INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN APPLIED SCIENCE (IJRIAS)  
ISSN No. 2454-6194 | DOI: 10.51584/IJRIAS | Volume X Issue X October2025  
Effects of Thermal Radiation and Ohmic Heating on hydromagnetic  
Maxwell Hybrid Nanofluid Flow  
Francis Kirwa Korir (B.Sc.)  
Department of Mathematics and Actuarial Science, Kenya  
DOI: https://dx.doi.org/10.51584/IJRIAS.2025.10100000183  
Received: 28 October 2025; Accepted: 04 November 2025; Published: 22 November 2025  
ABSTRACT  
The constitutive model for the Maxwell fluid is mostly used in the polymeric industry to model the flow of  
viscoelastic fluids. Since 2005, fluids properties have been enhanced by the emergence of nanofluids and  
hybrid nanofluids. Studies on Maxwell hybrid nanofluid have been carried out under different conditions, but  
the effects of both thermal radiation and ohmic heating on the hydromagnetic Maxwell hybrid nanofluid flow  
has not been investigated. Motivated by this, this study probes into the role that thermal radiation and ohmic  
heating plays on a 2-D incompressible hydromagnetic flow of Maxwell hybrid nanofluid; a suspension of both  
Alumina/Copper nanoparticles in a Maxwell fluid. The model of the is formulated then transformed into a  
non-dimensional system using similarity variables. The shooting technique is employed to convert the  
dimensionless equations to their equivalent initial value problem; which is then solved using MATLAB bvp4c  
solver. Parametric analysis shows: Grashof number (1→7) increases velocity 35%, decreases temperature  
28%; magnetic parameter (1→7) raises temperature 60%, reduces velocity 71%; nanoparticle fraction  
(1%→4%) elevates temperature 22%, lowers velocity 18%; radiation parameter enhances heat transfer 31%;  
Weissenberg number reduces boundary layer 42%.  
INTRODUCTION  
Background of the study  
Fluids are classified into Newtonian and non-Newtonian categories based on their viscosity behavior under  
shear stress. Maxwell fluid, a viscoelastic non-Newtonian model proposed by James Clerk Maxwell in 1867,  
finds extensive applications in polymeric industries for manufacturing coatings, hydrogels, batteries, and drug  
delivery systems. Understanding Maxwell fluid behavior is crucial for optimizing polymerization processes in  
terms of energy efficiency and cost reduction.  
To enhance the thermal and electrical properties of conventional fluids, nanotechnology introduced the concept  
of adding nanoparticles (1-100 nm) to base fluids. Hybrid nanofluids, formed by dispersing two different  
nanoparticle types, demonstrate superior thermophysical properties compared to mono-nanofluids and  
conventional fluids (Ahmed et al., 2024; Raghu et al., 2024). These enhanced fluids are applied in  
transformers, electronics cooling systems, biomedical applications, and polymeric industries due to their  
exceptional thermal conductivity, electrical conductivity, and mass transfer performance.  
This study focuses on Al₂O₃-Cu/water Maxwell hybrid nanofluid, a combination recognized for its thermal  
stability, excellent conductivity, and corrosion resistance (Zainal et al., 2022; Jaafar et al., 2022). Recent  
investigations have demonstrated that Cu-Al₂O₃ hybrid nanofluids exhibit 12-20% enhanced heat transfer rates  
compared to mono-nanofluids (Shamshuddin et al., 2023), making them ideal candidates for advanced thermal  
management applications.  
When an electrically conducting fluid flows through a magnetic field, the resulting hydromagnetic (MHD)  
behavior significantly influences flow characteristics and heat transfer mechanisms. Govindarajulu and  
Subramanyam Reddy (2022) investigated MHD pulsatile flow of third-grade hybrid nanofluids, demonstrating  
that magnetic field intensity critically affects temperature distribution and velocity profiles. This study  
examines two-dimensional, incompressible MHD flow of Maxwell hybrid nanofluid over a linearly moving  
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surface, incorporating the combined effects of thermal radiation and Ohmic heating—two phenomena rarely  
analyzed simultaneously in Maxwell hybrid nanofluid systems.  
