Conjugate Natural Convection of Al2O3-Water Nanofluid in Square Cavity
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International Journal of Mechanical Sciences 136 (2018) 200–219
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International Journal of Mechanical Sciences
journal homepage: www.elsevier.com/locate/ijmecsci
Conjugate natural convection of Al
2
O
3
–water nanouid in a square cavity
with a concentric solid insert using Buongiorno ’s two-phase model
A.I. Alsabery
a
,
∗
, M.A. Sheremet
b
,
c
, A.J. Chamkha
d
,
e
, I. Hashim
a
a
School of Mathematical Sciences, Universiti Kebangsaan Malaysia, UKM Bangi 43600, Selangor, Malaysia
b
Department of Theoretical Mechanics, Tomsk State University, Tomsk 634050, Russia
c
Institute of Power Engineering, Tomsk Polytechnic University, Tomsk 634050, Russia
d
Department of Mechanical Engineering, Prince Sultan Endowment for Energy and the Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi
Arabia
e
RAK Research and Innovation Center, American University of Ras Al Khaimah, Ras Al Khaimah, United Arab Emirates
Keywords:
Natural convection
Thermophoresis
Brownian diusion
Square cavity
Isothermal corner boundaries
Buongiorno model
The problem of conjugate natural convection of Al
2
O
3
–water nanouid in a square cavity with concentric solid
insert and isothermal corner boundaries using non-homogenous Buongiorno ’s two-phase model is studied numer-
ically by the nite dierence method. An isothermal heater is placed on the left bottom corner of the square cavity
while the right top corner is maintained at a constant cold temperature. The remainder parts of the walls are kept
adiabatic. Water-based nanouids with Al
2
O
3
nanoparticles are chosen for the investigation. The governing pa-
rameters of this study are the nanoparticle volume fraction (0 ≤ 𝜙≤ 0.04), the Rayleigh number (10
2
≤ Ra ≤ 10
6
),
thermal conductivity of the solid block ( 𝑘
𝑤
= 0 . 28 , 0.76, 1.95, 7 and 16) (epoxy: 0.28, brickwork: 0.76, granite:
1.95, solid rock: 7, stainless steel: 16) and dimensionless solid block thickness (0.1 ≤ D ≤ 0.7). Comparisons with
previously experimental and numerical published works verify good agreement with the proposed method. Nu-
merical results are presented graphically in the form of streamlines, isotherms and nanoparticles volume fraction
as well as the average Nusselt number and uid ow rate. The results show that the thermal conductivity ratio
and solid block size are very good control parameters for an optimization of heat transfer inside the partially
heated and cooled cavity.
© 2017 Elsevier Ltd. All rights reserved.
1. Introduction
Natural convection heat transfer in cavities is a signicant phe-
nomenon in engineering systems due to its widespread applications in
operations of solar collectors, cooling of containment buildings, heat ex-
changers, storage tanks, double pane windows, etc. Laminar uid ow
has become a favorite aspect in heat storage applications, as there has
been a rise in research activities regarding natural convection heat trans-
fer. Thermal uids are very important for heat transfer in many indus-
trial applications. The low thermal conductivity of conventional heat
transfer uids such as water and oils is a primary limitation in enhanc-
ing the performance and the compactness of many engineering elec-
tronic devices. An innovative and new technique to enhance heat trans-
fer is using solid particles in the base uid (i.e. nanouids) in the range
of sizes 10–50 nm. A nanouid is dened as a smart uid with sus-
pended nanoparticles of average sizes below 100 nm in conventional
heat transfer uids such as water, oil, and ethylene glycol [1] . Due
to small sizes and very large specic surface areas of the nanoparti-
∗ Corresponding author.
E-mail address: [email protected] (A.I. Alsabery).
cles, nanouids have superior properties like high thermal conductivity,
minimal clogging in ow passages, longterm stability and homogene-
ity. Also, nanoparticles are used because they stay in suspension longer
than larger particles. Thus, nanouid seems a good candidate for heat
removal mechanisms in practical, thermal, uid-based applications. The
thermal conductivity of nanoparticles is higher than that of traditional
uids. Thus, nanouids can be used in a large industrial applications
such as oil industry, nuclear reactor coolants, solar cells, construction,
electronics, renewable energy and many others. The solid particles are
usually metal or metal oxides such as copper (Cu), copper oxide (CuO),
aluminum oxide (Al
2
O
3
), titanium (TiO
2
) and silver (Ag).
