KING FAH

HD UNIVERSITY
OF
PE
ETROLEUM
M AND MIINERALS
Meechanical Enngineering Department
D

Mixeed-Conveection Fllow and Heat

H Traansfer in a Nanoflluid-Filleed
Lid--Driven Cavity
C w an Ellliptic Moving Lid
with d

Term
m Projecct
In
Adv
vanced Fluid
F Mechanics I Course (ME-532))
Sem
mester 1722

Sub
bmitted too

Dr. Rasheed Ben Maansour

Abstract

In this study, numerical simulation of mixed-convection flow and heat transfer inside a lid-
driven cavity is made using the finite volume method. The moving lid of the cavity has an
elliptical profile and its linear speed is constant. The cavity is filled with a water-Al2O3
nanofluid. The effects of different three parameters on the flow and heat transfer inside the cavity
are investigated. These parameters are: the aspect ratio of the elliptical moving lid, the
Richardson number, and the volume fraction of the nanoparticles. The Grashof number is kept as
constant at 2500 with changing only the Reynolds number. The results reveal that increasing the
aspect ratio of the moving lid enhances the flow circulation inside the cavity, and the same effect
is also found with decreasing the value of Richardson number. Regarding the effects on heat
transfer, the results also show that an enhancement in the average Nusselt number of the cavity is
achieved with increasing the aspect ratio of the moving lid, decreasing the value of Richardson
number, or increasing the volume fraction of the nanoparticles.

Keywords: Elliptical moving lid, Heat transfer, Lid-driven cavity, Mixed convection, Nanofluids

1. Introduction

The flow in a lid-driven cavity is the motion of a fluid inside a square or rectangular cavity
created by a constant translational velocity of one side of the cavity (called the lid) while the
other sides remain stationary. For many decades, the classical problem of the flow and heat
transfer inside a lid-driven cavity has been an interesting subject of various computational and
experimental studies. In such problems, it is now common to use nanofluids in order to achieve
more heat transfer enhancement.

Mixed-convection flow and heat transfer inside lid-driven cavities are important in many
industrial processing and technical applications, such as: Flexible blade coaters; material
processing, dynamics of lakes, metal casting and galvanizing, cooling of electronic equipment,
food processing, lubrication technologies [10, 12].

With the continuous development of the numerical methods, this problem has been studied
extensively by several authors through different studies. Over the past few years, many
researchers have dedicated their efforts trying to investigate the characteristics of the flow

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structures and the heat transfer in different geometrical cavities with two walls at different
temperatures [5, 8, 9, 13].

Low thermal conductivity of conventional fluid used inside cavities was a strong motivation to
develop advanced heat transfer fluid with a substantially higher thermal conductivity. These
fluids (called nanofluids) are a mixture of traditional fluids and fraction of nanoparticles which
have a unique physical and chemical characteristic. Tiwari et al. [14] proved through numerical
modeling of mixed convection in two sided square cavity that the nanoparticles have the
capability to increase the heat capacity of the working fluid. Aminossadati et al. [2] conducted a
numerical study describing the achievement of adding the nanoparticles on the natural
convection cooling of a heat source located at the bottom of a square cavity. A decreased heat
source of maximum temperature was noticed upon increasing the volume fraction of the
nanoparticles, especially at low Rayleigh numbers.

Nasrin et al. [11] showed through numerical solution, using the finite element technique, that
increasing the volume fraction of the nanoparticles has an impact on increasing the flow strength
inside the cavity. Furthermore, this will enhance the thermal conductivity and will, in turn,
increase the average rate of heat transfer. Boutra et al. [4] confirmed through a numerical study
for a mixed convection in a square cavity the enhancement of heat transfer by increasing the
concentration of the nanoparticles in the enclosure, however heat transfer was found to be a
decreasing function of Richardson number.

