Applied Mathematics, 2011, 2, 935-941
doi:10.4236/am.2011.28128 Published Online August 2011 (http://www.SciRP.org/journal/am)
Free Convection Flow between Vertical Plates Moving in
Opposite Direction and Partially Filled with
Porous Medium
Umesh Gupta1, Abhay Kumar Jha2, Rama Charan Chaudhary3
1Department of Mathematics, Institute of Engineering and Technology, JK Lakshmipat University, Jaipur, India
2Department of Mathematics, Apex Institute of Engineering and Technology, Jaipur, India
3Yagvalkya Institute of Technology, Jaipur, India
Received January 25, 2011; revised June 18, 2011; accepted June 25, 2011
Abstract
The laminar fully developed free-convection flow in a channel bounded by two vertical plates, partially filled
with porous matrix and partially with a clear fluid, has been discussed when both the plates are moving in
opposite direction. The momentum transfer in porous medium has been described by the Brinkman-extended
Darcy model. The affect of Darcy number on flow velocity has been discussed in fluid region, interface re-
gion and porous medium with the help of graphs. Analytic method has been adopted to obtain the expres-
sions of velocity and temperature. The skin-friction component has also been determined and presented with
the help of tables.
Keywords: Free Convection, Porous Medium, Skin-Friction
1. Introduction
Flow in a region, part of which is occupied by a clear
fluid and part by a fluid-saturated porous medium, has
recently attracted considerable attention due to its com-
mon occurrence in both geophysical and industrial en-
vironments, including engineering applications such as
thermal-energy storage system, a solar collector with a
porous absorber and porous journal-bearings. Flow me-
chanism at the fluid/porous interface, was first studied
by Beavers & Joseph [1] and it was investigated that the
velocity gradient on the fluid side of the interface is pro-
portional to the slip velocity at the interface. Taylor [2]
& Richardson [3] continued the investigation in which
they modeled fluid flow by Darcy number. Further,
Vafai & Kim [4] modeled the flow in the porous region
utilizing the so-called Brinkman-Forchheimer-extended
Darcy equation (Vafai & Kim [5] and Kuznetsov [6]).
Alazmi & Vafai [7], Valencia-Lopez and Ochoa-Tapia
[8] also inv estigated flu id flow and heat tran sfer interfa-
cial conditions of fluid and porous layers.
Furthermore, convection in porous media is applied in
utilization of geothermal energy, the control of pollutant
spread in groundwater, the design of nuclear reactors,
compact heat exchangers, solar power collectors, heat
transfer associated with the deep storage of nuclear
waste and high performance insulators for buildings.
Considerable progress in this area was made by Nield &
Bejan [9] and Kaviany [10]. Vafai & Tien [11] also
analyzed the effects of a solid boundary and the inertial
forces on flow and heat transfer in porous media.
The coupled fluid flow and heat transfer problem in a
fully developed composite region of two parallel plates
filled with Brinkman-Darcy porous medium was ana-
lytically investigated by Kaviany [12]. Rudraiah & Na-
graj [13] studied the fully developed free-convection
flow of a viscous fluid through a porous medium
bounded by two heated vertical plates. Beckerman [14]
studied natural convection in vertical enclosures con-
taining simultaneously fluid and porous layers. Recently,
Khalili et al. [15] studied the instability of superimposed
fluid and porous layers with vertical through-flow gov-
erned by Darcy-Forchheimer equation. Free convection
between vertical walls partially filled with porous me-
dium was investigated by Paul et al. [16]. Singh et al.
[17] analyzed heat and mass transfer phenomena due to

936 U. GUPTA ET AL.
natural convection in a composite cavity containing a
fluid layer overlying a porous layer saturated with the
same fluid, in which the flow in the porous region was
modeled using Brinkman-Forchheimer-extended Darcy
model that includes both the effect of macroscopic shear
(Brinkman effect) and flow inertia (Forchheimer effect).
In most of the flow studies in the channel, the plates
are stable. However, to the best of the author’s knowl-
edge, the effect of the same idea has not been studied yet,
when both the plates are moving. In this paper we ex-
tend the prob lem of [16], when both the plates are mov-
ing in the opposite direction.
2. Governing Equations
A channel of two vertical plates partially filled with
porous matrix and partially with a clear fluid having
interface is shown in Figure 1. The laminar fully de-
veloped free-conv ection flow bounded in the channel is
discussed, when both the plates are moving in opposite
direction and one plate is heated and other is cooled.
The
x
-axis is taken along one of the wall and
y
-
axis normal to it. The plate in the fluid region and the
plate in porous region are moving in opposite direction,
where
f
U and
p
U are the velocities in the direction
of
x
-axis. The temperature is also considered on the
walls 0yand
y
Has

f
ch
TTATT 
c
and

p
ch
TTBTT 
c
respectively.
Under usual Boussinesq approximation, the flow in
fluid and porous regions is governed by the following
equations:
Free Fluid Region:

2
2
d0
d
f
fc
Ug
TT
y


 (1)
Figure 1. Physical configuration of the system.

