Magneto-mixed convection in a lid driven cavity with nanofluid and rotating plate

Magneto-mixed convection in a lid driven partially
heated cavity equipped with nanofluid and rotating
flat plate
Mohammad Mokaddes Ali
a,*
, Rowsanara Akhter
b
, Md. Abdul Alim
c
a
Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail 1902, Bangladesh
b
Department of Electrical and Electronic Engineering, The International University of Scholars, Dhaka 1212, Bangladesh
c
Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh
Received 13 January 2021; revised 2 April 2021; accepted 1 May 2021
Available online 6 June 2021
KEYWORDS
Mixed convection;
Nanofluids;
Magnetic field;
Rotating flat plate;
Lid driven cavity;
Finite element method
Abstract In this study, mixed convection in a nanofluid filled cavity induced by thermal buoyancy
force, moving wall and rotating flat plate subjected to external magnetic field is numerically inves-
tigated. The cavity is partially heated from its bottom wall and cooled from top wall moving with
constant velocity in ±x direction and other walls are kept adiabatic. A counter-clockwise rotating
flat plate is placed at the centre of the cavity. The cavity is permeated by a transverse magnetic field.
Conservation equations are simulated through implementing finite element method. Numerical
results are presented using streamlines, isotherms and bar charts to explore the effects of physical
parameters on the flow and temperature fields. It is found that flow and thermal fields are impres-
sively affected with the variations in length and speed of rotating flat plate. Besides, higher length
and rotational speed of the plate causes maximum amount of heat transfer. Best heat transfer is
ensured while the direction of rotating plate is same as the direction of lid wall. Moreover, optimal
heat transfer performance is obtained up to 5% nanoparticles concentration which is 123.02% more
than base fluid. Higher magnetic field strength attenuates the fluid motion and hence heat transfer
rate significantly.
Ó2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria
University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/
licenses/by-nc-nd/4.0/).
1. Introduction
Among all the heat transfer mechanisms, mixed convection
heat transfer is immensely encountered in engineering applica-
tions such as heat exchangers, cooling of electronic and micro-
electronic equipment, nuclear reactors, solar collectors,
solidification and food processing, etc. Mixed convection is a
complex phenomenon in heat transfer processes which is acti-
vated due to the interaction of shear flow caused by moving
surface and thermal buoyancy flow. It has importance in
enhancing the flow mixing and heat transfer processes. Besides
this, improvement in heat transfer performance is an
indispensable issue due to the sophisticated developments in
*Corresponding author.
E-mail address: [email protected] (M.M. Ali).
Peer review under responsibility of Faculty of Engineering, Alexandria
University.
Alexandria Engineering Journal (2022) 61, 257–278
HOSTED BY
Alexandria University
Alexandria Engineering Journal
www.elsevier.com/locate/aej
www.sciencedirect.com
https://doi.org/10.1016/j.aej.2021.05.003
1110-0168 Ó2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

modern technologies. Scientists and researchers congregated to
develop innovative heat transfer fluids by amalgamating
nanometer-sized particles into conventional liquids named as
nanofluids. Nanofluids contain advanced properties like higher
thermal conductivity, stability and minimum clogging force,
etc. In this context, numerous studies have been conducted
on mixed convection in lid driven cavities with different ther-
mal conditions where nanofluids were either considered or
not. Some important studies have been presented here.
Basak et al. [1] studied lid driven mixed convection in
square cavity with different heating conditions at different
walls. In their study, Galerkin finite element with penalty
parameter was used to simulate non-linear governing equa-
tions and found that local and average Nusselt numbers at dif-
ferent walls are influenced by the heating conditions mentioned
as linearly heated, uniformly heated and cooled walls. They
also noted that variation in heat transfer depending on the val-
ues of Darcy number (Da) and Prandtl number (Pr) while Rey-
nolds number (Re) varies from 10 to 10
2
. Later on, Hassan and
Jamal [ 2] unitized finite volume method to investigate a similar
study [1] incorporating constant heat flux of different lengths
and demonstrated that convection effect was strengthened
with increase in Rayleigh number and temperature distribution
shows increasingly nonlinear behaviors for rising Richardson
number. After that, Ismael et al. [3] used USR finite difference
method to analyze the effect of partial slip on lid driven mixed
convection flow in a square cavity and reported that convec-
tion was declined at critical values of partial slip parameter.