Ben Mansour and Habib [3] have developed a computational model illustrating the performance
of the nanoparticles inside a cavity by studying the effect of the aspect ratio of the cavity and the
volume fraction of the nanoparticles on the flow pattern and on the heat transfer rate. Their
results emphasized a linear relationship between the average Nusselt number and the volume
fraction of the nanoparticles. The rate of heat transfer was increased as well as the volume
fraction increase from 2 to 10 % due to the high energy transport associated with the irregular
motion of the fine particles. Hemmat et al. [7] investigated through a numerical modeling, using
the finite element method, the thermal behavior and flow characteristics of water-Al2O3
nanofluid inside a square cavity. The results show that increasing Richardson number and the
nanoparticles diameter causes a decrement in the average Nusselt number. A minor change in the

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flow pattern and isotherms was observed by varying the nanoparticles diameter from 20 to 80 nm
while keeping Richardson number and the volume fraction constant.

In the current work, the finite volume method has been used through the well-known Fluent
Software for the simulation of the mixed-convection flow and heat transfer inside a lid-driven
cavity. The moving lid of the cavity has an elliptical profile and its linear speed is constant. It is
worth pointing out that cavities with an elliptical moving lid haven’t been studied before in this
area of research. The cavity here is filled with a water-Al2O3 nanofluid. The effects of different
three parameters on the flow and heat transfer inside the cavity have been investigated. These
parameters are: the aspect ratio of the elliptical moving lid, the Richardson number, and the
volume fraction of the nanoparticles.

2. Problem formulation

2.1. Physical model

Figure 1 shows a schematic view of the

geometry of the studied cavity and the
coordinate system. The cavity here is mainly a
square cavity of side length L with an elliptic
moving lid whose major and minor axes are 2b
and 2a, respectively, where 2a = L. All the
cavity walls are rigid, impermeable, and
motionless apart from the top one which
moves in its own plane with a constant total
linear speed of . The left and right vertical
walls are kept insulated, while the moving lid
and the bottom wall are at constant
temperatures Th and Tc, respectively, where Th
> Tc. The fluid in the cavity is Al2O3-water Figure 1. Schematic view of the lid-driven cavity with its
boundary conditions and coordinate system
nanofluid, and the gravitational force acts in
the vertically downward direction.

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2.2. Assumptions

The main assumptions made here are those commonly used, which are [6]:

 The base fluid and the nanoparticles are assumed to be in thermal equilibrium and they
all flow at the same velocity (i.e., no slip occurs between them);
 The nanoparticles are spherical;
 The nanofluid is Newtonian and incompressible;
 The flow is two-dimensional, laminar, and steady;
 For mathematical modeling, the nanoparticles are assumed to be uniformly dispersed
within the base fluid (i.e., the nanofluid fluid is assumed to be of single phase
continuum);
 The radiation heat transfer between the sides of the cavity is negligible when compared
with the other modes of heat transfer; and
 The viscous dissipation is assumed to be negligible.

2.3. Governing equations

According the prescribed assumptions above, the governing equations in dimensional form used
in the present study are:

Continuity equation:

0 (1)

x-momentum equation:

1
(2)

y-momentum equation:

1
(3)

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Energy equation:

(4)

The boundary conditions that correspond to the above equations are:

1. At the left and right walls: 0, and ⁄ 0;
2. At the bottom wall: 0, and ; and
3. At the top moving lid: , , , , and

Equations (1) to (4) can be converted to a non-dimensional form using the following
dimensionless parameters:

∗ ∗ ∗
; ; ; ; ;

After substituting the above dimensionless parameters into Equations (1) to (4), we get the
following non-dimensional equations:

Continuity equation:

∗ ∗
0 (5)

x-momentum equation:

∗ ∗ ∗ ∗ ∗
∗ ∗
1
(6)
Re

y-momentum equation:

∗ ∗ ∗ ∗ ∗
∗ ∗
1
Ri (7)
Re

Energy equation:

∗ ∗
1
(8)
Re. Pr

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where,

Gr
Ri ; Re ; Gr
Re

The corresponding dimensionless boundary conditions then take the following form:

∗ ∗ ⁄
1. At the left and right walls: 0, and 0;
∗ ∗
2. At the bottom wall: 0, and 0; and
∗ ∗ ∗ ∗
3. At the top moving lid: , , , , and 1

2.4. Properties of the nanofluid

The thermo-physical properties of water (the base fluid) and the solid nanoparticles at 25 ºC are
given in Table 1. Constant thermo-physical properties are considered for the nanofluid, whereas
the density variation in the buoyancy forces is determined using the Boussinesq approximation.

