EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6869
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Multi-Parametric Study of Convective Heat Transfer
in a Magnetized Wavy Cavity Using Hybrid

Nanofluids for Thermal Management Applications

Khamis Al Kalbani1, Nidal Ratib Anakira1, M. J. Uddin2,
Ali Fareed Al Jassar1, Ala Amourah1,3,∗, Tala Sasa4

1 Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman
2 Department of Mathematical and Physical Sciences, University of Nizwa, Nizwa, Sultanate of Oman
3 Jadara University Research Center, Jadara University, Jordan
4 Department of Mathematics, Faculty of Science, Applied Science Private University, Amman, Jordan

Abstract. This study presents a comprehensive numerical investigation of natural convective heat transfer in
a wavy-walled enclosure filled with a hybrid nanofluid consisting of copper ( Cu ) and aluminum oxide ( Al2O3 )
nanoparticles dispersed in a water-ethylene glycol (EG) base fluid. Natural convection in such enclosures is widely
encountered in electronics cooling, energy devices, and magnetic field-controlled thermal systems, which motivates
the present study. A range of water-EG mixture ratios ( 95%−5%, 90%−10%, 80%−20%, 60%−40%, 50%−50%
), including the limiting cases of pure water and pure EG, is considered to evaluate the influence of base fluid
composition on thermal performance. The aim of this work is to clarify how variations in base fluid ratio,
nanoparticle loading, and magnetic field strength affect convective transport. The governing equations are solved
numerically using the finite element method to capture coupled buoyancy and magnetohydrodynamic effects.
The nanoparticle volume fractions are systematically varied from 0.1% to 0.5% to capture the impact of particle
loading, while the Rayleigh number ranges from 103 to 106, and the Hartmann number from 0 to 40 , to assess
the effects of buoyancy and magnetic fields. The results show that increasing the Rayleigh number significantly
enhances convective heat transfer, while variations in base fluid composition lead to only marginal differences in
the average Nusselt number. For example, the average Nusselt number increases by nearly one order of magnitude
as Ra rises from 103 to 106, while nanoparticle addition yields up to ∼ 18% enhancement, with copper providing
the highest gains. In contrast, increasing Ha from 0 to 40 reduces the heat flux by as much as ∼ 22%. The
inclusion of Cu and Al2O3 nanoparticles improves thermal performance, with copper demonstrating a greater
enhancement due to its superior thermal conductivity. Furthermore, increasing the Hartmann number suppresses
convective currents and reduces the total heat flux, especially near regions of high thermal activity. The wavy
geometry intensifies convective mixing and promotes localized heat transfer, with observable peaks in the heat
flux distribution aligned with the undulations of the hot wall. These findings highlight the synergistic effects
of nanoparticle composition, base fluid selection, magnetic field control, and enclosure geometry on thermal
transport, providing valuable insights for designing advanced cooling systems in electronics, energy, and thermal
management applications. These insights highlight the relevance of the results for designing advanced cooling
systems in electronics, renewable energy devices, and thermal management technologies.

2020 Mathematics Subject Classifications: 76N10

Key Words and Phrases: Hybrid nanofluid, water-ethylene glycol mixture, wavy enclosure

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6869

Email addresses: KKalbani@su.edu.om (K. Al Kalbani),
Nanakira@su.edu.om (N. Anakira), AJassar@su.edu.om (A. Al Jassar),
AAmourah@su.edu.om (A. Amourah), t sasa@asu.edu.jo (T. Sasa)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 2 of 20

Introduction

The pursuit of enhanced heat transfer efficiency is a critical endeavor in thermal
engineering, propelled by the need for compact, energy-efficient systems in applications
such as power generation, automotive cooling, solar thermal collectors, and electronic
thermal management. Convective heat transfer, a key process in these systems, is often
limited by the thermophysical properties of conventional fluids, such as water, ethylene
glycol (EG), or their mixtures. To overcome these constraints, nanofluids-base fluids
infused with nanoparticles-have emerged as a groundbreaking approach, offering a 28%
increase in heat transfer coefficient with Cu − TiO2 hybrid nanofluids in a 50 : 50
water-EG mixture. This approach provides superior thermal conductivity and convective
performance.