D’Orazio et al. [4] implemented thermal Lattice Boltzmann
method to analyze mixed convection in an inclined rectangular
cavity with heat flux at the moving walls and observed that
heat transfer rate increases significantly for increasing cavity
inclination angle at higher Richardson number but it decreases
with the increase in Richardson number for a horizontal cav-
ity. Charttopadhyay et al. [5] simulated mixed convection in
a porous square cavity with moving vertical walls in presence
of sinusoidal thermal condition and they concluded that Darcy
number strongly affects the flow feature and increased ampli-
tude value causes enhancement in heat transfer rate. The prob-
lem of nanofluid mixed convection in a lid driven cavity with
sinusoidal temperature profiles was numerically investigated
by Arani et al. [6] using finite volume method with SIMPLER
algorithm and they suggested that heat transfer rate decreases
for increasing Richardson number, and increases for volume
fraction at constant Grashof number whereas it increases with
increase in Richardson number for constant Reynolds number.
Later on, Boutra et al. [7] developed a computer code based on
finite volume method to analyze mixed convection in a double
lid driven cavity with isothermal heating condition for two
types of nanofluids. In their study, average Nusselt number
was found in increasing function for both increasing nanopar-
ticle volume fraction and decreasing Richardson number. Kar-
eem et al. [8] used similar method and considered four types of
nanofluids to analyze mixed convection in lid driven trape-
zoidal cavity. They recommended that heat transfer rate
increases for volume fraction but decreases for increase in
diameter of nanoparticles. It was also noted that SiO
2
-water
nanofluid has best Nusselt number compared to others
Nomenclature
B0Magnitude of the applied magnetic field ðWbm
2Þ
cpSpecific heat at constant pressure ðJkg
1K1Þ
dLength of the rotating plate ðmÞ
Da Darcy number ðKL2Þ
gGravitational acceleration ðms
2Þ
Gr Grashof number ðgbfðThTcÞL3=mfafÞ
Ha Hartmann number ðB0Lffiffiffiffiffiffiffiffiffiffiffi
rf=lf
qÞ
LLength of the cavity ðmÞ
hLocal heat transfer coefficient ðWm
2K1Þ
kThermal conductivity ðWm
1K1Þ
Pr Prandtl number ðmf=afÞ
pDimensional pressure ðNm
2Þ
PDimensionless pressure
qwHeat flux ðWm
2Þ
Re Reynolds number ðU0L=mfÞ
Ri Richardson number ðGr =Re2Þ
Rs Rotating velocity ðms
1Þ
TDimensional temperature ðKÞ
u;vDimensional velocity components ðms
1Þ
U;VDimensionless velocity components
U0Lid velocity ðms
1Þ
x;yDimensional coordinates ðmÞ
X;YDimensionless coordinates
Greek symbols
aFluid thermal diffusivity ðm2s1Þ
bThermal expansion coefficient ðK1Þ
hDimensionless temperature
lDynamic viscosity ðNsm
2Þ
mKinematic viscosity ðm2s1Þ
qdensity ðkg m3Þ
XNon dimensional rotational velocity of cylinder
Subscripts
ffluid
hhot
ccold
nf nanofluid
Abbreviations
CW Clockwise
CCW Counter clockwise
CCWD Counter clockwise direction
FDM Finite difference method
FVM Finite volume method
FEM Finite element method
FP Flat plate
LW Lid driven wall
RFP Rotating Flat plate
SFP Stationary flat plate
258 M.M. Ali et al.

nanofluid containing Al
2
O
3
, TiO
2
and CuO. Esfe et al. [9] pre-
sented an in-depth review on nanofluid mixed convection flow
in different cavities with different conditions. They described
the effects of various geometries and key parameters on flow
and heat transfer mechanism to focus the role of mixed con-
vection in energy systems. The problem of mixed convection
in a rectangular cavity was studied by Ali et al. [10]. Their
results indicated that hybrid nanofluid and increased Richard-
son number cause augmentation in heat transfer rate but it
decreases for large cavity length.
Magnetic field associated with electrically conducting fluids
has received considerable attention due to its significance in
engineering applications, for instance, crystal growth in liq-
uids, purification of molten metal, electronic and microelec-
tronic devices, cooling of nuclear reactors and solar
collectors. In the case of mixed convection flow when magnetic
field effect is imposed, there are three body forces activated
simultaneously named as buoyancy force, shear force and Lor-
entz’s force. The interaction of these forces could affect the
fluid flow and heat transfer mechanisms. So, it is important
to analyze in details the transport phenomena of mixed con-
vection in the presence of magnetic field for better designing
in engineering equipment. Oztop et al. [11] analyzed magnetic
field effect on mixed convection flow in a lid driven cavity.