A comprehensive work on natural convection in cavities that are
partially occupied by nanouids was reported by Khanafer et al. [2] .
Jou and Tzeng [3] considered natural convective heat transfer in
nanouids occupying a rectangular cavity. The numerical simulation of
the uid and temperature distributions and the convective heat transfer
of the nanouid could be classied by two main approaches, namely a
single-phase model (homogenous) or a two-phase model [4] . The single-
https://doi.org/10.1016/j.ijmecsci.2017.12.025
Received 28 August 2017; Received in revised form 28 November 2017; Accepted 5 December 2017
Available online 21 December 2017
0020-7403/© 2017 Elsevier Ltd. All rights reserved.
A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219
Nomenclature
C
p
Specic heat capacity
d Width and height of the inner solid block
d
f
Diameter of the base uid molecule
d
p
Diameter of the nanoparticle
D Dimensionless thickness of the inner solid block,
𝐷 = 𝑑∕ 𝐿
D
B
Brownian diusion coecient
D
B 0
Reference Brownian diusion coecient
D
T
Thermophoretic diusivity coecient
D
T 0
Reference thermophoretic diusion coecient
g Gravitational acceleration
k Thermal conductivity
K
r
Wall to nanouid thermal conductivity ratio, 𝐾
𝑟
=
𝑘
𝑤
∕ 𝑘
𝑛𝑓
L Width and height of enclosure
Le Lewis number
N
BT
Ratio of Brownian to thermophoretic diusivity
𝑁𝑢 Average Nusselt number
Pr Prandtl number
Ra Rayleigh number
Re
B
Brownian motion Reynolds number
T Temperature
T
0
Reference temperature (310 K)
T
fr
Freezing point of the base uid (273.15 K)
v Velocity vector
V Normalized velocity vector
u
B
Brownian velocity of the nanoparticle
x, y and X, Y Space coordinates and dimensionless space coordi-
nates
Greek symbols
𝛼Thermal diusivity
𝛽Thermal expansion coecient
𝛿Normalized temperature parameter
𝜃Dimensionless temperature
𝜇Dynamic viscosity
𝜈Kinematic viscosity
𝜌Density
𝜑 Solid volume fraction
𝜑
∗ Normalized solid volume fraction
𝜙Average solid volume fraction
Subscripts
c Cold
f Base uid
h Hot
nf Nanouid
p Solid nanoparticles
w Inner solid block
phase approach considers the uid phase and the nanoparticles as being
in thermal equilibrium where the slip velocity between the base uid
and the nanoparticles is negligible. On the other hand, the two-phase
approach assumes that the relative velocity between the uid phase and
the nanoparticles may not be zero where the continuity, momentum and
energy equations of the nanoparticles and the base uid are handled us-
ing dierent methods. There are number of numerical studies used the
single-phase model for simulation of the nanouids. Hu et al. [5] studied
experimentally and numerically the natural convection heat transfer in
a square cavity lled with TiO
2
–water nanouids. They found that the
average Nusselt number increased with the addition of nanoparticles.
Karimipour et al. [6] reported a study on mixed convection in a shallow
inclined lid driven cavity lled with a Cu–water nanouid using
the lattice Boltzmann method. Using the same method Karimipour
et al. [7] investigated the problem of laminar forced convection in
a microchannel lled Cu–water nanouids. Sheremet et al. [8] and
Alsabery et al. [9] numerically investigated the natural convection heat
transfer of nanouid ow in dierent geometries. Ghalambaz et al.
[10] numerically studied the eects of the diameter and concentration
of nanoparticles on the natural convection of Al
2
O
3
–water nanouids
considering the variable thermal conductivity around a porous medium.
They found that the heat transfer rate decreased with an increase in the
volume fraction of nanoparticles or a decrease in the size of nanopar-
ticles. Zaraki et al. [11] theoretically analyzed the natural convection
heat transfer of nanouids for which various aspects of nanoparticles are
considered. Karimipour [12] developed a new correlation for Nusselt
number for the problem of convective heat transfer in a microchannel
lled with three types of nanouids by using lattice Boltzmann method.
