
Finally, we have
f 0 ( y)   3  y 4   2 e
g 0 ( y)   6 e
y
y
f 1 ( y )   26  y 25   24 e
y
 y
(2.2.21)
y
  5e
5 1 4  
ye
2  
  1e

(2.2.22)
y
 2 4
2
  23e
2
y e
 y
y

 5 

  2 4   2  3  ye
 2 



 1 4
2
2
y e
 y

y
7 2
e
12 2

    1 3

y

 8 2
e
12 2
y
(2.2.23)
g1 ( y)   39e
y
  33 y 2 e
  38e
 y
 y
  30 ye
y
  31 ye
 y
  32 y e
2
y
  29
(2.2.24)
where  , A1 to A4 and  1 to  39 are constants recorded in APPENDIX - 1.
2.3 DISCUSSION
30
We discuss the results for various values of s .
(i) s  0 is the case when the upper disk is stationary,
(ii) s  0 implies the rotation of both the disks in the same direction, and
(iii) s  0 means the rotation of the two disks is in opposite directions.
It is assumed that the permeabilities of the disks are small. It is noted here that large values of  i
corresponds to small permeabilities since i 
i H
.
ki
Velocity Profiles
For the numerical work we assume Re s  0.01 and N  30 .
A physical interpretation of N is given as follows
N
where Re R 
2


Re
R
Re
s
H2
is rotational Reynolds number. Therefore N defines the ratio of

Rotational Reynolds number to Squeeze Reynolds number.
Fig. 2.2 represents the profiles of f ( y) at 2  0.05 and s  0.5 . It is clear from the
figure that f  y  decreases with decrease in the permeability (as 1 increases) of the lower disk.
It is clear from the fig. 2.3 that keeping 1  0.05 fixed for the lower disk, f  y  decreases with
increase in the value  2 for the upper disk, that is, as permeability of the upper disk decreases.
Effect of varying rotational ratio s on f  y  is shown in fig. 2.4. Profiles have been drawn for
the values of parameters as 1  0.5, 2  0.5 , N  30 and Re s  0.01.
It is seen that with increase in the values of rotational ratio s in positive sense, the f  y 
decrease sharply. Fig. 2.5 describes f  y  with varying rotational ratio s at 1  0.5, 2  0.5
and Re s  0.01. Figure shows the decreasing behavior of the f  y  , when rotational ratio, s  0
and decreases continuously.
31
32
33
The effect of permeability of the porous region is shown in fig. 2.6. It is seen that
profiles of f ( y) decrease with increase in permeability k of the porous region. If permeabilities
of both the disks decrease at the same rate then profiles of f ( y) decrease as shown in fig. 2.7.
Fig. 2.8 analyzes the effect of permeability of the lower disk on the profiles of
g ( y) at
2  0.05 . It is also noted that g  y  increases with increase in the value of 1 , that is, with
decrease in permeability of the lower disk. g ( y ) decreases with decrease in permeability of the
upper disk as shown in fig. 2.9. Effect of permeability of the porous region on g ( y ) is shown in
fig. 2.10. It is noted that g ( y ) increases with increase in the value of k in the first half and it
shows reverse behavior in the remaining region. The concavity changes at y  0.58 .
34
35
36
Skin-friction
The shear stress on the upper disk is given by
 
r z y 1

 r
 
3/ 2
 2 H (1  t )

 f ( y )y 1


(2.3.1)
Let the coefficient of skin friction C f at the upper disk y  1 be defined as
C 
f
y 1

2(1   t ) 3 / 2
 r z y 1  r  f ( y)y 1
  r0
r0
(2.3.2)
where r0 is a characteristic distance along the radial direction.
Similarly the coefficient of skin friction on the lower disk y  0 is defined as
C 
f
y 0

