Journal of Molecular Liquids 382 (2023) 122003

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Journal of Molecular Liquids

journal homepage: www.elsevier.com/locate/molliq

Computational analysis of natural convection with water based nanofluid in

a square cavity with partially active side walls: Applications to
thermal storage
Shan Ali Khan a, Sumeira Yasmin a, Muhammad Imran a, Taseer Muhammad b,
Abdullah Alhushaybari c, Umar Farooq d, Hassan Waqas e, *
a
Department of Mathematics, Government College University Faisalabad, 38000, Pakistan
b
Department of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi Arabia
c
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
d
School of Mechanical & Manufacturing Engineering, National University of Sciences & Technology (NUST), Islamabad 44000, Pakistan
e
School of Energy and Power Engineering, Jiangsu University, Zhenjiang 212013, China

A R T I C L E I N F O A B S T R A C T

Keywords: Heat transport induced by buoyancy caused natural convection inside a segmented or selectively heated
TiO2-water nanofluid enclosure has grown in importance over the decades due to its significance in many fields in industrial imple­
Natural convection mentations such as heat source cooling, food sterilizers, crystal growth, geothermal power systems, solar thermal
Square cavity
collectors, melting and recrystallization procedures, microelectronic and nuclear industries, biomedical, and so
Partially active walls
Numerical analysis
forth. The novelty of current modern analysis is to scrutinize the natural convection of Titanium Oxide-water
Finite element method nanofluid in the cavity with partially active walls simulated by the finite element (FEM) method. The fluid
flow and the thermal transportation within the enclosure are scrutinized. Two cases are carried out in this novel
study, heat sink with constant temperature Tc consists in case 1 from the left and right partially active cavity
walls and isothermal heat source having temperature Th correspondingly, where(Th > Tc ). While upper and
lower walls with inactive portions of the left and right walls are kept insulated. In case 2 the left and right walls
of the cavity are heated while a partially active bottom wall is cold with other inactive parts insulated by the
cavity walls. The finite element method is applied to tackle the governing equations. The Significance of Rayleigh
number (100⩽Re⩽1e6) and volume fractions of nanopowders (0.01⩽ϕ⩽0.06) against fluid flow and heat transfer
is demonstrated through streamlines and isotherms. From the results it is concluded that velocity amount is
increased with escalating the values of Rayleigh number. The thermal results are enhanced with Rayleigh
number. Furthermore the heat transfer is rised with volume fraction of nanoparticles. Additionally, from the
analysis reveals that local Nusselt number is enhanced with Rayleigh number and nanoparticle volume fraction.

employed in microsystems and micro-devices. However, current nano­

1. Introduction technology allows for the production of nanomaterials with an average
size of 100 nm or less. These nanoparticles can be added in regular heat
Natural convection has become one of the essential aspects of transfer fluids, ethylene–glycol (EG), and oil, resulting in a novel type of
research in the last two decays due to its wide-range uses. These uses heat transfer fluid known as nanofluids [2]. Sheikholeslami & Vajravelu
include the crystallization process, cooling systems, oil extraction, solar [3] discussed the nanofluid behavior through a cavity in the existence of
thermal collectors, and other industry and economic applications. In the the variable electro-magnetic field. Shah et al. [4] analyzed the cavity
field of modern thermal sciences, researchers and engineers are striving behavior filled by a nanofluid by adopting a neural network.
to find novel approaches to meet industrial demands. Maxwell’s concept The fluid flow and heat transmission studies in cavities have emerged
of suspending metallic millimeters or µm in diameter particles in the as significant fields of research in heat storage technologies. As a result,
fluid to increase heat conductivity is well-known [1]. However, due to there has been a huge growth in convective heat transfer research ac­
major issues such as abrasion and clogging, such nanoparticles cannot be tivity. Natural convection flow is caused by fluid motion that is not

* Corresponding author.
E-mail address: hassan@ujs.edu.cn (H. Waqas).

https://doi.org/10.1016/j.molliq.2023.122003
Received 8 January 2023; Received in revised form 14 April 2023; Accepted 1 May 2023
Available online 6 May 2023
0167-7322/© 2023 Elsevier B.V. All rights reserved.
S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Mehmood et al. [10].