Table 1. Thermo-physical properties of water and nanoparticles at 25 ºC [10]

Thermo-physical properties Fluid (Water) Solid (Al2O3)

cp (J/kg-K) 4179 765
 (kg/m3) 997.1 3970
k (W/m-K) 0.613 25
β x 10-5 (1/K) 21 0.85
µ x 10-4 (kg/m-s) 8.9 ---

The effective density, the heat capacity, the thermal expansion coefficient, and the thermal
diffusivity of the nanofluid are defined, respectively, as follows [10]:

1 (9)

1 (10)

1 (11)

(12)

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The effective viscosity of the nanofluid is determined according to Brinkman model as [10]:

⁄ 1 . (13)

Regarding the thermal conductivity of the nanofluid, it is determined according to Maxwell–

Garnett’s (MG model) self-consistent approximation model [10]. That is:

2 2
(14)
2

2.5. Solution procedure

A user-defined-function is first made to model the motion of the elliptical moving lid of the
cavity. All the governing equations above are then solved using the Fluent Software that is based
on the finite volume method. After reaching a convergence of 10-6 for the continuity and
momentum equations, and a convergence of 10-10 for the energy equation, the spatial velocities
and temperatures inside the cavity are recorded.

The heat transfer coefficient of the cavity ( ) is then obtained by:

(15)

where, is the heat flux leaving the cavity and it is obtained from Fluent.

The Nusselt number of the cavity (Nu ) is then obtained by:

Nu (16)

3. Grid independency study

An extensive grid testing study has been conducted in order to guarantee a grid-independent
solution. In this study, Gr and Re numbers were set at 100 and 500, respectively, and the fluid
was considered as a clear water. Moreover, the aspect ratio of the elliptical moving lid ( ⁄ ) was
set at 0.2.

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 Five different grids were considered in this grid testing study, which are: 21 x 21, 41 x 41,
61 x 61, 81 x 81, and 101 x 101. Figure (2) shows the variation of the dimensionless x-velocity at
the mid section of the cavity along the vertical direction. As shown, it can be confirmed that the
81 x 81 grid ensures a grid-independent solution and there is no need to use the 101 x 101 grid to
eliminate the computation time. Consequently, all the work here was made using the 81 x 81 grid
and it is shown in Fig. (3) for an aspect ratio ( ⁄ ) of the elliptical moving lid equal to 0.2.

 
Figure 2. Variation of the dimensionless x-velocity at the mid section of the cavity ( ) along the vertical
direction for five different grids for clear water with Gr = 100 and Re = 500. Here, ⁄

Figure 3. Selected mesh for the current work (81 x 81 grid) – In this figure: b/a = 0.2
 

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4. Model validation

The current numerical solution has been validated by comparing the present code results with
the numerical results obtained by Abu-Nada et al. [1], whose work was on a square cavity with a
flat moving lid (i.e., 0 here) as shown in Fig. (4). This work was chosen for the validation
here because there is no information found in the literature about cavities with elliptical moving
lid.

Figure 4. Schematic view of the lid-driven cavity used for validation with
its boundary conditions and coordinate system

Figure (5) compares between the variation of the dimensionless x-velocity at the mid section
of the cavity along the vertical direction obtained by the current numerical model and that
obtained by Abu-Nada et al. [1]. As shown, there is a very good agreement between the two
results with a percentage of difference less than 2.5%.

5. Results

As introduced, the current work adopts the finite volume method for the simulation of the
mixed-convection flow and heat transfer inside a lid-driven cavity filled with a water-Al2O3
nanofluid and its moving lid has an elliptical profile. In this section, the effects of different three
parameters on the flow and heat transfer inside the cavity are investigated. These parameters are:
the aspect ratio of the elliptical moving lid ( ⁄ ), the Richardson number (Ri), and the volume

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fraction of the nanoparticles ( ). The values of ⁄ considered here are: 0 (flat lid), 0.2, 0.4, and
0.6. The values of Ri applied here are: 0.01, 0.1, 10, and 100, with changing only the Reynolds
number (Re) and keeping the Grashof number (Gr) as constant at 2.5 x 103. Finally, the values of
used in this study are: 0 (clear water), 2, 4, and 6%.