2(1   t )
  r0
3/ 2
 
r z y 0
37

r
 f ( y)y 0
r0
(2.3.3)
The effect of the rotational ratio s on the coefficient of skin friction at both the disks is
shown in fig. 2.11. As we increase the value of rotational ratio s , the coefficient of skin friction
increases at the lower disk and it also increases numerically on the upper disk.
Torque
The shearing stress component  z which acts in the plane of the disks, produces a force
in the  -direction. If edge effects are neglected, the torque on a disk of radius r0 is given by
r0
    z  2 r 2 dr
0
Thus torque on the upper and lower disks is given by
38
(2.3.4)
   r04
( ) upper 
g (1)
2 H (1   t ) 3 / 2
( ) lower 
(2.3.5)
   r04
g (0)
2 H (1   t ) 3 / 2
(2.3.6)
Variation of g (0) and g (1) for various values of rotational ratio s with N  30 and
s
Re  0.01 have been shown in Table 2.1. It is seen that the magnitude of torque on both the
disks increases with increase in the value of rotational ratio s .
Table – 2.1 Values of g (0) and g (1)
s
0
0.5
1
2
g (1)
-0.0014
-0.1533
-0.3053
-0.6091
g (0)
-0.3605
-0.3805
-0.4004
-0.4403
2.4 COMPARISON OF NUMERICAL METHOD AND
PERTURBATION METHOD
With a view to authenticate the analytical results we have solved the two point boundary
value problem in MATLAB using bvp4c solver. The bvp4c code uses Simpson’s method and the
solution is approximated over the entire interval with respect to the boundary conditions. Table 2.2 shows the comparison of the values of
f ( y) obtained by both the methods for
1  0.05, 2  0.05 and s  0.5 .
Table-2.2 Comparison of Numerical method and Perturbation Method.
39
y
0
0.2
0.4
0.6
0.8
1
Numerical
0.0000
0.0932
0.1907
0.2923
0.3958
0.5
Perturbation 0.0000
0.0997
0.1999
0.3001
0.4003
0.5
Method
Method
2.5 CONCLUSION
It is observed that the profiles of f ( y) decrease with decrease in permeabilities of both the
disks. These decrease with increase in the value of rotational ratio of the disks. On increasing the
permeability of the porous region, the profiles of f ( y) decrease, but the profiles of g ( y ) increase
in the first half and then decreases. The coefficient of skin friction increases at the lower disk and
it also increases numerically on the upper disk as the value of rotational ratio s increases. We
observed that torque on both the disks increases numerically with increase in the value of
rotational ratio s .
40
PART-II: SQUEEZING FLOW BETWEEN PARALLEL ROTATING
NATURALLY PERMEABLE DISKS WITH FLUID –
POROUS INTERFACE.
2.6 FORMULATION OF THE PROBLEM
We consider squeezing flow between two parallel rotating disks. Both the disks are made up
of a porous material with an impermeable surface at the bottom. The region between two disks is
divided into two parts. The upper part is a region-I, and is filled with viscous incompressible
fluid, while the lower part is a region-II, and is highly permeable porous medium.
z

 1/ 2
z  H (1   t )
k2
Region - I

z 0
Region - II

 1/ 2
z   H (1   t )
k1
Figure 2.12 Physical Model of the problem
The disks rotate in their own planes about the z  -axis with different angular velocities. Initially,

the disks are positioned at z   h(t ) , where h(t  )  H (1   t  )1 / 2 at any time t  . H and  1
denote the characteristic length, time respectively. The lower and the upper disks are assumed to
41
rotate with angular velocities 1 (1   t  ) 1 and  2 (1   t  ) 1 respectively. The permeabilities
of lower disk and upper disk are taken in the form k1 (1   t  ) and k 2 (1   t  ) respectively.
The flow in the free-fluid region - I is governed by equations
u

r

u

u

r
u 



w
r 
w
z

0

u 


z 
(2.6.1)
v 2
r

  2u 
1 u  u   2 u  





 r 2 r  r   r 2  z 2 
 r 


1 p
(2.6.2)
 1 v  v   2 v   2 v  
      2  2  2 
r
r
z 
 r r
(2.6.3)
  2 w
1 w  2 w 
u
w

   2  
 2 
 z 
r 
z 
r r 
z 
 r
(2.6.4)
u

v 
r 

w
v 

w

z 


u v
r
w

1 p




where u , v , w are the velocity components in the direction of r ,  and z .
The flow in the porous region –II is governed by the Brinkman equation
U
r


0

U
r



W
z


0
(2.6.5)
  2U 
1 U  U   2U    

 U





2 
 2
 r 
r  r  r  2
z   k 0
 r
1 P
(2.6.6)
 1 V  V   2V   2V    
 V
0    
 2 


2
2 
r
r 
z   k 0
 r r
(2.6.7)
  2W 
1 W   2W    

 W
0
 
 