Recently, Umavathi and Sheremet [13] numerically studied the ef-
fect of the temperature-dependent conductivity on natural convective
heat transfer in a vertical rectangular duct lled with a nanouid using
the nite-dierence method. They concluded that the heat transfer rate
increased at the left wall and decreased at the right wall as the aspect
ratio increased, whereas the heat transfer rate increased at both of the
walls as the solid volume fraction increased. Karimipour et al. [14] con-
sidered numerically the eect of indentation on ow parameters and
slow heat transfer in in a rectangular micro channel lled with Ag–
water nanouid using the nite volume method. Sheikholeslami et al.
[15] used the single phase model (Koo–Kleinstreuer–Li) of a nanouid to
study the natural convection heat transfer in a square cavity where they
found that the convection heat transfer is increased with the increase in
the volume fraction of nanoparticles. Most of these studies are used the
Maxwell-Garnett and Brinkman models to estimate the eective thermal
conductivity and viscosity of the nanouid. However, the study of Cor-
cione [16] questions the validity of these models and tended to proposed
a new model for estimating the eective thermal conductivity and vis-
cosity of the nanouid which appeared to be close to the experimental
data. The results showed that the heat transfer rate enhanced with the
relative concentration of nanouid. The experimental study of Wen and
Ding [17] found that the slip velocity between the base uid and par-
ticles may not be zero. Thus, the two-phase nanouid model observed
to be more accurate. Buongiorno [18] proposed a non-homogeneous
equilibrium model with the consideration of the eect of the Brownian
motion (movement of nanoparticles from high concentration site) and
thermophoresis (movement of nanoparticles from the high temperature
site to the low temperature site) as a two important primary slip mecha-
nisms in nanouid. He tended to introduce in this study seven slip mech-
anisms between nanoparticles and the base uid where he developed a
non-homogeneous two-component equations in nanouids showing the
importance of Brownian motion and thermophoresis compared to other
transport mechanisms. Sheikholeslami et al. [19] used the two-phase
model of the nanouid to investigate the thermal management for nat-
ural convection heat transfer in a 2D cavity. Esfandiary et al. [20] and
Motlagh and Soltanipour [21] investigated numerically the problem of
natural convection of nanouids in a square cavity using the two phase
model. The results of these studies indicated that the heat transfer rate
enhanced with the increasing of the concentration of the nanoparticles
up to 0.04.
Conjugate convective heat transfer for a regular uid has very im-
portant practical engineering applications in frosting practicalities and
refrigeration of the hot obtrusion in a geological framing. For exam-
ple, modernistic construction of thermal insulators which are formed of
two diverse thermal conductivities (solid and brous) materials can be
modeled by the partition length and conductivity model. Conjugate heat
transfer (CHT) contains heat exchange that happens simultaneously by
convection between a uid and an adjacent surface, and by conduction
over the solid. CHT is ubiquitous and very important in key applica-
tions that are predicted to be progressively relevant both in industrial
and domestic environments. CHT is a predominating process in dierent
201
A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219
external combustion motors, thermo-acoustic devices and other ma-
chines capable of utilizing immaculate and renewable energy styles like
solar, refuse and geothermal heat. Indeed, the prevalent and central ad-
vantage of these devices is their ingrained dependence on internally
thermal uid ows which result in CHT. However, developments to our
understanding of CHT and therefore, to the performance of numerous
of the above mentioned systems must come from exploring actual situ-
ations of CHT within a framework that can capture the varying inter-
play between uid ow and heat transfer, and account for the existence
of conjugate uid–solid interactions. There are some excellent studies
considering the impact of partition length and conductivity on the heat
transfer rate.