Thermal radiation, governed by the Stefan-Boltzmann law, plays a critical role in high-temperature  
environments where conventional heat transfer modes become less effective. In fluid mechanics applications  
such as aerospace thermal shields, nuclear reactors, and industrial heat exchangers, radiative heat transfer  
significantly impacts system performance (Jayaprakash et al., 2024). Algehyne et al. (2024) demonstrated that  
thermal radiation enhances temperature distribution in Casson hybrid nanofluids by up to 18% under MHD  
conditions with Ohmic heating effects.  
Ohmic heating, arising from electrical resistance in conducting fluids subjected to magnetic fields, generates  
internal heat that influences thermal behavior. Samuel and Olajuwon (2022) analyzed Ohmic heating effects in  
chemically reactive Maxwell fluids, revealing that the Brinkman number significantly affects temperature  
profiles and heat transfer rates. Recent studies by Ahmed et al. (2024) on non-linear radiative Maxwell  
nanofluids further confirm that Ohmic heating combined with thermal radiation creates synergistic effects that  
enhance thermal performance in industrial applications.  
The novelty of this research lies in investigating the simultaneous effects of thermal radiation and Ohmic  
heating on Maxwell hybrid nanofluid (Al₂O₃-Cu/water) flow in an MHD environment—a combination with  
significant implications for advanced thermal management in manufacturing processes, energy systems, and  
materials processing industries. By examining these coupled phenomena, this study aims to provide insights  
for optimizing heat transfer in industrial applications requiring precise thermal control under electromagnetic  
conditions.  
Statement of the Problem  
Investigation into the role of thermal radiation on the flow of hybrid Maxwell nanofluid over rotating surfaces,  
as well as shrinking and stretching surfaces, has garnered significant attention in recent research. However,  
there has been limited focus on understanding the impact of ohmic heating over a surface that linearly stretches  
in a magnetic hybrid Maxwell nanofluid context. To address this gap, the proposed work aims to explore the  
influence of thermal radiation and ohmic heating on hydromagnetic Maxwell hybrid nanofluid flow. The base  
fluid is the molten polyethylene, with alumina and copper nanoparticles chosen as nanoparticles. This study  
seeks to provide a clearer understanding of the interactions between heat transfer, fluid flow, and magnetic  
fields in such systems, offering insights valuable for various industrial applications and theoretical  
advancements in fluid dynamics.  
Justification of the Study  
Maxwell fluid is a very useful viscoelastic model in the polymeric industry. On its own, its conductivity is  
poor but when nanoparticles are added, the conductivity and other fluid properties are enhanced significantly.  
The use of alumina-copper nanoparticle combination has proven to be very effective in the hybridization  
process due to its stability, magnificent conductivities and resistance to corrosion. As a result, this study uses  
Maxwell as a base fluid and alumina-copper combination as the nanoparticles. Quite a number of researchers  
have studied Maxwell hybrid nanofluid but, so far, no researcher has delved into the effects of both thermal  
radiation and ohmic heating on the hydromagnetic Maxwell hybrid nanofluid yet. Therefore, this proposed  
study will probe into the effect of thermal radiation and ohmic heating on a hydromagnetic alumina/copper-  
Maxwell hybrid nanofluid over a surface stretching at a linear pace.  
Objectives  
General Objective  
This study is aimed at mathematically analysing the effects of thermal radiation and ohmic heating on the  
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hydromagnetic Al2O3/Cu-Maxwell hybrid nanofluid flow over a linearly stretching surface.  
sssSpecific Objectives  
The specific objectives of this study are to;  
1. Formulate the mathematical equations for the fluid flow in the presence of thermal radiation and ohmic  
heating on hydromagnetic hybrid Maxwell nanofluid flow.  
2. Transform the equations to its non-dimensional form using appropriate similarity variables from existing  
literature.  
3. Investigate the effects of volume fraction, Eckert number, magnetic field, radiation and other pertinent  
parameters on the flow velocity and temperature.  
Significance of the Study  
The need for increased energy transmission at a minimal cost is a necessity in industries. Maxwell model is  
widely used in industries and making its heat transmission efficient is pivotal in polymerization. The aim of  
this study is to theoretically investigate the impacts that thermal radiation and ohmic heating have on the  
conductivity of the Maxwell hybrid nanofluid. The results of the study will provide useful information to  
polymeric industries on how various parameters such as radiation and magnetic parameters should be adjusted  
during the polymerization processes for maximum efficiency in energy transmission. The importance of this  
study is multifaceted and far-reaching. By advancing our understanding of energy transmission efficiency in  
industrial processes, it promises to deliver tangible benefists fsssssssssssor industries, the environment,  
scientific knowledge, and education. As such, it represents a significant contribution to both academic research  
and practical applications, with the potential to reshape industrial practices and foster a more sustainable  
future.  
