Flow and heat transfer of micro polar and viscous

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 771
FLOW AND HEAT TRANSFER OF MICRO POLAR AND VISCOUS
FLUID IN A VERTICAL CHANNEL
Mahadev M Biradar
Associate Professor, Dept. of Mathematics Basaveshwar Engineering College (Autonomous) Bagalkot, Karnataka INDIA
587 102
Abstract
The flow nature of mixture of Newtonian and non-Newtonian (micro polar) fluid between vertical parallel plates is analyzed. Closed
form solutions are obtained for the governing equations. The effect of governing parameters such as the ratio of Grashof number to
Reynolds number, viscosity ratio, width ratio, pressure and material parameter on velocity and micro rotation velocity are depicted
graphically. The effect of material parameter is to reduce to velocity and micro rotation velocity.
Keywords: Newtonian and Non Newtonian, Convection Vertical, Heat transfer
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1. INTRODUCTION
Colloidal fluids, liquid crystals and fluids containing additives
are known to fall outside the domain of the classical theory of
Stokesian fluids. It has also been established (Eringen, 1965)
that the presence of a tiny amount of additives in the fluid
considerably lowers down the skin friction near a rigid body
and also the polymer concentration reduces the frictional drag.
This phenomenon is explained more adequately by the theory
of micropolar fluids, a sub class of simple microfluids
(Eringen, 1967). In the micropolar fluid theory, apart from the
classical velocity field, a microrotation vector and a gyration
parameter are introduced in order to investigate the kinematics
of microrotation.
Studies of external convective flows of micropolar fluids have
focused mainly on free, forced and mixed convection
problems. However, there are only few studies investigating
the effect of microstructure on the free convection heat
transfer in enclosures. (Chiu et.al., 1994). Natural convection
of an enclosed fluid is a long-standing classical subject.
Applications are found in a variety of engineering problems,
such as air conditioning of a room, solar energy collecting
devices, material processing and passive cooling of nuclear
reactors, to name a few. Chamakha et.al., (2002) studied the
fully developed free convection of a micropolar fluid in a
vertical channel. Fully developed natural convection heat
and mass transfer of a micropolar fluid in a vertical channel
with asymmetric wall temperatures and concentrations is
studied by Ching-Yang Cheng (2006).
The objective of this problem is to study the flow nature of the
mixture of Newtonian and non-Newtonian fluid (micropolar
fluid) between vertical parallel plates.
2. MATHEMATICAL FORMULATION
TW T1
(1) Y  h (2) Y  h
Fig-1: Physical Configuration
Consider a steady, laminar, incompressible micro polar fluid
through a vertical parallel plates. The region
1 0  y  h is
occupied by a micro polar fluid of density
1  , viscosity
1  ,
and material parameter  . The region
2 h  y  0 is
occupied by viscous fluid of density
2  , viscosity
2  . The
fluids are assumed to have constant properties except the
density in the buoyancy term in the momentum equation. The
fluid rises in the channel driven by buoyancy forces. The
transport properties of both fluids are assumed to be constant.
We assume the flow is steady, laminar and fully developed. It
is also assumed that the viscous dissipation is neglected.
Under these assumptions, the basic equations of the micro
polar fluid are
Region-I
Micro
polar
Region-II
Viscous
Y
X

__________________________________________________________________________________________
Region-I
   
 
     
2 1
1 1 1
2 1 0
d u dN P
g T T
dy dy x
  

      

(2.1)
2 1
2 2 0
d N du
N
dy dy
      (2.2)
2 1
2 0
d T
dy
 (2.3)
Region-II
 
 
     
2 2
2 2 2
2 2 0 0
d u P
g T T
dy x
  

   

(2.4)
2 2
2 0
d T
dy
 (2.5)
The appropriate boundary and interface conditions on velocity
in the mathematical form are:
(1) U  0 at
(1) Y  h
(2) U  0 at
(2) Y  h
(1) (2) U U (Continuity of velocity) atY  0
 
1 2
(1) (2) dU dU
N
dy dy
     (Continuity of
shear stress) at Y  0 (2.6a)
The walls are maintained at constant different temperatures Tw
and T1 at
1 y  h and
2 y  h respectively.
The boundary and interface conditions on temperature are
(1)
W T  T at
(1) Y  h
(2)
1 T  T at
(2) Y  h
(1) (2) T  T (Continuity of temperature)
atY  0
 
