MHD Jeffrey fluid Flow between Two Concentric
Rotating Cylinders When One of the Walls is provided
with Non-Erodible Porous Lining
K.Kumara Swamy Naidu
S.Sreenadh
A.Parandhama
Dept.of Mathematics
Sree Vidyanikethan
Engineering College
A.Rangampet, A.P
[email protected]
Dept. of Mathematics
Sri VenkateswaraUniversity
Tirupati, A.P
[email protected]
Dept.of Mathematics
Sree Vidyanikethan
Engineering College
A.Rangampet, A.P
[email protected]
Abstract
The flow of a Jeffrey fluid between two concentric cylinders under the influence of
a uniform radial magnetic field is investigated. The outer cylinder contains internally
porous lining of thickness δ. The inner and outer cylinders are rotating with angular
velocities ω and Ω respectively. The expressions for the velocity distribution, the shear
stress, the temperature distribution and the rate of heat transfer are obtained. The
effects of magnetic field, porosity, thickness of lining, product of Prandtland Eckert
numbers on the physical quantities are discussed. It is observed that the velocity
decreases close to the inner cylinder and increases near the porous lining region with
increasing Jeffrey parameter. It is also found that the temperature decreases with the
increase in the thickness of the porous lining whereas it increases with the increase in
the Biot number  .
Keywords: MHD, Jeffrey fluid,Rotating Cylinders, Non-Erodible Porous Lining,
Permeability, slip parameter, Magnetic parameter.
-----------------------------------------♦---------------------------------------------1.Introduction
Flow through and past a porous
medium has importantapplications in
petroleum engineering and biomechanics.
Hydromagnetic flows in rotating systems
continue to stimulate significant research
inthe engineering science and applied
mathematics fields. Such flows are
important in rocket propulsion control,
crystal growth technology, astrophysical
plasma fluid dynamics, tribological
regulation in moving machine parts and
also MHD energy generators. Flow in
rotating annular spaces frequently arises
for example in drilling operation of oil and
gas wells.
MHD flow in an annulus formed by
two concentric rotating cylinders is a
classical problem in literature(Hughes and
Young [1]). The effect of porous lining on
one of the cylindrical surfaces is an
interacting problems as it may serve as
one of the mathematical models for the
pumping flow from apetroleumreservoir in
the earth. Chennabasappa et al. [2]
studied the effect of porous lining in a
parallel plate channel is two cases,
namely, one of the walls being lined with
porous material and both the walls being
lined with porous material. This problem is
extended by Kumaraswamy Naidu et al.
[3] for Non-Newtonian fluid flows obeying
Jeffrey model. Bathaiah and Venugopal
[4] examined the MHD flow between two
concentric rotating cylinders under the
influence of a uniform magnetic field, with
non-erodible and non-conducting porous
lining on the inner wall of the outer
cylinder under the influence of a uniform
radial magnetic field.
Vajravelu and Kumar [5] studied
the effect of rotation on the twodimensional channel flow. They solved
the governing equations analytically and
numerically. Chakrabarti and Gupta [6]
studied the MHD flow of Newtonian fluids
over a stretching sheet at a different
uniform temperature. Vajravelu and
Hadjinicolaou [7] made analysis to flows
and heat transfer characteristics in an
electrically conducting fluid near an
isothermal sheet. In 1983, Borkakoti and
Bharali [8] studied the two-dimensional
channel flow with heat transfer analysis of
a magnetic hydro fluid where the lower
plate was a stretching sheet.
Several bio-fluids like blood and
industrial fluids such as gels and
emulsions are reported to behave as nonNewtonian fluids.
In view of this
considerable work has been done with
non-Newtonian biological / industrial fluids
to provide information for manufacturing
of artificial physiological systems. Jeffrey
fluid model is a significant generalization
of Newtonian fluid model as the latter one
can be deduced as a special case of the
former. Several researchers have studied
Jeffrey fluid flows under different
conditions. Ebaid et al. [9] have studied
the peristaltic transport in an asymmetric
channel through a porous medium.
Vajravelu et al. [10] investigated the
influence of heat transfer on peristaltic
transport of a Jeffrey fluid. Jyothi et al.
[11] have considered the pulsatile flow of
a Jeffrey fluid in a circular tube lined
internally with porous material. Ebaid [12]
has analyzed a mathematical model to
study the peristaltic transport of an
incompressible viscous fluid in an
asymmetric channel under the effect of
transverse magnetic field with slip
boundary conditions. Akbar et al. [13]
have studied the Jeffrey fluid model for
the peristaltic flow of chyme in the small
intestine with magnetic field. Abd-Alla et
al. [14] have investigated the peristaltic
flow of a Jeffrey fluid in an asymmetric
channel.
Recently,
Nallapu
and
Radhakrishnamacharya [15] studied a
two-fluid model for the flow of Jeffrey fluid
in tubes of small diameters.
Motivated by the above studies,
MHD flow of a viscous, non-Newtonian,
Jeffrey fluid through
an annular zone
between two concentric rotating cylinders
when one of the walls are provided with
non-erodible porous lining in the presence
of a radial magnetic field. The velocity
distribution, the shear stress, the
temperature distribution and the rate of
heat transfer at the interface walls are
analyzed for various physical parameters.
2. Formulation of the problem
We
consider
the
viscous
incompressible and slightly conducting
Jeffrey fluid flow between two concentric
rotating cylinders with non-erodible, nonconducting porous lining on the inner wall
of the outer cylinder under the influence of
a uniform radial magnetic field of the form
Br 
A
where A is constant (Hughes and
r
Young
cylinder
angular
cylinder
[2]). The inner impermeable
of radius a rotates with an
velocity  , whereas the outer
of radius b   a  rotating with
angular velocity  has a non-erodible
porous lining of thickness h on its inner
wall. The physical model explaining the
problem under study is shown in Fig. 1.
electrical conductivity, C p the specific heat
of the fluid at constant pressure, T the
temperature of the fluid, K T the thermal
conductivity,  the fluid density and
 du 2
  2  
 dr 
u
 