During simulation procedure, they used finite volume tech-
nique and found that reduction in heat transfer rate due to
Hartmann number is higher for high values of Grashof num-
ber. Al-Salem et al. [12] performed a similar numerical study
[11] while the cavity was linearly heated from its bottom wall.
In their study, heat transfer rate was decreased with increase in
Hartmann number for all parameters considered. Moreover,
direction of lid wall was found more effective in mixed convec-
tion dominating case than the case in forced convection.
Another numerical study for lid driven trapezoidal cavity
[13] demonstrated that local Nusselt number is maximum at
the edge and minimum at the center of the bottom wall. It
was also recorded that lid effect was negligible for higher Ray-
leigh number (Ra 10
5
). In addition, streamlines circulation
was observed stronger with increasing Rayleigh number and
hence convection becomes dominant inside the cavity. Togh-
raie [14] used finite volume method and Cu-water nanofluid
to extend the study of Ref. [13]. They observed that velocity
profiles were less perturbed for increased magnetic effect. Nus-
selt number also increased by adding nanopartices in base fluid
and it depends on the dimensionless parameters and tilted
angle studied. The problem of partially active magnetic field
on mixed convection in a lid driven cavity was numerical ana-
lyzed by Geridonmez and Oztop [15] utilizing pseudo spectral
method. They recommended that convective flow and heat
transfer were affected by the direction and length of the partial
magnetic field. Selimefendigil and Oztop [16] conducted a
numerical investigation of mixed convection nanofluid flow
in a lid driven cavity with flexible side wall and volumetric heat
generation in presence of magnetic field. They used Arbitrary-
Lagrangian-Eulerian method to describe fluid motion in the
fluid–structure interaction model. Their results suggested that
variation in absolute average heat transfer due to Richardson
number depends on values of Young’s modulus of elastic wall.
It was also found that local and average Nusselt number is
more effective at higher Richardson number for all volume
fractions. Bondareva et al. [17] utilized finite difference method
and heatline visualization technique to investigate nanofluid
free convection in a tilted open porous cavity with a corner
heater in presence of magnetic field. In their study, convective
flow and heat transfer were found attenuating with mutual
increase in magnetic field effect and its inclination angle. Sim-
ilar behavior was found for increasing Hartmann number and
cavity inclination angle. Later on, Astanina et al. [18] imple-
mented similar method to analyze combined natural convec-
tion and entropy generation in a nanofluid filled open
trapezoidal cavity having a porous layer and ferrofluid layer
in presence of magnetic field. They found a growth of oscilla-
tions amplitude in average Nusselt number and entropy gener-
ation for increasing Hartmann number. They also noticed
unstable phenomena in heat and fluid flow while magnetic
inclination angle was at a=p/2. After that, three dimensional
forced convection flow in a rectangular channel with a baffle
was numerically studied by Benzenine et al. [19] using finite
volume method with SIMPLE algorithm. They observed better
performance in heat transfer for using perforated baffle than
solid baffle at the lower wall of the channel studied. Thereafter,
Aich et al [20] performed a computational study to show the
effect of buoyancy force on air-flow and temperature pattern
in a three dimensional prismatic greenhouse with ventilation
process. The flow structure was found sensitive relating to
Rayleigh number and heat transfer enhancing with increasing
Rayleigh number. Sivasankaran et al. [21] numerically studied
heat and mass transfer of double diffusive mixed convection in
a lid driven cavity with non-uniform heating of the vertical
walls. They recommended that phase deviation and amplitude
affect the heat and mass transfer rate for all Richardson num-
ber. In addition, heat transfer rate was found increasing with
the amplitude of wall temperature, and heat and mass transfer
rate was increased more while both sidewalls were
non-uniformly heated compared to one sidewall was
non-uniformly heated. Abu-Hamdeh et al. [22] developed finite
volume technique based computer code to analyze mixed con-
vection in a lid driven cavity with one side opening wall filled
with porous media. They found complex behaviors of heat
transfer and flow field for lid and open side walls and also hea-
ter. They also noted that heat transfer rate enhances for Gra-
shof number and heater length but decreases for Darcy
number. Later on, Jakeer et al. [23] used Cattaneo-Christov
heat flux pattern to investigate lid driven mixed convection
in a hybrid nanofluid filled porous cavity in presence of mag-
netic field. In their study, local Nusselt number was found
decreasing for higher Ha whereas heat transfer rate was higher
with the increase in width of the obstacle. It was also showed
that heat transfer rate is better in hybrid nanofluid than nano-
fluid. The finite element solution [24] demonstrated that heat
transfer rate decreases by 30.66% when Hartmann number
varies from 0 to 50 for the cease of ferrofluid with concentra-
tion of 5%. In addition, isothermal distribution strongly
increased with the increase in corner heater length. Another
finite difference method based numerical analysis [25] recom-
mended that best heat transfer occurs at maximum amount
of sink power. Bakar et al. [26] used finite volume technique
to expose a significant effect of magnetic field on flow and tem-
perature field within a lid driven rectangular cavity. They also
indicated that both the flow convection and heat transfer rate
decline with increased Ha.
