Journal of Modern Physics, 2011, 2, 62-71

doi:10.4236/jmp.2011.22011 Published Online February 2011 (http://www.SciRP.org/journal/jmp)

Copyright 2011 SciRes. JMP
Natural Convective Boundary Layer Flow over a
Horizontal Plate Embedded in a Porous Medium
Saturated with a Nanofluid
Rama Subba Reddy Gorla
1
, Ali Chamkha
2

1
Cleveland State University, Cleveland, USA
2
Public Authority for Applied Education and Training, Shuweikh, Kuwait
E-mail: r.gorla@csuohio.edu, achamkha@yahoo.com
Received September 5, 2010; revised October 8, 2010; accepted October 11, 2010
Abstract

A boundary layer analysis is presented for the natural convection past a horizontal plate in a porous medium
saturated with a nano fluid. Numerical results for friction factor, surface heat transfer rate and mass transfer
rate have been presented for parametric variations of the buoyancy ratio parameter Nr, Brownian motion pa-
rameter Nb, thermophoresis parameter Nt and Lewis number Le. The dependency of the friction factor, sur-
face heat transfer rate (Nusselt number) and mass transfer rate on these parameters has been discussed.

Keywords: Natural Convection, Porous Medium, Nanofluid
1. Introduction

The study of convective heat transfer in nanofluids is
gaining a lot of attention. The nanofluids have many ap-
plications in the industry since materials of nanometer
size have unique physical and chemical properties. Nan-
ofluids are solid-liquid composite materials consisting of
solid nanoparticles or nanofibers with sizes typically of
1-100 nm suspended in liquid. Nanofluids have attracted
great interest recently because of reports of greatly en-
hanced thermal properties. For example, a small amount
(<1% volume fraction) of Cu nanoparticles or carbon
nanotubes dispersed in ethylene glycol or oil is reported
to increase the inherently poor thermal conductivity of
the liquid by 40% and 150%, respectively [1,2]. Conven-
tional particle-liquid suspensions require high concentra-
tions (>10%) of particles to achieve such enhancement.
However, problems of rheology and stability are ampli-
fied at high concentrations, precluding the widespread
use of conventional slurries as heat transfer fluids. In
some cases, the observed enhancement in thermal con-
ductivity of nanofluids is orders of magnitude larger than
predicted by well-established theories. Other perplexing
results in this rapidly evolving field include a surpris-
ingly strong temperature dependence of the thermal
conductivity [3] and a three-fold higher critical heat flux
compared with the base fluids [4,5]. These enhanced
thermal properties are not merely of academic interest. If
confirmed and found consistent, they would make nan-
ofluids promising for applications in thermal manage-
ment. Furthermore, suspensions of metal nanoparticles
are also being developed for other purposes, such as
medical applications including cancer therapy. The in-
terdisciplinary nature of nanofluid research presents a
great opportunity for exploration and discovery at the
frontiers of nanotechnology.
Porous media heat transfer problems have several en-
gineering applications such as geothermal energy recov-
ery, crude oil extraction, ground water pollution, thermal
energy storage and flow through filtering media. Cheng
and Minkowycz [6] presented similarity solutions for
free convective heat transfer from a vertical plate in a
fluid-saturated porous medium. Gorla and co-workers
[7,8] solved the nonsimilar problem of free convective
heat transfer from a vertical plate embedded in a satu-
rated porous medium with an arbitrarily varying surface
temperature or heat flux. Chen and Chen [9] and Mehta
and Rao [10] presented similarity solutions for free con-
vection of non-Newtonian fluids over horizontal surfaces in
porous media. Nakayama and Koyama [11] studied the
natural convection over a non-isothermal body of arbitrary
geometry placed in a porous medium. All these studies
were concerned with Newtonian fluid flows. The bound-
ary layer flows in nano fluids have been analyzed re-
R. S. R. GORLA ET AL.

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63
cently by Nield and Kuznetsov and Kuznetsov [12] and
Nield and Kuznetsov [13]. A clear picture about the nan-
ofluid boundary layer flows is still to emerge.
The present work has been undertaken in order to
analyze the natural convection past an isothermal hori-
zontal plate in a porous medium saturated by a nanofluid.
The effects of Brownian motion and thermophoresis are
included for the nanofluid. Numerical solutions of the
boundary layer equations are obtained and discussion is
provided for several values of the nanofluid parameters
governing the problem.