 
 
 
1 2
0 0
dT dT
dy dy
 (Continuity of heat flux)
atY  0 (2.6b)
It is convenient to non-dimensionalize the governing
equations using the variables,
 
 
 
 
 
 
     
0
2
, , ,
/
, 1,2
/
i i
i i i i
i
s
i i i
y u T T
y U
h U Tw T
P x
P i
u h



  

 
 
 
 
 
1
1
1 2
, ,
2
,
U
N N j
h
j h
 


 
   
 
 

 
 
 
 
 
 
 
 
1 2 2 2
2 1 1 1
, , ,
h
m h
h
  
 
  
   
(2.7)
Where, U is the characteristic velocity
Region-I
 
 
 
2 1
1
2 1 0
d u dN
GR P
dy dy
      (2.8)
2 1
2 1 2 0
2
d N du
N
dy dy

 
 
     
 
(2.9)
2 1
2 0
d
dy

 (2.10)
Region-II
 
 
2 2
2 2 2
2 0
d u
mr h GR mh P
dy
     (2.11)
2 2
2 0
d
dy

 (2.12)
The non-dimensional form of the velocity, temperature
boundary and interface conditions becomes

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N  0 at y  1
0
dN
dy
 at y  0 (2.13a)
(1) u  0 ; at y 1
(2) u  0 ; at y  1
(1) (2) u  u at y  0
 
1 2 1
1
du du
N
dy mh dy
   at y  0 (2.13b)
(1)  1 at y 1
(2)   0 at y  1
    (1) (2)  0  0 at y  0
 
 
 
 
1 2
0 0
d d
dy dy
 
 at y  0 (2.13c)
3. SOLUTIONS
Solving analytically equations (2.8) to (2.12) using boundary
and interface conditions as given by equations 2.13 (a, b, c),
are
Region-I
2
5 6 1 2 3 4 N  C cosh ay C sinh ay  d y  d y  d  d A
(2.14)
1 3 2
5 5 5 6 1 2
7 8
sinh cosh
(2.15)
u d C ay d C ay l y l y
C y C
   
 
1
1 2   C y C (2.16)
Region-II
2 3 2
3 4 9 10 u  l y  l y C y C (2.17)
2
3 4   C y C (2.18)
4. RESULTS AND DISCUSSION
Equations (2.8) to (2.12) subject to boundary and interface
condition (2.13a) to (2.13c) have been solved in the closed
form. Solutions are given by equations (2.14) to (2.18) and the
results are depicted graphically in figures 2 to 11.
It is seen from figures 2 and 3, the effect of GR (ratio of
Grashof number to Reynolds number) on velocity u and micro
rotation velocity N, that both u and N increases as GR
increases for Newtonian  0 and for micro polar fluid
1 . The magnitude of promotion is more for Newtonian
fluid compared to micro polar fluid. It is interesting to note
that there is a flow reversal for micro polar fluid at the plate y
= -1 for only velocity profiles (figure 2) but not for micro
rotation velocity (figure 3).
The effect of viscosity ratio m on velocity u and micro rotation
velocity N is show, in figures 4 and 5. As m increases, u and N
also increases. The velocity u is large for viscous fluid
compared to micro polar fluid when small m<1 and the
velocity is large for micro polar fluid compared to viscous
fluid for m>1 .Figure 7.5 shows that there is a flow reversal
for micro rotation velocity N when the viscosity ratio is less
than 0.5 at the interface. We observe that effect of width ratio
h on velocity u and micro rotation velocity N show the similar
nature as that of viscosity ratio m (figures 6 and 7).
Figure 8 and 9 show the effect of pressure gradient P on
velocity u and micro rotation velocity N. As P (>0) increases
both u and N increases and as P (<0) decreases both velocity
and micro rotation velocity decreases. In the absence of
pressure gradient the magnitude of u and N lies in between
P>0 and P<0.
Effect of material parameter κ on velocity and micro rotation
velocity is to reduce the velocity profiles which is the similar
result obtained for one fluid model (Chamakha et. al., 2002) as
seen in figures 10 and 11.
Where
 
2 2
1
a




; G
GR =
Re
;
  1 4 2
GR
d



    2 2 2 2
GR P
d
 
 
 
;
 
  3
1
4 2
GR
d

 



;
 