Hybrid nanofluids-fluids containing multiple nanoparticle types-have gained consid-
erable attention due to their ability to synergistically enhance thermal properties be-
yond those of single-component nanofluids. Copper nanoparticles are renowned for their
high thermal conductivity, while Al2O3 nanoparticles provide stability and affordabil-
ity, making the Cu Al2O3 hybrid an attractive option for practical applications. The
water-EG base fluid, widely employed in heat exchangers and cold-climate systems, ex-
hibits adjustable properties-such as viscosity, density, and thermal conductivity-based
on its composition. Recent studies have highlighted the promise of hybrid nanofluids
in enhancing thermal performance. For example, Yaser et al. [1] reported a mixture,
while Sidik et al. [2] observed that the thermal characteristics of the hybrid nanofluid
were higher compared to the base fluid and the fluid containing single nanoparticles,
respectively. However, most prior works have focused either on straight channels, simple
cavities, or single-fluid nanofluids, with relatively fewer investigations addressing hybrid
nanofluids in enclosures under combined buoyancy and magnetic field effects. In partic-
ular, systematic studies on the role of water-EG ratio in hybrid nanofluid-based wavy
geometries remain scarce.

The wavy enclosure introduces a distinctive flow domain where natural convection
is influenced by factors such as the Rayleigh number, nanoparticle volume fraction, and
wall waviness (amplitude and wavelength). This geometry enhances heat transfer by
increasing surface area and inducing flow perturbations that promote mixing. Bhardwaj
et al. [3] found that wavy cavities can improve heat transfer compared to flat enclosures,
while Al Kalbani and Uddin [4] noted an increase in the total heat flux at the heated
wall by increasing the ripple parameter N. This research harnesses these geometric ben-
efits, pairing them with the Cu − Al2O3 hybrid nanofluid in the specified water-EG
ratios to optimize thermal performance. The motivation stems from the demand for
efficient, versatile thermal systems that can operate across a range of conditions, from
low-temperature refrigeration to high-temperature industrial processes.

Recent literature provides a strong foundation for this investigation. Tokgoz et al.
[5] explore heat transfer enhancement in turbulent channel flow using corrugated ducts
and alumina-water nanofluids. Results show that corrugation increases turbulence and
heat transfer, and the nanofluid further improves thermal performance compared to pure
water. Similarly, Said et al. [6] examine the thermophysical properties of nanodiamond



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 3 of 20

+Fe3O4 hybrid nanofluids (in water and water-EG mixtures) using experiments and
predictive models, including MLR and MLRI. They found that the MLRI ( R = 0.9996
), MLR ( R = 0.99989 ), MLR ( R = 0.9999998 ), and MLRI ( R = 0.9857 ) mod-
els demonstrated excellent predictive accuracy, with correlation coefficients indicating
strong agreement between predicted and experimental values. Studies on wavy enclo-
sures, such as those by Abdulkadhim et al. [7] and Cheng, C. Y.[8], emphasize geometry’s
role in amplifying buoyancy-driven convection. Despite these contributions, a direct con-
nection between hybrid nanofluid composition (e.g., Cu−Al2O3 with variable water-EG
ratios), magnetic field strength, and wavy enclosure design has not been comprehensively
established in the literature. Further relevant research has been reported in [9–29].

This research employs a methodical approach to assess heat transfer enhancement
by testing the Cu−Al2O3 hybrid nanofluid across the specified water-EG ratios: 95%−
5%, 90% − 10%, 80% − 20%, 60% − 40%, and 50% − 50%, as well as pure water and
pure EG. Key variables include nanoparticle volume fractions (e.g., 0.1% to 0.5% ),
base fluid composition, and operating conditions (e.g., temperature gradients driving
natural convection). The wavy enclosure is designed to maximize convective effects, and
the study utilizes advanced experimental and/or numerical techniques to capture fluid
flow and thermal behavior. Specifically, this work aims to investigate the heat transfer
enhancement for pure water, pure ethylene glycol (EG), and various mixture ratios
while also examining the effect of the wavy wall on flow patterns and streamlines. The
objectives are to measure the enhancement in convective heat transfer (e.g., via Nusselt
number) across these conditions and to analyze how the wavy geometry influences flow
dynamics, providing insights into the interplay between fluid composition, nanoparticle
effects, and enclosure design for next-generation thermal management strategies. In
summary, the novelty of this study lies in combining Cu−Al2O3 hybrid nanofluids with
variable water - EG ratios inside a wavywalled enclosure under magnetic field effects an
area where literature remains limited. This work fills the gap by quantifying the coupled
influence of nanoparticle loading, base fluid composition, and wall geometry on natural
convection, offering new guidelines for designing efficient thermal management systems.

Physical Modelling

In the current study, we used nanofluidic flow to analyze time-independent, incom-
pressible, laminar, and two-dimensional flow. The hydromagnetic and opposing gravi-
tational forces were used to build a nano-fluidic model, where the equation space is a
square of length L and filled with a Cu − Al2O3 /water - EG two-component hybrid
nanofluid. The y-axis is normally next to the left wall, and the x-axis measures along
the bottom surface in dimensional coordinates. The equations for wavy walls are:

xleft = 0.05 cos

(
Nπy

L

)
, xright = 0.05 cos

(
Nπy

L

)
+ L, ;N = 4 (1)

Regarding the coordinate system, a magnetic field with a strength of B0 is applied.
The upper and underside walls are insulated, and the right and left side walls are used to



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 4 of 20

isothermally cool and heat nanofluids at uniform temperatures of TC and TH correspond-
ingly. It is always maintained that, TH > TC . The Cu − Al2O3 Hybrid nanoparticles
and a water-ethylene glycol mixture were used as the base fluid in the current study.
Fig. 1 illustrates the geometry and coordinate system schematically.