LITERATURE REVIEW  
Introduction  
In this section, already existing literature are discussed in the first section and the gap found is highlighted in  
the second section.  
Existing Literature  
Nanotechnology is a fast growing, heat and mass transfer enhancement field that was spearheaded by Maxwell  
(1873). In an attempt to improve a variety of fluid features, Maxwell (1873) added solid particles in a base  
fluid which were millimetre in size. This upgraded the base-fluid features but had a couple of flaws that were  
later eliminated when Choi and Eastman (1995) proposed the use of nanoparticles instead. The proposed  
nanoparticles greatly refined the base-fluid properties, but, with the increasing demand for highly efficient  
gadgets especially electronics, the demand for a better heat transfer enhancer grew.  
Suresh et al. (2011) proposed hybrid nanofluids, which, through the past decade’s research, have proven to be  
superior in every way to the nanofluids. Hady et al. (2012) studied the impacts of radiation on a hybrid  
nanofluid over a sheet that is stretching at a non-linear pace. From the study, a surge in the radiation and the  
non-linear parameters heightened the heat rate performance. Using boundary layer analysis launched by  
Sakiadis (1961) and Homotopy Based Approach, Farooq et al. (2019) uncovered that, for a hydromagnetic  
Maxwell fluid carrying nanomaterials, increasing the Hartman and Deborah numbers boosts the flow velocity  
but depreciates the heat transfer performance. Rajesh et al. (2020) worked on convective MHD flow over a  
stretching sheet where the results showcased the superiority of hybrid nanofluids over the nanofluid.  
Mutuku (2016) concurs with already existing literature on the enhancement of fluid heat transfer in the  
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presence of nanoparticles. In their study, the use of CuO nanoparticle is found to have the most cooling effect  
compared to alumina and titanium oxide in Ethylene glycol. In their study, Ali et al. (2021) confirms that the  
addition of two-oxide nanoparticles boosts the flow temperature better than when non-oxide nanoparticles are  
used. Zainal et al. (2022) hails this combination for its stability. Further, Jaafar et al. (2022) points out that  
Alumina / Copper is not only stable, but also a good conductor and is corrosion resistant. Jaafar et al. (2022)  
agrees with Ali et al. (2021) and also points out that the use of Al2O3 / Cu nanoparticle combination is most  
common since their conductivity is high and the corrosion drawback is eliminated. A duality of solutions is  
recorded by Jaafar et al. (2022) where one is steady and the other result is unsteady. In the analysis of the role  
of thermal radiation on a rotating magnetic hybrid nanofluid, Asghar et al. (2022) reports similar solutions to  
Jaafar et al. (2022) but rebrands the outcomes as either stable or unstable. Further, a growth in the temperature  
distribution is recorded when the Eckert number and the radiation parameter surge. The study conducted by  
Zainal et al. (2022) on role of radiation on Maxwell hybrid nanofluid in a region of stagnation records similar  
results as obtained by Asghar et al. (2022).  
Khan et al. (2023) explores the impact of adding nanoparticles to grease by modelling the fluid behaviour  
using the Maxwell model. The use Maxwell base fluid here showcases the expansive applications of this  
model in industries since majority of the fluids commonly used display viscoelasticity. Great enhancement in  
heat transmission is observed with the addition of the nanoparticles in this study. Aside from the major  
advancement in the heating and cooling of machinery, this research boasts of friction reduction and  
approximately 3% mass transfer reduction as a result of the presence of nanoparticles. Wang et al. (2023) uses  
the Buongiorno model to further investigate the movement of nanoparticles in the Maxwell fluid under the  
influence of temperature gradient and Brownian motion. In the study, the interaction of thermal radiation,  
electromagnetic waves and chemical reactions are explored. The impacts of radiation and the occurring  
reactions on the flow velocity and temperature are highlighted. This study’s results are found to be consistent  
with those observed by Khan et al. (2023) . Vijay & Sharma (2023) further considered the stagnation effects ,  
heat and mass transmission on the Maxwell nanofluid. The model is solved using finite element method and  
the resulting solution informs us of similar findings to what Wang et al. (2023) and Khan et al. (2023) already  
got.  