Viskanta and Kim [22] studied a rectangular cavity formed by nite
conductance walls of dierent void fractions and aspect ratios. They con-
sidered the outer edge of the wall and the opposing vertical side were
isothermal at high and low temperatures, respectively, and the two hori-
zontal sides were insulated. They showed that for the high Grashof num-
ber and decreasing wall conductivity, two dimensional eects on con-
duction in the wall were non-negligible. House et al. [23] investigated
the eect of a centered heat-conducting body on the natural convection
heat transfer in a square cavity. The two vertical walls were maintained
at two dierent constant temperatures and the horizontal walls were
adiabatic. The results showed that the heat transfer decreased with the
increase of the solid body. Ha et al. [24] investigated the eect of un-
steady natural convection processes in similar vertical cavities with a
centered heat-conducting body. Zhao et al. [25] studied the eect of
a centered heat-conducting body on the conjugate natural convection
heat transfer in a square enclosure. The results show that the thermal
conductivity ratio has strong inuence on the ow within the square
cavity. Saeid [26] investigated a dierentially-heated vertical square
porous cavity where the conducting wall is next to the hot side. He
found in most of the cases that either increasing the Rayleigh number
and the thermal conductivity ratio or decreasing the thickness of the
bounded wall can increase the average Nusselt number.
Mahmoodi and Sebdani [27] used the nite volume method to inves-
tigate the conjugate natural convective heat transfer in a square cavity
lled with nanouid and containing a solid square block at the center.
They concluded that the heat transfer rate decreased with an increasing
of the size of the inner block for low Rayleigh numbers and increased
at high Rayleigh numbers. Mahapatra et al. [28] numerically used the
nite volume method to investigate the CHT and entropy generation
in a square cavity in the presence of adiabatic and isothermal blocks.
They found that the heat transfer enhanced with the low Rayleigh num-
bers and for a critical block sizes. Chamkha and Ismael [29] studied the
eect of conjugate natural convection heat transfer in a porous square
cavity lled with nanouids and heated by a thick triangular wall. Their
study showed that the heat transfer was signicantly enhanced at low
Rayleigh number with the increase of the nanoparticles volume fraction.
Using the nite volume method, Esfe et al. [30] investigated the prob-
lem of natural convection in a 2D cavity lled with dierent types of
nanouids and containing a heated cylindrical block. Recently, Alsabery
et al. [31] used the nite dierence method to study the unsteady natu-
ral convective heat transfer in nanouid-saturated porous square cavity
with a concentric solid insert and sinusoidal boundary condition. Very
recently, el malik Bouchoucha et al. [32] considered numerically the
problem of natural convection and entropy generation in a nanouid
square cavity with a non-isothermal heating thick bottom wall. They
concluded that an increasing of the thicknesses of the bottom solid wall
tended to reduce the heat transfer rate. Garoosi and Rashidi [33] used
the nite volume method to investigate the two phase model of conju-
gate natural convection of the nanouid in a partitioned heat exchanger
containing several conducting obstacles. They found that the heat trans-
fer rate was signicantly inuenced by changing the orientation of the
conductive partition from vertical to horizontal mode.
Based on the previously mentioned papers and to the authors ’best
knowledge, there have been no studies of conjugate natural convection
Fig. 1. Physical model of convection in a square cavity together with the coordinate
system. (For interpretation of the references to colour in this gure legend, the reader is
referred to the web version of this article.)
of Al
2
O
3
–water nanouid in a square cavity with a concentric solid in-
sert and corner heater using Buongiorno ’s two-phase model. Thus, we
believe that this work is valuable. The aim of this study is to investi-
gate the conjugate natural convection of Al
2
O
3
–water nanouid in a
square cavity with a concentric solid insert and corner boundaries us-
ing Buongiorno ’s two-phase model. Solid inner blocks can be used to
control heat transfer as passive element in various shaped cavities lled
with nanouids or pure liquids. This application can be seen in building
design, electronic equipment, heat exchangers and solar energy systems
[34] . The interaction between the shear-driven ow and natural con-
vection in closed enclosures is one of the most interesting topics which
can be used in design and analysis of many industrial heating or cool-
ing systems such as: indoor ventilation with radiators, nuclear reactors,
cooling of electronic components and heat exchangers and so on. The
considered geometry can be found in all of the above mentioned elds
where an internal solid block can be considered like an additional inter-
nal system element (pipe, bracket, electrode, electronic element and so
on).