2. Analysis

We consider the steady free convection boundary layer
flow past a horizontal plate placed in a nano-fluid satu-
rated porous medium. The co-ordinate system is selected
such that x-axis is in the horizontal direction. We con-
sider the two-dimensional problem. Figure 1 shows the
coordinate system and flow model. At the surface, the
temperature T and the nano-particle fraction take con-
stant values T
W
and
W
, respectively. The ambient val-
ues, attained as y tends to infinity, of T and are denoted
by T

and

, respectively.
The Oberbeck-Boussinesq approximation is employed
and the homogeneity and local thermal equilibrium in the
porous medium are assumed. We consider the porous
medium whose porosity is denoted by and permeability
by K. The Darcy velocity is denoted by v

. The follow-
ing four field equations embody the conservation of total
mass, momentum, thermal energy, and nano-particles,
respectively. The field variables are the Darcy velocity
v

,
the temperature T and the nano-particle volume fraction.
.v 0 V =

(1)
( ) ( ) ( ) { }
1 1
f
p f
v
P v T T g
t K

|
c

c
(
= V + +

c

(2)
( ) ( )
( )
2
m f
T
m B
p
T
c c v T
t
D
k T c D T T T
T

c

c
+ V =
c
(
V + V V + V V
(

(3)


Figure 1. Coordinate system and flow model.
2 2
1
T
B
D
v D T
t T

c

c
+ V = V + V
c

(4)
We write
( ) v u , v =

.
Here
f
, and are the density, viscosity and volu-
metric volume expansion coefficient of the fluid;
p

the density of the particles; g the gravitational accelera-
tion;
( )
m
c
the effective heat capacity and k
m
effective
thermal conductivity of the porous medium and D
B
the
Brownian diffusion coefficient and D
T
the thermopho-
retic diffusion coefficient. The flow is assumed to be
slow so that an advective term and a Forchheimer quad-
ratic drag term do not appear in the momentum equation.
The boundary conditions are taken to be
0 0
W W
v , T T , , at y , = = = = (5)
0 u , T T , , as y

= (6)
We consider the steady state flow. In keeping with the
Oberbeck-Boussinesq approximation and an assumption
that the nano-particle concentration is dilute, the mo-
mentum equation may be written as:
( )( ) ( ) ( )
0
1
P f f
P v
K
T T g

|

= V +
(
+

(7)
We now make the standard boundary layer approxi-
mation based on a scale analysis and write the governing
equations.
0
u v
x y
c c
+ =
c c
(8)
P
u
x K
c
=
c
(9)
( ) ( ) ( ) ( ) 1
f P f
P
g T T g
y
|

c
(
=

c

(10)
2
2 T
m B
D T T T T
u v T D
x y y y T y

o t

(
| || | c c c c c
+ = V + + (
| |
c c c c c
( \ . \ .


(11)
2 2
2 2
1
T
B
D T
u v D
x y T y y

c

| | | | c c c c
+ = +
| |
c c c c
\ . \ .
(12)
where
( )
( )
( )
p
m
m
f f
c
k
,
c c
c
o t

= =
(13)
One can eliminate P from Equations (9) and (10) by
cross-differentiation. At the same time one can introduce
a stream line function such that the continuity is auto-
matically satisfied:
R. S. R. GORLA ET AL.

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64
u , v
dy dx
c c
= =
(14)
We are then left with the following three equations.
( ) ( )
2
2
1
P f f
gK gK
T
x x y
|




c c c
= +
c c c
(15)
2
2 T
m B
D T T T T
T D
y x x y y y T y

o t

(
| || | c c c c c c c
= V + + (
| |
c c c c c c c
( \ . \ .


(16)
2 2
2 2
1
T
B
D T
D
y x x y T y y

c

| | | | c c c c c c
= +
| |
c c c c c c
\ . \ .
(17)
Proceeding with the analysis we introduce the follow-
ing dimensionless variables:
1
3
x
y
Ra
x
q =
( ) 1
f
x
m
gKAx
Ra
|
o