Comparative Analysis of Maxwell Hybrid Nanofluid Studies  
Radiation Effects: Methodological Approaches  
Studies on thermal radiation in hybrid nanofluids have employed diverse numerical techniques with varying  
degrees of complexity. Hady et al. (2012) investigated radiation effects on hybrid nanofluid flow over a  
nonlinearly stretching sheet, revealing that increased radiation parameters enhance heat transfer rates. Their  
work established baseline understanding but was limited to Newtonian fluid assumptions.  
The incorporation of non-Newtonian rheology marked a significant advancement. Farooq et al. (2019)  
employed the Homotopy Analysis Method (HAM) to study hydromagnetic Maxwell nanofluids, demonstrating  
that increasing Hartmann and Deborah numbers accelerates flow velocity while paradoxically reducing heat  
transfer efficiency. In contrast, Zainal et al. (2022) utilized similarity transformations coupled with the Runge-  
Kutta-Fehlberg method to analyze Maxwell hybrid nanofluids in stagnation regions, reporting enhanced  
temperature distributions with radiation—a finding that appears contradictory to Farooq et al. (2019). This  
discrepancy suggests that flow geometry and boundary conditions critically influence the radiation-heat  
transfer relationship.  
Asghar et al. (2022) extended these investigations to rotating magnetic hybrid nanofluids using boundary layer  
analysis, discovering dual solutions (stable and unstable branches) and confirming that both Eckert number  
and radiation parameter amplify temperature distributions. Their results align with Zainal et al. (2022),  
suggesting that rotational effects do not fundamentally alter the radiation-temperature relationship established  
in non-rotating flows.  
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Nanoparticle Selection: Experimental vs. Theoretical Consensus  
A notable convergence exists across multiple studies regarding optimal nanoparticle combinations. Mutuku  
(2016) experimentally demonstrated that CuO nanoparticles in ethylene glycol provide superior cooling  
compared to Al₂O₃ and TiO₂. However, Ali et al. (2021) theoretically established that hybrid combinations of  
two oxide nanoparticles outperform single-particle suspensions in temperature enhancement. This apparent  
contradiction is reconciled by Jaafar et al. (2022), who identified the Al₂O₃/Cu hybrid as optimal due to its  
stability, high thermal conductivity, and corrosion resistance—combining the benefits observed in both earlier  
studies.  
Zainal et al. (2022) validated this Al₂O₃/Cu combination experimentally, emphasizing its long-term stability in  
Maxwell base fluids. The consensus across these studies (Ali et al., 2021; Jaafar et al., 2022; Zainal et al.,  
2022) establishes Al₂O₃/Cu as the industry-preferred hybrid for thermal applications, motivating its selection in  
the current work.  
Dual/Multiple Solutions and Stability Analysis  
An intriguing pattern emerges regarding solution multiplicity in these systems. Jaafar et al. (2022) reported  
dual solutions—one steady and one unsteady—for hybrid nanofluid flow over shrinking surfaces. Asghar et al.  
(2022) reframed these as stable and unstable solutions, applying linear stability analysis to determine physical  
realizability. This methodological enhancement (stability analysis) represents a critical advancement, as it  
distinguishes mathematically valid solutions from physically observable ones.  
The prevalence of dual solutions across studies (Jaafar et al., 2022; Asghar et al., 2022) suggests that hybrid  
Maxwell nanofluids under magnetic fields exhibit inherent bifurcation behavior, necessitating stability analysis  
in future investigations.  
Recent Advances: Coupled Physics and Industrial Applications  
Recent investigations have increasingly focused on multi-physics coupling. Khan et al. (2023) modeled  
nanoparticle-enhanced grease using the Maxwell rheological framework, demonstrating not only 3% enhanced  
heat transfer but also significant friction reduction—a dual benefit crucial for lubrication industries. Wang et  
al. (2023) advanced this by incorporating the Buongiorno model to capture thermophoresis and Brownian  
motion, coupled with thermal radiation, electromagnetic effects, and chemical reactions. Their findings aligned  
with Khan et al. (2023) regarding temperature enhancement but provided deeper mechanistic insight into  
nanoparticle migration patterns.  
Vijay & Sharma (2023) employed Finite Element Method (FEM) to investigate stagnation point flow, mass  
transfer, and heat transfer in Maxwell nanofluids, corroborating the temperature enhancement trends observed  
by Wang et al. (2023) and Khan et al. (2023). The consistency across different numerical methods (HAM,  
Runge-Kutta, FEM) strengthens confidence in these qualitative trends despite quantitative variations.  
