Mixed Convection Flow of Nanofluids over a Vertical Surface
Embedded in a Porous Medium with Temperature Dependent Viscosity
SAHAR M. ABDEL-GAIED AND MOHAMED R. EID 1
1 Department of Science and Mathematics, Faculty of Education, Assiut University, The New Valley
72111, Egypt.
[email protected] Abstract
A boundary layer analysis is presented for the mixed convection past a vertical plate in a porous medium
saturated with a nanofluid in the presence of variable viscosity effect. The governing partial differential
equations are transformed into a set of similar equations and solved numerically by an efficient, implicit,
iterative, finite-difference method. A parametric study illustrating the influence of various physical parameters
is performed. Numerical results for the velocity, temperature, and nanoparticles volume fraction profiles, as
well as the friction factor, surface heat and mass transfer rates have been presented for parametric variations of
the mixed convection parameter λ , the variable viscosity parameter eθ , the buoyancy ratio parameter Nr, the
Brownian motion parameter Nb, the thermophoresis parameter Nt, and the Lewis number Le. The dependency
of the friction factor, surface heat transfer rate, and mass transfer rate on these parameters has been discussed.
Keywords: Nanofluid, Mixed convection, Variable viscosity, Porous medium
Nomenclature
a constant
C nanoparticles volume fraction
wC nanoparticles volume fraction at
the vertical plate
∞C ambient nanoparticles volume
fraction attained as ∞→y
BD Brownian diffusion coefficient
TD thermophoretic diffusion
coefficient
f dimensionless stream function
g gravitational acceleration
mk effective thermal conductivity
K permeability of the porous
medium
Le Lewis number
Nr buoyancy ratio
Nb Brownian parameter
Nt thermophoresis parameter
xPe Peclet number
xRa local Rayleigh number
T temperature
wT temperature at the vertical plate
∞T ambient temperature attained as
∞→y
vu, velocity components along x-
and y-directions, respectively
∞U free stream velocity
yx, Cartesian coordinates along the
plate and normal to it,
respectively
Greek symbols
mα thermal diffusivity
β volumetric expansion
coefficient
ε porosity
γ thermal property of the fluid
η similarity variable
θ dimensionless temperature
eθ the viscosity/temperature
parameter
λ mixed convection parameter
µ effective viscosity
υ kinematic viscosity
ρ density
fρ density of the base fluid
pρ nanoparticles mass density
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ISBN: 978-1-61804-178-4 163
( )fcρ heat capacity of the fluid
( )mcρ effective heat capacity of the
porous medium
( )pcρ effective heat capacity of the
nanoparticles material
φ Rescaled nanoparticles volume
fraction
ψ stream function
Subscripts
w condition at the wall
∞ condition at the infinity
I. Background he study of convective heat transfer in
nanofluids is gaining a lot of attention.
The nanofluids have many applications in the
industry since materials of nanometer size have
unique physical and chemical properties.
Nanofluids 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 enhanced 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]. Conventional particle-liquid
suspensions require high concentrations
(>10%) of particles to achieve such
enhancement. However, problems of rheology
and stability are amplified at high
concentrations, precluding the widespread use
of conventional slurries as heat transfer fluids.
In some cases, the observed enhancement in
thermal conductivity of nanofluids is orders of
magnitude larger than predicted by well-
established theories. Other perplexing results
in this rapidly evolving field include a
surprisingly 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 nanofluids promising for applications in
thermal management. Furthermore,
suspensions of metal nanoparticles are also
being developed for other purposes, such as
medical applications including cancer therapy.
The interdisciplinary nature of nanofluid
research presents a great opportunity for
exploration and discovery at the frontiers of
nanotechnology.
The porous media heat transfer problems
have numerous thermal engineering
applications such as geothermal energy
recovery, crude oil extraction, thermal
insulation, ground water pollution, oil
extraction, thermal energy storage, thermal
insulations, and flow through filtering devices.
Excellent reviews on this topic are provided in
the literature by Nield and Bejan [6], Vafai [7],
Ingham and Pop [8] and Vadasz [9]. Recently,
Cheng and Lin [10] examined the melting
effect on mixed convective heat transfer from a
permeable vertical flat plate embedded in a
liquid-saturated a porous medium with aiding
and opposing external flows. EL-Kabeir et al.