Fig. 1. Physical model’s schematic view.

It is also expected that the reference solution and the nanomaterials are in thermo-
dynamic equilibrium and that there is no slippage between the two media.

Table 1. Thermo-physical characteristics of solid nanoparticles (Oztop and Abu-Nada
[30] ) and the base fluid (pure water, pure EG, and mixtures of water–EG) as reported
in [30], (Peyghambarzadeh et al. [31]), and (Bargal et al. [32]).

Items cp ρ k µ α β Prandtl number

Unit J/(kg K) kg/m3 W/(mK) N s/m2 m2/s 1/K –

Pure water 4174 992 0.633 6.5× 10−4 1.5× 10−7 2.1× 10−4 4.286

Pure EG 2382 1101 0.256 9.5× 10−3 9.8× 10−8 5.7× 10−4 88.395

Water–EG (95%–5%) 4157 995 0.615 1.01× 10−3 1.49× 10−7 3.8× 10−4 6.827

Water–EG (90%–10%) 4090 1002 0.600 1.65× 10−3 1.46× 10−7 3.95× 10−4 11.248

Water–EG (80%–20%) 4020 1008 0.580 1.9× 10−3 1.43× 10−7 4.2× 10−4 13.169

Water–EG (60%–40%) 3500 1050 0.380 3× 10−3 1.03× 10−7 1.25× 10−3 27.632

Water–EG (50%–50%) 3340 1062 0.360 3.9× 10−3 1.015× 10−7 1.32× 10−3 36.183

Cu 385 8933 401 – 1.17× 10−4 1.67× 10−5 –

Al2O3 765 3970 40 – 1.32× 10−5 0.85× 10−5 –

Despite the density change in the body force element of the momentum equation
estimated using the Boussinesq assumption, the physical properties of the nanofluids
were assumed to be static. The acceleration due to gravity has a negative y-direction.
Since all fixed boundaries are assumed to be rigid, the walls do not move. Table 1 is a
listing of the thermophysical properties of nanofluids.



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 5 of 20

Mathematical Modelling

Assuming that the flow is continuous and laminar and that the nanofluid is New-
tonian, a set of basic equations for nanofluids has been modified. The incompressible
Navier-Stokes equations are used to model two-dimensional flow. The variation in den-
sity is analyzed using the Boussinesq approximation. Conservation of mass, momentum,
and energy is addressed through the relevant mathematical equations. Given the above
premises, the governing equations [33]for this model are presented as follows:

∂u

∂x
+

∂v

∂y
= 0 (2)

u
∂u

∂x
+ v

∂u

∂y
= − 1

ρhnf

∂p

∂x
+

µhnf

ρhnf

(
∂2u

∂x2
+

∂2u

∂y2

)
(3)

u
∂v

∂x
+ v

∂v

∂y
= − 1

ρhnf

∂p

∂y
+

µhnf

ρhnf

(
∂2v

∂x2
+

∂2v

∂y2

)
+ g (T − TC)

(ρβ)hnf
ρhnf

(4)

u
∂T

∂x
+ v

∂T

∂y
= αhnf

(
∂2T

∂x2
+

∂2T

∂y2

)
(5)

The parameters of the hybrid nanofluid are defined as follows ([33],[34]):

ρhnf = ρbf (1− φhnf ) + ρn1φn1 + ρn2φn2 (6)

(ρβ)hnf = (ρβ)bf (1− φhnf ) + (ρβ)n1φn1 + (ρβ)n2φn2 (7)

Where φhnf = φn1 + φn2

αhnf =
khnf

(ρcp)hnf
, (8)

(ρcp)hnf = (1− φn1 − φn2) (ρcp)bf + φn1 (ρcp)n1 + φn2 (ρcp)n2 (9)

µhnf =
µbf

(1− φn1)
2.5 (1− φn2)

2.5 (10)

khnf =
kn2 + 2knf − 2φn2 (knf − kn2)

kn2 + 2knf + φn2 (knf − kn2)
(knf ) (11)

knf =
kn1 + 2kbf − 2φn1 (kbf − kn1)

kn1 + 2kbf + φn1 (kbf − kn1)
(kbf ) (12)