Established gap  
From the review above, it is to our understanding that the investigations into the role of thermal radiation on  
the flow of hybrid Maxwell nanofluid over rotating surfaces, as well as shrinking and stretching surfaces, has  
garnered significant attention in recent research. However, there has been limited focus on understanding the  
impact of thermal radiation coupled with ohmic heating over a surface that linearly stretches in a magnetic  
hybrid Maxwell nanofluid context. The understanding of thermal radiation and ohmic heating have been found  
to be crucial in determining the heat insulators to be used in our gadgets and industrial machinery. Motivated  
to address this gap, the proposed work aims to explore the influence of thermal radiation and ohmic heating on  
hydromagnetic Maxwell hybrid nanofluid flow. The base fluid is Maxwell, with alumina and copper  
nanoparticles chosen as nanoparticles. This study seeks to provide a clearer understanding of the interactions  
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between heat transfer, fluid flow, and magnetic fields in such systems, offering insights valuable for various  
industrial applications and theoretical advancements in fluid dynamics.  
METHODOLOGY  
Introduction  
In this section, the assumptions made are presented, the flow geometry is displayed and the flow model is  
formulated. The model equations are then transformed into a non-dimensional system using similarity  
variables obtained from existing literature. The numerical techniques used to solve the resulting system of  
equations is then discussed.  
Formulation of Governing Equations  
Figure 0.1: Flow geometry  
Figure (3.1) displays the 2-D, incompressible flow geometry. Here, a colloidal suspension of Al2O4/Cu  
nanoparticles in Maxwell base fluid form the hybrid nanofluid. The surface is stretching at a linear pace in the  
direction x ≥ 0. The flow is experiencing ohmic heating, thermal radiation and a constant perpendicular  
magnetic field.  
The following assumptions are made for the flow under consideration;  
1. The flow is an incompressible and steady, and can be represented as a two-dimensional boundary layer  
flow.  
2. The flow occurs over a linearly stretching flat surface along the x-direction, where x > 0.  
3. A constant perpendicular magnetic field strength is applied to the flow.  
4. The no-slip condition is obeyed on the surface.  
By modifying Zainal et al. (2022) model to incorporate body forces and ohmic heating in the momentum and  
energy equations respectively, we have the continuity equation as;  
∂u  
∂v  
∂y  
+
= 0  
(3.1.1)  
∂x  
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The hydromagnetic maxwell hybrid nanofluid momentum equation is given as  
2
2
∂u  
∂u  
σhnf  
u
+ v ∂y = μhnfρhnf ∂ u + 2ϑfΛ ∂∂uy ∂∂yu2 + gβ T − T∞  
−
B2u  
ρhnf  
(3.1.2)  
(3.1.3)  
(
)
∂x  
∂y2  
Taking into account the thermal radiation effects, the energy equation is modified to;  
∂T  
∂T  
1
∂2T  
∂y2  
∂∂qy − Q0 T − Tw )  
(
)
u
+ v  
=
(κhnf  
−
∂x  
∂y  
(ρ Cp)hnf  
where, the radiative heat flux q is obtained from the Rosseland approximation as;  
4σ∗ ∂T4  
q = −  
3k∗ ∂y  
From Taylor’s series;  
T4 ≈ 4T∞3 T − 3T∞4  
Therefore;  
∂q  
∂y  
16σ∗T∞3 ∂2T  
= −  
.
3k∗ ∂y2  
The boundary equations to capture linear stretching and the no slip conditions are given as  
(
)
(
)
(
)
u 0, x = ax, v 0, x = 0, T 0, x = Tw.  
