Journal of Electronics Cooling and Thermal Control, 2013, 3, 111-130
http://dx.doi.org/10.4236/jectc.2013.33013 Published Online September 2013 (http://www.scirp.org/journal/jectc)
Mixed Convection Heat Transfer for Nanofluids in a
Lid-Driven Shallow Rectangular Cavity Uniformly Heated
and Cooled from the Vertical Sides: The Opposing Case
Hassan El Harfi1, Mohamed Naïmi1*, Mohamed Lamsaadi2,
Abdelghani Raji1, Mohammed Hasnaoui3
1Physics Department, Faculty of Sciences and Technologies, Sultan Moulay Slimane University,
Laboratory of Flows and Transfers Modelling (LAMET), Beni-Mellal, Morocco
2Polydisciplinary Faculty, Sultan Moulay Slimane University, Interdisciplinary Laboratory of
Research in Sciences and Technologies (LIRST), Beni-Mellal, Morocco
3Physics Department, Faculty of Sciences Semlalia, Cadi Ayyad University, Laboratory of Fluid
Mechanics and Energetics (LMFE), Marrakech, Morocco
Email: *naimi@fstbm.ac.ma; *naimima@yahoo.fr
Received May 31, 2013; revised July 1, 2013; accepted July 10, 2013
Copyright © 2013 Hassan El Harfi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
An investigation on flow and heat transfer due to mixed convection, in a lid-driven rectangular cavity filled with Cu-
water nanofluids and submitted to uniform heat flux along with its vertical short sides, has been conducted numerically
by solving the full governing equations with the finite volume method and the SIMPLER algorithm. In the case of a
slender enclosure, these equations are considerably reduced by using the parallel flow concept. Solutions, for the flow
and temperature fields, and the heat transfer rate, have been obtained depending on the governing parameters, which are
the Reynolds, the Richardson numbers and the solid volume fraction of nanoparticles. A perfect agreement has been
found between the results of the two approaches for a wide range of the abovementioned parameters. It has been shown
that at low and high Richardson numbers, the convection is ensured by lid and buoyancy-driven effects, respectively,
whereas between these extremes, both mechanisms compete. Moreover, the addition of Cu-nanoparticles, into the pure
water, has been seen enhancing and degrading heat transfer by lid and buoyancy-driven effects, respectively.
Keywords: Component Nanofluids; Mixed Convection; Heat Transfer; Lid-Driven Cavity; Parallel Flow Assumption;
Finite Volume Method
1. Introduction
Heat transfer in fluidic systems has often been the subject
of ambitious research in order to enhance it considering
its importance in several industrial processes. However,
with conventional fluids, such as water and oil, whose
thermal conductivity is inherently poor, heat transfer is
limited, which is a crucial problem to challenge. Also,
current design solutions already push available technol-
ogy to its limits, and an innovative way should be taken.
In such a context, Choi of Argonne National Laboratory
[1] developed the novel concept of nanofluids as a route
to improve the performances of heat transfer fluids cur-
rently available. This new class of advanced heat transfer
fluids is engineered by dispersing solid nanoparticles
(metallic, non-metallic or polymeric), smaller than 100
nm in diameter, in base fluids (aqueous or organic host
liquids), which confers a large thermal conductivity on
these ones and makes them potentially useful in engi-
neering equipments involving heat transfer. To know
about nanoparticles, nanofluids, their production and ap-
plications, see, for instance, the report of Yu et al. is cur-
rently available in [2].
During the last decade, nanofluids have attracted lots
of researchers, who are encouraged by their critical impor-
tance and promising role, as new advanced heat transfer
fluids, to take up challenges. Therefore, numerous stud-
ies, on convection heat transfer, have been conducted, and
most of them have dealt with forced convection, indicat-
ing that nanoparticle suspensions have unquestionably a
great potential for heat transfer enhancement, as reported
*Corresponding author.
C
opyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
112
in a recent paper by Corcione [3]. In contrast, although
the investigations concerned with buoyancy-driven con-
vection are relatively few, they have seen a gradual in-
crease lately, leading to contradictory findings, thus leav-
ing still unanswered question, if the use of nanoparticle
suspensions for natural convection applications is actu-
ally advantageous with respect to pure liquids [3]. At the
same time, mixed convection has not received either less
attention in view of the number of the related works re-
cently done. Among them, flow and heat transfer prob-
lem in lid-driven cavities, which finds applications in
industrial processes such as food processing, float glass
production [4], thermal-hydraulics of nuclear reactors [5],
dynamics of lakes [6], crystal growth, flow and heat
transfer in solar ponds [7], lubrication technologies [8]
and so on. The interaction of the shear driven flow due to
the lid motion and natural convective flow due to the
buoyancy effect are quite complex, which necessitates a
comprehensive analysis to understand the physics of the
resulting flow and heat transfer process. In this respect,
different configurations and combinations of thermal and
dynamical boundary conditions have been considered
and analyzed by some investigators. The contributions
can be divided in two cases:
1) Steady state case where all boundary conditions are
time independent. In this regard, it is advisable to men-
tion the work of Tiwari and Das [8], who studied heat
transfer enhancement in a nanofluid-filled square cavity,
with the vertical sides moving and differentially heated,
while the horizontal ones are insulated and motionless.
Three situations, depending on the direction of the mov-
ing walls, were examined, and a model taking into ac-
count the solid volume fraction of nanoparticles was de-
veloped to analyze the nanofluids behavior. With only
one uniformly moving wall, from left to right, first, it is
to bring up the research of Abu-Nada and Chamkha [9]
deal with mixed convection flow in an inclined square
enclosure filled with a nanofluid. The left and right walls
are kept insulated while the bottom and the moving top
ones are maintained at constant cold and hot tempera-
tures, respectively. It was found that significant heat
transfer enhancement can be obtained due to the presence
of nanoparticles and that this is accentuated by inclina-
tion of the enclosure at moderate and large Richardson
numbers. Mahmoodi [10] investigated mixed convection
fluid flow and heat transfer in rectangular enclosures
filled with a nanofluid. The left and right walls as well as