2
2 
 z 
r

r
z  k 0
 r
(2.6.8)
1 P
The permeability of the porous medium is taken as k 0  k (1   t  ) .
The effect of permeability enters through the slip boundary conditions.
At z  h(t )
*
*
u
*
z
*

  2u *
k 2*
,
2  *
2r* 

,

v 
* 
* 
z *
1


t
k2 

v
*
*
At z  0
42
w 
*
dh
dt *
w*  W * , u *  U * , v *  V *
 ezr I

  ezr

II
,
 ez I

  ez

II
At z *  h(t * )
U
*
z *

 1U *
,
k1*
1 r * 
V 
 , W *  0, dP  0

*
*

z
1   t 
dz *
k1* 
V

1 
*

*



(2.6.9)

The velocity components (u , v , w ) and (U , V , W ) are taken as

u ( y) 

 r  F f ( y)
U ( y) 
2(1   t  )


, v ( y) 
 r  F p ( y )

2(1   t )
r  G f ( y)
(1   t  )

, w ( y) 
  H F f ( y)


, V ( y) 
r  G p ( y)

(1   t )

, W ( y) 
1 t
  H Fp ( y)
1 t

(2.6.10a)
(2.6.10b)
where
y
z

H 1 t
(2.6.11)

With a view to maintain uniformity, we replace r   r ,
z   z and t   t .
Prime denotes derivative with respect to y.
Solution for region-I
Using equations (2.6.10a), (2.6.11) in (2.6.2) to (2.6.4), we get
 F f

r 2
1 p

 2 F f F f  ( F f ) 2  N 2 G 2f 
2 
s
 r 4(1   t )  Re

G f
Re
s
 2 F f G f  2 F f G f  0
(2.6.13)
2
3
1 p   H  F f 2 F f F f 




 y 2(1   t )  Re s
H 
where Re (Squeeze Reynolds number) 
s
(2.6.12)
(2.6.14)
 H 2
2
and N 
.
2

43
Equation (2.6.14) implies that
2 p
0
yr
(2.6.15)
Equation (2.6.12) gives
 r
2 p

yr 4(1   t ) 2
2
 F fiv

2
 s  2 F f F f 2 N G f G f 
 Re

(2.6.16)
Equation (2.6.15) and (2.6.16) imply that

F f  Re  2F f F f  2 N G f G f
iv
s
2

(2.6.17)
Solution for region-II
Using equations (2.6.10b) in (2.6.6) to (2.6.8), we get
r 2
1 P

 r 4(1   t ) 2
G p
Re
s


k
 F p  F p 
 s 

k 
 Re
(2.6.18)
Gp  0
(2.6.19)
 2 H  F p  F p 
1 P




 y 2(1   t )  Re s
k 
(2.6.20)
where   2 / 
Equation (2.6.20) implies that
 P
0
yr
2
(2.6.21)
Equations (2.6.18) give
 r
 P

yr 4(1   t ) 2
2
2
 F p iv  F p
 s 

k 
 Re
(2.6.22)
Equations (2.6.17) and (2.6.18) show that
44
Fp 
iv
 Re s
k
F p
(2.6.23)
The boundary conditions take the form
F f(1)  2 F f (1), G f (1)  2 {G f (1)  s}, F f (1)  1 / 2
F f (0)  Fp (0), G f (0)  G p (0),
F f (0)  Fp (0), F f(0)  1 Fp(0), G f (0)  1G p (0)
Fp(1)  1 Fp (1), G p (1)  1{G p (1)  1}, Fp (1)  0,
P(1)  0
(2.6.24)
The slip conditions are given by i   i H / k i where  i , for i  1, 2 represents the slip
parameter for the lower and upper disks respectively. Ratio of the rotational velocities of the
disks is defined as s   2 / 1 .
2.7 METHOD OF SOLUTION
Governing equations are solved using regular perturbation technique. The approximate
solutions for the coupled non-linear equations are derived for Re  o (1) . f , g , F and G are
s
expressed in the following forms
F
G
f
f

