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Abstract
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The problem of fully-developed natural-convective heat and mass transfer through
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porous medium in a vertical channel is investigated analytically. One region filled
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with micropolar fluid and the other region with a viscous fluid or both regions filled
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by viscous fluids. Using the boundary and interface conditions, the expressions for
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linear velocity, micro-rotation velocity, temperature and mass have been obtained.
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Numerical results are presented graphically for the distribution of velocity, micro-
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rotation velocity, temperature and mass fields for various values of physical
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parameters such as the ratio of Grashof number to Reynolds number, viscosity
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ratio, channel width ratio, conductivity ratio and micropolar fluid material
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parameter. It is found that the effect of the micropolar fluid material parameter
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suppress the velocity whereas it enhances the micro-rotation velocity. The effect of
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the ratio of Grashof number to Reynolds number is found to enhance both the linear
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velocity and the micro-rotation velocity. The effects of the width ratio and the
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conductivity ratio are found to enhance the temperature distribution.
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Heat and Mass Natural-Convective Flow of Micropolar and Viscous
Fluids through Porous Medium in a Vertical Channel
Maen Menayzel Al- Rashdan,
Mechanical Engineering Department, Al-Huson University College
Al-Balqa Applied University, P.O.Box 50, Irbid-Jordan
e-mail: [email protected]
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Keywords: Natural-convection, micropolar fluid, porous medium
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Introduction
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The research area of micropolar fluids has been of great interest because the Navier-
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Stokes equations for Newtonian fluids can not successfully describe the characteristic of
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fluid with suspended particles. There exist several approaches to study the mechanics of
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fluids with a substructure. Ericksen (1960) derived field equations which account for the
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presence of substructures in the fluid. It has been experimentally demonstrated by Hoyt
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and Fabula (1964) and Vogel and Patterson (1964) that fluids containing small amount of
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polymeric additives display a reduction in skin friction. Eringen (1966) formulated the
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theory of micropolar fluids which display the effects of local rotary inertia and couple
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stresses. This theory can be used to explain the flow of colloidal fluids, liquid crystal,
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animal blood, etc. Eringen (1972) extended the micropolar fluid theory and developed the
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theory of thermo-micropolar fluids. Extensive reviews of the theory and application can
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be found in the review articles by Ariman et al. (1974) and the recent books by
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Lukaszewicz (1999) and Eringen (2001).
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Physically, micropolar fluids may be described as non-Newtonian fluids
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consisting of dumb-bell molecules or short rigid cylindrical element, polymer fluids, fluid
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suspension, etc. The presence of dust or smoke, particularly in a gas, may also be
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modeled using micropolar fluid dynamics. The theory of micropolar fluids first proposed
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by Eringen (1966) is capable of describing such fluids.
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Studies of external convective flows of micropolar fluids have focused mainly on
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free, forced and mixed convection problems. Applications are found in variety of
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engineering problems, such as air conditioning of a room, solar energy collecting devices,
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material processing and passive cooling of nuclear reactors. Studies of the flows of heat
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convection in micropolar fluids have focused mainly on flat plate Ahmadi (1976); Jena
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and Mathur (1982); Ÿucel (1989); and Rahman et al. (2009) or on regular surfaces
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Balram and Sastry (1973); Lien et al. (1990). Chamkha et al. (2002) analyzed numerical
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and analytical solutions of the developing laminar free convection of a micropolar fluid in
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vertical parallel plate channel with asymmetric heating. The subject of two-fluid flow and
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heat transfer has been extensively studied due to its importance in chemical and nuclear
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industries. The design of two-fluid heat transport system for space application requires
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knowledge of heat and mass transfer processes and fluid mechanics under reduced gravity
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conditions. Identification of the two-fluid flow region and determination of the pressure
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drop, void fraction, quality reaction and two-fluid heat transfer coefficient are of great
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importance for the design of two-fluid systems. Lohsasbi and Sahai (1988) studied two-
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phase MHD flow and heat transfer in a parallel plate channel with the fluid in one phase
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being electrically conducting. Malashetty and Leela (1992) have analyzed the Hartmann
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flow characteristic of two fluids in horizontal channel. The study of two-phase flow and
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heat transfer in an inclined channel has been studied by Malashetty and Umavathi (1997)
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and Malashetty et al. (2001). Fully-developed free-convective flow of micropolar and
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viscous fluids in a vertical channel is investigated by Kumar et al. (2009). Kumar and
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Gupta (2009) considered the unsteady MHD and heat transfer of two viscous immiscible
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fluids through a porous medium in a horizontal channel.
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The aim of this paper is to investigate the fully-developed heat and mass natural-
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convective flow through porous medium in a vertical channel with asymmetric wall
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temperature distribution.
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Formulation of the Problem
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The geometry under consideration illustrated in Figure 1, consists of two infinite vertical
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parallel walls maintained at different or equal constant temperatures extending in x  -and
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z  - directions. y  -direction is taken normal to the nonconducting walls. The region
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 h1  y   0 is occupied by micropolar fluid of density 1 , viscosity 1 , vortex viscosity
1
k, thermal conductivity k1 , thermal expansion coefficient 1
2
and the region 0  y   h2
is occupied by viscous fluid of density  2 , viscosity  2 , thermal conductivity k 2
3
thermal expansion coefficient  2
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The fluids are assumed to have constant properties except the density in the
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1   0 1  1 T1  T0  and  2   0 1   2 T2  T0  in momentum
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equation, where T0 is the mean temperature. Let us assume that the walls of the channel
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are isothermal. In particular, the temperature and concentration of the boundary at
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buoyancy term