  4
1
2
d





1 2 3 4
1
2
C  C  C  C  ; 2
6
d
C
a
 

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  5 6 1 2 3
1
sinh
cosh
C c C a d d d
a
    
5 4
1
cosh
C cc d
a
  ; C5  C5 c C5 cc A
  5 1
d
a


 

   
1
1
1
1 3 Re 12 1
d G
l

 
  
 
;
     
2
2 1 2 4 1 2 1
d GR P
l

  
   
  
;
2
3 12
mrh GR
l

  ;
2 2
4 4 2
mrh GR mh P
l

  
 
 
 
5 5 5 6
10
1 2 5 6
3 4 3
sinh cosh
1
1 1
1
mh d C c a d C a
C c
mh l l d C
l l d
mh



   
    
      
  
 
  10 5 5 4 1 sinh
1 1
mh
C cc d C cc ha d
mh
 

      
10 10 10 C  C c C ccA, 8 10 5 6 C  C  d C
9 10 3 4 C  C  l  l ,
7 5 5 5 6 1 2 5 6 10 C c  d C csinh a  d C c cosh a  l  l  d C C c
7 5 5 10 C cc  d C ccsinh a C cc
7 7 7 C  C c C ccA
 
     
3
7
4 7
1
1 1 1
d
A C c
d C cc
  
    
      
 
   
-1.0
-0.5
0.0
0.5
1.0
0 5 10 15 20 25
k = 0
k = 1
50 100
u
50 100 GR = 0
y
Fig. 2 Effects of GR on velocity profiles u
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0
y
100
GR = 0 50
N
Fig. 3 Effect of GR on microrotation velocity profiles N
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4 5
k = 0
k = 1
1.0 1.0
0.5
0.1
0.5
2.0 2.0
m = 0.1
Fig. 4 Effect of viscosity ratio m on the velocity profiles u
u
y
0.0
0.2
0.4
0.6
0.8
1.0
-0.1 0.0 0.1 0.2 0.3 0.4 0.5
m = 2.0
m = 1.0
m = 0.5
m = 0.1
Fig. 5 Effect of viscosity ratio m on microrotation velocity profiles N
N
y

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-1.0
-0.5
0.0
0.5
1.0
0 2 4 6 8
k = 0
k = 1
0.1
2.0
1.0
0.5
2.0
1.0
0.5
h = 0.1
Fig. 6 Effect of width ratio h on the velocity profiles u
u
y
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8
2.0
0.5 1.0
h = 0.1
Fig. 7 Effect of width ratio h on microrotation velocity profiles N
N
y
-1.0
-0.5
0.0
0.5
1.0
-1 0 1 2 3 4
k = 0
k = 1
5.0
P = -5.0 -5.0 P = 0.0 P = 5.0
Fig. 8 Effect of Pressure gradient P on velocity profiles u
u
y
0.0
0.2
0.4
0.6
0.8
1.0
-0.2 0.0 0.2 0.4
P = -5.0 P = 0.0 P = 5.0
Fig. 9 Effect of Pressure gradient P on microrotation velocity profiles N
N
y
-1.0
-0.5
0.0
0.5
1.0
0 1 2 3 4
2.0 1.0 0.5 0.1  = 0.0
Fig. 10 Effect of material parameter  on the velocity profiles u
u
y
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4
0.5 1.0 2.0
 = 0.1
Fig. 11 Effect of material parameter  on microrotation velocity profiles N
N
y

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REFERENCES [1]. Eringen, A. C. (1965) Theory of micropolar fluids continua, in; proceedings of the Ninth Mid-western, Mechanics conference pp-23. [2]. Eringen, A. C. (1967) Simple micro fluids, J. Engg. Mech. Div. ASCE, vol. 93 pp 5375. [3]. Chiu C. P, Chou H. M, (1994) Transient analysis of natural convection along a vertical wavy surface in micropolar fluids, Int. J. of Engg. Sci. vol. 32, pp 19-33. [4] Chamakha A. J, Grosan T, Pop I, (2002) Fully developed free convection of a micropoalr fluid in a vertical channel, Int. Comm. in Heat and Mass transfer vol.29 pp 1119-1127. [5]. Ching-Yang Cheng (2006) Fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with asymmetric wall temperatures and concentrations, Int. Comm. in Heat and Mass transfer vol.33, pp 627-635. [6]. Ahmadi G, (1976) Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite flat plate. Int. J. Engng. Sci. vol. 14, pp 639-646.

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