2. Mathematical formulation
The steady two-dimensional natural convection problem in a square
cavity with length L and with the cavity center inserted by a solid square
with side d , as illustrated in Fig. 1 . The Rayleigh number range chosen
in the study keeps the nanouid ow incompressible and laminar. The
corner heater and cooler have isothermal boundaries in both vertical
and horizontal directions with length 0.4 L , which are shown by thick
red and blue lines, respectively. While the remainder of these walls are
kept adiabatic. The boundaries of the annulus are assumed to be imper-
meable, the uid within the cavity is a water-based nanouid having
Al
2
O
3
nanoparticles. The Boussinesq approximation is applicable, the
nanouid physical properties are constant except for the density. By
considering these assumptions, the continuity, momentum, energy and
volume fraction equations for the laminar and steady state natural con-
vection of incompressible ow can be written as follows [18] :
Continuity equation:
∇ ⋅𝑣 = 0 (1)
Momentum equation:
𝜌𝑛𝑓
𝑣 ⋅∇ 𝑣 = −∇ 𝑝 + ∇ ⋅𝜇𝑛𝑓
∇ 𝑣
+ ( 𝜌𝛽)
𝑛𝑓
( 𝑇 − 𝑇
𝑐
) 𝑔 (2)
Energy equation:
( 𝜌𝐶
𝑝
)
𝑛𝑓
𝑣 ⋅∇ 𝑇
𝑛𝑓
= ∇ ⋅𝑘
𝑛𝑓
∇ 𝑇
𝑛𝑓
− 𝐶
𝑝.
𝐽
𝑝
⋅∇ 𝑇
𝑛𝑓
(3)
202
A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219
Volume fraction equation:
𝑣 ⋅∇ 𝜑 = −
1
𝜌𝑝
∇ ⋅𝐽
𝑝
(4)
The energy equation of the inner solid wall is:
∇ 𝑇
𝑤
= 0 , (5)
where ∇ is the velocity vector, g is the acceleration due to gravity, 𝜑
is the local volume fraction of nanoparticles and J
p
is the nanoparticles
mass ux. Based on Buongiorno ’s model nanoparticles mass ux can be
written as:
𝐽
𝑝
= 𝐽
𝑝,𝐵
+ 𝐽
𝑝,𝑇
, (6)
where J
p, B
and J
p, T
represent the mass ux due to Brownian motion and
thermophoresis eect. J
p, B
is the drift ux due to the Brownian motion
which can be dened as:
𝐽
𝑝,𝐵
= − 𝜌𝑝
𝐷
𝐵
∇ 𝜑, (7)
where the Brownian motion is described by the Brownian diusion co-
ecient, D
B