1 3 /
m x
S
Ra

o
=

W
T T
T T
u

W
f

(18)
We assume that T
w
and

are constants.
Substituting the expressions in Equation (18) into the
governing Equations (15)-(17) we obtain the following
transformed equations:
| |
2
0
3
r
S N f q u '' ' ' = (19)
( )
2 1
0
3
b t
S N f N u u u u '' ' ' ' ' + + + = (20)
1
0
3
t
e
b
N
f L S f
N
u '' ' '' + + = (21)
where the four parameters are defined as:
( )( )
( )( ) 1
P f W
r
f W
N
T T

|



=

,
( ) ( )
( )
B W
P
b
m
f
c D
N
c
c
o

= ,
( ) ( )
( )
T W
P
t
m
f
c D T T
N
c T
c
o

= ,
m
e
B
L
D
o
c
=

(22)
The transformed boundary conditions are:
0 0 1 1
0 0 0
: S , , f
: S , , f
q u
q u
= = = =
' = = =
(23)
The local friction factor may be written as
( )
0
2
2 0
2
y x
x
x
u
y
Ra .S"
Cf
Re .Pr U

=
| | c
|
c
\ .
= =
The heat transfer rate at the surface is given by:
0
W f
y
T
q k
y
=
| c
=
|
c
.

The heat transfer coefficient is given by:
( )
W
W
q
h
T T

Local Nusselt number is given by:
( )
1
3
0
x x
f
h x
Nu Ra ,
k
u

' = =
(24)
The mass transfer rate at the surface is given by:
( )
0
W m W
y
N D h
y

=
| c
= =
|
c
.

where
m
h = mass transfer coefficient,
The local Sherwood number is given by:
( )
1
3
0
m
x
h x
Sh Ra f ,
D

' = = (25)

3. Results and Discussion

3.1. Numerical Method

The system of Equations (19)-(21) with the boundary
conditions (23) is solved numerically by means of an
efficient, iterative, tri-diagonal implicit finite-difference
method discussed previously by Blottner [13]. Equations
(19)-(21) are discretized using three-point central differ-
ence formulae with S' replaced by another variable V.
The q direction is divided into 196 nodal points and a
variable step size is used to account for the sharp
changes in the variables in the region close to the surface
where viscous effects dominate. The initial step size used
is
1
0 001 . Aq = and the growth factor 1 037 K . = such
that
1 n n
K Aq Aq

= (where the subscript n is the number
of nodes minus one). This gives
max
q 35 which repre-
R. S. R. GORLA ET AL.

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65
sents the edge of the boundary layer at infinity. The or-
dinary differential equations are then converted into lin-
ear algebraic equations that are solved by the Thomas
algorithm discussed by Blottner [14]. Iteration is em-
ployed to deal with the nonlinear nature of the governing
equations. The convergence criterion employed in this
work was based on the relative difference between the
current and the previous iterations. When this difference
or error reached 10
-5
, the solution was assumed con-
verged and the iteration process was terminated.
Equations (19)-(21) were solved numerically to satisfy
the boundary conditions (23) for parametric values of Le,
Nr (buoyancy ratio number), Nb (Brownian motion pa-
rameter) and Nt (thermophoresis parameter) using finite
difference method. Tables 1-5 indicate results for wall
values for the gradients of velocity, temperature and
concentration functions which are proportional to the
friction factor, Nusselt number and Sherwood number,
respectively. From Tables 1-3, we notice that as Nr and
Nt increase, the friction factor increases whereas the heat
transfer rate (Nusselt number) and mass transfer rate
(Sherwood number) decrease. As Nb increases, the fric-

Table 1. Effects of N
r
on S"(0), '(0) and f'(0) for N
b
= 0.3,
N
t
= 0.1 and Le = 10.
N
r
S"(0) '(0) f'(0)
0 2.514805E-05 3.279025E-01 1.498672
0.1 8.547099E-05 3.263273E-01 1.484164
0.2 1.189362E-04 3.246233E-01 1.468161
0.3 2.209482E-04 3.224377E-01 1.452664
0.4
2.975905E-05
3.209329E-01 1.436392
0.5 1.645268E-04 3.185953E-01 1.419499

Table 2. Effects of N
t
on S"(0), '(0) and f'(0) for N
b
= 0.3,
N
r
= 0.5 and Le = 10.
N
t
S"(0) '(0) f'(0)
0.1 1.645268E-04 3.185953E-01 1.419499
0.2 1.337430E-07 3.052335E-01 1.416536
0.3 3.323112E-07 2.933325E-01 1.416866
0.4 3.482237E-05 2.817253E-01 1.421582
0.5 7.448123E-05 2.709439E-01 1.429226