Nomenclature Traditional energy transfer fluids, such as water, have poor thermal
conductivity, which has limited designers. Fluids including nano­
u, v Components of velocity (ms− 1 ) structure solid particles may provide a solution to this issue. The effi­
ρnf nanofluid density (kgm− 3 ) cient heat efficiency of the nanofluid is better than that of the working
p Pressure (Pa) host fluid. Modern engineering questions such as cooling applications,
g Gravitational acceleration(ms− 2 ) geothermal energy extraction, cooling of electronics, solar concentrator,
nanofluids in fuels, computer chips cooling, nano drug delivery, cancer
αnf Thermal diffusivity (m2 s− 1 )
therapeutic applications, sensing and imaging, and others requirement
θ Dimensionless temperature
an essential exploration of convective processes affected by this category
H Enclosure height (m)
of nanofluid flows. Alsaedi et al. [11] analyzed the hybrid nanofluid
cp specific heat (kJkg− 1 K− 1 )
flow through concentric cylinders. The obtained results concluded that
Pr Prandtl number (-)
increment in magnetic and Brinkman parameters to increase in hybrid
Nuavg Average Nusselt number (-)
nanofluid temperature. The 3D flow of Powell-Eyring nanofluid flow
U* , V * Dimensionless velocity components with activation was disclosed by Muhammad et al. [12]. There results
μnf Dynamic viscosity of nanofluid (kgm− 1 s− 1 ) concluded that temperature distribution is improved with higher ther­
P* Dimensionless pressure mal radiation parameter. Mahanthesh [13] scrutinized the importance
β Coefficient of thermal expansion (K− 1 ) of radiative heat flux on flow as well as heat transfer of nanomaterial.
T Temperature (K) The outcomes of their analysis showed that temperature is enhanced
h Heat transfer coefficient with nanoparticles and heat flux at plate is diminish. Mahanthesh et al.
L Length of cavity (m) [14] introduced the consequence of thermal radiation on Maxwell
ϕ Nanoparticles volume fraction (-) nanofluid. It was analyzed as the radiation parameter rises, the tem­
Ra Rayleigh number (-) perature field is upsurges. Gireesha et al. [15] scrutinized the non-
k Thermal conductivity (Wm− 1 K− 1 ) uniform heat source/sink behavior in dual phase flow of rate type
fluid. Krupalakshmi et al. [16] studied the impact of magnetic field on
Maxwell dusty fluid flow over stretching sheet. Ghahremanian et al.

Fig. 1(a-b. ). physical description of natural convection in Cavity together with (a) Case 1: Left and Right partially active walls are cold and heated, inactive portions
are adiabatic (b) Case 2: sided partially active walls are heated; partially active portion of lower walls is cold while the portions which inactive of the whole cavity are
kept adiabatic.

induced by any external source. Such a mechanism can be found in a [17] disclosed the nanofluidic flow on nanochannel. Mosavi et al. [18]
range of engineering fields, with cooling implementations, confinement scrutinized the Ar/Cu nanofluid flow. Shang et al. [19] examined H2O/
building cooling, room ventilation, cooling systems, reservoirs, solar CuO nanofluid with dynamic simulation. Aman et al. [20] investigated
collectors, etc. Because of nanofluids’ outstanding and adjustable the carbon nanotubes Maxwell nanofluids.
characteristics, such as greater thermal efficiency, clogging in flow The microelectronic devices cooling have been a critical problem in
passages due to tiny size and shape and very high specific surface areas recent decades as a result of technological advances, attracting the
of submicron particles, many researchers have been interested in using attention of scientists. Many recent researches have demonstrated that
them in natural convection flow. Islam et al. [5] analyzed the natural employing nanofluids, which have good thermal stability, can enhance
convectional flow of nanoliquid through enclosure. The convectional heat transference performance. The investigation of heat transfer inside
flow across dual plate was investigated by Kumar & Premachandran [6]. cavities of various shapes, including as square-shaped, U-shaped, T-
Nazir et al. [7] introduced the magnetized nanofluid in the cavity shaped, C-shaped, L-shape, and incinerator-shaped, has been done in
problem. Ghalambaz et al. [8] illustrated the hybrid nanofluid inside a recent years, and the results reveal that employing a nanofluid than a
square cavity. Noghrehabadi et al. [9] dissected the nanofluid convec­ regular fluid improves convection heat flow. Hatami et al. [21] disclosed
tional flow across cone-shaped geometry. The convectional flow of the nanofluidic through a T-shaped cavity. The obtained outcomes
water-based nanofluid flow across a wavy cone was scrutinized by designated that heat transfer enhances by introducing nanoparticles in