r
  dv v  2
  
  dr r 
2
(2.5)
The boundary conditions are
u  0 at r  a and r  b  h
v  a at r  a
v  vB at r  b  h
(2.5a)
(2.5b)
(2.5c)
T  T1 at r  a
(2.5d)

 dT 

(TB  T0 )
 dr 
K
r b  h
(2.5e)
where vB
is the slip velocity obtained by
using Beavers and Joseph condition

 dv 
(2.5f)

(vB  Q)
 dr 
r b  h
The fluid flow in the annulus is
governed by Jeffrey model [10]. Following
Chandrasekhar [16] and assuming that
the flow is caused by rotation of the
cylinders, the basic equations of the flow
in the annulus (in cylindrical coordinates)
are
 du v 2 
dp
  d 2 u 1 du u 
  


(2.1)
dr 1  1  dr 2 r dr r 2 
 dr r 
 u
  d v 1 dv u   0 A v
 dv uv 
 u   
 2   2 (2.2)
 2
r
 dr r  1  1  dr r dr r 
du u
 0
(2.3)
dr r
dT KT  d 2T 1 dT 

Cpu




(2.4)


2
dr
  dr
r dr   (1  1 )
where u and v are the radial and
azimuthal components of the velocity, r
the radial distance, P the fluid pressure,
 the coefficient of viscosity,  0 the
2
2
K
p  p1 at r  a
(2.5g)
 is theslipparameter
K is the porosity of the lining material,
 is the Biot number,
TB is the temperature at the permeable
bed,
T0 the ambient temperature, and
Q is, the Darcy velocity in the permeable
lining
given by
Q  r  E
(2.6)
Here r is the velocity in the porous
medium due to the rotation of the porous
medium with angular velocity  and E is
given by [1]
2
E
K (1  1 )

b
0
bh
2
 2 r (rd dr )
b
 
rd dr
0 bh

2  K  (1  1 )  3b 2  3bh  h 2 


3
2b  h


2
(2.7)
From the continuity equation (2.3) and the
boundary condition (2.5a) we obtain
(2.8)
u0
We introduce the following nondimensional quantities:
v
r
a
b
, r '  ,   , * 
b
b
b
K
2

h
b

P
 *  ,   , Re 
, P' 

b
v
 b2 2
v' 
1
D1 
(2.9)
D2 
 A2 ' T  T0
M  0
,T 

T1  T0
1
1 1
 * 1 N 1  1  vB (1   ) N
 2N
N
1 1
1 1
(1   ) N
 (1   )
1 1
2N
1 1
 (1   )
*
(1   )
2 N 1 1

N 1 1
 vB  N
1 1

2 N 1 1
where N= 1  M (Magnetic parameter).
In view of (2.8) and (2.9), Eqs. (2.1) and
(2.4) reduce to (after dropping the
primes),
dp v 2
(2.10)

dr r
d 2v 1 dv  1  M


dr 2 r dr  r 2
to the boundary conditions (2.13a)
and (2.13b), we get the velocity as
N (1  )
 N (1  )
(2.14)
v  D1r
 D2 r
where

 (1  1 )v  0

(2.11)
2
P E  dv
d 2T
dT

(2.12)
r
 r
r v
2
dr
dr
(1  1 )  dr

The boundary conditions in the nondimensional form are
at r  
(2.13a)
v   *
at r  1  
(2.13b)
v  vB
r2
Using (2.13c), we obtain the slip velocity
vB as