Insertion of separated abstraction of different shapes either
in stationary state or in rotation within closed or open enclo-
sures can have impact in controlling fluid motion and thermal
Magneto-mixed convection in a lid driven partially heated cavity 259

characteristics. Such configurations could be found in heat
exchangers, electric equipment, building designs, solar systems,
etc. Mustafizur et al. [27] conducted a numerical study on
mixed convection in a ventilated rectangular cavity containing
a heat conducting solid cylinder. They pointed out that flow
and thermal characteristics strongly depended on the values
of Richardson number and aspect ratio. Later on, Gupta
et al. [28] extended the study [27] for a square cavity and
observed that flow and heat transfer characteristics were influ-
enced significantly due to effects of governing parameters men-
tioned in their study. Afluq et al. [29] presented a numerical
model of mixed convection heat transfer in a vented lid driven
rectangular cavity with triangular obstacle on the top wall and
found 26% increment in Nusselt number for increasing height
ratio of the triangular with three blocks. Alsabery et al. [30]
considered Buongiorno’s two-phase model to simulate mixed
convection in a nanofluid filled enclosure with inner solid body
and horizontal moving walls. Their result indicated a notice-
able enhancement in heat transfer for nanofluid strategy in
the cavity. It was also revealed that a larger size of solid body
causes heat transfer augmentation at higher values of Richard-
son and Reynolds numbers. Boulahia et al. [31] utilized finite
volume discretization method and SIMPLE algorithm to ana-
lyze mixed convection flow of different nanofluids in a double
lid driven cavity in presence of triangular heaters. In their
study, the heat transfer rate was increased with the reduction
in nanoparticles diameter and Richardson number. Later on,
Mehmood et al. [32] incorporated magnetic field effect and
isothermally heated square hollow cylinder in a lid driven cav-
ity filled with nanofluid to investigate mixed convective flow
and heat transfer characteristics. They highlighted that stream-
lines strength and heat transfer rate decline with the increase in
magnetic field strength. It was also reflected that nanoparticle
volume fraction enhances average Nusselt number and kinetic
energy. Azizul et al. [33] conducted a numerical study of mixed
convection in a nanofluid filled wavy bottom cavity with a
solid body. They found better convective heat transfer and
fluid flow for high values of Rayleigh number. They also con-
firmed the maximum heat transfer for high Grashof number.
The problem of mixed convection heat transfer phenomena
over a rotating heated cylinder within a trapezoidal cavity
was numerically investigated by Khan et al. [34]. Their results
indicated that heat transfer characteristics were more affected
due to inclination of side walls and rotating cylinder compared
to motionless heated cylinder. After that, Khanafer et al. [35]
simulated mixed convection for two rotating cylinders within
a square cavity. In their study, heat transfer rate was found
strongly dependent on cylinder rotation speed, Reynolds and
Richardson numbers. They also showed that at higher
Richardson number (Ri = 10), heat transferring is indepen-
dent of cylinder rotation speed. Another simulated results
[36] demonstrated that 14.2% more heat transfer augmenta-
tion occurs for the case of rotating cylinder (X=10) com-
pared to motionless case (X= 0). It also found 17% more
heat transfer in natural convection dominant regime
(Ri = 10) than mixed convection dominant regime (Ri = 1).
Kimura et al. [37] first conducted an experimental study of
heat transport characteristics for a rotating plate within a ver-
tical cavity. They clarified the heat transfer rate depends
mostly on the ratio of Grashof number to the square of Rey-
nolds number (Gr=Re2), and used rotating plate in regulating
the heat transferring largely. Later on, Lee et al. [38] employed
implicit virtual boundary method to investigate mixed convec-
tion in an air filled square cavity having a rotating plate. They
observed thermal oscillations while Rayleigh number exceeds
0.55 10
6
, and heat transferring increases with rotor while
Rayleigh number was lower than 0.13 10
6
but a suppressing
occurred beyond this critical Rayleigh number.