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

2. Materials and methods

2.1. Physical description

In this analysis, buoyancy-driven fluid flow and thermal trans­

portation in partially active side walls of square-shaped cavity con­
taining TiO2 -water nanofluid were analyzed. The physical configuration
and coordinate system of the developed model in the current scruti­
nization is disclosed in Figs. 1. The width and the altitude of the
enclosure has a cavity are represented byL. The Cartesian coordinate
system is chosen in such direction, x − axis is taken along the length side
and y − axis with enclosure height side. There are two different cases are
discussed such as:

Case 1:

Fig. 1c. Normal element size Mesh distribution of the square cavity. It is considered that on the left and right sides of the cavity the heat
sinks with constant temperature Tc and isothermal heat source with
temperature Th respectively. Consider heat source temperature is higher
than the heat sinkTh > Tc . The length of the heat sink and the heat
source isL/2. It is supposed that the above and lowest walls of the cavity
with inactive parts of the side walls are insulated.

Case 2:

In this case partially active side walls (left & right) of the enclosure
are heated walls with temperature Th and the partially active bottom
wall is cooled with temperatureTc , as shown in Fig. 2. The above cavity
wall is considered thermally insulated.

2.1.1. Computational mesh

In computational fluid dynamics, mesh is one of the most essential
components. The quality of the mesh determines the rate of convergence
and the accuracy of the solution. Finer element size mesh gives more
Fig. 1d. Finer element size Mesh distribution of the square cavity. precise results and provides stronger performance than the normal
element size. Figs. 1c and 1d depict normal and finer element size
base fluid. Dogonchi et al. [22] examined water-based nanofluid in the meshes, and it was analyzed that mash was greater refined along the
occurence of Cu nanoparticles. They demonstrated that the average walls of the square cavity, while less refined far away from the walls of
Nusselt number is increased with reducing the radius of semicircle in the the cavity. Table 1 depicts the test of multiple grid adaption levels. The
absence of magnetic field. The performance of hybrid nanofluid inside L- mesh independence test was guaranteed by considering mesh level, the
shaped cavity was scrutinized by Armaghani et al. [23]. They examined mesh element, quality and calculating Nusselt number on the boundary
that higher heat transfer enactment is obtained with maximum sink as shown in Table 1. As a result, finer mesh is considered for current
power. Rashid et al. [24] disclosed the natural convection of nanofluid simulation.
in cavity having circular obstacle with different shape factors of nano­
particles. They concluded that lamina shape nanoparticle shown larger
performance of heat transfer. Hassan et al. [25] searched the behavior of 2.2. Problem’s mathematical representation
MHD Newtonian fluid flow inside rectangular cavity. They demon­
strated that surface velocity magnitude of Newtonian fluid is up surged The TiO2 -water nanofluid flow is investigated across a square cavity
under a magnetic field. through partially active walls. The cavity is packed by TiO2 -water-based
In main novelty of the study is to scrutinize the natural convection nanofluid which is considered to be Newtonian. The nanopowders are
heat transmission of TiO2 -water nanofluid inside a partially heated considered to be uniform in shape and size. Furthermore the consider­
square cavity. The flow of fluid and transfer are both analyzed in this ations are that nanofluid flow is laminar under the zero density. By
analysis. In this article two different cases are discussed for partially considering the above-mentioned assumptions, the partial differential
heated walls of cavity. The Galerkin finite element method (GFEM) is equations with Boussinesq approximation are expressed as follows
applied to calculate the mathematically developed model. The impacts [26,28]:
of prominent parameters are discussed through streamlines and iso­ ux + vy = 0, (1)
therms. The impacts of natural convection heat transportation inside
( ) ( )
cavity has many applications like solar collector, computer chips cool­ ρnf uux + vuy = − px + μnf uxx + uyy , (2)
ing, cooling of nuclear reactor, polymer, cooling of electronics, solar
concentrator, fuel cells, Nano drug delivery, sensing and imaging etc. ( ) ( )
ρnf uvx + vvy = − py + μnf vxx + vyy + g(ρβ)nf (T − Tc ), (3)
( )
uTx + vTy = αnf Txx + Tyy . (4)

With boundary conditions for current model are:

Case 1:

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 2a. Outline of the finite element method.