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 6 of 20

The following are the different types of boundary conditions for the current investi-
gation:

u = v = 0, T = TH for xleft, 0 ≤ y ≤ L, (a)

u = v = 0, T = TC for xright, 0 ≤ y ≤ L, (b)

u = v = 0,
∂T

∂y
= 0 for y = 0, L, 0 ≤ x ≤ L. (c)

(13)

The following transformations can be used to make the aforementioned equations
dimensionless:

X =
x

L
, Y =

y

L
,U =

uL

αbf
, V =

vL

αbf
, P =

pL2

ρbfα2
bf
, θ =

T − TC

TH − TC
(14)

With equation (14), the governing equations (2) through (5) can be expressed in the
dimensionless form as follows:

∂U

∂X
+

∂V

∂Y
= 0 (15)(

ρhnf

ρbf

) (
U ∂U

∂X + V ∂U
∂Y

)
= − ∂P

∂X + Pr
(1−φn1)

2.5(1−φn2)
2.5

(
∂2U
∂X2 + ∂2U

∂Y 2

)
,(

ρhnf
ρbf

)(
U
∂V

∂X
+ V

∂V

∂Y

)
(16)

= −∂P
∂Y + Pr

(1−φn1)
2.5(1−φn2)

2.5

(
∂2V
∂X2 + ∂2V

∂Y 2

)
+ (Ra)(Pr)

(ρβ)hnf

ρbfβbf
θ

−Ha2 PrV (17)

U
∂θ

∂X
+ V

∂θ

∂Y
=

αhnf

αbf

(
∂2θ

∂X2
+

∂2θ

∂Y 2

)
(18)

Where Pr =
νbf
αbf

is the Prandtl number, Ra =
gβbf (TH−TC)L3

αbfνbf
is the Rayleigh number

and Ha = B0L
√

σbf

µbf
is the Hartmann number.

The boundary conditions’ dimensionless forms include:

U = V = 0, θ = 1 for Xleft, 0 ≤ Y ≤ 1, (a)

U = V = 0, θ = 0 for Xright, 0 ≤ Y ≤ 1, (b)

U = V = 0,
∂θ

∂Y
= 0 for Y = 0, 1, 0 ≤ X ≤ 1. (c)

(19)

At the heated bottom wall, it is estimated that the local Nusselt number is:

Nuhnf = −
(
khnf
kbf

)(
∂θ

∂X

)
X=0

(20)



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 7 of 20

Furthermore, the local Nusselt number is integrated to produce the average Nusselt
number:

Nuhnf =

∫ 1

0
NuhnfdY (21)

Numerical Solution and Validation

Various mesh combinations of the geometry were tested to determine the optimal
heat transfer rate and flow fields. During the grid test, a mixture of water and EG (
90%− 10% ) was selected as the base fluid for the experiment. The following parameter
set combinations were used: Ra = 105, φn1 = 0.02, φn2 = 0.02,Ha = 20,Pr = 11.25, and
N = 4. Four different non-standard grid systems were utilized in the simulation system
to assess the expected number of elements for the constrained area. The arithmetic
pattern is completed to be extremely accurately input in the typical Nusselt number (
Nuav ) for the elements mentioned above, as shown in Table 2, in order to understand
the mesh size. We noticed that the average Nusselt number ( Nuav ) for 5554 elements
showed a slight difference, in contrast to the results for the 7990 elements. Therefore,
the grid sizes of 5554 and 7990 elements can be obtained to get accurate results, and
7990 triangle elements were considered in the simulation to generate the results.

Table 2. Grid sensitivity test at Ra = 105, φn1 = 0.02, φn2 = 0.02, Pr = 11.25, Ha = 20
and N = 4 for Cu-(Al2O3)/water(90%)-EG(10%) nanofluid at the left wavy wall of the
cavity.

Elements 3440 5554 7990 18120

Nuav 3.023 3.005 3.002 2.989

We compared our results for the predetermined conditions with previously published
results by Ghasemi et al. [35]. For this comparison, we selected: φn1 = 0, φn2 =
0, 0.02, 0.06, N = 0 for calculating the average Nusselt number presented in Table 3.
In both studies, the average Nusselt number is calculated at the heated cavity wall for
similar buoyancy forces. The current simulation results and the results of Ghasemi et
al. [35] agree very closely. The numerical results of the study are now more credible and
have been further validated.

Table 3. Comparison between the data of Ghasemi et al. [35] and the present work

Ra
φ = 0 φ = 0.02 φ = 0.06

Present Ghasemi et al. [35] Present Ghasemi et al. [35] Present Ghasemi et al. [35]

103 1.002 1.001 1.060 1.160 1.184 1.184

104 1.181 1.179 1.224 1.302 1.319 1.291

105 3.137 3.151 3.180 3.138 3.259 3.108

106 7.82 7.907 7.965 7.979 8.230 8.098



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1. Results and Discussion

To investigate the thermal and flow behavior of hybrid nanofluids in a wavy en-
closure, a series of numerical simulations was conducted under various physical pa-
rameters. Isotherm distributions were analyzed for Rayleigh numbers ranging from
104 to 106, considering different base fluid compositions of water and ethylene glycol:
50% − 50%, 60% − 40%, 80% − 20%, 90% − 10%, and 95% − 5%. The mixture with
equal proportions ( 50%−50% ) was adopted as the reference base fluid for comparative
analysis.