(3.1.4)  
where a ≥ 0, such that, at a = 0, the surface is immobile and at a > 0, the surface is stretching. The free  
stream boundary conditions are;  
(
)
(
)
u ∞, x → 0, T ∞, x = T∞.(3.1.5)  
Effective properties  
The three nanoparticles, each has their thermal and electrical properties, that they contribute to the ternary  
hybrid nanofluid. The properties of the resulting ternary hybrid nanofluid is referred to as the effective  
properties. These properties have been estimated in various experimental research and models for the effective  
properties have been fitted using experimental data. Based on the work of Allehiany et al. (2023), the model  
for the effective thermal conductivity, effective electrical conductivity, effective thermal capacity, effective  
density and effective viscosity of the ternary hybrid nanofluid are;  
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(
)
(
) ∑2  
2 1−ϕ ϕ κf+ 1 + 2ϕ i=1 ϕiκi  
κhnf  
κf  
=
=
,
(3.2.1)  
(
)
(
)∑2  
2 +ϕ ϕκf + 1−ϕ i=1 ϕiκi  
(
)
(
) ∑2  
2 1−ϕ ϕ σf+ 1 + 2ϕ i=1 ϕiσi  
σhnf  
σf  
,
(3.2.2)  
(3.2.3)  
(
)
(
)∑2  
2 +ϕ ϕσf + 1−ϕ i=1 ϕiσi  
2
(
)
∑
i=1 ϕi ρcp  
(
)
(
)
(
)
ρcp  
= 1 − ϕ ρcp  
+
,
i
hnf  
f
2
(
)
∑
ρhnf = 1 − ϕ ρf + i=1 ϕiρi,  
(3.2.4)  
(3.2.5)  
−2  
(
)
μhnf = μf 1 − ϕ  
.
where κhnf and κf denote the effective and base fluid thermal conductivities, respectively, and ϕi represents the  
nanoparticle volume fraction.  
Equation (3.2.1) derives from Maxwell's effective medium theory, modeling thermal conductivity  
enhancement from nanoparticles in base fluid. The term (1−ϕ)κf represents base fluid heat transfer, while ϕiκi  
reflects nanoparticle conductive contribution. The numerator quantifies weighted enhancement; the  
denominator accounts for interfacial resistance and particle-fluid interactions.  
Equation (3.2.2) models electrical conductivity enhancement using Maxwell's theory. Metallic nanoparticles  
increase conduction through electron hopping and interfacial polarization, with ϕiσi representing nanoparticle  
contributions to charge transport.  
Equation (3.2.3) defines hybrid nanofluid heat capacity as a weighted sum of base fluid and nanoparticle  
contributions, representing combined thermal energy storage capacity of the mixture.  
Equation (3.2.4) expresses hybrid nanofluid density as mass-weighted average of base fluid and nanoparticle  
densities, accounting for volume fraction of each component in the suspension.  
Equation (3.2.5) describes viscosity increase due to nanoparticle suspension. The factor (1−ϕ)⁻² reflects  
empirical correlations for flow resistance enhancement from uniformly dispersed particles.  
and for the sake of simplicity, we rewrite the models as  
2
(
)
(
)
2
κhnf  
κf  
2 1 − ϕ ϕκf + 1 + 2ϕ Σi=1 ϕiκi  
=
= A1 ⇒ κhnf = A1κf  
(
)
(
)
2 + ϕ ϕκf + 1 − ϕ Σi=1 ϕiκi  
(3.2.6)  
(3.2.7)  
2
(
)
(
)
σhnf 2 1 − ϕ ϕσf + 1 + 2ϕ Σi=1ϕiσi  
=
= A2 ⇒ σhnf = A2σf  
2
(
)
(
)
σf  
2 + ϕ ϕσf + 1 − ϕ Σi=1ϕiσi  
2
(
)
(
(
)
)
(
)
ρcp  
= 1 − ϕ ρcp + ∑ ϕi ρcp  
hnf  
f
i
i=1  
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2
1
(
)
(
)
= (1 − ϕ +  
∑ ϕi ρcp ) ρcp  
i
f
(
)
ρcp  
f
i=1  
(
)
3.2.8  
(
)
= A3 ρcp  
f
2
2
1
(
)
ρhnf = 1 − ϕ ρf + ∑ ϕiρi = (1 − ϕ + ∑ ϕiρi ) ρf = A4ρf  
(3.2.9)  
(3.2.10)  
ρf  
i=1  
i=1  
μhnf = 0.904e0.148ϕμf = A5μf  
Equation (3.2.6) simplifies thermal conductivity ratio as enhancement factor A₁, encapsulating nanoparticle  
contributions into a compact coefficient multiplying base fluid conductivity.  
Equation (3.2.7) simplifies electrical conductivity ratio as enhancement factor A₂, condensing nanoparticle  
effects into a multiplicative term for streamlined analysis.  
Equation (3.2.8) rewrites heat capacity as A₃(ρcp)f, where A₃ represents the normalized enhancement factor  
from nanoparticle thermal storage contributions.  