22
2
d
d0
dd
fc
fTT
U
yy








 (2)
Porous Region:

2
2
d0
d
pp
pc
UU g
TT
k
y



 (3)
222
2
dd(
0
dd
pc
pp
UU TT
yk y


 )




 (4)
The corresponding boundary conditions are


 
20
20
0: ;
:;
dd dd
:; ;;
dd dd
hc
fc hc
f
hc
pc hc
p
fp
fp
fp
fp
gH TT u
yU TTATT
gH TT u
yHUTTBTT
UU TT
ydU UT T
yy yy







 




 







(5)
Introduci ng following non - di mensional qua nt i t ies:
2
k
Da
H
; y
y
H

; d
d
H
;

2
f
f
hc
U
U
g
HT T


;

2
p
p
hc
U
U
g
HTT


;


f
c
f
hc
TT
TT


;



p
c
p
hc
TT
TT


;

22 4hc
g
HT T
N




(6)
Equations (1) to (4) become
Fluid Region:
2
2
d0
d
f
f
U
y
 
(7)
2
2
2
dd
0
d d
ff
U
Ny
y






(8)
Porous Region:
2
2
d0
d
pp
p
UU
Da
y


 (9)
2
22
2
dd 0
d d
pp
p
UN
N
yDa
y

 U



  (10)

U. GUPTA ET AL.
937
In Equation (9), The momentum transfer in porous
domain is described based on Brinkman-extended Darcy
model [18].
Where Da is the Darcy number, d the distance of in-
terface from the plate , g the acceleration due to
gravity, H the distance between vertical plates,
0y
k the
permeability of the porous matrix, the thermal con-
ductivity, N the Buoyancy parameter,
 the coefficient of
thermal expansion,  the dynamic viscosity,  the kine-
matic viscosity,
 the density, and  is the temperature.
The subscripts f, represent fluid layer, p porous layer, h
hot plate and c, the cold plate.

The boundary and matching conditions (5) in dimen-
sionless form are:
0
0
0: ;
1: ;
dd
:;
dd
dd
:;
dd
ff
pf
f
p
fp
fp
fp
yUu A
yU uB
UU
ydU Uyy
yd yy






 
 
(11)
where, the matching conditions for velocity are due to
continuity of velocity and shear stress at the interface.
The continuity of temperature and heat flux at the inter-
face has been considered as matching conditions for
temperature.
3. Solution
It can be observed that problem is non-linear due to vis-
cous and Darcy dissipation terms. This problem can be
tackled by using a perturbation method as N is small in
most of the practical problems. Accordingly we assume,
for small N, the expansions:




2
01
2
01
2
01
2
01
ff f
pp p
ff f
pp p
UUNU ON
UUNU ON
NON
NON

 

 
 
 
(12)
Substituting (12) in the Equations (7) to (10) gives
200
2
d0
d
f
f
U
y


 (13)
211
2
d0
d
f
f
U
y


 (14)
20
2
d0
d
f
y


(15)
2
210
2
dd 0
d d
ff
U
y
y






(16)
2000
2
d10
d
p
pp
UU
Da y



(17)
2111
2
d10
d
p
pp
UU
Da y


 (18)
20
2
d0
d
p
y

 (19)
2
210 2
0
2
dd 10
d d
pp
p
UU
yDa
y





  (20)
The correspondin g bou ndary condition s are
001 01
0:;0;; 0
ffff
yUuU A



;
001 01
1:;0;; 0
pppp
yUuU B



;
001
0011
0011
0011
:
dddd
;; ;
dddd
dddd
;;;
dddd
1
f
pf
fpfp
fpfp
fpfp
yd
UUUU
UUUUyyy
yyyy


p
y

 
 
(21)
Solving Equations (13) to (20) using boundary condi-
tions (21) gives the following velocity and temperature
components

23
00
26
f
Ay y
UABu 5 Ky


04
2ee
my my
p
ABAy
UK
m



3
K


00fp
A
BAy  
 
265
5
1
322 24 5513
120 2012
12 32
f
4
A
ByAABy KABy
AK yKy Ay Ky


 
 

 


22
22 22 22
3 4
124
243
4
222
4
322
312 11
2
e e
22 22
2e
12 3
2e
2e
2e
my my
p
my
my
my
my
BAy
K KAy
mm
BAy ABAyKBA
Am
mmm
KBAy
KBA
Am
mm
KBAy Ky K
m