the top one are maintained at a constant cold temperature.
The moving bottom is kept at a constant hot temperature.
A parametric study was performed and the effects of the
Richardson number, the aspect ratio of the enclosure and
the volume fraction of the nanoparticles on the fluid flow
and heat transfer were analysed. It was found that for the
selected values of the Richardson number, the average
Nusselt number increases with the nanoparticles volume
fraction, and seems to be higher with tall enclosures than
with shallow ones. In the case of a nanofluid-filled squ-
are cavity with cold sides, a partially heated (with con-
stant heat flux heater) and insulated bottom, and a mov-
ing cold top, Mansour et al. [11] examined the effects of
Reynolds number, type of nanofluids, size and location
of the heater and the volume fraction of the nanoparticles
in their study related to mixed convection. They observed
that the heat transfer enhances with all the above men-
tioned parameters. Muthtamilselvan et al. [12] studied
heat transfer enhancement of nanofluids in rectangular
enclosures, where the moving top is at higher constant
temperature than the bottom, whereas the left and right
boundaries are insulated. They found that at higher as-
pect ratios, the heat transfer rate increases strongly with
the nanoparticles volume fraction. Nemati et al. [13] in-
vestigated heat transfer performance of a moving top
square cavity, filled with nanofluids and subject to dif-
ferent side wall temperatures. They reported that an in-
crease of nanoparticles volume fraction enhances heat
transfer, but such an effect reduces with the Reynolds
number. As for Talebi et al. [14], they conducted an in-
vestigation on mixed convection flows in a square lid-
driven cavity, having left and right sides heated and
cooled, respectively, and moving top and bottom both
adiabatic, utilizing nanofluids. These authors showed that,
at given Rayleigh and Reynolds numbers, an increase of
the nanoparticles concentration favours the flow and heat
transfer. Finally, like Tiwari and Das [8], Sheikhzadeh et
al. [15] were interested in laminar mixed convection of a
nano-fluid in two sided lid-driven enclosures. The mov-
ing left and right walls are maintained at constant cold
and hot temperatures, respectively, while the horizontal
ones are insulated. The effect of moving direction of
walls on mixed convection is studied for various Ri-
chardson numbers, aspect ratios and nanoparticles vol-
ume fractions, and was found to affect mainly the flow
field, temperature gradient and heat transfer. In addition,
increasing the volume fraction of nanoparticles resulted
in a linear increase of the average Nusselt number, as an
index of heat transfer rate improvement, for all the con-
sidered cases.
2) Unsteady state case, where some boundary condi-
tions are time dependent as in the only work done, in this
subject, by Karimipour et al. [16], where periodic mixed
convection of a nanofluid inside a rectangular cavity,
with insulted vertical sides and hot temperature bottom
kept at rest and cold temperature top horizontally oscil-
lating, was carried out. The effects of Richardson number
and volume fraction of nanoparticles on the flow and
thermal behaviour of the nanofluid were examined. It
was observed that the best heat transfer is obtained with a
Richardson number lower than unit and that the higher
value of this parameter corresponds to the lower ampli-
tude of the oscillation of the heat transfer rate in the
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL. 113
steady periodic state. In addition, heat transfer was found
to be improved by nanoparticles presence.
All the above mentioned studies are of numerical na-
ture, using a finite volume method (for the most), a finite
difference method or Lattice Boltzmann method to solve
the governing equations and various single-phase models
to describe effective conductivity and viscosity of the
considered nanofluids, which are principally or
Cu-water.
23
Al O
To the best of our knowledge, the problem of mixed
convection heat transfer of nanouids in a lead-driven
enclosure subject to Neumann boundary conditions for
temperature (i.e. boundaries subject to heat fluxes) is not
yet analyzed. So, in order to know more about the effect
of the boundary conditions kind on flow and heat transfer
within nanofluids, the present paper focuses on such a
problem within a two-dimensional shallow rectangular
enclosure, filled with Cu-water nanofluids, whose short
vertical sides are submitted to uniform heat fluxes while
the long horizontal ones are maintained adiabatic with
the top moving in the opposite direction to the heat flux.
A numerical solution of the full governing equations is
obtained via a finite volume method. An analytical one,
based on the parallel flow approximation, is also pro-
posed. The results are presented, in terms of streamlines,
isotherms, stream function and temperature profiles and
heat transfer rates, and discussed for various values of
the dimensionless parameters, controlling the problem,
which are the Reynolds, Re, and Richardson, Ri, num-
bers, and the solid volume fraction of nanoparticles, Φ.
2. Mathematical Formulation
The studied configuration is sketched in Figure 1. It is a
shallow rectangular enclosure of height
H
q
and length
, filled with Cu-water nanofluids. The long horizontal
walls are adiabatic, while the vertical short ones are sub-
mitted to a uniform density of heat flux, . All these
boundaries are rigid, impermeable and motionless apart
from the top one which moves in its own plane from
right to left at uniform velocity, 0. The main assump-
tions made here are those commonly used, i.e.:
L
U
H
L
q
q
0
y
T
0
0
y
T
0
U
v ,y
u,x 
Figure 1. Schematic view of the geometry and coordinates
system.
The base uid and the nanoparticles are in thermal
equilibrium and they ow at the same velocity (i.e. no
slip occurs between them or the nanoparticles are uni-
formly dispersed within the base uid so that the result-
ing nanofluid can be considered a single-phase fluid).
The nanoparticles are spherical;
The nanouid is Newtonian and incompressible;
The thermophysical properties of the considered nan-
ofluids are constant (taken at the reference tempera-
ture, 0
T
) except for the density in the buoyancy term
(containing the gravitational acceleration, g), which
obeys the Boussinesq approximation;
The ow is two-dimensional, laminar and steady;
The radiation heat transfer between the sides of the
cavity is negligible when compared with the other
mode of heat transfer.
Therefore, the equations describing the conservation of
mass (1), momentum (2)-(3) and energy (4), written in
terms of velocity components
, pressure
,uv