 
Figure 5. Variation of the dimensionless x-velocity at the mid section of the cavity ( . ) along the vertical
direction for flat moving lid for clear water with Gr = 100 and Re = 400. Here, ⁄

5.1. Effects of the aspect ratio of the elliptical moving lid

Figure (6) shows the variation of the dimensionless x-velocity at the mid section of the cavity
along the vertical direction for different values of b/a with φ = 4 % at Ri = 0.01 and at Ri = 10.
The corresponding contours of the dimensionless velocity and temperatures are shown in Figs.
(A.1) and (A.2) and Figs. (B.1) and (B.2), respectively. As shown, it can be noted that increasing
the value of b/a increases the momentum transfer to the fluid inside the cavity and hence
improves the flow circulation inside the cavity, which in turn enhances the heat transfer inside
the cavity. This effect of b/a value decreases as the value of Ri increases, where the buoyancy
effects dominate.

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 (a)

(b)

Figure 6. Variation of the dimensionless x-velocity at the mid section of the cavity ( ) along the vertical
direction for different values of b/a with φ = 4 %: (a)- Ri = 0.01 and (b)- Ri = 10

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5.2. Effects of the Richardson number

Figure (7) shows the variation of the dimensionless x-velocity at the mid section of the cavity
along the vertical direction for different values of Ri with b/a = 0.4 at φ = 0 % and at φ = 4 %.
The corresponding contours of the dimensionless velocity and temperatures are shown in Figs.
(A.3) and (A.4) and Figs. (B.3) and (B.3), respectively. As expected, the results indicate that
increasing the value of Ri leads to decreasing the momentum transfer to the fluid inside the
cavity and hence decreasing the flow circulation inside the cavity, which in turn reduces the heat
transfer inside the cavity.

5.3. Effects of the volume fraction of the nanoparticles

Figure (8) shows the variation of the dimensionless x-velocity at the mid section of the cavity
along the vertical direction for different values of φ with Ri = 0.01 at b/a = 0 and at b/a = 0.4.
The corresponding contours of the dimensionless velocity and temperatures are shown in Figs.
(A.5) and (A.6) and Figs. (B.5) and (B.6), respectively. As shown, changing the value of φ has
unnoticeable effect on the flow behavior inside the cavity, but however, this change has a
considerable effect on the heat transferred through the cavity since the addition of the
nanoparticles into the fluid increases its thermal conductivity.

5.4. Summarized results in terms of Nusselt number

The combined effects of the different three parameters considered in the current study (b/a,
Ri, and φ) can be represented in terms of the average Nusselt number of the cavity whose values
are given in Table (2). As expected, it was found that the value of Nu increases by increasing the
value of b/a or increasing the value of φ. While increasing the value of Ri leads to decreasing the
value of Nu.

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 (a)

(b)

Figure 7. Variation of the dimensionless x-velocity at the mid section of the cavity ( ) along the vertical
direction for different values of Ri with b/a = 0.4: (a)- φ = 0 % and (b)- φ = 4 %

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(a)

(b)

Figure 8. Variation of the dimensionless x-velocity at the mid section of the cavity ( ) along the vertical
direction for different values of φ with Ri = 0.01: (a)- b/a = 0 and (b)- b/a = 0.4

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 Table 2. Values of the average Nusselt number of the cavity for the different cases studied
in the present work

b/a
Ri φ [%]
0.0 0.2 0.4 0.6
4.051 4.502 4.839 5.007 0
4.204 4.671 5.021 5.194 2
0.01
4.356 4.839 5.201 5.380 4
4.507 5.007 5.381 5.568 6
2.094 2.371 2.637 2.886 0
2.185 2.470 2.743 2.997 2
0.1
2.276 2.569 2.849 3.110 4
2.367 2.668 2.955 3.224 6
--- --- --- --- 0
--- --- --- --- 2
10
--- --- --- 1.001 4
--- --- --- 1.018 6
--- --- --- --- 0
--- --- --- --- 2
100
--- --- --- --- 4
--- --- --- --- 6
 