Subsequent results show the total heat flux magnitude along the heated wall for dif-
ferent Rayleigh numbers and nanoparticle concentrations. Copper and aluminum oxide
nanoparticles were introduced in different volume fractions to evaluate their respective
influences on heat transfer characteristics.
Moreover, the effect of an externally applied magnetic field on flow structure and heat
transfer was examined. The average Nusselt number was computed for each base fluid
composition across a range of Rayleigh numbers and subsequently evaluated for varying
concentrations of one type of nanoparticle while keeping the other type constant. These
comprehensive simulations aim to elucidate the interplay between thermal buoyancy,
magnetic forces, and nanoparticle-enhanced conductivity in complex enclosures.

Fig. 2 depicts the streamlines within the wavy enclosure for different Rayleigh num-
bers (Ra = 104, 105, 106) and varying water–ethylene glycol compositions, while keeping
the nanoparticle volume fractions constant (ϕn1 = 0.02 for Cu, ϕn2 = 0.02 for Al2O3)
and maintaining a fixed Hartmann number (Ha = 20).

For Ra = 104, the streamlines are concentrated along all four walls, with a higher
density near the right and left vertical walls, indicating the dominance of conduction
with weak convection. A single clockwise rotating cell forms at the cavity center as
buoyancy forces begin to overcome viscous resistance, establishing a stable and symmet-
ric circulation pattern. When Ra increases to 105, the buoyancy effect becomes stronger
relative to viscous and magnetic damping forces, leading to thinner thermal and velocity
boundary layers and an increase in streamline density along the vertical walls. The cen-
tral vortex elongates diagonally, reflecting enhanced convective transport and stronger
coupling between the upward and downward streams adjacent to the hot and cold walls.
As Ra reaches 106, the flow becomes highly convective, with two secondary vortices
developing near the upper-right and lower-left corners, and irregular motion appearing
in the central region. This behavior signifies the transition toward a multi-cellular and
partially unstable regime, where buoyancy-driven inertia starts to dominate despite the
magnetic damping.

Furthermore, varying the water–EG mixing ratio did not noticeably alter the stream-
line topology, implying that the overall flow structure is primarily governed by buoyancy
and magnetic effects rather than by modest variations in fluid thermophysical proper-
ties. The Rayleigh number dictates the strength and pattern of convection, while the
Hartmann number (Ha = 20) acts to moderate the flow intensity without changing the
fundamental circulation structure.



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Fig. 3 illustrates the temperature contours (isotherms) within the wavy enclosure for
various Rayleigh numbers (Ra = 104, 105, 106) and different compositions of the water–
ethylene glycol base fluid, while maintaining constant nanoparticle volume fractions
(ϕn1 = 0.02 for Cu, ϕn2 = 0.02 for Al2O3) and a fixed Hartmann number (Ha = 20).

As the Rayleigh number increases, the isotherms transition from a nearly vertical
alignment to a more horizontal orientation. This transformation reflects the shift from
conduction-dominated heat transfer at lower Ra to convection-dominated regimes at
higher Ra. In low Rayleigh number cases, thermal diffusion governs the transport mech-
anism, resulting in stratified temperature fields. However, at higher Ra, buoyancy forces
become more significant, inducing stronger convective currents that distort the temper-
ature field and enhance thermal mixing.

Fig 2. Streamlines for different Rayleigh numbers (Ra) and different percentages of
water-ethylene glycol mixture when φn1 = 0.02, φn2 = 0.02, Ha = 20.



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 10 of 20

Fig 3. Isotherms for different Rayleigh numbers (Ra) and different percentages of water-
ethylene glycol mixture when φn1 = 0.02, φn2 = 0.02, Ha = 20.

Additionally, the condensation of isotherms near the hot wall becomes more pro-
nounced with increasing Ra, indicating intensified thermal gradients and localized heat
transfer enhancement due to stronger convective activity adjacent to the heated bound-
ary.
Across all configurations, elevated temperatures are observed near the hot wall and the
upper region of the cavity. This is attributed to the upward movement of the heated
fluid due to buoyancy, which accumulates along the top surface, especially under stronger
convection.
Variations in the water-ethylene glycol mixture ratios result in only minor changes in the
overall temperature distribution. This limited sensitivity is likely due to the relatively
small difference in thermophysical properties (such as thermal conductivity and viscos-
ity) between the considered mixtures, which exert only a secondary influence compared
to the dominant effects of Rayleigh number and nanoparticle loading.