[11] applied the group theoretical method to
solve the problem of coupled heat and mass
transfer by natural convection boundary layer
flow for water-vapor around a permeable
vertical cone embedded in a non uniform
porous medium in the presence of magnetic
field and thermal radiation effects. Rashad [12]
studied the combined effect of MHD and
thermal radiation on heat and mass transfer by
free convection over vertical flat plate
embedded in a porous medium. Ibrahim et al.
[13] studied the effect of chemical reaction on
free convection heat and mass transfer for a
non-Newtonian power law fluid over a vertical
flat plate embedded in a fluid-saturated a
porous medium has been studied in the
presence of the yield stress and the Soret
effect. Hady et al. [14] applied the effect of
yield stress on the free convective heat transfer
of dilute liquid suspensions of nanofluids
flowing on a vertical plate saturated in a porous
medium under laminar conditions is
investigated considering the nanofluid obeys
the mathematical model of power-law. Abdel-
Gaied and Eid [15] presented a numerical
analysis of the free convection coupled heat
and mass transfer is presented for non-
Newtonian power-law fluids with the yield
stress flowing over a two-dimensional or
axisymmetric body of an arbitrary shape in a
fluid-saturated a porous medium. Hady et al.
[16] focused on the natural convection
boundary-layer flow over a downward-pointing
T
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ISBN: 978-1-61804-178-4 164
vertical cone in a porous medium saturated
with a non-Newtonian nanofluid in the
presence of heat generation or absorption.
There have been several studies of the effect
of temperature-dependent viscosity on
free/mixed boundary layer flow. Ling and
Dybbs [17] studied the forced convection with
variable viscosity over flat plate in a porous
medium. Kafoussias and Williams [18] and
Kafoussias et al. [19] investigated the effect of
temperature-dependent viscosity on mixed
convection flow of a non-porous fluid past a
vertical isothermal flat plate. Hady et al. [20]
presented the influences of variable viscosity
and buoyancy force on laminar boundary layer
flow and heat transfer due to a continuous flat
plate. Kumari [21] studied the effect of mixed
convection boundary layer flow over heated
flat plate embedded in a porous medium with
variable viscosity. Hossain et al. [22-23]
studied the free convection flow of a viscous
and incompressible fluid past a vertical cone
and vertical wavy surface, respectively, with
viscosity inversely proportional to the linear
function of temperature. Recently, Molla et al.
[24] considered a problem of natural
convection flow from an isothermal horizontal
circular cylinder. Kim and Choi [25] presented
theoretical analysis of thermal instability
driven by buoyancy forces under a time-
dependent temperature field of conduction is
conducted in an initially quiescent, horizontal
liquid layer. The dependency of viscosity on
temperature is considered and the propagation
theory is employed for the stability analysis.
Pantokratoras [26] considered the forced
convection boundary layer flow of a viscous
and incompressible fluid past a wedge with
temperature-dependent viscosity, while Ali
[27] considered the effect of variable viscosity
on mixed convection along a vertical moving
surface. The steady mixed convection
boundary layer flow over a vertical
impermeable surface embedded in a porous
medium when the viscosity of the fluid varies
inversely as a linear function of the
temperature is studied. Both cases of assisting
and opposing flows are considered by Chin et
al. [28].
Nanofluids have been found to possess
enhanced thermophysical properties such as
thermal conductivity, thermal diffusivity,
viscosity, and convective heat transfer
coefficients compared to those of base fluids
like oil or water. It has demonstrated great
potential applications in many fields. The aim
of the present study is to investigate the effect
of variable viscosity or temperature-dependent
viscosity on mixed convection boundary layer
flow over a vertical surface embedded in a
porous medium saturated by a nanofluid. It is
assumed that viscosity of the fluid varies
inversely as a linear function of temperature.
Both cases of assisting and opposing flows will
be considered. The governing partial
differential equations are transformed into
ordinary differential equations which are then
solved numerically using an efficient, implicit,
iterative, finite-difference method for some
values of the physically governing parameters.
Flow and heat transfer characteristics are
presented in some tables and figures, and a
corresponding discussion is made. Quantitative
comparison with the existing results as
reported by Chin et al. [28]. All comparisons
of the existing and present results show
excellent agreements.
II. Problem formulation
We consider the steady two-dimensional free
convection boundary layer flow past a vertical
plate placed in a nanofluid saturated a porous
medium. The coordinate system is selected
such that x-axis is in the vertical direction. Fig.
1 shows the coordinate system and flow model.