Table 3. Effects of N
b
on S"(0), '(0) and f'(0) for N
r
=0.5,
N
t
=0.1 and Le=10.
N
b
S"(0) '(0) f'(0)
0.1 -5.269319E-06 3.679768E-01 1.327454
0.2 5.612235E-05 3.433554E-01 1.393615
0.3 1.645268E-04 3.185953E-01 1.419499
0.4
2.058221E-05
2.942436E-01 1.435464
0.5 1.521388E-04 2.724658E-01 1.447720
Table 4. Effects of Le on S"(0), '(0) and f'(0) for N
b
=0.3,
N
r
= 0.5, and N
t
= 0.1.
Le S"(0) '(0) f'(0)
1 9.363982E-05 2.782226E-01 2.825313E-01
10 1.645268E-04 3.185953E-01 1.419499
100 -2.860076E-04 3.123462E-01 4.712465
1000 -5.822286E-04 3.077888E-01 15.029030

Table 5. Effects of Le on S"(0), '(0) and f'(0) for N
b
= 0,
N
r
= 0, and N
t
= 0.
Le S"(0) '(0) f'(0)
1 1.871019E-04 4.303957E-01 4.303957E-01
10 1.871019E-04 4.303957E-01 1.483679
100 1.871019E-04 4.303957E-01 4.732183
1000 1.871019E-04 4.303957E-01 14.978180

tion factor and surface mass transfer rates increase
whereas the surface heat transfer rate decreases. Results
from Table 4 indicate that as Le increases, the heat and
mass transfer rates increase. From Table 5, we observe
that the nano fluids display drag reducing and heat and
mass transfer rate reducing characteristics.
Figures 2-4 indicate that as Nr increases, the velocity
decreases and the temperature and concentration increase.
Similar effects are observed from Figures 5-10 as Nt and
Nb vary.
Figure 11 illustrates the variation of velocity within the
boundary layer as Le increases. The velocity increases as
Le increases. From Figures 12 and 13, we ob- serve that
as Le increases, the temperature and concentration within
the boundary layer decrease and the thermal and concen-
tration boundary later thicknesses decrease.
The influence of nanoparticles on natural convection is
modeled by accounting for Brownian motion and ther-
mophoresis as well as non-isothermal boundary condi-
tions. The thickness of the boundary layer for the mass
fraction is smaller than the thermal boundary layer
thickness for Large values of Lewis number Le. The
contribution of N
t
to heat and mass transfer does not de-
pend on the value of Le. The Brownian motion and
thermophoresis of nano particles increases the effective
thermal conductivity of the nanofluid. Both Brownian
diffusion and thermophoresis give rise to cross diffusion
terms that are similar to the familiar Soret and Dufour
cross diffusion terms that arise with a binary fluid dis-
cussed by Lakshmi Narayana et al. [15].

4. Concluding Remarks

In this paper, we presented a boundary layer analysis for
the natural convection past a non-isothermal vertical
R. S. R. GORLA ET AL.

Copyright 2011 SciRes. JMP
0 1 2 3 4 5 6 7 8
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
1. 2
N
r
=0, 0. 1, 0. 2, 0. 3, 0. 4, 0. 5
N
b
=0. 3
N
t
=0. 1
Le=10
S
'
q

Figure 2. Effects of N
r
on velocity profiles.

0 1 2 3 4 5 6
0. 00
0. 25
0. 50
0. 75
1. 00
N
b
=0. 3
N
t
=0. 1
Le=10
N
r
=0, 0. 1, 0. 2, 0. 3, 0. 4, 0. 5
u
q

Figure 3. Effects of N
r
on temperature profiles.

0. 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0
0. 00
0. 25
0. 50
0. 75
1. 00
N
b
=0. 3
N
t
=0. 1
Le=10
N
r
=0, 0. 1, 0. 2, 0. 3, 0. 4, 0. 5
f
q

Figure 4. Effects of N
r
on volume fraction profiles.
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67
0 1 2 3 4 5 6 7 8
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
1. 2
N
b
=0. 3
N
r
=0. 5
Le=10
N
t
=0. 1, 0. 2, 0. 3, 0. 4, 0. 5
S
'
q

Figure 5. Effects of N
t
on velocity profiles.

0 1 2 3 4 5 6
0. 00
0. 25
0. 50
0. 75
1. 00
N
b
=0. 3
N
r
=0. 5
Le=10
N
t
=0. 1, 0. 2, 0. 3, 0. 4, 0. 5
u
q

Figure 6. Effects of N
t
on temperature profiles.