p
and temperature
T
, are:
0
uv
xy



 (1)
22
22
1nf
nf nf
uuuPuu
uv
txy x
x
y


  
 

 

 
 

(2)


22
0
22
11
nf
nf
nf nfnf
vvv
uv
txy
Pvv
g
TT
yxy

 









 




(3)
22
22
nf
TTTTT
uv
txy xy

 


 
 
 

(4)
To close the problem, the following appropriate bound-
ary conditions are applied:
0 and0for0 and
nf
Tq
uvx xL
xk



(5)
0 and0for0
T
uv y
y


(6)
00 and0for
T
uUvy H
y
 
 
(7)
To model the effective physical properties of the nano-
fluid, appearing in the above equations, the following
formulas are used:
1
nff np

 (8)
for the effective density, as shown in [2];
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
114

2.5
1
f
nf
 (9)
for the effective dynamic viscosity, which is due to
Brinkman [2];
 
1
nff np

 

(10)
for the thermal expansion coefficient [17];
1
nff np
CpCp Cp

 
(11)
for the heat capacity [2];

22
2
npff np
nf
fnpff np
kk kk
k
kkk kk

 (12)
for the effective thermal conductivity, due to Maxwell-
Garnett, which is a restriction of the Hamilton-Crosser
model to spherical nanoparticles [2];

nf
nf
nf
k
Cp
(13)
for the thermal diffusivity [18].
Note that the Subscripts f, nf and np stand for the base
fluid, nanofluid and nanoparticles, respectively.
On the other hand, using the characteristic scales ,
H
2
0,
fU
0,
H
U

0 and ,U
f
qH k
, corresponding to
length, pressure, time, velocity and temperature, respect-
ively, the dimensionless governing equations and the cor-
responding boundary conditions are:
0
uv
xy


 (14)
22
22
1uu
uuuP
uv
txy x
Re
x
y


 
 

 

(15)
22
22
1
vvv
uv
txy
vv
PRiT
yRe xy







 



(16)
22
22
TT
TTT
uv
txy
Pe
x
y



 

 

(17)
10for 0and
T
uvx A
xk
 
(18)
0for 0
T
uvy
y
 
(19)
10for
T
uv y
y
 
where nf f
kkk, nf f

, nf f

,
nf f
 
and nf f

are parameters
depending on Φ, according to models given above. In
addition, to analysis the flow structure, the stream func-
tion, ψ, related to the velocity components via

andwith0 on all boundaries
ψψ
u v
yx

 

(21)
is used.
The above equations let appears some dimensionless
parameters that govern the problem, namely, the solid
volume fraction Φ, the aspect ratio of the enclosure, A,
the Peclet, Pe, Reynolds, Re, and Richardson, Ri, num-
bers. For the last four, the expressions are
2
00
2
0
,, f
ff f
g
qH
UH UH
L
APeReand Ri
HkU

 
 
(22)
Note that
2
and Gr Ra
PePrReRiPeRe
Re
 (23)
where
4
2,and
ff
f
ff
gqH
GrPrRa PrGr
k



(24)
are the Grashof, Prandtl and Rayleigh numbers, respect-
ively.
The local heat transfer, through the nanofluid-filled
cavity, can be expressed in terms of the local Nusselt
number defined as

*1
ff
hLq LLTA
Nu ykTkHTTT

 

A
(25)
where h is the heat exchange coefficient,
*
f
TqHk
 a characteristic temperature and
Δ0, ,TTyTAy the side to side dimensionless
local temperature difference. This definition is based on
the thermal conductivity of the base fluid,
f
k, which
seems logical since, according to Corcione [3], Nu that
would describe the thermal performance of the enclosure,
with immediacy, should vary in the same manner as h
and vice versa. However, Equation (25) is notoriously
inaccurate owing to the uncertainty of the temperature
values evaluated at the two vertical walls (edge effects).
Instead, Nu is calculated on the basis of a temperature
difference between two vertical sections, far from the end
sides, as suggested by Lamsaadi et al. [19]. Thus, by
analogy with Equation (25), and considering two infini-
tesimally close sections, Nu can be expressed by
 
00
2
11
lim lim
xx
xA
x
Nu yTTxTx



 
 (26)
1 (20)
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL. 115
where
x
is the distance between two symmetrical sec-
tions with respect to the central one.
The corresponding average Nusselt number is calcu-
lated, at different locations, from