At the surface, the temperature T and the
nanoparticles fraction take constant values Tw
and wC , respectively. The ambient values
attained as y tends to infinity of T and C are
denoted by T∞ and ∞C , respectively. The
Oberbeck-Boussinesq approximation is
employed and the homogeneity and local
thermal equilibrium in the porous medium are
assumed.
We consider a porous medium whose
porosity is denoted by ε and permeability by
K . The Darcy velocity is denoted by v. The
following four field equations embody the
conservation of total mass, momentum,
thermal energy, and nanoparticles,
respectively. The field variables are the Darcy
velocity v, the temperature T and the
nanoparticles volume fraction C . The basic
steady conservation of mass, momentum and
thermal energy equations for nanofluid by
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ISBN: 978-1-61804-178-4 165
using usual boundary-layer approximations can
be written in Cartesian coordinates x and y as:
,0=∂
∂+
∂
∂
y
v
x
u (1)
( ) ( )( )[ ],)(1 fpf ∞∞∞∞∞∞ −−−−−+= CCgTTCgK
Uu ρρβρµ
(2)
,
2
2
2
∂
∂+
∂
∂
∂
∂+
∂
∂=
∂
∂+
∂
∂
∞ y
T
T
D
y
T
y
CD
y
T
y
Tv
x
Tu T
Bm τα
(3)
.1
2
2
2
2
y
T
T
D
y
CD
y
Cv
x
Cu T
B∂
∂+
∂
∂=
∂
∂+
∂
∂
∞ε (4)
Where u and v are the velocity
components along x and y coordinates,
respectively, µ and β are the density,
viscosity, and volumetric volume expansion
coefficient of the fluid while pρ is the density
of the particles. The gravitational acceleration
is denoted by g . We have introduced the
effective heat capacity mc)(ρ , and the effective
thermal conductivity mk of the porous
medium. The coefficients appeared in Eqs. (3)
and (4) are the Brownian diffusion coefficient
BD and the thermophoretic diffusion
coefficient TD . The corresponding boundary
conditions are defined as follows:
,,,
,0,,0
∞→→→→
====
∞∞∞ yasCCTTUu
yatCCTTv ww
(5)
Where f)( c
kmm ρ
α = , f
p
)(
)(
c
c
ρρε
τ = . (6)
We assume that the dynamic viscosity µ has
the following form (see Chin et al. [28]).
)(
1
)(1 eTTaTT −=
−+=
∞
∞
γµ
µ , (7)
where γ and ∞µ are the thermal property of
the fluid and the ambient fluid viscosity,
respectively, which is a constant. Also, a and
eT are constants, given by:
γµγ 1
, −=−= ∞∞
TTa e . (8)
The continuity equation is automatically
satisfied by defining a stream function
),( yxψ such that:
. and x
vy
u∂∂
−=∂∂
=ψψ
(9)
The dimensionless variables are:
( )
( ).
1,
-
-)( ,)(
),(2 ,2
w
2
12
1
∞∞
∞
∞
∞
∞
∞
−−=
−
−==
−
−=
=
=
TTaTT
TT
CC
CC
TT
TT
fPePe
x
y
ww
ee
w
xmx
θηφηθ
ηαψη
(10)
Where m
xUPe
α∞= (11)
is the Peclet number for a porous medium and
eθ is the viscosity/temperature parameter and
its value is determined by the
viscosity/temperature characteristics of the
fluid and the operating temperature difference
∞−=∆ TTT w .
Substituting the expressions in Eq. (10) into
the governing Equations (1)-(5) we obtain the
following transformed equations:
( )φθθθ
λ Nrfe
−
−+=′ 11 (12)
02 =′+′′+′++′′ θφθθθ NtNbf (13)
,0Nb
Nt Le =′′+′+′′ θφff (14)
With the boundary conditions:
.0,0,0: as
,1,1,0:0at
===′∞→
====
φθηφθη
f
f (15)
In the above equations primes denote
differentiation with respect to η and λ is the
constant mixed convection parameter which is
given by:
( )( ) ( )( )
,1
,1
m
wx
x
xw xCTTgKRa
Pe
Ra
U
CTTgK
αυβ
υβ
λ∞
∞∞
∞∞
∞∞ −−==
−−=
(16)
where Rax being the local Rayleigh number for
a porous medium and where the four
parameters Nr, Nb, Nt and Le are defined by:
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ISBN: 978-1-61804-178-4 166
( )( )( )
( ).,
,)(
,1
)(
f
fp
B
m
m
wT
m
wB
w
w
DLe
T
TTDNt
CCDNb
TTC
CCNr
εα
ατ
ατ
βρ
ρρ
=−
=
−=
−−
−−=
∞
∞
∞
∞∞∞
∞∞
(17)
It should be mentioned that 0>λ
corresponds to a heated plate (assisting flow),
0<λ corresponds to a cooled plate (opposing
flow) and 0=λ corresponds to the forced
convection flow. We notice that the effect of
variable viscosity can be neglected if eθ is
large (≫1), i.e., if either the constant a or
(Tw−T∞) is small. If eθ is small then either the
fluid viscosity changes with the temperature or
the temperature difference is high. It may be
noted that eθ is negative for liquids and
positive for gases. It is also noted that if
∞→eθ , Eqs. (8) and (9) reduce to those for a
constant viscosity case found by Merkin [29]
or Aly et al. [30] for an isothermal flat plate.