On the side wall of cavity at partially active portion:u = 0, v = 0, Here the thermally expansion coefficient for mono nanoliquid can be
T = Tc written as
On active right cavity wall:u = 0, v = 0, T = Th
(ρβ)nf = (1 − ϕ)(ρβ)f + ϕ(ρβ)s . (7)
On insulated portions of walls:u = 0, v = 0, Tn = 0
Where βf represents the base fluid coefficient of thermal expansion and
Case 2:
βs indicates the solid particle coefficient of thermal expansion.
Nanofluid thermal diffusivity can be expressed as:
On active wall portions of cavity along left and right sides:u = 0,v =
0, T = Th knf
αnf = ( ) . (8)
On the lower partially active wall of cavity:u = 0, v = 0, T = Tc ρc p nf
On insulated walls:u = 0, v = 0, Tn = 0
The nanofluid thermal conductivity is define as
2.2.1. Thermophysical characteristics of nanofluid (
knf ks + 2kf − 2ϕ kf − ks
)
Many formulations for the nanofluid properties are developed in = ( ). (9)
kf ks + 2kf + ϕ kf − ks
literature. In the current problem, the relation depends on volume
friction of nanoparticles to obtain the properties of nanofluid is obeying The nanofluid dynamic viscosity is
[27,29].
knf
The nanofluid density can be evaluated by following relation μnf = . (10)
(1 − ϕ)2.5
ρnf = (1 − ϕ)ρf + ϕρs . (5)
The presented thermophysical features of base fluid and particles are
Here, ϕ denotes the volume fraction, ρf be the base fluid density and ρs described in Table 2.
represents the particle density.
The heat capacitance is defined as 2.2.2. Dimensionless scales
( ) ( ) ( ) Dimensionless variables for converting the above-mentioned gov­
ρcp nf = (1 − ϕ) ρcp f + ϕ ρcp s . (6) erning equations:

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 2b. Streamlines and Isotherms outcomes matches with Sheikhzadeh et al. [26].

( )
Table 1 ∂V * ∂V * ∂P* μnf ∂2 V * ∂2 V *
Grid adaptation study atϕ = 0.04, Ra = 1E5 andPr = 6.7.
U* *
+ V * * = − ρf /ρnf * + +
∂X ∂Y ∂Y αρnf ∂X *2 ∂Y *2 (14)
Level Mesh Mini. Average Nuavg CPU
+(ρβ)nf /βf ρnf RaPrθ,
elements Quality Quality Times (s)

Normal 1494 0.3408 0.7741 0.7165 16 ( )

∂θ ∂θ αnf ∂2 θ ∂2 θ
Fine 2520 0.3521 0.7541 1.2345 18 U* + V* * = , (15)
Finer 6586 0.347 0.774 2.3465 23 ∂X * ∂Y αf ∂X *2 ∂Y *2
Extra fine 17,094 0.295 0.8019 3.3212 40
Extremely 26,210 0.3103 0.8437 4.4522 59
The non-dimensional boundary constrains are.
fine
Case 1:

X * = x/L, Y * = y/L, U * = uL/αf , V * = vL/αf , On the partially active left wall:U* = V * = 0, θ = 0 (heat sink)
* 2
P = pL /ρnf α 2
= θ(Th − Tc ) + Tc , Pr = νf /αf ,
f,T (11) On the partially active right wall:U* = V * = 0, θ = 1 (heat source)
Ra = gρf βf (Th − Tc )L3/μf αf On insulated walls: U* = V * = 0, ∂θ/∂n = 0.

As results, our nondimensional form of equations form is follows:

Table 3
∂U ∂V *
*
+ = 0, (12) Evaluation between current outcomes and other published results for average
∂X * ∂Y * Nusselt number.
( )
∂U * * ∂U
*
∂P* μnf ∂2 U * ∂2 U * Ra = 103 Ra = 104 Ra = 105 Ra = 106
U* + V = − ρ /ρ + + , (13)
∂X * ∂Y * f nf
∂X * αρnf ∂X *2 ∂Y *2 Current work 1.150 2.325 4.658 9.045
Sheikhzadeh et al. [26] 1.146 2.312 4.652 9.011
Khanafer et al. [27] 1.119 2.244 4.521 8.806

Table 2
Thermophysical features of host fluid with nanoparticle.
( ) ( ) ( ) ( )
Properties μ Ns/m2 ρ kg/m3 cp (J/kg.K) K(W/m.K) β K− 1 α m2 /s

Water 8.55 × 10− 4 997 4179 0.613 2.761 × 10− 4

1.47 × 107
TiO2 4250 686.2 8.9538 0.9 × 10− 4 30 × 107

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 3. Streaamlines and Isotherms at different Rayleigh number when ϕ = 0.04 for Case 1.

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 3. (continued).