1
NuNu y0dy (27)
3. Numerics
17) associated with Equations (18)-(20)
Equations (14)-(
have been solved by using a finite volume method and
SIMPLER algorithm in a staggered uniform grid system
[20]. A second order back-wards finite difference scheme
has been employed to discretise the temporal terms ap-
pearing in Equations (15)-(17). A line-by-line tridiagonal
matrix algorithm with relaxation has been used in con-
junction with iterations to solve the nonlinear discretised
equations. The convergence has been considered as
reached when 151
,,,
10
kk k
ij ijij
,,ij ij
f
ff



, where ,
k
ij
f
stands for the value of u, v, p or T at the kth iteration level
and grid location (i, j) in the plane (x, y). The mesh size
has been chosen so that a best compromise between run-
ning time and accuracy of the results may be found. The
procedure has been based on grid refinement until the
numerical results agree, within reasonable accuracy, with
the analytical ones, obtained from the parallel flow ap-
proach developed in the next section. Hence, as shown in
Table 1, a uniform grid size of 160 40 has been se-
lected for 8A (value used for the numerical compu-
tations) andeen estimated sufficient to model accu-
rately the flow and temperature fields within the cavity.
The time step size, t
has b
, has been varied in the range
74
10 10t

 , depeing on the values of the govern-
4. Approximate Parallel Flow
nd
ing parameters.
Analytical
As from Figures 2-4, displaying streamlines
Solution
can be seen
(left) and isotherms (right), the flow and temperature
fields exhibit a parallel aspect and a linear stratification,
respectively, in the most part of the cavity, for 8
A
and various values of Re, Ri and Φ. Accordingly, th-
lowing simplifications
e fol
 
,,uxy uy v,0,,
and ,2
xy ψxy ψy
TxyCxAθy

  (28)
where C is unknown constant temperature gradient in the
x-direction, are possible, which leads to the ordinary non-
dimensional governing equations:
3
duT
3
dReRi ReRiC
x
y

 
(29)
Table 1. Accuracy tests conducted with Re = 1, 3
various values of . Ri = 10and
Grids (160 20)
Φ c
Nu
0.0 0.4429 0
0.4179 3.6429
s (120 40) (200
4.720
0.1
0.2 0.3533 2.9500
Grid (160 40) 400)
Φ c
Nu c
Nu c
Nu
0.0 0.460.4427 4 0.0
0.41693.5940.4171 3.5952 0.41713.594
s
427 4.6444.6444428 4.644
0.1 89
0.2 0.35042.90180.3504 2.9024 0.35062.9018
Grid (160 60)
Φ c
Nu
0.0 0.4427 4
0.4179 3.5952
4.644
0.1
0.2 0.3502 2.9024
2
d

2
dCu
Pe y (30)
with
dd
0for 0and 10for1
dd
uyu
yy

y
  (31)
as boundary, return
tions, respectively.

1
0
d0uy y
1
(32)

0
d0yy
flow and me
(33)
an temperature condi-
Using such an approach, the solution of Equations (29)
and (30), satisfying Equations (31)-(33), is


32 2
23 32
12
uyReRiC yyyyy
  (34)

543
2
43
11
121046 120
1
4330
yyy
yRaC
PeC y y

 






The expression of the stream function,
(35)

ψ
y
, can be
deduced by integration of Equation (21)nto ac-
co nditi
Eq
, taking i
unt of the corresponding boundary coons and
uation (34), which gives:
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
Copyright © 2013 SciRes. JECTC
116
(a
(c)
(b)
)
(1)
(b)
(c)
(a)
(2)
(b)
(c)
(a)
(3)
(a)
(b)
(c)
(4)
Figure 2. Streamlines (left) and isotherms (right) for Re = 0.1 ((a) = 0.1, (b) = 0.1 and (c) = 0.2)
and Ri ((1) Ri = 1, (2) Ri = 10, (3) Ri = 102 and (4) Ri = 103). and various values of


42
yy

33
2
1222
yReRiCy yy
  

 (36)
where

 . Therefore, the flow intensity is
max min
,
cSup

(37)
where max
and min
are the extrema of
ψ
y
at
the cevertical section of the enclosure ntral

2xA.
natuThey correspond tointensities of forced andral
convections, respectively.
the
On the other hand, according to Bejan [21], the energy
balance in x-direction is
111
dd
TPe T
yu
Ty



 0 or
000
xA
xx



In particular, in the pa
d
y
(38)
rallel flow region and with the
application of Equation (18), Equation (38) becomes:
1
0k
1
d
Pe
Cuy

(39)
ves
the following transcendental equation:
which, when substituted to Equations (34) and (35), gi
H. EL HARFI ET AL. 117
(c)
(b)
(a)
(1)
(c)
(b)
(a)
(2)
(a)
(c)
(b)
(3)
(c)
(b)
(a)
(4)
Figure 3. Streamlines (left) and isotherms (right) for Re = 1 and various values of ((a) = 0.1, (b) = 0.1 and (c) 0.2)
and Ri ((1) Ri = 1, (2) Ri = 10, (3) Ri = 102 and (4) Ri = 103). =
222
2
2
10
3360 362,880
105 CC C
k