Fig. 4 presents the distribution of total heat flux magnitude along the heated wavy
wall for various Rayleigh numbers, with fixed nanoparticle volume fractions of φn1 = 0.02
(Cu) and φn2 = 0.02 (Al2O3). The analysis is conducted for multiple water–ethylene
glycol base fluid compositions, all of which exhibit qualitatively similar trends.



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 11 of 20

Fig. 4. Total heat flux magnitude for Rayleigh number (Ra) when φn1 = 0.02, φn2 =
0.02, Pr = 36.183, Ha = 20 (Cu–Al2O3/water(50%)–EG(50%)).

At a low Rayleigh number ( Ra = 103 ), the heat flux profile remains nearly uniform
with slight undulations that mirror the geometrical variations of the wavy boundary.
This behavior is characteristic of conduction-dominated regimes, where heat transfer is
primarily governed by temperature gradients and is not significantly influenced by fluid
motion. The minor fluctuations in flux are attributed to the increased local surface area
introduced by the wall corrugation.
As the Rayleigh number increases, the total heat flux along the heated wall exhibits a no-
table rise, especially toward the lower sections of the wall. This enhancement corresponds
to the transition from conduction to convection-dominated heat transfer. In higher Ra
regimes, buoyancy-driven flow enhances thermal transport by actively carrying heated
fluid away from the hot wall and replacing it with cooler fluid. The bottom portion of
the wavy wall experiences greater convective interaction due to the upward circulation
of thermal plumes originating from that region, resulting in a localized increase in the
magnitude of heat flux.

Despite varying the water-ethylene glycol mixing ratios, the overall distribution pat-
terns remain consistent. This suggests that the influence of the Rayleigh number and
nanoparticle loading on thermal convection outweighs the relatively modest variations
in the base fluid’s thermophysical properties. Nonetheless, fluids with higher water con-
tent, having lower viscosity and higher thermal conductivity, may support slightly more
vigorous convection, further contributing to enhanced heat transfer.
Fig. 5 displays the distribution of total heat flux magnitude along the heated wavy wall
for varying volume fractions of aluminum oxide nanoparticles (ϕn2 ranging from 0 to 0.05)
while maintaining a constant volume fraction of copper nanoparticles (ϕn1 = 0.02). The
analysis is conducted under fixed thermal and flow conditions to isolate the effect of ϕn2

on local heat transfer performance.

The resulting heat flux profiles exhibit a characteristic wavy pattern with two pronounced
peaks: one located just above the lower segment of the wall and another near the mid-
height of the wall. These peaks correspond to zones of intensified convective activity



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driven by the local geometry and flow recirculation patterns within the enclosure. The
distribution curves for different ϕn2 values largely overlap along the wall, indicating
minimal variation in overall flux except near the peak regions.

Fig. 5. Total heat flux magnitude for different values of φn2 when φn1 = 0.02, Ra = 105,
Pr = 36.183, Ha = 20 (Cu–Al2O3/water(50%)–EG(50%)).

At these peak locations, a clear enhancement in the total heat flux magnitude is
observed with increasing φn2, highlighting the role of aluminum oxide nanoparticles in
boosting local thermal conductivity and, consequently, the rate of energy transfer to the
fluid. This enhancement is attributed thermophysically to the higher effective thermal
conductivity of the hybrid nanofluid, which increases as the concentration of Al2O3 rises.
Although aluminum oxide has a lower thermal conductivity than copper, its synergistic
interaction with Cu nanoparticles and the base fluid contributes positively to the overall
heat transfer, especially in regions with steep temperature gradients.

A similar trend is observed when the role of φn1 and φn2 are interchanged-i.e., keep-
ing φn2 fixed and varying φn1. The corresponding results demonstrate analogous peak
enhancement behavior with minor differences in magnitude and position, which will be
discussed in detail in a subsequent section.
Fig. 6 presents the variation of total heat flux magnitude along the heated wavy wall
for different Hartmann numbers (Ha = 0, 10, 20, 30, 40), illustrating the influence of
magnetic field strength on heat transfer performance in the presence of electrically con-
ducting hybrid nanofluids.

Across all cases, the heat flux profiles exhibit a characteristic wavy decreasing pat-
tern, with two distinct local maxima—one located near the lower section and the other
near the mid-height of the wall. These peak regions correspond to zones of enhanced
convective motion and intensified thermal gradients induced by the enclosure’s wavy
geometry.