Case 2: ∫ Y2
⃒
hL knf ∂θ ⃒⃒
Nuavg = =2 dY. (18)
On the partially active left and right walls of cavity:U* = V * = 0, kf kf Y1 ∂X ⃒X=0
θ = 1 (heat source) Here Y1 = y1 andY2 = y2 .
L L
On the partially active lower wall of cavity:U* = V * = 0, θ =
0 (heat sink)
3. Numerical approach
On insulated wall: On insulated walls:U* = V * = 0, ∂θ/∂n = 0.
In the current novel Scrutinization, the nonlinear coupled partial
2.2.3. Nusselt number calculation
differential equations (PDE’s) with subjective boundary restrictions are
The Nusselt number which is associated with thermal transportation
numerically tackled utilizing FEM (finite element method). To achieve
of square cavity is mathematically defined as:
this, we transmit the nonlinear PDEs into a linear system using Newton’s
Lqw linearization method. Because to the fact that a big system is divided
Nuloc = ,
kf (Th − Tc ) into a finite number of smaller components, known as finite elements,
FEM computations are capitalised in the majority of complex domains.
where
Discretizing the full flow regime or solution zone into numerous smaller
( )
∂T triangular components is the first step. The pressure of Equations (13)–
qw = − knf (16) ( * *) ( )
∂y (14) is addressed by the equation P = (γ ∂∂UX* + ∂∂UY* ), where γ ≈ 107 is
⃒ denoted as a penalty factor. The solution procedure’s convergence limit
knf ∂θ ⃒⃒ is10− 6 . The outline of the finite element method is shown in Fig. 2a. To
Nuloc = − . (17)
kf ∂Y ⃒Y * =0
*
validate our model, we have made a comparative analysis with analysis
of Sheikhzadeh et al. [26]. The streamlines and Isotherms in Fig. 2b of
The average Nusselt number (Nuavg ) is computed through the following our results and Sheikhzadeh et al. [26] are mentioned. Here it is
expression: observed a same pattern for our current result with published work. In
comparative analysis we develop a streamlines isotherms forRa = 103 .

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 4. Streaamlines and Isotherms at different Rayleigh number when ϕ = 0.04 for Case 2: when active part of Left and Right wall of cavity are hot but active part of
cavity lower wall part is cold.

Table 3 depicts the comparison analysis of current results with other to larger buoyancy forces. The Rayleigh numbers are influenced by the
investigators results for average Nusselt number. There is fair agreement rise in buoyancy force and convection intensity, which enhances the
between present result with findings of Sheikhzadeh et al. [26] and strength of the flow rotation. The Rayleigh numbers are influenced by
Khanafer et al. [27]. the rise in buoyancy force, which improves the power of the flow rota­
tion. The strength of the streamlines, as well as the number and size of
4. Result’s interpretation the cells, rises as the Rayleigh number (RN) improves owing to the
strong buoyancy forces contrasted to the viscous forces. Furthermore, it
Current pagination analyzes convection flow of mono nanofluid via is analyzed that escalating the Rayleigh number improves the flow ve­
square cavity. The current portion performs numerical results via locity of the nanofluid, which is owing to an increment in the influence
streamlines, temperature contour and isotherm against different values of buoyancy force inside the enclosure, particularly near the walls.
of Rayleigh number parameters at ϕ = 0.04 for both cases. For both Similarly, the impact of the buoyancy force in the cavity caused by a
cases the heat sink and heat source length are equals, is half of length of augmentation in the Rayleigh number is magnified by a modification in
square cavity. the temperature contours and isotherm distribution.
The significance of Rayleigh number on streamlines and isotherms
for the TiO2 − water nanofluid in cavity with side walls partially active
4.1. Significance of Rayleigh number for case 1 at selected amount Ra = 1000 at ϕ = 0.04 is shown in Fig. 3 (b). The
influence of convection thermal transfer becomes increasingly sub­
Fig. 3 (a-e) illustrates the decoration of streamlines and isotherms stantial as the Rayleigh number grows. The influence of convection
inside the partially active walls of square enclosure for Rayleigh number thermal transportations becomes increasingly substantial as the Ray­
when . Fig. 3 (a) reveals the consequence of Rayleigh number for . An leigh number grows, and different boundary layers form along the
antic clockwise vertex are appears in inside the cavity for this case. It is enclosure partially active walls. By increasing Rayleigh number, the
observed the streamlines intensity escalates with Rayleigh number due

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 4. (continued).

isotherms close to energy source and sink region. At larger Rayleigh cavity. However, one can significantly raise the buoyancy forces and
number the flow nature and thermal distribution are appreciably outweigh the viscous forces by raising the Rayleigh number. High
exaggerated as display in Fig. 3 (c-d). From these figures we concluded Rayleigh numbers favor convection currents. Additionally, the iso­
that isotherms intensity improves with escalating Rayleigh number. therms exhibit distortion at larger Rayleigh numbers due to the better
Here streamlines and isotherms are more crowded at active surface of convection impacts brought on by an increment in the cavity’s flowing
cavity at larger Rayleigh number, as shown Fig. 3 (e). At Ra = 1e6 the velocity.
streamlines are close to partially heat source wall appear on vertical of

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 5. Velocity magnitude for case 1.