(1   )  2 N 1 1  (1   ) 2 N 1 1


*2
2
2
 3 (  3  2)  2 Re (1  1 )(  3  3)

N 1 1
* * 1 N 1 1
(1   )
 6 N (2   ) 1  1   

vB 
3 * ( 2  3  2)  2 N 1 1  (1   ) 2 N 1 1


2 N 1 1
2 N 1 1
*
 (1   )
 3 N 1  1  (2   ) 

















(2.15)
Radial Pressure
where the non-dimensional slip velocity
vB is given by
2 Re ( 2  3  3)(1  1 )
 dv 
  *vB   * (1   ) 
 
3 * (2   )
 dr r 1
p
(2.13c)
r 
(2.13d)
 dT 
  *TB
 
 dr r 1
(2.13e)
T 1
Solving (2.10) subject to the boundary
condition (2.13f) the radial pressure is
given by
at
(2.13f)
p  p1 at r  
where  *
is the permeability
parameter and TB
is the nondimensional temperature at the
permeable wall.Solving (2.11) subject
 D12 r 2 N 11  D2 2 r 2 N 1 1 

 A
2 N 1  1  4 D1 D2 N 1  1 log r 
1
(2.16)
where
 D12  2 N 1 1  D22  2 N 1 1 


A  p1 
2 N 1   1  4 D1 D2 N 1  1 log    
1
Shear Stress
The shearing stress can be calculated
using
 
r
d v
1  1  dr  r 
and it is given by




N 1 1
N 1 1
(1   )
 ( N 1  1  1)



  * (1   ) N 1 1  vB  N 1 1





N
1



1
*
1


 2 N 1 1 
  ( N 1    1)   

r
1
N 1 1


vB (1   )






N 1 1  2 N 1 1
2 N 1 1 
r

 (1   )
1  1 








(2.17)
If  1 and  2 are the shearing stresses
at the inner cylinder and the permeable
interface respectively,
then





 N 1    1  N 1 1 (1   ) N 1 1
1

 *
N 1 1
N 1 1 1
 vB 

  (1   )

 N 1  1  1  *  N 11 1  vB (1   ) N 11  N
1  
 2 N 11  (1   )2 N 11  1  1 










(2.18)
 N 1    1  N 11 (1   ) N 11
1

 *
N 1 1
N 1 1
 vB 

  (1   )

 N 1  1  1 (1   ) N 11  * N 11 1  vB (1   ) N
2  
1  1   2 N 11  (1   )2 N 11  (1   )








1 1 1


1 1








TB 


 1     D 2 ( N  1) 2 (1   ) 2 N 
Pr E log 
 1






2
2
2 N
1 

  D 2 ( N  1) (1   ) 
2 N (1  1 ) 



4 MND1 D2 log(1   )













2
2
2N
2N
 D 1 ( N  1) ((1   )   )



Pr E

2
2
2 N
2 N  


D
(
N

1)
((1


)


)
2

 4 N 2 (1   ) 
1 


2


 4 MN 2 D D  log(1   )    

 
1 2 

2

(log  )
 




1  
1   * (1   ) log 

  
(2.21)
Rate of Heat Transfer
The rate of heat transfer at the wall
of the inner cylinder is given by
(1   )  *TB
 T 
q
 

 r r 


 D12 ( N  1) 2 (1   ) 2 N   2 N  


Pr E
  D22 ( N  1) 2 (1   ) 2 N   2 N 


2 N  (1  1 ) 


1  
 4MND1 D2 log 


  


(2.22)
The rate of heat transfer at the interface is
given by
 T 
q*  
  *TB


r

r 1
(2.19)
Temperature Distribution
Solving Eq.(2.12) by using the boundary
conditions (2.13d) and (2.13e), we obtain
the temperature distribution as
2
2N
r 2

Pr E log    D 1 ( N  1) (1   )

r





*
2
2
2 N 
T  1  (1   )  TB log   
  D 2 ( N  1) (1   ) 
   2 N 1  1 
4MND1 D2 log(1   ) 


 D 21 ( N  1)2 (r 2 N   2 N ) 

 2

Pr E
2 2 N
2 N
 2
 D 2 ( N  1) (r   )