The quantities of physical interest to be
considered in this study are the local skin
friction coefficient )0(f ′′ , heat transfer
coefficient )0(θ ′− , mass transfer coefficient
)0(φ′− , the non-dimensional temperature
profiles )(ηθ , and the non-dimensional
concentration profiles )(ηφ for both cases of
the assisting and the opposing flows.
III. Results and discussion By using the similarity transformations and
the similarity representation of the effect of
variable viscosity on steady mixed convection
boundary layer flow over a vertical surface
embedded in a porous medium saturated a
nanofluid is obtained and it shown in equations
(12)-(15). The similarity reduction corresponds
to constant viscosity can be easy obtain if we
put ( ∞→eθ ). We have obtained the same
result as in Chin et al. [28]. Equations (12)-
(14) with the boundary layer (15) have been
solved numerically by using an efficient,
implicit, iterative, finite-difference method.
The basic idea is to introduce new variables,
one for each variable in the original problem
plus one for each of its derivatives up to one
less than the highest derivative appearing. We
used the finite-difference method (we used
MATLAB package for this purpose) for the
solution of boundary value problem of the
form:
bxayxfy ≤≤=′ ),,( (18)
subject to general nonlinear, two-point
boundary conditions:
( ) .0)().( =byayg (19)
Finite-difference equations are set up on a
mesh of points and estimated values for the
solution at the grid points are chosen. Using
these estimated values as starting values a
Newton iteration is used to solve the Finite-
difference equations. The accuracy of the
solution is then improved by deferred
corrections or the addition of points to the
mesh or a combination of both. The absolute
error tolerance for this method is 410−.
Computations are carried out for various values
of the mixed convection parameter λ , the
viscosity/temperature parameter eθ , the
buoyancy ratio parameter Nr, the Brownian
motion parameter Nb, the thermophoresis
parameter Nt, and the Lewis number Le. Both
assisting ( 0>λ ) and opposing ( 0<λ ) flows
are considered.
Tables 1 and 2 show the values of skin
friction coefficient )0(f ′′ and heat transfer
coefficient )0(θ ′− for various values of the
mixed convection parameter λ and the
viscosity/temperature parameter eθ . The
results calculated for 100,50,10,1=λ and
∞−−−−−∞= ,8,4,2,1,1,2,4,8,eθ . Also, in
these tables the comparisons of the present
work with Chin et al. [28]. It is found that the
agreement between both results is excellent.
Therefore, we believe that this comparison
supports very well the validity of the present
work. The results show that for each eθ when
λ increases, the values of the magnitude of
)0(f ′′ and )0(θ ′− increase. Physically, this is
because the nanofluid velocity increases when
the buoyancy force increases and hence
increases the skin friction and it means the
temperature of the plate is increased. Tables 3-
6 indicate results for wall values for the
temperature and concentration functions which
are proportional to the Nusselt number and
Sherwood number, respectively. From Tables
3-5, we notice that as Nb and Nt increase, the
heat transfer rate (Nusselt number) )0(θ ′− and
mass transfer rate (Sherwood number) )0(φ ′−
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ISBN: 978-1-61804-178-4 167
decrease for each eθ at the critical value of λ
( 354.1−=cλ ). As Nr increases, the surface
heat transfer and the surface mass transfer rates
increase for each eθ at the critical value of λ
( 354.1−=cλ ). Results from Table 6 indicate
that as Le increases, the heat transfer rate
decreases whereas the mass transfer rate
increases this for each positive value of eθ
while the negative value of eθ is different at
the critical value of cλ .