6. Conclusion

In this work, mixed-convection flow and heat transfer inside a lid-driven cavity has been
simulated numerically using the finite volume method. The moving lid of the cavity was with
elliptical profile and its linear speed was constant. The cavity was filled with a water-Al2O3
nanofluid. The effects of different three parameters on the flow and heat transfer inside the cavity
have been investigated. These parameters were: the aspect ratio of the elliptical moving lid
(values from 0 to 0.6 have been considered), the Richardson number (values from 0.01 to 100
have been applied), and the volume fraction of the nanoparticles (values from 0 to 6 % have been
studied). The Grashof number was kept as constant at 2500 with changing only the Reynolds
number. It has been found that increasing the aspect ratio of the moving lid enhances the flow

Page 15 of 30 
 
circulation inside the cavity, and the same effect was found with decreasing the value of
Richardson number. Regarding the effects on heat transfer, an enhancement in the average
Nusselt number of the cavity has been recorded with increasing the aspect ratio of the moving
lid, decreasing the value of Richardson number, or increasing the volume fraction of the
nanoparticles.

7. Recommendations for future work

The following are the points recommended for future studies on the mixed-convection flow and
heat transfer inside lid-driven cavities with an elliptical moving lid:

i. Rectangular cavities need to be added to this study in the future to investigate the
effects of the cavity shape on the flow and heat transfer inside the cavity together
with the effects of other three parameters studied here.
ii. Other types of nanofluids and more different values of Richardson numbers should be
also considered.
iii. The effects of the inclination angle of the cavity should be studied.
iv. The flow and heat transfer on an obstacle inside the cavity should be investigated.
v. Different thermal boundary conditions at the different sides of the cavity should be
studied.
vi. Higher values of Grashof number and radiation effects should be studied.

References

[1] Abu-Nada, Eiyad, and Ali J. Chamkha. "Mixed convection flow in a lid-driven inclined square
enclosure filled with a nanofluid." European Journal of Mechanics-B/Fluids, vol. 29 no. .6, pp.: 472-
482, 2010.
[2] Aminossadati S. M., and B. Ghasemi. “Natural convection cooling of a localised heat source at the
bottom of a nanofluid-filled enclosure,” Eur. J. Mech. B/Fluids, vol. 28, no. 5, pp. 630–640, 2009.
[3] Ben-Mansour R., and M. A. Habib. “Use of nanofluids for improved natural cooling of discretely
heated cavities.” Advances in Mechanical Engineering, vol. 5, 2013.
[4] Boutra A., K. Ragui, N. Labsi, and Y. K. Benkahla. “Lid-driven and inclined square cavity filled with
a nanofluid: Optimum heat transfer.” Open Engineering, vol. 5, no. 1, pp. 248–255, 2015.
[5] Chang T. S., and Y. L. Tsay. “Natural convection heat transfer in a enclosure with a heated backward
step,” International Journal of Heat and Mass Transfer, vol. 44, no. 20, pp. 3963–3971, 2001.

Page 16 of 30 
 
[6] Elharfi, Hassan, et al. "Mixed convection heat transfer for nanofluids in a lid-driven shallow
rectangular cavity uniformly heated and cooled from the vertical sides: The cooperative case." ISRN
Thermodynamics 2012.
[7] Hemmat Esfe M., M. Akbari, and A. Karimipour. “Mixed convection in a lid-driven cavity with an
inside hot obstacle filled by an Al2O3–water nanofluid.” Journal of Applied Mechanics and
Technical Physics, vol. 56, no. 3, pp. 443–453, 2015.
[8] Hsu T., and S. Wang. “Mixed convection of micropolarfluids in a cavity.” International Journal of
Heat and Mass Transfer, vol. 43, pp. 1563–1572, 2000.
[9] Liaqat A., and A. C. Baytas. “Conjugate natural convection in a square enclosure containing
volumetric sources.” International Journal of Heat and Mass Transfer, vol. 44, no. 17, pp. 3273–
3280, 2001.
[10] Mohammad Hemmat, et al. "Mixed-convection flow and heat transfer in an inclined cavity equipped
to a hot obstacle using nanofluids considering temperature-dependent properties." International
Journal of Heat and Mass Transfer, vol. 85, pp. 656-666, 2015.
[11] Nasrin R., M. A. Alim, and A. J. Chamkha. “Buoyancy-driven heat transfer of water–Al2O3
nanofluid in a closed chamber: Effects of solid volume fraction, Prandtl number and aspect ratio,”
International Journal of Heat and Mass Transfer, vol. 55, no. 25–26, pp. 7355–7365, 2012.
[12] Omari, Reyad. "CFD simulations of lid driven cavity flow at moderate Reynolds number." European
Scientific Journal, ESJ 9.15 (2013).
[13] Oztop, H.F., and I. Dagtekin. “Naturalconvection heat transfer by heated partition within enclosure.”
International Communication of Heat Mass Transfer, vol. 28, no. 6, pp. 823–834, 2001.
[14] Tiwari R. K., and M. K. Das. “Heat transfer augmentation in a two-sided lid-driven differentially
heated square cavity utilizing nanofluids,” International Journal of Heat and Mass Transfer, vol. 50,
no. 9–10, pp. 2002–2018, 2007.