T   Tw1
y   h1 is Tw1 and C w1 , while the temperature at y  h2 is Tw 2 and C w 2 with w 2
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and C w 2  C w1 . A fluid rises in the channel driven by buoyancy forces. The transport
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properties of both the fluids are assumed to be constant. It should be mentioned here that
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the micropolar and viscous fluids are immiscible (that is, no mixing between the fluids
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exists) and the constitutive equations for micropolar and viscous fluids are different. Also,
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the viscosity of both fluids is different. For instant, Synovial fluid which is a clear
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thixotropic lubrication fluid is a good example of micropolar fluids and water is a good
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example of viscous fluids and it is well known that a Synovial fluid and water cannot be
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mixed. Since our model is general, one can choose any two different fluids which are
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immiscible.
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
It is assumed that the only non-zero component of the velocity q along x
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is u i i  1,2 . Thus, as a consequence of the mass balance equation, we obtain
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u i
0
x 
So that u i 1,2 
(1)
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depends only on y  .
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Under these assumptions, the momentum, energy and mass equations are given by
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Region-I
1
d 2 u1
1 
d 


1  k  2  k   1 g1T T1  T0   u1  1 g1C C1  C0  0
dy
dy
K




(2)
2
(3)
3
d 2T1
0
dy 2
(4)
4
d 2 C1
0
dy 2
(5)
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
  du1 
d 2 
 2     0

k
dy 2
dy 

Region-II
2
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d 2 u 2
 2 




g

T

T

u 2   2 g 2C C 2  C0  0
2
2T
2
0
dy 2
K




(6)
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d 2T2
0
dy 2
(7)
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d 2 C 2
0
dy 2
(8)
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where   is the component of micro-rotation vector normal to the plane x  y  , g the
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acceleration due to gravity,  the coefficient of electrical conduct
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permeability of porous medium and  the spin gradient viscosity. To solve the above set
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of differential equations from (2) to (6), six boundary conditions are required for velocity,
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four boundary conditions for temperature, and four boundary conditions for mass. The
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first two boundary conditions are obtained from the fact that there is no slip near the wall.
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Next condition is obtained by assuming the continuity of velocity and the last four
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conditions are obtained from the equality of stresses at the interface and constant cell
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rotational velocity at the interface as proposed by Ariman et al. (1973). Thus, the
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appropriate boundary and interface conditions on velocity in the mathematical form are
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u1  0 at y   h1, u 2  0 at y   h2, u1 0  u 2 0,

1  k  du1
dy
 k    2
du1
at y   0
dy 
(9)
1
d 
 0 at y   0,    0 at y   h1

dy
For the corresponding temperature and mass boundary conditions, it is assumed
that the temperatures and heat fluxes are continuous at the interface
2
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T1  Tw1 at y   h1 , T2  Tw2 at y   h2 ,
T1 0   T2 0 , 1
dT1
dT2


at y   0
2


dy
dy
C1  Cw1 at y   h1 , C2  Cw 2 at y   h2 ,
C1 0   C2 0 , 1
(10)
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dC1
dC2