Fig. 2 indicates that, as Nr increases, the
velocity decreases, and the temperature and
concentration increase. Similar effects are
observed from Figs. 3 and 4 as Nb and Nt vary.
Fig. 5 illustrates the variation of velocity
within the boundary layer as Le increases. The
velocity increases as Le increases. As Le
increases, the temperature and concentration
within the boundary layer decrease and the
thermal and concentration boundary later
thicknesses decrease. Fig. 6 shows that as the
mixed convection parameter λ increases, the
velocity increases, whereas temperature and
concentration decrease. Fig. 7 focuses the
influence of eθ on the velocity, the
temperature and the concentration. We observe
that the velocity increases, whereas
temperature and concentration decrease and for
fixed values of λ , the values of )0(θ ′− and
)0(φ ′− for the base fluid is gas ( 0>eθ ) is
always higher than the values of )0(θ ′− and
)0(φ ′− for the base fluid is liquid ( 0<eθ ).
Even when the critical value cλ , we find the
same influence, which may vary
slightly at some values because the dual
solutions but only the first solutions have the
importance if compared with the second
solutions, which depicts in Tables 3-5.
Figs. 8, 9, 10, 11 and 12 display results for
wall values for the temperature, and
concentration functions which are proportional
to the Nusselt number, and Sherwood number,
respectively. From Figs. 8 and 10, we notice
that as Nr and Nt increase, the heat transfer rate
(Nusselt number) and mass transfer rate
(Sherwood number) decrease. As Nb increases,
the surface mass transfer rates increase
whereas the surface heat transfer rate decreases
as shown by Fig. 9 as we expect before. Fig. 11
indicates that as Le increases, the heat transfer
rate decreases whereas the mass transfer rate
increases. From Fig. 12, we observe that, as the
mixed convection parameter λ increases, the
heat and mass transfer rates increase. From
these Figs. we note that with variable viscosity,
the separation of boundary layer is delayed for
0>eθ than 0<eθ . This indicates that for
cλλ > , the boundary layer separates from the
plate surface, and therefore solutions based on
the boundary layer approximations are not
valid anymore.
It can also be summarize that decreasing λ
from 0>λ (assisting flow) to 0<λ
(opposing flow) leads to an increase of the
thermal boundary layer thickness and the
concentration boundary layer thickness. It is
also observed from the numerical results that
for fixed values of eθ , decreasing λ leads to
an increase of the thermal boundary layer and
the concentration thicknesses. It is also showed
that the changes of the thermal boundary layer
and concentration boundary layer thicknesses
with the changes of λ are much more cleared
than the changes with eθ . Therefore, this refers
that the parameter eθ has smaller effect on the
thicknesses of thermal boundary layer and
concentration boundary layer compared to that
of the mixed convection parameter λ . The
influence of nanoparticles on mixed convection
is modeled by accounting for Brownian motion
and thermophoresis as well as non-isothermal
boundary conditions. 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 Nt to heat and mass transfer
does not depend on the value of Le. The
Brownian motion and thermophoresis of
nanoparticles 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.
IV. Conclusions
The Nanofluids have attracted more
and more attention. The main driving force for
nanofluids research lies in a wide range of
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 168
applications. The important technique to
enhance the stability of nanoparticles in fluids
is the use of surfactants. However, the
functionality of the surfactants under high
temperature is also a big concern, especially
for high-temperature applications. Two-step
method is the most economic method to
produce nanofluids in large scale, because
nanopowder synthesis techniques have already
been scaled up to industrial production levels.
Due to the high surface area and surface
activity, nanoparticles have the tendency to
aggregate.
In this study, we presented a boundary
layer analysis for the mixed convection past a
vertical surface embedded in a porous medium
saturated with a nanofluid ( two step method)
in the presence of the effect of variable
viscosity. The transformed equations are
solved numerically using the efficient, implicit,
iterative, finite-difference method. We have
quantitatively compared our present results
with those of Chin et al. [28], and the
agreement between both results is excellent.
Numerical results for surface heat transfer rate,
and mass transfer rate have been presented for
parametric variations of the mixed convection
parameter λ , the viscosity/temperature
parameter eθ , the buoyancy ratio parameter
Nr, Brownian motion parameter Nb,
thermophoresis parameter Nt, and Lewis
number Le. The numerical results are
influenced by mixed convection parameter λ
and the variable viscosity parameter which
defines the effect of variable viscosity of the
fluid eθ . It is observed that the separation of
boundary layer is delayed for 0>eθ than
0<eθ . In the assisting flow, the values of
)0(θ ′− and )0(φ ′− increase as λ increases. In
the opposing flow case, dual solutions exist
and the separation of boundary layer occurs.