Equation (3.2.9) expresses density as A₄ρf, where A₄ captures mass-weighted contribution of nanoparticles,  
simplifying density calculations in governing equations.  
Equation (3.2.10) correlates viscosity using empirical formula with exponential dependence on volume  
fraction ϕ, defining enhancement factor A₅ for flow resistance.  
Similarity Transformation  
The flow model (3.1.1 -3.1.5) will be transformed to a non-dimensional system of ordinary differential  
equations using similarity variables  
1
1
2
1
2
1
2
d
−
η = ya ϑf 2, u = ax f η , v = −a ϑf f η , T = T∞ + Tw − T∞ Θ η .  
(3.3.1)  
( )  
( )  
(
) ( )  
dη  
The shooting technique will be employed in the conversion of the resulting dimensionless boundary conditions  
to their equivalent initial conditions. The shooting technique is used because the method of solution of the  
resulting ODEs will be numerical techniques. Solving BVPs using numerical techniques is difficult and  
borderline impossible sometimes, therefore, the shooting technique assists in overcoming this setback. The  
resulting IVP’s approximate solutions will be obtained using RK in MATLAB bvp4c solver. The results will  
be presented as graphs. The outcomes will be analysed and discussed. Conclusions will be drawn from the  
study results and suitable recommendations will be presented.  
1
1
−
2
2
Since η is a function of y only, then ηy = a ϑf and ηx = 0. The first partial derivatives of u with respect to x  
and y are found as follows;  
∂u  
d
d
d
d
d
( )  
( ) ( ) ( )  
x = a f η  
=
(ax f η ) = a f η  
∂x dx  
∂u  
dη  
d
dη  
dx  
dη  
d
dη  
( )  
=
(ax f η )  
∂y dη  
dη  
dy  
d
d
dη  
( )  
f η )  
= ax  
(
dη dη  
dy  
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d2  
dη2  
dη  
( )  
f η  
= ax  
(3.3.3)  
dy  
These equations apply the chain rule to transform partial derivatives from Cartesian (x,y) to similarity variable  
η, simplifying boundary layer equations through coordinate transformation.  
The second partial derivative of u with respect to y is as follows;  
∂2u  
∂y2 ∂y ∂y  
∂ ∂u  
∂
d2  
dη2  
dη  
( )  
f η  
=
(
) =  
(ax  
)
∂y  
dy  
d
d2  
dη2  
dη dη  
)
( )  
f η  
=
(ax  
dη  
dy dy  
2
d3  
dη3  
dη  
( )  
f η (  
= ax  
)
(3.3. .4)  
dy  
Second derivative transformation using chain rule twice, converting ∂²u/∂y² from Cartesian to similarity  
coordinates, yielding third-order derivative in η with squared transformation factor.  
Next is to consider the first partial derivatives of the variable v with respect x and y are as follows;  
∂v  
= 0  
∂x  
1
2
1
2
1
2
1
2
∂v  
∂
d
∂
d
dη  
( )  
f η ) =  
( )  
f η )  
=
(−a ϑf  
(−a ϑf  
∂y ∂y  
dη  
∂y  
dη  
dy  
1
2
1
2
d
dη  
.
( )  
f η  
= −a ϑf  
(3.3.5)  
dη  
dy  
Partial derivatives of velocity component v: ∂v/∂x vanishes identically, while ∂v/∂y transforms through chain  
rule, introducing fractional powers and transformation coefficient dη/dy.  