at y   0
2


dy
dy
Also we assume that


k
2

   1   j  1 1 

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k1 
j
2
(11)
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where j is the micro-inertia density and k1  k 1 , is the micropolar fluid material
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parameter of Region-I. We notice that k1  0 describes the case of a viscous or
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Newtonian fluid. Relation (11) expresses the fact that the micropolar fluid field can
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predict the correct behavior in the limiting case when the micro-structure effects become
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negligible and total spin reduces to the angular flow velocity or flow vorticity. Relation
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(9) was established by Ahmadi (1976) and Kline (1977) and it has been used by many
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researchers, for examples, Rees and Bassom (1996), Gorla (1988), and Rees and Pop
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(1998).
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Method of Solution
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We introduce the following non-dimensional quantities
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yi 
u
T   T0
C   C 0
h
y
K
, ui  i ,  i  i
, i  i
,   1 ,   2 ,
hi
U0
T
C
U0
h1
g1C Ch13
U h
g1T Th13
Gr
GrT 
, GrC 
, Re  0 1 , GRT  T ,
2
2
v1
Re
v1
v1
GRC 
(12)
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GrC
k
, k1 
Re
1
where Gr the Grashof number, Re the Reynolds number, GR the mixed convection
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parameter, j  h12 the characteristic length, T and C the characteristic temperature
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and
concentration
C  C w1  C w 2
which
is
defined
as
T  Tw1  Tw2
if Tw1  Tw2
and
if C w1  C w 2
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Using (12), equations (2) to (6) in non-dimensional form become
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Region-I
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GRC
d 2 u1
k d
GRT
1

u1  1

1 
1
2
K 1  k1 
1  k1 dy
1  k1
1  k1
dy
d 2

2k1 
du 
 2  1   0
2  k1 
dy 
(13)
10
(14)
11
d 21
0
dy 2
(15)
12
d 21
0
dy 2
(16)
13
dy
2
Region-II
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d 2u2 h 2
 u 2  mbh 2 GRT  2  mbh 2 GRC  2
2
K
dy
(17)
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d 2 2
0
dy 2
(18)
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7
d 2 2
0
dy 2
(19)
1
with the boundary conditions
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u1  1  0, u 2 1  0, u1 0   u 2 0 ,
du1 0 
k
du 2 0 
1
 1  0  
,
dy
1  k1
mh1  k1  dy
d 0 
 0,   1  0,
dy
Tw1  T0
T   T0
 m1 ,  2 1  w 2
 m2 ,
T
T
d
1 d 2
1 0    2 0, 1 
dy h dy
1  1 
(20)
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C w1  C 0
C w 2  C 0
1  1 
 n1 ,  2 1 
 n2 ,
C
C
d
1 d 2
1 0   2 0, 1 
dy h dy
where
h
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h1




, m  1 ,   1 ,   1 , and b  1 . Are the channel width ratio, viscosity
h2
2
2
2
2
ratio, thermal conductivity ratio, density ratio and thermal expansion ratio, respectively.
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Solution
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On solving coupled linear differential equations from (13) to (19) under boundary and
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interface conditions (20), we have the solutions
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1  c1 y  c2
(21)
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1  c3 y  c4
(22)
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 2  c5 y  c 6
(23)
13
 2  c7 y  c8
(24)
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c1  
1
h
h
h
, c2 
, c5  
, c6 
1  h
1  h
1  h
1  h
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Region I
1

1
1
 y    c1 KGRT  c3 KGRC  c11e
2
2
L1  R
y
2 L2
 c12 e
 c13e
L R
 1
y
2 L2

1 
u1  y  
L4 y  L5  L6 e
8L2 k1 

L1  R
y
2 L2
(25)
 c14 e
L1  R
y
2 L2
2
L1  R
y
2 L2
 L7 e
L1  R
y
2 L2






3
Region II
u 2  y   c9 e
4
h
K
y
 c10 e

h
K
y
 mbKGRC c7 y  c8   mbKGRT c5 y  c6 
(26)
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R  4  4k1  16 Kk1  32 Kk12  k12  12 Kk13  16 K 2 k12  16 K 2 k13  4 K 2 k14
where L1  2  k1  4 Kk1  2 Kk12
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L2  K 2  k1 1  k1 
L3  8  8k1  32 Kk1  64 Kk12  2k12  24 Kk13  32 K 2 k12  32 K 2 k13  8K 2 k14
7
L4  24k12 GRC K 2 c3  8k13GRC K 2 c3  24k12 GRT K 2 c1  8k13GRT K 2 c1
8
 16k1GRT K 2 c1  16k1GRc K 2 c3
L5  24k12 GRT K 2 c2  8k13GRT K 2 c2  24k12 GRC K 2 c4  8k13GRC K 2 c4
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 16k1GRT K 2 c2  16k1GRc K 2 c4