The results indicate that, as Nr and Nt increase,
mass transfer rate (Sherwood number)
decrease. As Nb increases, the surface mass
transfer rates increase, whereas the surface heat
transfer rate decreases. As Le increases, the
heat transfer rate decreases, whereas the mass
transfer rate increases.
V. Competing interests This is just the theoretical study, every
experimentalist can check it experimentally
with our consent.
VI. Authors' information
S. M. Abdelgaied Assistant Professor
and
M. R. Eid mathematics lecturers.
VII.Endnotes
This is just a theoretical study;
every experimentalist can check it
experimentally with our consent.
VIII. References
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and L.J. Thompson, Anomalously
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[2] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E.
Lockwood and E.A. Grulke, Anomalous
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[3] H.E. Patel, S.K. Das, T. Sundararajan, A.
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Table 1:Comparison of skin friction coefficient )0(f ′′ with 0=== NtNbNr for various values λ
and eθ :
eθ Chin et al. [28] Present study
λ 1 10 50 100 1 10 50 100
∞−
-1.0191 -21.445 -225.67 -633.26 -1.0191 -21.446 -225.67 -633.28
-8 -1.2963 -27.835 -294.22 -826.12 -1.2963 -27.833 -294.23 -826.12
-4 -1.5820 -34.592 -367.02 -1031.1 -1.5820 -34.593 -367.04 -1031.1
-2 -2.1779 -49.134 -524.56 -1474.8 -2.1779 -49.135 -524.58 -1474.8
-1 -3.4625 -81.969 -882.79 -2484.7 -3.4625 -81.969 -882.79 -2484.7
1 0.8603 12.908 120.16 330.92 0.8603 12.908 120.17 330.93
2 0 0 0 0 0 0 0 0
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ISBN: 978-1-61804-178-4 171
4 -0.4911 -9.8446 -102.41 -286.91 -0.4911 -9.8446 -102.41 -286.91
8 -0.7506 -15.440 -161.63 -453.23 -0.7506 -15.440 -161.63 -453.23
∞ -1.0191 -21.445 -225.67 -633.26 -1.0191 -21.445 -225.67 -633.26
Table 2:Comparison of heat transfer coefficient )0(θ ′− with 0=== NtNbNr for various values λ
and eθ :
eθ Chin et al. [28] Present study
λ 1 10 50 100 1 10 50 100
∞−
1.0191 2.1445 4.5133 6.3326 1.0191 2.1443 4.5118 6.3322
-8 1.0370 2.2268 4.7075 6.6089 1.0370 2.2267 4.7061 6.6089
-4 1.0546 2.3061 4.8937 6.8738 1.0546 2.3060 4.8924 6.8705
-2 1.0889 2.4567 5.2456 7.3741 1.0889 2.4566 5.2448 7.3718
-1 1.1542 2.7323 5.8853 8.2822 1.1542 2.7323 5.8853 8.2826
1 0.8603 1.2908 2.4032 3.3092 0.8603 1.2905 2.4008 3.2990
2 0.9435 1.7748 3.6288 5.0705 0.9435 1.7746 3.6267 5.1429
4 0.9821 1.9689 4.0963 5.7382 0.9821 1.9687 4.0945 5.7382
8 1.0008 2.0587 4.3101 6.0431 1.0008 2.0585 4.3084 6.1593
∞ 1.0191 2.1445 4.5133 6.3326 1.0191 2.1443 4.5118 6.4934
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 172
Table 3:Values related to the heat transfer and the mass transfer coefficients for different values of the
governing parameters eθ and Nr with 1.0== NtNb , 10=Le and 354.1−=cλ :
eθ Nr )0(θ ′− )0(φ ′−
-8
0.1 0.1499 0.0252
0.2 0.2737 0.3470
0.3 0.3246 0.6463
0.4 0.3626 0.8984
0.5 0.3945 1.1160
-4
0.1 0.0003 0.0000
0.2 0.2117 0.1042
0.3 0.2836 0.4118
0.4 0.3285 0.7002
0.5 0.3646 0.9525
4
0.1 0.3941 1.0080
0.2 0.4204 1.1714
0.3 0.4434 1.3158
0.4 0.4640 1.4466
0.5 0.4830 1.5670
8
0.1 0.3389 0.6932
0.2 0.3758 0.9198
0.3 0.4056 1.1095
0.4 0.4313 1.2755
0.5 0.4542 1.4247
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ISBN: 978-1-61804-178-4 173