 (40)
whose solution, via Newton-Raphson method, for
Pe, Ra and Φ, leads to C.
3
1PePeRa Ra


  
given
Finally, taking into account of Equations (26) and (27),
the Nusselt number is constant and can be expressed as
1
Nu NuC
 (41)
5. Results and Discussion
With boundary conditions of Neumann kind (uniform
heat flux imposed to vertical walls) the flow and thermal
fields, and thermo-convective ch
parallel, stratified and independent on the enclosure as-
this parameter tends to
aracteristics become
pect ratio, A, respectively, when
be large enough. In our situation, this has been occurred
with 8
A in the limit of the explored values of Re
0.1 10Re , Ri
3
110Ri ,

00.2 ,
and 7Pr
(water based mixtures). Consequently, the
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
118
(b)
(a)
(c)
(1)
(b)
(a)
(c)
(2)
(b)
(a)
(c)
(3)
(b)
(a)
(c)
(4)
Figure 4. Streamlines (left) and isotherms (right) for Re = 10 and various values of ((a) = 0.1, (b) = 0.1 and (c) = 0.2)
and Ri ((1) Ri = 1, (2) Ri = 10, (3) Ri = 102 and (4) Ri = 103).
tours of streamlines (left) and isotherms (right)
the flow is parallel to the horizontal boundaries and the
temperature is linearly stratified in the horizontal direc-
en from Figure 2
mixed convection flow developed within the enclosure is
governed only by four dimensionless parameters, namely,
e, Ri, in accordance with Equations (22)-(24), and
are presented in Figures 2-4 for each Re and various Ri
and Φ. First of all, remember that, except in the end sides,
R
possibly the type of nanoparticles, even if the present
study is limited to Cu-water nanofluid, with the thermo-
physical properties of Cu and water given in Table 2
[22]. The effects of these parameters on the flow and
thermal fields and the resulting heat transfer will be now
discussed.
5.1. Flow and Thermal Patterns
Typical con
tion. On the other hand, as can be se
corresponding to Re = 0.1, the shear effect due to moving
top wall is dominall (=1), relatively small (= ant for Ri sm
10) and moderate (=10²), since the flow is unicellular and
counterclockwise with streamlines crowded near this
boundary causing the lost of their own symmetry with
respect to the horizontal mid-plane and this, whatever the
value of Φ. In contrast, for relatively high Ri (=10³), a
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL. 119
clockwise cell, whose size reduces with an increasing Φ,
is generated by buoyancy effect in the lower part of the
cavity, i.e. under that due to lid-driven effect. The influ-
ence of Φ can be explained by the fact that an increase of
this parameter leads to an increase of the effective vis-
cosity, which, in its turn, tends to decelerate the buoy-
ancy flow and accelerate the shear one. In fact, the vis-
cosity is a manifestation of surface phenomena and there-
fore can only favour the shear flow. As for the isotherms,
they remain almost vertical synonymous of pseudo-con-
duction regime (i.e. low convection regime, Re = 0.1).
An increase of Re to 1 (Figure 3) anticipates the buoy-
ancy flow, since the related cell takes place at Ri = 10²,
with a decreasing size with Φ like before (i.e. Re = 0.1).
However, such a cell tends to be large when passing to Ri
= 10³, which is normal if one refer to Equation (22). In
such a situation, the isotherms, which are tilted, with res-
pect to y-axis, in the counterclockwise direction (i.e. that
of the shear flow), gradually change direction of tilting as
the buoyancy flow takes importance (i.e. while increas-
ing Ri). But, with the presence of nanoparticles (i.e. with
increasing Φ), such a change seems to occur slowly, de-
pending on Ri. Finally, for Re = 10 (Figure 4), two small
clockwise eddies develop in the bottom left and right cor-
ners of the enclosure, for Ri = 10 (Figure 4(2a)), as a re-
sult of buoyancy driven effect, although these structures
may disappear wile introducing nanoparticles in the base
fluid. As before, a progressive increase of Ri to 10² (Fig-
ure 4(3)) makes stronger buoyancy effect and the two
tiny eddies, observed previously, join each other and
grow to give rise to a large buoyancy cell, whose size
reduces slightly with Φ, for the reasons given above. A
further increase of Ri to 10³ (Figure 4(4)) makes bigger
enough buoyancy cell so that the effect of Φ, on this,
cannot be detected, which is obvious since, according to
Equation (22), inertia effects due to gravity prevail over
those associated with top moving wall. At the same time,
the isotherms appear to be nearly tilted in the direction of
the strongest flow (shear flow for Ri = 1 and 10, and
buoyancy flow for Ri = 10³). Between the two case, i.e.
for Ri = 10², both the tendencies exist and the shape of
the isotherms is strongly distorted. Here also, it is easy to
see that an increase of Φ does not affect strongly the
shape of isotherms.
In addition to that, the similarity notion is not re-
spected in the present problem since a change of Re may
cause big changes in flow and thermal fields for given Ri
Table 2. Thermophysical properties of base fluid (H2O) and
nanoparticles (Cu) [22].