L6  4c17 Kk1 2 L2 L1  R  c16 k1 2 L2 L1  R



 2c17 Kk12 2 L2 L1  R  4c16 Kk1 2 L2 L1  R


L L L  R   c k
k K 2 L L  R 


2 L L  R 



 c17 L2 L3 L1  R  2c16 2 L2 L1  R  2c17 2 L2 L1  R
 c16
 2c16
2
3
1
2
1
17 1
2
2

10
1
1






2 L L  R    2c L L  R 
Kk 2 L L  R   2c k K 2 L L  R 
L7  c15 L2 L3 L1  R  2c18 2 L2 L1  R  4c1 8Kk1 2 L2 L1  R


 c18 L2 L3 L1  R  k1c15


 c18 k1 2 L2 L1  R  4c15

 2c18 k12 K 2 L2 L1  R

2
1
1
15
2
1
2
2
15 1
1
2
11
1
9
where ci constants of integration, not included here for the sake of brevity.
1
Limiting Case
2
For a Newtonian fluid k1  0 , the solution of equations (13) and (19) using
boundary and interface conditions (20) are
u 2  y   c9 e
e
u1  y   c15
h
K
1
K
y
y
e
 c10

e
 c16

h
K
1
K
y
y
3
4
 mbKGRC c7 y  c8   mbKGRT c5 y  c6 
(27)
5
 KGRC c3 y  c4   KGRT c1 y  c2 
(28)
6
where ci constants are constants of integration, not included here for the sake of brevity.
7
8
Results and Discussion
9
An analytical solution for the problem of Heat and Mass Fully-developed Natural-
10
Convective Flow of Micropolar and Viscous Fluids through Porous Medium in a Vertical
11
Channel is investigated. The analytical solutions are evaluated numerically for different
12
values of governing parameters and results are presented through graph by assuming that
13
at second wall, the temperature is alike of mean temperature i.e. Tw2  T0 , so that
14
m1 , n1  1 and m2 , n2  0.
15
The effect of the mixed convection parameter or Grashof to Reynolds numbers
16
ratio GRT and GRC on the linear velocity and micro-rotation velocity are shown in
17
Figures. 2 and 5. Increase in the mixed convection parameter means an increase of the
18
buoyancy force which supports the motion. It is also observed from Figure 2 that if the
19
micropolar fluid is replaced by the clear viscous fluid, the effect of mixed convection
20
parameter GRT and GRC is still retained. But the magnitude of promotion is large for
21
viscous–viscous fluids system compared to micropolar–viscous fluids system. Figures 3
22
and 5 show the effect of mixed convection parameter on micro-rotation velocity. It is
23
10
observed from the figure that an increase of buoyancy force reduces the magnitude of
1
micro-rotation velocity.
2
Figures 6 and 7 display the effect of the viscosity ratio m  1  2 on the linear
3
velocity and micro-rotation velocity. As the viscosity ratio m increases, the linear velocity
4
increases, but the magnitude of promotion is large for k1 = 0 (Newtonian fluid) compared
5
with k1 = 1 (micropolar fluid). The effect of the viscosity ratio m is found to reduce the
6
micro-rotational velocity.
7
The effect of the channel width ratio h on the linear velocity and micro-rotation
8
velocity is shown in Figures 8 and 9, respectively. As the width ratio h increases, both the
9
linear velocity and the micro-rotation velocity increase for viscous– viscous k1 = 0 and
10
micropolar–viscous k1 = 1 fluids systems. The effect of the width ratio h is also found to
11
promote the temperature and mass fields as seen in Figures. 15 and 17.
12
The effect of the permeability parameter K on the linear velocity and micro-
13
rotational velocity is presented through Figures 10 and 11. It is clear from figure 10 that
14
an increase in the permeability parameter promotes the linear velocity. It is also observed
15
that if the micropolar fluid is replaced by the clear viscous fluid, the effect of permeability
16
parameter is still maintain, but the magnitude of promotion is large for viscous-viscous
17
fluids system compared to micropolar-viscous fluids system. Figure 11 shows the effect
18
of permeability parameter on micro-rotation velocity. It is observed from the figure that
19
an increase of permeability parameter reduces the magnitude of micro-rotation velocity.
20
The effect of the conductivity ratio  on the linear velocity and micro-rotational
21
velocity are presented through Figures 12 and 13. The effect of conductivity ratio is
22
predicted to increase the linear velocity for micropolar-viscous and viscous-viscous
23
(Figure 12) fluids system, but the effect of the conductivity ratio 
24
the micro-rotational velocity as seen in Figure 13.
25
11
The effects of the conductivity ratio  on the temperature and mass fields are
1
shown in Figures 14 and 16. The effect of the conductivity ratio  is predicted to
2
increase both temperature and mass fields, i.e. the larger the conductivity of the
3
micropolar fluid compared to the viscous fluid, the larger the flow nature.
4
5
Conclusions
6
We consider the fully-developed laminar natural-convective flow through porous medium
7
in a vertical channel with one region is filled with micropolar fluid and the other region
8
with a viscous fluid. It is found that effects of the Grashof to Reynolds number ratio,
9
channel width ratio, conductivity ratio and permeability parameter are to promote the
10
linear velocity where as the micropolar fluid material parameter suppressed the velocity.
11
Whereas the Grashof to Reynolds number ratio, channel width ratio, conductivity ratio,
12
micropolar fluid material parameter and permeability parameter repressed the micro-
13
rotational velocity. The effect of the width and conductivity ratio parameters promotes the
14
temperature and mass fields.
15
16
12
Nomenclature
1
b
thermal expansion coefficient ratio,  2 1
2
C
concentration
3
g
acceleration due to gravity
4
GrT
Grashof number for heat transfer
5
GrC
Grashof number for mass trasnfer
6
GRT
Grashof to Reynolds numbers ratio for heat transfer, GrT Re
7
GRC
Grashof to Reynolds numbers ratio for mass transfer, GrC Re
8
h
channel width ratio, h1 h2
9
h1
Height of Region-I
10
h2
Height of Region-II
11
j
micro-inertia density
12