Table 4:Values related to the heat transfer and the mass transfer coefficients for different values of the
governing parameters eθ and Nb with 1.0== NtNr , 10=Le and 354.1−=cλ :
eθ Nb )0(θ ′− )0(φ ′−
-8
0.1 0.1499 0.0252
0.2 0.1749 0.0064
0.3 0.0672 0.0000
0.4 0.0039 0.0001
0.5 0.0038 0.0001
-4
0.1 0.0003 0.0000
0.2 0.0003 0.0000
0.3 0.0003 0.0000
0.4 0.0003 0.0000
0.5 0.0003 0.0000
4
0.1 0.3941 1.0080
0.2 0.3597 1.0153
0.3 0.3287 1.0067
0.4 0.3002 0.9945
0.5 0.2742 0.9812
8
0.1 0.3389 0.6932
0.2 0.3073 0.6741
0.3 0.2789 0.6480
0.4 0.2531 0.6203
0.5 0.2295 0.5921
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 174
Table 5:Values related to the heat transfer and the mass transfer coefficients for different values of the
governing parameters eθ and Nt with 1.0== NbNr , 10=Le and 354.1−=cλ :
eθ Nt )0(θ ′− )0(φ ′−
-8
0.1 0.1499 0.0252
0.2 0.1284 0.0305
0.3 -0.0542 -0.0408
0.4 0.0910 0.0172
0.5 0.3597 0.2423
-4
0.1 0.0003 0.0000
0.2 0.0003 0.0001
0.3 0.1250 0.0530
0.4 -0.0011 -0.0007
0.5 0.0002 0.0001
4
0.1 0.3941 1.0080
0.2 0.3729 0.9884
0.3 0.3528 0.9862
0.4 0.3337 0.9990
0.5 0.3156 1.0246
8
0.1 0.3389 0.6932
0.2 0.3180 0.6843
0.3 0.2981 0.6897
0.4 0.2790 0.7071
0.5 0.2608 0.7340
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 175
Table 6:Values related to the heat transfer and the mass transfer coefficients for different values of the
governing parameters eθ and Le with 1.0=== NtNbNr and 354.1−=cλ :
eθ Le )0(θ ′− )0(φ ′−
-8
1 0.1592 0.2146
5 0.1178 0.0546
10 0.1499 0.0252
100 0.1738 0.0024
-4
1 0.1457 0.2330
5 -0.0644 -0.2509
10 0.0003 0.0000
100 0.0973 0.0010
4
1 0.4331 0.3341
5 0.4073 0.6907
10 0.3941 1.0080
100 0.3734 2.4863
8
1 0.3770 0.3138
5 0.3513 0.5480
10 0.3389 0.6932
100 0.3225 0.8376
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 176
Fig. 1 Flow model and coordinate system.
Fig. 2 Velocity, temperature and concentration profiles for various values of buoyancy ratio Nr.
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 177
Fig. 3 Velocity, temperature and concentration profiles for various values of Brownian motion
parameter Nb.
Fig. 4 Velocity, temperature and concentration profiles for various values of thermophoresis
parameter Nt.
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 178
Fig. 5 Velocity, temperature and concentration profiles for various values of Lewis number Le.
Fig. 6 Velocity, temperature and concentration profiles for various values of mixed convection
parameter λ .
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 179
Fig. 7 Velocity, temperature and concentration profiles for various values of variable viscosity
parameter eθ .
Fig. 8 Heat transfer rate and mass transfer rate for various values of buoyancy ratio Nr.
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 180
Fig. 9 Heat transfer rate and mass transfer rate for various values of Brownian motion parameter Nb.
Fig. 10 Heat transfer rate and mass transfer rate for various values of thermophoresis parameter Nt.
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 181
Fig. 11 Heat transfer rate and mass transfer rate for various values of Lewis number Le.
Fig. 12 Heat transfer rate and mass transfer rate for various values of mixed convection parameter λ .
Recent Advances in Mathematical Methods and Computational Techniques in Modern Science
ISBN: 978-1-61804-178-4 182