Based on the open literature review, it is apparent that
nanofluid mixed convection flow in presence of magnetic field
is of great interest to the researchers and has applicability in
different fields of engineering sciences. Though, a lot of studies
have been conducted relating to mixed convection in different
geometries filled with nanofluids where either detached
obstruction was considered or not and either moving or stag-
nant. Moreover, different shapes of obstruction like square,
circular, triangular, rectangular etc are available in the litera-
ture but rotating flat plate within confined enclosure is rarely
used focusing on mixed convection flow. Additionally, in the
literature, we have come across different thermal conditions
like isothermal temperature, uniform or non-uniform temper-
ature, heat flux, etc. Among them partially heated thermal
condition has a noticeable impact on the flow and heat trans-
port characteristics. But there is no work found in the open lit-
erature survey where all these issues (mixed convection,
nanofluids, magnetic field, partially heated domain, moving
wall, and rotating flat plate) were considered in a combined
manner. Therefore, we attempted to investigate the flow and
thermal performance considering mixed convection in a par-
tially heated lid driven cavity equipped with nanofluid and
rotating flat plate in presence of transverse magnetic field. This
problem may be found while developing appropriate engineer-
ing equipment especially during mixing fluids and enhancing
convective heat transfer as well. The governing equations are
simulated by using a very popular mathematical technique
finite element method. The flow field is characterized by
streamlines while thermal field is characterized by isotherm
contours and bar charts of average Nusselt number. The pre-
sent study explores the effects of length and rotational speed
of flat plate, Hartmann number and nanoparticle volume frac-
tion on flow and thermal fields.
This study has been organized as: Section 1 represents
exclusive literature survey, Table of literature review (see
Table 1) and novelty of this investigation. Section 2 describes
the physical model. Mathematical model with appropriated
boundary conditions, numerical procedure, grid independency
test and validation of numerical procedure are outlined in
Section 3.Section 4 represents the detailed discussion based
on simulated results. Finally, the prediction has been summa-
rized in Section 5.
2. Physical model
The physical model of this investigation is presented in Fig. 1.
A square cavity of length L is considered whose top wall is
moving with constant velocity from either left to the right at
u=U
0
or right to the left at u = -U
0
. The cavity is cooled
from its top wall at temperature T
c
and partially heated at tem-
perature T
h
from its bottom wall covering 40% of its length
while other walls are maintained at adiabatic condition. A
counter clockwise rotating flat plate of length d is placed at
the center of the cavity. Rectangular co-ordinate system is
260 M.M. Ali et al.

Table 1 Summary of literature review relating to the present study.
Sl.
No.:
Researchers /
investigators
Methods Type of fluids Parametric ranges Geometry of
study
Physical model
[1] Basak et al. [1]
Galerkin
finite element
method
Different model
fluids,
(0.015 Pr 10)
10
-5
Da 10
-3
10
3
Gr 10
5
1Re 10
2
Lid driven
porous square
with linearly
heated side wall
(s)
[2] Hassan and
Jamal [ 2]
Finite volume
method Air
10
3
Ra 10
6
0.5 Ri 8
0.2 e0.8
Lid driven
square cavity
with constant
heat flux
[3] Ismael et al. [3]
USR finite
difference
method
water
0.01 Ri 100
0S20(1)
Lid driven
square cavity
with partial slip
condition
[4] Orazio et al. [4]
Lattice
Boltzmann
method
Air
0.01 Ri 10
0
0
c90
0
Re = 200
Inclined lid
driven
rectangular
cavity with heat
flux boundary
condition
[5] Charttopadhyay
et al. [5]
Compact
finite
difference
scheme
Air
10
-5
Da 10
-1
10 Re 100
0.1 Ri 100
Double lid
driven
sinusoidally
heated porous
square cavity
[6] Arani et al. [6]
Finite volume
method and
SIMPLER
algorithm
Cu-water nanofluid
0.0001 Ri 10
0
0
c90
0
Lid driven
square cavity
with sinusoidal
heating walls
[7] Boutra et al. [7]
Computer
code based on
Finite volume
method
Cu-water
nanofluids,
Ag-water nanofluids
0.1 Ri 100
0.0 /0.10 Lid driven
square cavity
[8] Kareem et al. [8] Finite volume
method
Al
2
O
3,
CuO, SiO
2
,
TiO
2
-water
nanofluids
0.1 Ri 10
100 Re 1200
30U60
1%/4%
0.5 A2
Inclined lid
driven
trapezoidal
enclosure
(continued on next page)
Magneto-mixed convection in a lid driven partially heated cavity 261