0 1 2 3 4 5
0. 00
0. 25
0. 50
0. 75
1. 00
N
b
=0. 3
N
r
=0. 5
Le=10
N
t
=0. 1, 0. 2, 0. 3, 0. 4, 0. 5
f
q

Figure 7. Effects of N
t
on volume fraction profiles.
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68
0 1 2 3 4 5 6 7 8
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
1. 2
N
r
=0. 5
N
t
=0. 1
Le=10
N
b
=0. 1, 0. 2, 0. 3, 0. 4, 0. 5
S
'
q

Figure 8. Effects of N
b
on velocity profiles.

0 1 2 3 4 5 6
0. 00
0. 25
0. 50
0. 75
1. 00
N
r
=0. 5
N
t
=0. 1
Le=10
N
b
=0. 1, 0. 2, 0. 3, 0. 4, 0. 5
u
q

Figure 9. Effects of N
b
on temperature profiles.
0 1 2 3 4
0. 00
0. 25
0. 50
0. 75
1. 00
N
r
=0. 5
N
t
=0. 1
Le=10
N
b
=0. 1, 0. 2, 0. 3, 0. 4, 0. 5
f
q

Figure 10. Effects of N
b
on volume fraction profiles.
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69
0 1 2 3 4 5 6 7 8
0. 0
0. 2
0. 4
0. 6
0. 8
1. 0
1. 2
Le=100, 1000
Le=10
N
b
=0. 3
N
r
=0. 5
N
t
=0. 1
Le=1
S
'
q

Figure 11. Effects of Le on velocity profiles.
0 1 2 3 4 5 6 7 8
0. 00
0. 25
0. 50
0. 75
1. 00
N
b
=0. 3
N
r
=0. 5
N
t
=0. 1
Le=1, 10, 100, 1000
u
q

Figure 12. Effects of Le on temperature profiles.

0 1 2 3 4 5 6 7 8 9 10
0. 00
0. 25
0. 50
0. 75
1. 00
N
b
=0. 3
N
r
=0. 5
N
t
=0. 1
Le=1, 10, 100, 1000
f
q

Figure 13. Effects of Le on volume fraction profiles.

R. S. R. GORLA ET AL.

Copyright 2011 SciRes. JMP
plate in a porous medium saturated with a nano fluid.
Numerical results for friction factor, surface heat transfer
rate and mass transfer rate have been presented for pa-
rametric variations of the buoyancy ratio parameter Nr,
Brownian motion parameter Nb, thermophoresis pa-
rameter Nt and Lewis number Le. The results indicate
that as Nr and Nt increase, the friction factor increases
whereas the heat transfer rate (Nusselt number) and mass
transfer rate (Sherwood number) decrease. As Nb in-
creases, the friction factor and surface mass transfer rates
increase whereas the surface heat transfer rate decreases.
As Le increases, the heat and mass transfer rates increase.
Nano fluids display drag reducing and heat and mass
transfer rate reducing characteristics.

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R. S. R. GORLA ET AL.

Copyright 2011 SciRes. JMP
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Nomenclature

D
B
Brownian diffusion coefficient
D
T
thermophoretic diffusion coefficient
f rescaled nano-particle volume fraction
g gravitational acceleration vector
k
m
effective thermal conductivity of the porous medium
K permeability of porous medium
Le Lewis number
Nr Buoyancy Ratio
Nb Brownian motion parameter
Nt thermophoresis parameter
Nu Nusselt number
P pressure
q wall heat flux
Ra
x
local Rayleigh number
Re Reynolds number
S dimensionless stream function
T temperature
T
W
wall temperature of the vertical plate
T

ambient temperature
U reference velocity
u, v Darcy velocity components
(x,y) Cartesian coordinates


Greek Symbols:

m
thermal diffusivity of porous medium
volumetric expansion coefficient of fluid
porosity
dimensionless distance
dimensionless temperature
viscosity of fluid

f
fluid density

p
nano-particle mass density
(c)
f
heat capacity of the fluid
(c)
m
effective heat capacity of porous medium
(c)
p
effective heat capacity of nano-particle material
parameter defined by equation (13)
nano-particle volume fraction

W
nano-particle volume fraction at the wall of the vertical plate
ambient nano-particle volume fraction
stream function