, which is dened based on the model of Einstein–Stokes:
𝐷
𝐵
=
𝑘
𝑏
𝑇
3 𝜋𝜇𝑓
𝑑
𝑝
, (8)
J
p, T
represents the drift ux due to thermophoretic eects which can be
dened as:
𝐽
𝑝,𝑇
= − 𝜌𝑝
𝐷
𝑇
∇ 𝑇 , (9)
where the thermophoresis eect is described by the thermal diusion
coecient, D
T
, which is dened as:
𝐷
𝑇
= 0 . 26
𝑘
𝑓
2 𝑘
𝑓
+ 𝑘
𝑝
𝜇𝑓
𝜌𝑓
𝑇
𝜑. (10)
The thermo-physical properties of the nanouid can be determined
as follows:
The heat capacitance of the nanouids ( 𝜌C
p
)
nf
is given as
( 𝜌𝐶
𝑝
)
𝑛𝑓
= (1 − 𝜑 )( 𝜌𝐶
𝑝
)
𝑓
+ 𝜑 ( 𝜌𝐶
𝑝
)
𝑝
. (11)
The eective thermal diusivity of the nanouids 𝛼nf
is given as
𝛼𝑛𝑓
=
𝑘
𝑛𝑓
( 𝜌𝐶
𝑝
)
𝑛𝑓
. (12)
The eective density of the nanouids 𝜌nf
is given as
𝜌𝑛𝑓
= (1 − 𝜑 ) 𝜌𝑓
+ 𝜑𝜌𝑝
. (13)
The thermal expansion coecient of the nanouids 𝛽nf
can be deter-
mined by:
( 𝜌𝛽)
𝑛𝑓
= (1 − 𝜑 ) ( 𝜌𝛽)
𝑓
+ 𝜑 ( 𝜌𝛽)
𝑝
. (14)
The dynamic viscosity ratio of water–Al
2
O
3
nanouids for 33 nm
particle-size in the ambient condition was derived in reference [16] as
follows:
𝜇𝑛𝑓
𝜇𝑓
= 1∕
1 − 34 . 87
𝑑
𝑝
∕ 𝑑
𝑓
−0 . 3
𝜑
1 . 03
. (15)
The thermal conductivity ratio of water–Al
2
O
3
nanouids is calculated
by Corcione et al. model [16] is:
𝑘
𝑛𝑓
𝑘
𝑓
= 1 + 4 . 4 𝑅𝑒
0 . 4
𝐵
Pr
0 . 66
𝑇
𝑇
𝑓𝑟
10
𝑘
𝑝
𝑘
𝑓
0 . 03
𝜑
0 . 66
, (16)
where Re
B
is dened as
𝑅𝑒
𝐵
=
𝜌𝑓
𝑢
𝐵
𝑑
𝑝
𝜇𝑓
, (17)
𝑢
𝐵
=
2 𝑘
𝑏
𝑇
𝜋𝜇𝑓
𝑑
2
𝑝
, (18)
where 𝑘
𝑏
= 1 . 380648 ×10
−23 (J/K) is the Boltzmann constant. 𝑙
𝑓
=
0 . 17 nm is the mean path of uid particles. d
f
is the molecular diam-
eter of water given as [16]
𝑑
𝑓
=
6 𝑀
𝑁𝜋𝜌𝑓
, (19)
where M is the molecular weight of the base uid, N is the Avogadro
number and 𝜌f
is the density of the base uid at standard temperature
(310 K). Accordingly, and basing on water as a base uid, the value of
d
f
is obtained:
𝑑
𝑓
=
6 ×0 . 01801528
6 . 022 ×10
23
×𝜋× 998 . 26
1∕3
= 3 . 85 ×10
−10
m . (20)
Now, we introduce the following non-dimensional variables:
𝑋 =
𝑥
𝐿
, 𝑌 =
𝑦
𝐿
, 𝑉 =
𝑣𝐿
𝜈𝑓
, 𝑃 =
𝑝𝐿
2
𝜌𝑛𝑓
𝜈2
𝑓
, 𝜑
∗
=
𝜑
𝜙, 𝐷
∗
𝐵
=
𝐷
𝐵
𝐷
𝐵0
,
𝐷
∗
𝑇
=