3
kg m
Jkg k
P
C
Wmkk
5
101 k
O
2
H 997.1 4179 0.613 21
Cu 8933 385 401 1.67
and Φ (compare, for example, Figures 2(3a)-4(3a) cor-
responding to Ri = 102 and Φ = 0). Also, for the case Ri =
1, although buoyancy and lid-driven effects have the
sam magnitude, th shear flows dominan because of
thall depthe encl (shallow cavity).
5.2. Validaf the Aoximaarallew
erature
seen the two types of results is
ee it
e smth of osure
tion opprte Pl Flo
Analytical Solution
In order to validate the approximate analytical solution,
the numerical results (full circles) are compared to those
obtained analytically (solid lines), as displayed in Fig-
ures 5-7 giving stream function (left) and temp
(right) profiles along the y-axis at the mid-length of the
cavity,
(A/2, y) and T(A/2, y), respectively. As can be
, the agreement between
quite perfect, which confirms the existence of an ana-
lytical solution and validates mutually the parallel flow
hypothesis and the elaborated computing code.
Moreover, computed and calculated values of the
stream function at the vertical central section of the cav-
ity, c
, and mean Nusselt number, Nu, presented in
Figures 8-10, show also good agreement between the
analytical and numerical results for a wide range of Ri,
and various values of Re and Φ.
5.
Se
those discussed in sub-
increase
and
natuhere
Φ reost without influence due to mixing effects
e extremum) corre-
sp
from the left hot side and cools the top (y = 1) after pass-
3. Stream Function and Temperature
Distributions along the Vertical Central
ction
Although the results (Figures 5-7) of subsection 5.2 are
related to the core region, where the parallel flow con-
cept is valid, they confirm mostly
section 5.1 and demonstrate that, in general, an
of Φ amplifies and reduces the strengths of forced
ral flows, respectively, except in some cases w
mains alm
that occur at Re = 10 (Figure 7).
The presence of a maximum in the stream function
profile indicates that the flow is unicellular counter-
clockwise, driven mainly by the moving top wall. When
this profile presents, simultaneously, two extrema, maxi-
mum and minimum, this means that both the shear and
buoyancy driven flows coexist and the flow regime is
bicellular. The maximum (positiv
onds to shear flow, which is counterclockwise, and the
minimum (negative extremum) is related to buoyancy
flow that is clockwise.
Generally, the temperature profile presents two or
three zones with positive and negative signs, namely
(+,), (,+,) and (,+), depending on the flow nature
and the competition between lid and buoyancy driven
effects. Thus, with a dominant lid-driven effect, the shear
flow warms the bottom (y = 0) by transporting the heat
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
Copyright © 2013 SciRes. JECTC
120
0.16
0.02
0.04
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12
Ri = 1
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
y0.0 0.2 0.4 0.6 0.8 1.0
-0 .0 4
-0 .0 2
0.00
TAnalytical solution
Numerical solution
= 0.0
= 0.1
Ri = 1
= 0.2
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.04
0.08
0.12
0.16
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 10
y0.0 0.2 0.4 0.6 0.8 1.0
-0.04
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
Numerical solution
= 0.0
= 0.1
= 0.2
Analytical solution
Ri = 10
T
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.03
0.06
0.09
0.12
0.15
Numerical solution
= 0.0
= 0.1
= 0.2
Analytical solution
Ri = 102
y0.0 0.2 0.4 0.6 0.8 1.0
-0.03
-0.02
-0.01
0.00
0.01
0.02
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 102
T
y
0.0 0.2 0.4 0.6 0.8 1.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15 Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 103
y0.0 0.2 0.4 0.6 0.8 1.0
-0.03
-0.02
-0.01
0.00
0.01
0.02
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 103
Analytical solution
T
y
Figure 5. The stream function (left) and temperature (right) profiles at mid-length of the cavity, along the vertical coordinate,
for Re = 0.1 and various values of and Ri.
H. EL HARFI ET AL. 121
0.0 0.2 0.40.6 0.8 1.0
0.00
0.04
0.08
0.12
0.16
Ri = 1
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
y0.0 0.2 0.4 0.6 0.8 1.0
-0.3
-0.2
-0.1
0.0
0.1
0.2
Ri = 1
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
T
y
0.0 0.2 0.4 0.6 0.8 1.0
-0.02
0.00
0.04
0.08
0.12
0.16
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 10
y0.0 0.2 0.40.6 0.8 1.0
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 10
T
y
0.0 0.2 0.40.6 0.8 1.0
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12
Numerical solution
= 0.0
= 0.1
= 0.2
Analytical solution
Ri = 102
y0.0 0.2 0.40.6 0.8 1.0
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 102
T
y
0.0 0.20.4 0.6 0.81.0
-0.2
-0.1
0.0
0.1
0.2
Numerical solution
= 0.0
= 0.1
= 0.2
Analytical solution
Ri = 103
T
y
0.0 0.20.4 0.6 0.8 1.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1 Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 103
y
Figure 6. The stream function (left) and temperature (right) profiles at mid-length of the cavity, along the vertical coornate,
for Re = 1 and various values of and Ri. di
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
122
0.06
0.16
0.0 0.2 0.4 0.6 0.8 1.0
-0.02
0.00
0.04
0.08
0.12
Ri = 1
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
y0.0 0.2 0.4 0.6 0.8 1.0
-0. 0 9
-0. 0 6
-0. 0 3
0.00
0.03
Analytical solution
Ri = 1
Numerical solution
= 0.0
= 0.1
= 0.2
T
y
0.0 0.2 0.4 0.6 0.8 1.0
-0.02
0.00
0.04
0.08
0.12
0.16
Numerical solution
= 0.0
= 0.1
= 0.2
Analytical solution
Ri = 10
y0.0 0.2 0.4 0.6 0.8 1.0
-0.08
-0.04
0.00
0.04
0.08
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 10
T
y
0.0 0.2 0.4 0.6 0.8 1.0
-0.09
-0.06
-0.03
0.00
0.03
0.06
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 102
y0.00.20.40.60.81.0
-0.10
-0.05
0.00
0.05
0.10
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 102
T
y
0.0 0.2 0.4 0.6 0.8 1.0
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
Numerical solution
= 0.0
= 0.1
= 0.2
Analytical solution
Ri = 103
y0.0 0.2 0.4 0.6 0.8 1.0
-0.050
-0.025
0.000
0.025
0.050
Analytical solution
Numerical solution
= 0.0
= 0.1
= 0.2
Ri = 103
T
y
Figure 7. The stream function (left) and temperature (right) profiles at mid-length of the cavity, along the vertical coordinate,
for Re = 10 and various values of and Ri.
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
Copyright © 2013 SciRes. JECTC
123
0
.
7
0.0015001000 1500 2000 25003000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(a)
Analytical solution
Numerical solution
= 0.2
= 0.1
= 0.0
Ri
c
0.0015001000 15002000 2500 3000
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Analytical solution
Numerical solution
(b)
= 0.2
= 0.1
= 0.0
Nu
Ri
Figure 8. Flow intensity (a) and heat transfer rate (b) versus Ri, for Re = 0.1 and various values of .
ing near the right cold side. This is the case of a tem-
perature profile of signs (+,). With competing shear and
buoyancy flows, the temperature sign is such that the
corresponding counterclockwise and clockwise cells act
so that the interface between them is warm, the bottom is
cold and the top is warm (,+) or cold (,+,) depending
on whether shear or buoyancy effect is dominant, and on
Φ.
5.4. Flow Intensity and Heat Transfer Rate
For further analysis of the problem, the flow intensity
(top), , and heat transfer rate (bottom), Nu
c
, are re-