 
 

 











938 U. GUPTA ET AL.
 
287
5
1
524 3 26 551322
6720 840360
360 60246
f
6
A
ByAA ByKA By
U
AK yKyKy
Ay Ky

 
 









22 22 3
4
121 2022
22
22 2
62
44 2244
22
6
34
45
3
22
e e
ee
66
12
21
8
12 24 2
()
e2 16
3
()
e2 1
24
e
221
my my
my my
p
my
my
my
my
K K
UK Kmm
AB BA
my A
mmm
BA
K
mymyA m
m
AB Ay
mym y
m
BA
K
2
K
BAe
Amy
m
mm
KBA
my my




 








 





 
 





5
22 12 11
2
4
221
m
Ky K
my mym


where

1/2
mDa

;

3
2
10
23
ABd
Ad
K
u

 ;
 
(1 )(1 )
21e 1e
md md
Kmd md

  ;

31
22
2
1e1e
mmd
AB
KK md
Kmm

 
 
 

 

0
u



;
2
403
2ee
mm
B
Ku K
m


 

 ;



543 2
3
2
0
1ee
;
26
md md
A
BAd
KKK
dm
ABd
Ad u






 



 

254
23
6
3
522
55
204 3
;
3
ABd AABd
A
d
K
KABdAK dKd

 


 

265
24
7
4322
555
12020 12
;
123 2
ABd AABd
A
d
K
KABdAK dKd

 






2
22 22 2
3 4
824
2
4
222
4
322
3
2
e e
22
22
2e
12 3
2
2e
2e
;
m m
m
m
m
m
BA
K KA
Kmm
BA ABAKBA
Am
mmm
KBAe
KBA
Am
mm
KBA
m




 
 

 










 
 
22 22 22
3 4
92
22
24
422
3 422
43
22
e e
22 2
2123
2e 2e
2e2e
;
md md
md md
md md
K KAd
Km
BAdBAd ABAd
mmm
K KBA BA
AA
mm mm
KBAdKBAd
mm



 
 


 

 
 


3




 


2
2
2222
10 4324
232
4
22
4
3
3
ee
2e
3
2e
2e
2e
;
md md
md
md
md
md
BAd
Ad
KmK mKmm
BAdABAdAK
m mm
KBAd
AK
mm
KBAd
m




 

 






1110 69 7
K
KKdKK

;
128 11
K
KK

 ;
1310612
K
KKK

 ;








 



2
22 22 22
3 4
14 2262
2
42 4
84
23
64
4
5
3 22
5
12 11
2
e e12
662
12 24
12 2
e2 16
32
e21 4
221 221
4
;
m m
m
my
AB
K K
KmA
mmm m
AB BA
K
mm A
m mm
AB AK
mm
mm
BA KBA
Am
mm
KBA
mm mm
m
KK
m




  







 


 



 



U. GUPTA ET AL.
939
 

287
15
65
26
55
243
513
6720 840
360360 60
;
24 6
ABd AABd
K
KABd
A
Kd
Ad
KdK d








 

 




22
22 22
3
4
16 226
22
244
28
446
22 34
43
22
55
22 1
e e12
662
12 24
12
e(21)
23
6e2
2
ee
221
44
221
md
md
md
md
md md
K K
Kmd
mmm
AB BA
Amdm
mm
BA ABAd
KAmd
m mm
BA
K
md Amd
m m
KBA KBA
md md
mm
K
md md



 








 



 





22
1
d
211
2
dK
m

  
276 5
5
17
423 2
25 5513
840 12060
;
60 1262
A
BdAA BdKA Bd
K
AK dKdKd Ad

 
 






 





2
22 22 2
3 4
18 42
2
22 22
66
3 43
4
4
22
44
322
4
e
e
33
6
3
e(21)
2
e(12) 2
()e
e2 122 1
4
e221
4
md md
md
md
md
md
md
AB
K
Kd
KA
mm
mm
BA ABA
dmdmd
mm
BA K
KAmd
m m
BA KBA
Amd
mm
KBA
mdm dmd
m
KBA md md
m
K





  






 


 




 




3
2
2m



312
42
e21 ;
2
md
BA K
md
mm


 
191416 151817
1e
md m
KmdKK KdKK

 
e


;
2019 2
K
KK;
2
21 1420
ee
mm
KKK

 ;
2221201817
ee
md md
K
mKmKKK

 .
Skin-Friction Components
At plate 0y

:
15 22
K
NK



At plate 1y

:




 

 





243
2
2
22 22 2
3
442
2
22
4
66
33
44
44
3
22
4
ee
e
e1
33
()
62
32
e21 2
()e
e12e2 1
24
e
221 2
4
mm
m
m
m
m
mm
m
BA mK mK
m
AB
K
K
NA
mm mm
BA K
AB A
mm
mmm
BA BA
K
AmA
mm m
KBA KB A
mm
mm
KBA
mm m
m