As the Hartmann number increases, a systematic reduction in the total heat flux mag-
nitude is observed. This attenuation in heat transfer can be attributed to the mag-
netohydrodynamic (MHD) damping effect: the presence of a transverse magnetic field
introduces Lorentz forces that act as a resistive force against fluid motion. These forces



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 13 of 20

suppress the buoyancy-driven convective currents that are responsible for enhancing heat
transport, thereby shifting the dominant heat transfer mechanism toward conduction,
especially at higher Ha values.

Fig. 6. Total heat flux magnitude for the Hartmann number (Ha) when φn1 = 0.02,
φn2 = 0.02, Pr = 36.183, Ra = 105 (Cu–Al2O3/water(50%)–EG(50%)).

Notably, the divergence between the flux curves becomes more noticeable at the peak
regions. This suggests that the localized convective enhancements, which are highly
sensitive to fluid motion, are more significantly damped under stronger magnetic fields.
In contrast, the regions between peaks, where conduction plays a relatively larger role,
are less affected.

Overall, the results confirm that increasing magnetic field strength exerts a stabilizing
and suppressive influence on convective transport, leading to reduced thermal efficiency
along the heated wall. This highlights the critical interplay between electromagnetic
control and thermal performance in MHD applications involving hybrid nanofluids.

Table 4 presents the average Nusselt number for various Rayleigh numbers (Ra =
103, 104, 105, 106) and different water–ethylene glycol base fluid compositions: 50%–
50%, 60%–40%, 80%–20%, 90%–10%, and 95%–5%. The concentrations of copper and
aluminum oxide nanoparticles are fixed at φn1 = 0.02 and φn2 = 0.02, respectively.

As expected, the average Nusselt number increases significantly with the rising
Rayleigh number for all base fluid mixtures. This trend reflects the transition from
conduction-dominated to convection-dominated heat transfer regimes. At low Ra, ther-
mal energy is primarily transferred through molecular diffusion, resulting in lower Nusselt
numbers. With increasing Ra, the buoyancy-driven flow becomes stronger, enhancing
thermal transport via fluid motion and leading to a substantial increase in convective
heat transfer rates.
In contrast, the variation in average Nusselt number among the different water–ethylene
glycol mixtures is relatively small across all Rayleigh numbers. This limited sensitiv-
ity can be attributed to the moderate changes in thermophysical properties—such as
thermal conductivity, viscosity, and specific heat—across the selected mixing ratios. Al-
though increasing the water content generally improves thermal conductivity and reduces



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 14 of 20

50− 50 60− 40 80− 20 90− 10 95− 05

Ra = 103 0.963 0.963 0.963 0.964 0.964

Ra = 104 1.345 1.345 1.345 1.345 1.345

Ra = 105 3.719 3.720 3.721 3.722 3.722

Ra = 106 8.584 8.593 8.595 8.597 8.602

Table 4. Average Nusselt number for different percentages of Water–Ethylene Glycol
base fluid when φn1 = 0.02, φn2 = 0.02, Ha = 20.

viscosity (thus promoting convection), the impact of these variations is overshadowed by
the dominant influence of the Rayleigh number and the presence of thermally conductive
nanoparticles.

These findings suggest that, within the examined range of base fluid compositions,
the enhancement in heat transfer is governed more strongly by buoyancy effects and
nanoparticle loading than by the specific ratio of water to ethylene glycol.

Fig. 7 illustrates the variation of the average Nusselt number with increasing nanopar-
ticle volume fractions of copper ( φn1 = 0 to 0.05 ) and aluminum oxide ( φn2 = 0 to 0.05
), under a fixed Rayleigh number of Ra = 105. In the first case (red curve), the volume
fraction of Al2O3 is held constant at φn2 = 0.02, while φn1(Cu) varies. In the second
case (blue curve), the volume fraction of Cu is fixed at φn1 = 0.02, and φn2 (Al2O3) is
varied.
Both trends exhibit a monotonic increase in the average Nusselt number with increasing
nanoparticle volume fraction, confirming the positive effect of nanoparticle loading on
convective heat transfer. However, the enhancement is more pronounced in the case of
copper nanoparticles. For equivalent volume fractions, the red curve lies consistently
above the blue curve, indicating that increasing φn1(Cu) results in greater thermal per-
formance compared to increasing φn2 ( Al2O3 ).
This difference can be thermophysically explained by the superior thermal conductivity
of copper relative to aluminum oxide. The inclusion of highly conductive Cu nanopar-
ticles significantly boosts the effective thermal conductivity of the hybrid nanofluid,
facilitating more efficient heat transport. Although Al2O3 contributes to thermal en-
hancement, its lower intrinsic conductivity yields a more modest improvement when
added in the same proportions.

These results highlight the importance of nanoparticle selection in hybrid nanofluid
formulations and underscore the dominant role of copper in enhancing convective heat
transfer under moderate to high Rayleigh number conditions.