ratio of thermal conductivities, 1  2
13
1
thermal conductivity of the fluid in Region-I
14
2
thermal conductivity of the fluid in Region-II
15
K
permeability parameter
16
k1
micropolar fluid material parameter
17
m
ratio of viscosities, 1  2
18
Re
Reynolds number
19
T0
average temperature
20
T
temperature
21
T1 ,T2 temperature of the boundaries
U0
22
average velocity
23
13
U
velocity
1
x  , y  space coordinates
2
Greek letters
3

coefficient of thermal expansion
4

spin gradient viscosity
5
j
vortex viscosity
6

micro-rotational velocity
7

viscosity
8
1
density of Region-I
9
2
density of Region-II
10

ratio of densities, 1  2
11
T
difference in temperature
12
i
dimensionless temperature
13
i
dimensionless mass
14
Subscript
15
1, 2
reference quantities for Region-I and Region-II, respectively
16
w
condition at the wall
17
18
14
References
1
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2
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3
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4
5
18
1
2
3
4
Figure 1. Geometrical configuration
19
1
2
3
Figure 2. Velocity distribution for different values of GRT
20
1
2
3
Figure 3. Micro-rotational velocity distribution for different values of GRT
21
1
2
3
Figure 4. Velocity distribution for different values of GRC
22
1
2
3
Figure 5. Micro-rotational velocity distribution for different values of GRC
23
1
2
Figure 6. Velocity distribution for different values of viscosity ratio m
24
Figure 7. Micro-rotational velocity distribution for different values of viscosity
ratio m
25
1
2
3
4
5
1
2
3
Figure 8. Velocity distribution for different values of width ratio h
26
Figure 9. Micro-rotational velocity distribution for different values of width ratio
h
27
1
2
3
4
5
Figure 10. Velocity distribution for different values of permeability parameter K
28
1
2
3
4
5
6
1
2
3
4
Figure 11. Micro-rotational velocity distribution for different values of
permeability parameter K
29
1
2
Figure 12. Velocity profiles for different values of conductivity ratio  .
30
Figure 13. Micro-rotational velocity profiles for different values of conductivity
ratio  .
31
1
2
3
1
2
3
Figure 14. Temperature profiles for different values of conductivity ratio  .
32
1
2
Figure 15. Temperature profiles for different values of width ratio h.
33
1
2
3
Figure 16. Mass profiles for different values of conductivity ratio  .
34
1
2
Figure 17. Mass profiles for different values of width ratio h.
35