 
 Re G

s
s 2
, Fp   F f 0 , Fp 0  Re F f 1 , F f 1  O (Re ) ,

,Gp  G f 0 ,Gp0
s
f1

, G f 1  O(Re )
s
2
(2.7.1)
(2.7.2)
Using (2.7.1) and (2.7.2) into the equations (2.6.13), (2.6.17), (2.6.19) and (2.6.23), we get
Ff 0  0
iv
(2.7.3)
F f 1  2F f 0 F f0  2 N G f 0 G f 0
(2.7.4)
G f0  0
(2.7.5)
G f1  2F f 0 G f 0  2F f 0 G f 0
(2.7.6)
Fp 0  0
(2.7.7)
iv
2
iv
Ff 1 
iv

k
F p0
(2.7.8)
Corresponding boundary conditions reduce to
45
1
F f 0 (1)  ,
2
F f0 (1)   2 F f 0 (1),
F f 1 (1)  0,
F f1 (1)  2 F f 1 (1),
G f 0 (1)  2 { G f 0 (1)  s}, G f 1 (1)  2 G f 1 (1),
F f 0 (0)  Fp 0 (0),
F f 1 (0)  Fp1 (0), G f 0 (0)  G p 0 (0), G f 1 (0)  G p 1 (0)
F f 0 (0)  Fp 0 (0),
F f 1 (0)  Fp 1 (0),
F f0 (0)  1 Fp0 (0),
F f1 (0)  1 Fp1 (0)
G f 0 (0)  1G p 0 (0), G f 1 (0)  1G p 1 (0),
Fp 0 (1)  0,
Fp 1 (1)  0,
Fp0 (1)  1 Fp 0 (1),
Fp1 (1)  1 Fp1 (1),
G p 0 (1)  1{ G p 0 (1)  1}, G p 1 (1)  1G p 1 (1),
F p0 (1)  0,
F p1 (1) 

k
F p 0 (1)
(2.7.9)
Solution of equations (2.7.3) - (2.7.8) with boundary conditions (2.7.9) is given by
2
F f 0 ( y) 
3
m y
m4 y
 5  m6 y  m7
6
2
3
(2.7.10)
2
m y
m y
F p 0 ( y )  8  5  m6 y  m7
6
2
(2.7.11)
G f 0 ( y)  m11 y  m12
(2.7.12)
G p 0 ( y)  m13 y  m12
(2.7.13)
6
F f 1 ( y)  
4
3
5
  m8 y
m5 y 4  m41 y 3 m40 y 2
F p 1 ( y)  


 m37 y  m39

k  120
24 
6
2
3
G f 1 ( y) 
G p 1 ( y) 
2
2 7
5
m m y
m y
m y
m y
m4 y
m y
 4 5  14  15  38  36  m37 y  m39
2520
360
120
24
6
2

k 
6
(2.7.15)
2
5
4
mm y
m y
m4 m11 y
m y
 42  5 12  43  m50 y  m49
30
12
3
2
3
  m13 y
(2.7.14)
m12 y 2 