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 15 of 20

Fig. 7. Average Nusselt number for different values of φn1 and φn2 when Ha = 20,
Pr = 36.183, Ra = 105 (Cu–Al2O3/water(50%)–EG(50%)).

Conclusion

A comprehensive numerical investigation was conducted to analyze the convective
heat transfer characteristics of a hybrid nanofluid consisting of copper ( Cu ) and alu-
minum oxide ( Al2O3 ) nanoparticles, which were dispersed in various mixtures of water
and ethylene glycol as the base fluid. This analysis took place within a wavy-walled
enclosure under the influence of magnetic fields. The simulations considered a wide
range of Rayleigh numbers, nanoparticle volume fractions, and Hartmann numbers to
evaluate the effects of thermal buoyancy, nanoparticle loading, base fluid composition,
and magnetic field intensity on heat transfer performance.

The key findings can be summarized as follows:

• Rayleigh Number Influence: The transition from conduction-dominated to con-
vectionenhanced regimes was clearly observed with increasing Rayleigh number,
as reflected in both the isotherm patterns and the significant rise in the average
Nusselt number.

• Base Fluid Composition: Variations in water-ethylene glycol ratios exhibited only
minor influence on the thermal behavior, indicating that the base fluid’s ther-
mophysical properties play a secondary role compared to buoyancy forces and
nanoparticle effects.

• Nanoparticle Volume Fraction: Both Cu and Al2O3 nanoparticles contributed to
enhanced heat transfer. However, Cu demonstrated a more substantial effect,
attributable to its higher thermal conductivity. Hybrid nanofluids with increasing
Cu volume fraction achieved superior Nusselt numbers compared to those with
increasing Al2O3 content under fixed Rayleigh conditions.

• Magnetic Field Effects: The application of a transverse magnetic field (modeled
through varying Hartmann numbers) suppressed convective motion due to the



K. Al Kalbani et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6869 16 of 20

Lorentz force, resulting in decreased total heat flux along the heated wall. The
dampening effect was especially pronounced in regions of high thermal activity.

Overall, the study confirms the potential of Cu − Al2O3 /water-ethylene glycol hy-
brid nanofluids to enhance thermal performance in thermally driven enclosures. These
insights are particularly valuable for the development of advanced cooling technolo-
gies in applications such as electronic device thermal management, energy systems, and
microfluidic heat exchangers, where precise thermal control is critical under variable
operating conditions.

Directions for Future Research

While this study offers valuable numerical insights, it highlights the need for further
multi-faceted research to advance the field. Experimental validation of the predicted
thermal performance remains a critical priority to verify model accuracy under practi-
cal conditions. Future work should also explore a wider range of nanoparticle materi-
als - including graphene-based composites - and non-spherical morphologies to optimize
thermal conductivity without undue viscosity increases. Expanding simulations to three-
dimensional and transient flows with realistic boundary conditions would greatly improve
practical relevance. Additionally, thorough investigation into nanofluid stability, rheo-
logical behavior, and pumping power requirements is essential to assess overall energy
efficiency. Finally, integrating multi-physics interactions such as electro hydrodynamic
or thermoelectric effects could enable next-generation thermal management solutions for
advanced microelectronics and microfluidic devices.

Acknowledegements

The authors gratefully acknowledge the financial support provided by the Ministry
of Higher Education, Research and Innovation (MoHERI), Oman, through the Block
Funding Program at Sohar University (SU Ref: SU/BFP/RG/2024/07). This support
was essential for the successful completion of this research.

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Nomenclature

Dimensional quantities
A amplitude [m]
B0 magnetic field strength [kg/s2. A]
cp specific heat at constant pressure [J/kgK]
g gravitational acceleration [ m/s2 ]
k thermal conductivity [W/mK]
L enclosure length [m]
N undulation parameter [m]
p dimensional fluid pressure [Pa]
T Temperature [K]
u, v dimensional velocity components [ m/s ]
x, y dimensional coordinates [m]
Dimensionless quantities
Ha Hartmann number [-]
Nu Nusselt number [-]
P dimensionless fluid pressure [-]
Pr Prandtl number [-]
Ra Rayleigh number [-]



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U, V dimensionless velocity components [-]
X,Y dimensionless coordinates [-]
Greek symbols
α thermal diffusivity [ m2/s ]
β thermal expansion coefficient [1/K]
θ dimensionless temperature [-]
µ dynamic viscosity [ Ns/m2 ]
v kinematic viscosity [ m2/s ]
ρ density [ kg/m3 ]
σ electrical conductivity [ S/m ]
φ nanoparticle volume fraction [-]
Subscript
av average
bf base fluid
C cold wall
H hot wall
hnf hybrid nanofluid
n1 copper nanoparticle
n2 aluminum oxide nanoparticle
nf nanofluid