4 N (1  1 ) 
2
2
2 
 4 MN D1 D2 (log r )  (log  ) 
(2.20)
The temperature at the permeable wall is
given by
Results and Discussions
The effect of non-erodible porous
lining on the flow of a Jeffrey fluid
between two concentric rotating cylinders
is analyzed. The velocity and temperature
distributions are numerically evaluated for
various values of physical parameters
such as Pr E , magnetic parameter N , Biot
number  and the results are shown in
figures 2 to 10.
Fig. 2 gives the velocity profiles for
different values of N . We have observed
that the velocity v decreases with the
1.8
increase in N at points sufficiently close
to the inner cylinder and this trend gets
reversed at points sufficiently close to the
porous lining. Fig.3 shows the velocity
profiles for different values of  . We
observed that velocity increases with the V
increase in  . Fig. 4 shows the velocity
profile for different values of permeable
parameter  * . We observed that velocity
decreases with the increase in  * . Fig. 5
gives the velocity profiles for different
values of Jeffrey parameter 1 . We have
r
observed that the velocity v decreases Fig.2. Velocity profiles for different values of N, for
with the increase in 1 at points   0.6,  *  10,   1,  *  0.5,
sufficiently close to the inner cylinder and Pr E  4, M  1.5,   0.1, 1  0.5
this trend gets reversed at points
sufficiently close to the porous lining.
N=1
N=1.6
N=2.2
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
0.3
1.4
0.4
0.5
0.6
0.7
0.8
0.9
1
fixed
=0.02
=0.06
=0.1
1.2
Fig.6 shows the temperature profile for
*
different values of  . We have observed
that the temperature increases at points
V
sufficiently close to the inner cylinder while
*
it decreases with the increase in  at points
sufficiently close to the interface. Figs.7 and
8 gives the temperature profiles for different
values of  and  . We have observed that
the temperature decreases with the
r
increase in  whereas it increases with Fig.3.Velocity profiles for different values of  ,
*
*
increase in  . Fig. 9shows the temperature   0.6,   10,   1,   0.5,
profiles for different values of 1 .We have Pr E  4, M  1.5, N  1, 1  0.5
seen that the temperature decreases with
1 at points sufficientlyclose to the inner
cylinder while it increases with the increase
in 1 at pointssufficiently close to the
interface. Fig. 10 shows the heat transfer
at the interface profiles  q* against  for
different values of 1 . We observed that  q*
1
0.8
0.6
0.4
0.2
0
0.2
decreases
with
the
increasing 1 .
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
forfixed
1.5
1.2
=6
=6
=8
=10
=8
1
 = 10
0.8
1
0.6
T
V
0.4
0.2
0.5
0
-0.2
0
0.2
0.3
0.4
0.5
r
0.6
0.7
0.8
0.9
1
Fig. 4.Velocity profiles for different values of  ,forfixed
*
  0.1,   0.6,   1,   0.5,
Pr E  4, M  1.5, N  1, 1  0.5
0.3
0.4
0.5
r
0.6
0.7
0.8
0.9
Fig. 6. Temperature profiles for different values of
1

*
*
M  1.5, 1  0.5, N  1,   0.2, Re  0.2
1 =0.1
1.6
=0.02
=0.06
=0.1
1 =0.5
1.2
for
fixed   0.5,  0.6,   1,  0.1, Pr E  4,
*
1.4
-0.4
0.2
1 =1
1.4
1.2
1
1
0.8
0.8
T
V
0.6
0.6
0.4
0.4
0.2
0
0.2
-0.2
0
0.2
0.3
0.4
0.5
r
0.6
0.7
0.8
0.9
1
Fig. 5. Velocity profiles for different values of 1 , for fixed
  0.1,   0.6,   1,  *  0.5,
Pr E  4, M  1.5, N  1,  *  10
-0.4
0.2
0.3
0.4
0.5
r
0.6
0.7
0.8
0.9
Fig. 7. Temperature profiles for different values of
  0.5,   0.6,   1, N  1, Pr E  4,
*
fixed
1  0.5,  *  10,   0.2, Re  0.2
1

for
1.2
1.2
=0.6
=0.8
1
1=0.1
1=0.5
=1
1=1.0
1.1
0.8
0.6
1
0.4
-q*
T
0.9
0.2
0
0.8
-0.2
0.7
-0.4
-0.6
0.2
0.3
0.4
0.5
r
0.6
0.7
0.8
0.9
1
Fig. 8. Temperature profiles for different
valuesof
*
 for fixed   0.5,  0.6,   0.1, N  1, Pr E  4,
1  0.5,  *  10,   0.2, Re  0.2
0.02
0.03
0.04
0.05

0.06
0.07
0.08
0.09
Fig. 10. Heat transfer at interface for different values of
for fixed
0.1
1
Pr E  4,   0.6,   0.1,   1, N  1,
 *  0.5,  *  10,   0.2, Re  0.2
References:
3.5
1=0.1
1=0.5
3
1=1
2.5
2
1.5
T
1
0.5
0
-0.5
-1
-1.5
0.2
0.3
0.4
0.5
r
0.6
0.7
0.8
Fig. 9. Temperature profiles for different values of
fixed
0.9
1 for
Pr E  4,   0.6,   0.1, N  1,
  1,  *  0.5,  *  10,   0.2, Re  0.2
1
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