  m48 y  m49
2 
(2.7.16)
(2.7.17)
where m1 to m50 are constants recorded in APPENDIX - II.
2.8 RESULTS AND DISCUSSION
46
In this section, we discuss fluid velocity in porous and free fluid region. Numerical values
have been calculated by taking
Squeeze Reynolds number Re  0.01, N  40,   0.02 , 1  2 .
s
Permeability parameter 1 of lower disk,  2 of upper disk, ratio of the rotational velocities of
the disks s and permeability of the porous medium k have been taken arbitrarily. For
convenience, F ( y) represents F p ( y ) in porous region for  1  y  0 and F f ( y ) in free fluid
region for 0  y  1 .
Velocity Profiles
Figs. 2.13 and 2.14 show the profiles of F ( y) for various values of the ratio of the
rotational velocities of the disks s at 1  2  0.5 . It is seen from fig. 2.13 that if both the disks
rotate in the same direction, and with increase in the value of the ratio of the rotational velocities
of the disks, the profiles of F ( y) in both the regions decrease. Similar behavior is seen from
fig. 2.14, when both the disks rotate in opposite direction.
Figs. 2.15 and 2.16 show the impact of permeabilities of both the disks on profiles of
F ( y) in both the regions. The large values of  i corresponds to small permeabilities. Fig. 2.15
depicts that when permeability of the lower disk is fixed, and on increasing the permeability of
the upper disk (  2 decreases) the profiles of F ( y) decrease in both the regions. Permeability of
the upper disk has less impact on the profiles on porous region. On the other hand, with fixed
permeability of upper disk, the profiles of F ( y) decrease with decrease in permeability of the
lower disk ( 1 increases) as shown in fig. 2.16.
Fig. 2.17 represent the effect of permeability k of the porous region on F ( y) for
1  2  0.5 and s  0.5 . It is clear from the figure that profiles of F ( y) in porous region
decrease with increase in permeability k of the porous region.
47
48
49
From equation (2.6.10a) and (2.6.10b), radial velocity in free fluid and porous region are
expressed in term of F f ( y ) and F p ( y ) respectively. Fig. 2.18 depict that, when both the disks
rotate in same direction, the profiles of F ( y) decrease in porous region with decrease in
permeability of the upper disk (  2 increases) and then start increasing with decrease in
permeability of the upper disk in free-fluid region. Similar behavior is seen from fig. 2.19, when
permeability of the lower disk varies.
Fig. 2.20 represents the effect of permeability k of the porous region on F ( y) . It is seen
from the figure that profiles of F ( y) decrease with increase in permeability k of the porous
region. Effect of the ratio of the rotational velocities of the disks on F ( y) is shown in fig. 2.21.
It is observed that on increasing the value of s the value of F ( y) decreases and maxima of the
profiles shifted towards the upper disk.
50
51
52
Figs. 2.22 and 2.23 show the effect of permeabilities of the disks on profiles of G( y) ,
where G( y) represents G p ( y) in porous region for  1  y  0 and G f ( y ) in free-fluid region
for 0  y  1 . It is seen from fig. 2.22 that profiles of
G( y) decrease with decrease in
permeability of the upper disk (  2 increases). Profiles of G( y) increase with decrease in
permeability of the lower disk ( 1 increases) as shown in fig. 2.23. Fig. 2.24 shows that on
increasing the value of ratio of the rotational velocities of the disks, profiles of G( y) increase.
53
54
Skin-friction
The shear stress on the upper disk is given by
 
r z y 1

 r
 
3/ 2
 2 H (1  t )

 f ( y )y 1


(2.8.1)
Let the coefficient of skin friction C f at the disk y  1 be defined as
C 
f
y 1
2(1   t ) 3 / 2
 r z y 1  r  f ( y)y 1

  r0
r0
(2.8.2)
where r0 is a characteristic distance along the radial direction.
Similarly the coefficient of skin friction on the lower disk y  0 is defined as
C 
f
y 0
2(1   t ) 3 / 2
 r z y 0  r  f ( y)y 0

  r0
r0
(2.8.3)
Fig. 2.25 depicts the effect of rotation ratio s on the coefficient of skin friction. It is observed
that the magnitude of coefficient of skin friction increases on the upper disk with increase in the
value of rotation ratio s , while it decreases on the lower disk with increase in rotation ratio s .
Torque
The shearing stress component  z which acts in the plane of the disks, produces a force in the
 -direction. If edge effects are neglected, the torque on a disk of radius r0 is given by
r0
    z  2 r 2 dr
(2.8.4)
0
Thus torque on the upper disk is
u 
   r04
g (1)
2 H (1   t ) 3 / 2
55
(2.8.5)
Thus torque on the lower disk is
l 
   r04
G (1)
2 H (1   t ) 3 / 2
(2.8.6)
Variation of g (1) and G (1) for various values of rotational ratio s with N  40 and
s
Re  0.01 have been shown in Table 2.3. It is seen that torque on both the disks increases with
increase in the value of rotational ratio s .
Table 2.3 Values of g (1) and G (1)
56
s
0
1
2
3
G (1)
-0.0777
-0.0013
0.0751
0.1515
g (1)
-0.1480
-0.0395
-0.0013
0.0751
2.9 CONCLUSION
It is concluded that profiles of F ( y) decrease with increase in permeability of the upper
disk, while profiles of F ( y) increase with increase in the permeability of the lower disk. It is
also observed that profiles of F ( y) in porous region decrease with increase in permeability of
the porous region. Profiles of G( y) increase with decrease in permeability of the lower disk,
while decrease with decrease in permeability of the upper disk. Torque on both the disks
increases with increase in the value of rotational ratio s .
57