 
 



 




 
 
 
 
 






3

312 21 20
42
21
e21e e
2
m
mm
m
KBA K
mmKmK
mm







4. Discussions and Conclusion
Natural convection in a vertical channel partially filled
with porous medium has been discussed in the preceding
sections when it is assumed that plates are moving in
opposite direction. The governing equations having non-
linear nature have been solved by analytical method.
Different types of interfacial conditions between a po-
rous medium and fluid layer are analyzed in detail. Three
primary regions were found likewise, fluid region (near
wall 0y

), interface region and porous region (near
the wall 1y

). The affect of Darcy number on the flow
has been discussed.
When viscous and Darcy dissipation are zero, velocity
profiles between vertical plates have been illustrated in
Figure 2, for the cases when 1
A
, (plate 1B0y

is heated and plate 1y

is cooled) and 0A

, 1B

(plate 0y

is cooled and is heated). It is ob-
served that in first region, the fluid velocity increases
with increasing Darcy number (Da), whereas the reversal
phenomenon occurs in third region. Near the interface
region in porous medium, the velocity is constant for Da
= 10–3 and effect of Brinkman term is almost negligible,
and flow is only characterized by classical Darcy
1y

940 U. GUPTA ET AL.
Figure 2. Graph of U0 against y.
law [19], but for Da = 10–2, the velocity does not show
the constant nature. It is also observed that the fluid ve-
locity near the heated wall 0y

(i.e.,1
A
,0B

) is
more than the cooled wall 0y

(i.e., ,0BA 1

), i.e.,
heat in the porous medium exerts the restraining force on
the fluid.
Figure 3 and 4 show the effect of viscous and Darcy
dissipation terms on the velocity and temperature respec-
tively. The governing equations become non-linear be-
cause of existence of these terms whereas very small
influence of these terms on both temperature and veloc-
ity fields, is observed.
The numerical values of skin-friction on both the
plates are also obtained and shown in Tables 1 and 2 for
and respectively. τ1 is the skin-friction
00u00u
Figure 3. Graph of  against y.
Figure 4. Graph of U1 against y.
Table 1. Values of skin-friction for U0 = 0.
Table 2. Values of skin-friction for U0 = 0.5.
A = 1.0, B = 0.0 A = 0.0, B = 1.0
NDad1 1
0.3–0.73864631.836859 –0.9040495 1.9791916
0.5–0.58607691.8926462 –0.7598024 2.0320331 0.1 0.7–0.57395951.8944114 –0.7456908 2.0370888
0.3–1.07228035.0123941 –1.2198258 5.0923165
0.5–0.60179965.0225739 –0.7792456 5.1018567
0
0.01 0.7–0.39343165.0685534 –0.5767524 5.1473955
0.3–0.73586371.839478 –0.9014259 1.9816794
0.5–0.58279511.8955703 –0.7567184 2.0348034 0.1 0.7–0.57008951.8980549 –0.7420603 2.0405343
0.3–1.07082655.0132408 –1.2184213 5.0931465
0.5–0.59971825.023457 –0.7772541 5.1027177
0.1
0.01 0.7–0.39061995.0697341 –0.5740707 5.148541
when 1
A

,0B

and τ2 is the skin friction when
0A

,1.B

Similar effects of Darcy number on skin-
friction are observed as mentioned in [16] for 00u

, as
A = 1.0, B = 0.0 A = 0.0, B = 1.0
NDad1 1
0.30.26013170.0772881 0.0947285 0.2196207
0.50.29608250.0916772 0.122357 0.2310641
0.1 0.70.32215760.1221194 0.1504263 0.2647968
0.30.18 0.0101185 0.0324544 0.090041
0.50.24277590.011647 0.0653299 0.0909297
0
0.010.70.29407340.0249912 0.1107526 0.1038333
0.30.26016310.077311 0.0947545 0.2196449
0.50.29612610.0917077 0.1223924 0.231094
0.1 0.70.32221150.1221609 0.1504735 0.2648389
0.30.18000620.0101198 0.032457 0.0900422
0.50.24279570.0116499 0.065338 0.0909317
0.1
0.010.70.29411410.025 0.1107776 0.1038401

U. GUPTA ET AL.
941
clearly shown in Table 1. Increasing values of skin- fric-
tion are observed with increasing width of fluid layer in
Table 2. It is also found in Table 2, that increasing
Darcy number and dissipation results a very small in-
crement in skin-friciton. The effect of temperature on
skin-friction on both the plates is also studied and found
that the skin friction on both the plates increases when
those are heated.
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