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 APPENDIX A

Dimensionless Velocity Contours

Figure A.1. Contours of the dimensionless velocity of the flow inside the cavity for different values of b/a with
φ = 4 % and Ri = 0.01: (a)- b/a = 0, (b)- b/a = 0.2, (c)- b/a = 0.4,and (d)- b/a = 0.6

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Figure A.2. Contours of the dimensionless velocity of the flow inside the cavity for different values of b/a with
φ = 4 % and Ri = 10: (a)- b/a = 0, (b)- b/a = 0.2, (c)- b/a = 0.4,and (d)- b/a = 0.6

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Figure A.3. Contours of the dimensionless velocity of the flow inside the cavity for different values of Ri with
φ = 0 % and b/a = 0.4: (a)- Ri = 0.01, (b)- Ri = 0.1, (c)- Ri = 10,and (d)- Ri = 100

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Figure A.4. Contours of the dimensionless velocity of the flow inside the cavity for different values of Ri with
φ = 4 % and b/a = 0.4: (a)- Ri = 0.01, (b)- Ri = 0.1, (c)- Ri = 10,and (d)- Ri = 100

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Figure A.5. Contours of the dimensionless velocity of the flow inside the cavity for different values of φ with
Ri = 0.01and b/a = 0: (a)- φ = 0 %, (b)- φ = 2 %, (c)- φ = 4 %,and (d)- φ = 6 %

Page 22 of 30 
 
 

Figure A.6. Contours of the dimensionless velocity of the flow inside the cavity for different values of φ with
Ri = 0.01 and b/a = 0.4: (a)- φ = 0 %, (b)- φ = 2 %, (c)- φ = 4 %,and (d)- φ = 6 %

Page 23 of 30 
 
 APPENDIX B

Dimensionless Temperature Contours

Figure B.1. Contours of the dimensionless temperature of the flow inside the cavity for different values of b/a with
φ = 4 % and Ri = 0.01: (a)- b/a = 0, (b)- b/a = 0.2, (c)- b/a = 0.4,and (d)- b/a = 0.6

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Figure B.2. Contours of the dimensionless temperature of the flow inside the cavity for different values of b/a with
φ = 4 % and Ri = 10: (a)- b/a = 0, (b)- b/a = 0.2, (c)- b/a = 0.4,and (d)- b/a = 0.6

Page 25 of 30 
 
  

Figure B.3. Contours of the dimensionless temperature of the flow inside the cavity for different values of Ri with
φ = 0 % and b/a = 0.4: (a)- Ri = 0.01, (b)- Ri = 0.1, (c)- Ri = 10,and (d)- Ri = 100

Page 26 of 30 
 
 

Figure B.4. Contours of the dimensionless temperature of the flow inside the cavity for different values of Ri with
φ = 4 % and b/a = 0.4: (a)- Ri = 0.01, (b)- Ri = 0.1, (c)- Ri = 10,and (d)- Ri = 100

Page 27 of 30 
 
  

Figure B.5. Contours of the dimensionless temperature of the flow inside the cavity for different values of φ with
Ri = 0.01 and b/a = 0: (a)- φ = 0 %, (b)- φ = 2 %, (c)- φ = 4 %,and (d)- φ = 6 %

Page 28 of 30 
 
 

Figure B.6. Contours of the dimensionless temperature of the flow inside the cavity for different values of φ with
Ri = 0.01 and b/a = 0.4: (a)- φ = 0 %, (b)- φ = 2 %, (c)- φ = 4 %,and (d)- φ = 6 %

Page 29 of 30