ported, against Ri, in Figures 8-10, for each Re and var-
ious Φ.
It is easy to observe that c
depen
a dec
exhibits in general two
tendencies, whose expanseds on Re and Φ. The
first one is characterized byrease of c
with Ri,
expressing a reduction of thrength of tear flow
until a minimum reached at particular value Ri, which
increases and decreases with and Re, respectively. The
second one corresponds toncrease of
e st
Φ
an i
he sh
of
c
with Ri,
beyond the minimum obserd, and expr aug-
veesses an
H. EL HARFI ET AL.
124
0.5
0.001200 400 600 8001000
0.0
0.1
0.2
0.3
0.4
(a)
= 0.2
= 0.1
= 0.0
Ri
c
Analytical solution
Numerical solution
0.001200 400 600 8001000
0
1
2
3
4
5
(b)
Analytical solution
Numerical solution
= 0.2
= 0.1
= 0.0
Nu
Ri
Figure 9. Flow intensity (a) and heat transfer rate (b) versus Ri, for Re = 1 and various values of .
mentation of the strength of the buoyancy flow which
reduces with Φ. It can be seen, also, that an increase of
Re leads, first, to an increase of c
and, second, to a
decrease of this quantity (compare Figures 8-10, ob-
tained for Re = 0.1, 1 and 10, respectively). This is due,
probably, to a change of the dominant role from one re-
gime to another.
With regard to Nu , a slight increase of this quantity,
with Ri, is observed for Φ = 0, but such a tendency dis-
appears, as Φ increases, leading to a constant . This
behaviour is the consequence of an increase of the effect-
e mo-
ion is
of
Nu
ive viscosity with Φ, which acts to slow down th
tion, particularly, for Re = 0.1 where mixed convect
weak (Figure 8). Moreover, for Re = 1 and 10, two
trends evolution appear for Nu , as foc
r
, since
Nu decreases and increases with Ri on both sides of a
minimum depending on Φ (Figures 9 and 10e first
tre
driven flow. In contrast, the second trend, whiretches
). Th
ch st
nd, which corresponds to a short range of Ri, whose
expanse increases with Φ, is related to a dominant lid-
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL. 125
0.25
0.001 2004006008001000
0.05
0.10
0.15
0.20
(a)
Analytical solution
Numerical solution
= 0.2
= 0.0
= 0.1
Ri
c
0.001200400 600 8001000
0
20
40
60
80
(b)
Analytical solution
Numerical solution
= 0.2
= 0.1
= 0.0
Nu
Ri
Figure 10. Flow intensity (a) and heat transfer rate (b) versus Ri, for Re = 10 and various values of .
over a large range of Ri, is associated with a prevailing
buoyancy-driven flow. Last, Figures 8-10 show also the
quite obvious increase of Nu with Re, conveying the
favorable role of the lid-driven flow to heat transfer.
In order to examine the influence of Φ on flow inten-
sity and heat transfer rate, the quantities c
(top) and
Nu (bottom) are plotted against Φ for various Ri and
each Re, in Figures 11-13. For Re = 0.1 (Figure 11),
except for Ri = 103 where a decrease of c
,
with Φ, due
probably to the reduction of buoyancy effect by nanopar-
are, in
general, increasing functions of Φ. For Re = 1, Figure 12
shows different variations of c
and Nu , with Φ, de-
ticles, is observed for Φ < 7.5%, c
andNu
pending on Ri. Hence, for Ri = 1 and 10, c
is almost
constant (very weak slop), ncreawhile the ise of Nu is
is not notabr Ri = , clear, although itle. Fo10² c
pre-
send nts a decrease aan increase on both sides of Φ =
6.25%, whereas the increase of Nu is monotonic. For
Ri = 10³ the tendency is reversed since c
andNu
are decreasing functions of Φ. At last, for Re = 10 (Fig-
ure 13), c
seems to be quite unconcerned about any
variatioΦ n of
, whilst Nu undconsta
c
nt ergoes
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
126
0.14
0.16
0.00 0.02 0.040.06 0.08 0.10 0.120.14 0.16 0.18 0.20
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Ri = 1
Ri = 102
Ri = 10
Analytical solution
(a)
Numerical solution
Ri = 103
c
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
1
1.2
1.5
1.8
(b)
Ri = 1,10,102,103
Analytical solution
Numerical solution
Nu
Figure 11. Flow intensity (a) and heat transfer rate (b) versus , for Re = 0.1 and various values of Ri.
a diminution with this parameter, at least for the selected
values of Ri.
It is clear
fluids lead to contradictory conclusions, depending on
uall
ven
flo
happens, which is paradoxical when nanoflds are ex-
pected to improve heat transfer. This can be related, es-
nductivity
and v
that the results related to heat transfer in nan- sentially, to the conflict between effective co
o
the flow nature, thus leaving still unanswered the ques-
tion if the use of nanoparticle suspensions for mixed con-
vection applications is acty advantageous with re-
spect to pure liquids. In fact, with dominant lid-dri
w, heat transfer enhances with nanoparticles, whereas
with dominant buoyancy-driven flow the opposite effect
iscosity with the complicity of the cavity aspect
ratio, which is large and favours the effect of viscosity
and disfavours that of conductivity.
5.5. Onset of the Bicellular Flow
In Figure 14 is depicted the evolution of c
Ri , corre-
ui
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL. 127
0.4
0.5
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.0
0.1
0.2
0.3
Ri = 1
(a)
Analytical solution
Numerical solution
Ri = 103
Ri = 102
Ri = 10
c
0.00 0.02 0.04 0.06 0.080.10 0.12 0.14 0.16 0.18 0.20
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Ri = 1
Ri = 102
Ri = 10(b)
Ri = 103
Analytical solution
Numerical solution
Nu
Figure 12. Flow intensity (a) and heat transfer rate (b) versus , for Re = 1 and various values of Ri.
sponding to the onset of buoyancy-driven flow, which
gives rise to bicellular flow, versus Re, for various values
of Φ. As can be seen, decreases and increases on
both sides
alue, , of Re suc, depending on
flow tends s precocious,
so
the presence of nanoparticles delays the onset of buoy-
ancy-driven flow and then opposes to the corresponding
effect, confirming the deterioration observed, efore, for
c
Ri
h that
to
of a minimum corresponding to a particular heat transfer associated with this kind of flow.
vm
Φ. This trend indicates that the appearance of the buoy-
ancy-driven
Re 15
m
Re
be sometime
metimes late around m
Re . In addition to that, the more
Φ increases the more c
Ri increases, which means that
6. Conclusions
In this paper, a numerical and analytical study on mixed
convection in a two-dimensional horizontal shallow
b
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL.
128
0.25
0.00 0.02 0.040.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.05
0.10
0.15
0.20
Ri = 1
(a)
Analytical solution
Numerical solution
c
Ri = 103
Ri = 102
Ri = 10
0.00 0.02 0.04 0.06 0.08 0.100.120.140.16 0.18 0.20
5
10
20
30
40
50
60
70
80
90
Ri = 10
Ri = 102
Ri = 1
(b)
Ri = 103
Analytical solution
Numerical solution
Nu
Figure 13. Flow intensity (a) and heat transfer rate (b) versus , for Re = 10 and various values of Ri.
enclosure, of aspect ratio , filled with a nanofluid,
has been conducted in the case where both short vertical
sides are submitted to uniform heat fluxes while the long
horizontal ones are assumed adiabatic, with the top one
uniformly moving in the opposite direction to heat flux.
The full partial differential equations, governing the
problem, ha
olume method. The computations, which have been lim-
respectively, in the ranges, 0
and .2
8A
ve been solved numerically using a finite
v
ited to Cu-water mixtures, with 7Pr , have been car-
ried out with governing parameters, Re, Ri and Φ, varying,
0.1 10Re 3
11Ri
00