Fig. 6. Velocity magnitude for case 2.

4.2. Significance of Rayleigh number for case 2 isotherms plot are plotted at. From this we concluded that here two
contour rotating vortices are appeared with a reverse way within in the
The significance of Rayleigh number (RN) for streamlines contours enclosure of cavity. Left vortex move with clockwise while right move
and isotherm lines for cavity filled by nanofluid. In this case the partially with anti-clockwise. At lower Rayleigh number floatability forces im­
active side walls (left & right) and partially bottom wall are heated and proves, the streamlines condense near the wall. From the figure it is
cold respectively. scrutinized that by growing the variations of Rayleigh number the
Fig. 4 (a-c) presents represents streamlines (LHS) and isotherms floatability effect improved as a result the vortex circulation speed de­
(RHS) for dissimilar magnitudes of Rayleigh number in cavity filled by creases within the enclosure. The convective heat transmission becomes
nanofluid with volume fraction. In Fig. 4 (a) the streamlines and more noteworthy by mounting the Rayleigh number. From the figure we

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

Fig. 7. Local Nusselt number (LNN) estimation for Rayleigh number.

Fig. 8. Local Nusselt number (LNN) estimation for solid nanoparticle fraction.

observed that alongside the active walls of cavity the streamlines and condensed near the active walls. From Fig. 4 (e) on critical values of
isotherms examine the larger velocities and strong convective heat Rayleigh number we observed isotherms are much packed on partially
transfer in these portions of cavity. Furthermore, it can be observed that active cold walls. Hence the convection heat transfer is improved. From
by escalating the Rayleigh number the streamlines and isotherms are the figure it is concluded that the streamlines are highly packed next to
more crowded at partially active walls. From the Fig. 4(b-e) it can be the active hot wall of the enclosure. Here two oval shape vortices are
analyzed from the collection of Rayleigh number to the streamlines are seen in the cavity. Physically, the symmetry state effluents and the
closely packed on active walls such as sided heated walls and lower centralization of the streamlines in the cavity’s core tend to the hot wall
partially cold wall of length. It is analyzed the isotherms are more as the Rayleigh number rises. Rayleigh number improvement leads to a

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S.A. Khan et al. Journal of Molecular Liquids 382 (2023) 122003

boost in the thermal transfer process. 7. Authors statement

In Figs. 5 and 6 the flow magnitude for various magnitudes of Ray­
leigh number is illustrated via Arc length in both cases 1 and 2. Fig. 5 All authors state that they have read the revised manuscript and
signifies the larger velocity magnitude via Rayleigh equally contributed.
number1E3⩽Ra⩽1E6. From Fig. 6,it is to be noticed that the velocity
magnitude is also boosted via Rayleigh number 1E3⩽Ra⩽1E6 for Case 2
when left and right walls are heated while partially active lower. Declaration of Competing Interest

4.3. Nusselt number The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
Fig. 7 examines the consequence of Rayleigh number (RN) via local the work reported in this paper.
Nusselt number (Heat transfer rate). It is concluded that the Heat
transfer rate is improved by growing the values of the Rayleigh number. Data availability
The consequence of nanoparticle fraction on the Heat transfer rate is
depicting in Fig. 8. Here, it is analyzed that local Nusselt number (Heat Data will be made available on request.
transfer rate) is declined by collective the magnitude of volume fraction.
The correlation for the average Nusselt number in case 1 of paramters Acknowledgments
Rayleigh number and nanoparticles volume fraction is addressed as:
(
Nuavg = 0.4568ϕ + 0.3343Ra0.25
)
(19) The authors extend their appreciation to the Deanship of Scientific
Research at King Khalid University, Abha, Saudi Arabia for funding this
work through Large Groups Project under grant number RGP.2/51/44.
5. Tables
The researchers would like to acknowledge Deanship of Scientific
Research, Taif University for funding this work.
See Tables 1-3.

6. Conclusions References

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