𝐷
𝑇
𝐷
𝑇0
, 𝛿=
𝑇
𝑐
𝑇
ℎ
− 𝑇
𝑐
, 𝜃𝑛𝑓
=
𝑇
𝑛𝑓
− 𝑇
𝑐
𝑇
ℎ
− 𝑇
𝑐
, 𝜃𝑤
=
𝑇
𝑤
− 𝑇
𝑐
𝑇
ℎ
− 𝑇
𝑐
. (21)
This then yields the following dimensionless governing equations:
∇ ⋅𝑉 = 0 , (22)
𝑉 ⋅∇ 𝑉 = −∇ 𝑃 +
𝜌𝑓
𝜌𝑛𝑓
𝜇𝑛𝑓
𝜇𝑓
∇
2
𝑉 +
( 𝜌𝛽)
𝑛𝑓
𝜌𝑛𝑓
𝛽𝑓
1
Pr
𝑅𝑎 ⋅𝜃𝑛𝑓
, (23)
𝑉 ⋅∇ 𝜃𝑛𝑓
=
( 𝜌𝐶
𝑝
)
𝑓
( 𝜌𝐶
𝑝
)
𝑛𝑓
𝑘
𝑛𝑓
𝑘
𝑓
1
Pr
∇
2
𝜃𝑛𝑓
+
( 𝜌𝐶
𝑝
)
𝑓
( 𝜌𝐶
𝑝
)
𝑛𝑓
𝐷
∗
𝐵
Pr ⋅𝐿𝑒
∇ 𝜑
∗
⋅∇ 𝜃𝑛𝑓
+
( 𝜌𝐶
𝑝
)
𝑓
( 𝜌𝐶
𝑝
)
𝑛𝑓
𝐷
∗
𝑇
Pr ⋅𝐿𝑒 ⋅𝑁
𝐵𝑇
∇ 𝜃𝑛𝑓
⋅∇ 𝜃𝑛𝑓
1 + 𝛿𝜃𝑛𝑓
, (24)
𝑉 ⋅∇ 𝜑
∗
=
𝐷
∗
𝐵
𝑆𝑐
∇
2
𝜑
∗
+
𝐷
∗
𝑇
𝑆𝑐 ⋅𝑁
𝐵𝑇
⋅
∇
2
𝜃𝑛𝑓
1 + 𝛿𝜃𝑛𝑓
, (25)
∇ 𝜃𝑤
= 0 , (26)
where 𝐷
𝐵0
=
𝑘
𝑏
𝑇
𝑐
3 𝜋𝜇𝑓
𝑑
𝑝
is the reference Brownian diusion coecient,
𝐷
𝑇0
= 0 . 26
𝑘
𝑓
2 𝑘
𝑓
+ 𝑘
𝑝
𝜇𝑓
𝜌𝑓
𝜃𝜙is the reference thermophoretic diusion coef-
cient, 𝑆𝑐 = 𝜈𝑓
∕ 𝐷
𝐵0
is Schmidt number, 𝑁
𝐵𝑇
= 𝜙𝐷
𝐵0
𝑇
𝑐
∕ 𝐷
𝑇0
( 𝑇
ℎ
− 𝑇
𝑐
)
is the diusivity ratio parameter (Brownian diusivity/thermophoretic
diusivity), 𝐿𝑒 = 𝑘
𝑓
∕( 𝜌𝐶
𝑝
)
𝑓
𝜙𝐷
𝐵0
is the Lewis number, 𝑅𝑎 = 𝑔𝜌𝑓
𝛽𝑓
( 𝑇
ℎ
−
𝑇
𝑐
) 𝐿
3
∕( 𝜇𝑓
𝛼𝑓
) is the Rayleigh number for the base uid and Pr = 𝜈𝑓
∕ 𝛼𝑓
is the Prandtl number for the base uid. The dimensionless boundary
conditions of Eqs. (22) –(26) are:
𝑈 = 𝑉 = 0 ,
𝜕𝜑
∗
𝜕𝑛
= −
𝐷
∗
𝑇
𝐷
∗
𝐵
⋅
1
𝑁
𝐵𝑇
⋅
1
1 + 𝛿𝜃𝑛𝑓
𝜕𝜃𝑛𝑓
𝜕𝑛
, 𝜃𝑛𝑓
= 1
on the horizontal bottom wall ,
0 ≤ 𝑋 ≤ 0 . 4 , 𝑌 = 0 and on the vertical left wall ,
0 ≤ 𝑌 ≤ 0 . 4 , 𝑋 = 0 (27)
𝑈 = 𝑉 = 0 ,
𝜕𝜑
∗
𝜕𝑛
= 0 ,
𝜕𝜃𝑛𝑓
𝜕𝑛
= 0 ,
(for the adiabatic parts of the remainder walls) (28)
𝑈 = 𝑉 = 0 ,
𝜕𝜑
∗
𝜕𝑛
= −
𝐷
∗
𝑇
𝐷
∗
𝐵
⋅
1
𝑁
𝐵𝑇
⋅
1
1 + 𝛿𝜃𝑛𝑓
𝜕𝜃𝑛𝑓
𝜕𝑛
, 𝜃𝑛𝑓
= 0
on the horizontal top wall ,
0 . 6 ≤ 𝑋 ≤ 1 . 0 , 𝑌 = 1 and on the vertical right wall ,
0 . 6 ≤ 𝑌 ≤ 1 . 0 , 𝑋 = 1 (29)
203
A.I. Alsabery et al. International Journal of Mechanical Sciences 136 (2018) 200–219
Fig. 2. Streamlines (a), Das and Reddy [35] (left), present study (right), isotherms (b), Das and Reddy [35] (left), present study (right) for 𝐾
𝑟
= 0 . 2 (top) and 𝐾
𝑟
= 5 (bottom) at 𝑅𝑎 = 10
6
,
𝜙= 0 and 𝐷 = 0 . 5 .
204
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