s of a paralle
nclosure.
be summar
In the limit o
numerical ones
. Analytice
basi l flow assu
the eThe main findi
can ized as follows
f the selected th
, which
al solution is d
mption in the
ngs of such an
:
values of
validates mutually bot
rived on the
core region of
investigation
e governing
ell with the
h the cor-
parameters, analytical results, agree very w
responding approaches.
Flow and temperature fields strongly depend on the
Copyright © 2013 SciRes. JECTC
H. EL HARFI ET AL. 129
103
0.1 110
10
102
Analytical solution
Numerical solution
= 0.0
Re
Figure 14. Limit of the onset of the bicellular flow, for various values of .
Richardson number, measuring the relative impor-
tance of both lid and buoyancy-driven effects.
Increasing the Richardson number is, in general, as-
sociated with decreasing of heat transfer rate due to
shear flow and increasing of that due to buoyancy-
driven flow.
The addition of Cu-nanoparticles into the pure water
leads to an enhancement of lid-driven convection heat
transfer.
Against all odds, the addition of Cu-nanoparticles into
the pure water results in a degradation of buoyancy-
driven convection heat transfer. Therefore, although
prospects of nanofluids are very promising, there is
still a dearth of enough research in this area.
The onset of buoyancy-driven flow, giving rise to bi-
cellular flow, depends strongly on the Reynolds num-
ber and nanoparticles volume fraction.
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