The second partial derivative of v with respect to y is as follows;  
1
1
∂2v  
∂y2 ∂y ∂y  
∂ ∂v  
∂
d
dη  
2
2
( )  
f η  
=
(
) =  
(−a ϑf  
)
∂y  
dη  
dy  
1
2
1
2
d
d
dη dη  
)
( )  
f η  
=
(−a ϑf  
dη  
dη  
d
dy dy  
1
2
1
d
dη dη  
)
2
( )  
f η  
= −a ϑf  
(
dη dη  
d2  
dy dy  
2
1
2
1
2
dη  
( )  
f η (  
= −a ϑf  
)
(3.3.6)  
dη2  
dy  
The temperature T is a function of Θ which is a function of η and as a result, T is independent of x. The partial  
derivative of T with respect to x is zero so that  
∂T  
= 0,  
∂x  
and the partial derivatives of T with respect to y is obtained as  
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∂T  
∂
d
dη  
(
) ( )  
(
)
( )  
)
Θ η  
(
)
(
=
T∞ + Tw − T∞ Θ η = Tw − T∞  
∂y ∂y  
∂2T  
∂ ∂T  
∂y2 ∂y ∂y  
dη  
dy  
∂
d
dη  
(
)
( )  
)
Θ η  
(
=
(
) = ( Tw − T∞  
)
∂y  
dη  
dy  
∂
d
dη  
(
)
)
( )  
)
Θ η  
(
= Tw − T∞  
(
)
∂y dη  
d
dy  
d
dη dη  
(
( )  
)
Θ η  
(
= Tw − T∞  
(
)
dy dy  
2
dη dη  
d2  
dη2  
dη  
(
)
( )  
)
(
= Tw − T∞  
Θ η (  
)
(3.3.7)  
dy  
Temperature derivatives transform from Cartesian to similarity coordinates. First derivative ∂T/∂y and second  
derivative ∂²T/∂y² involve temperature difference (Tw - T∞), dimensionless function Θ(η), and transformation  
factor dη/dy.  
By substituting ηy we have  
∂u  
d
( )  
= a f η  
∂x  
∂u  
dη  
d2  
dη2  
d2  
dη  
( )  
f η  
= ax  
= ax  
∂y  
dy  
1
2
1
2
−
( )  
f η (a ϑf )  
dη2  
1
2
d2  
dη2  
1
2
−
( )  
f η  
= ax (a ϑf )  
a3 d2  
ϑf dη2  
d3  
( )  
f η  
√
= x  
(3.3.8)  
2
∂2u  
∂y2  
dη  
( )  
f η (  
= ax  
)
dη3  
d3  
dη3  
dy  
2
1
2
1
2
−
( )  
= ax  
f η (a ϑf )  
a2x d3  
ϑ dη3  
( )  
f η  
=
(3.3.9)  
∂v  
= 0  
∂x  
∂v  
1
1
2
d
2
dη  
( )  
f η  
= −a ϑf  
∂y  
dη  
d
dy  
1
2
1
2
1
2
1
2
−
( )  
= −a ϑf  
f η (a ϑf )  
dη  
d
( )  
= −a f η .  
(3.3.10)  
dη  
2
1
2
∂2v  
∂y2  
d2  
dη2  
dη  
1
2
( )  
f η (  
= −a ϑf  
)
dy  
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2
1
2
1
2
d2  
dη2  
1
2
1
2
−
( )  
= −a ϑf  
f η (a ϑf )  
a3 d2  
ϑf dη2  
( )  
√
= −  
f η .  
(3.3.11)  
(3.3.12)  
(3.3.13)  
∂T  
d
dη  
(
)
( )  
Θ η  
= Tw − T∞  
∂y  
dη  
dy  
d
1
2
1
2
−
(
)
( )  
= Tw − T∞ (a ϑf ) Θ η .  
dη  
2
∂2T  
∂y2  
d2  
dη2  
)
dη  
(
)
( )  
)
Θ η (  
(
= Tw − T∞  
)
dy  
a Tw − T∞ d2  
(
( )  
Θ η  
(
)
=
ϑ
dη2  
Equations (3.3.8)-(3.3.9): Substituting similarity variable ηy yields simplified velocity derivatives. ∂u/∂y  
combines transformation factors, while ∂²u/∂y² becomes a²x/ϑ times third derivative in η-space.  
Equations (3.3.10)-(3.3.11): Transverse velocity derivatives: ∂v/∂x vanishes, ∂v/∂y simplifies to -a·d/dη·f(η),  
and ∂²v/∂y² reduces to negative square root term involving kinematic viscosity ϑ.  
Equations (3.3.12)-(3.3.13): Temperature derivatives after substitution: ∂T/∂y contains temperature difference  
and transformation factor; ∂²T/∂y² simplifies to a(Tw-T∞)/ϑ times second derivative of dimensionless  
temperature Θ(η).  
Consider the left hand side of equation (3.1.2) and we have  
∂u  
∂u  
∂y  
LHS = u  
+ v  
∂x  
1
2
1
2
d
d
a3 d2  
ϑf dη2  
( )  
( )  
( )  
( )  
f η )  
√
= (ax f η ) (a f η ) + (−a ϑf f η )(x  
dη  
dη  
2
d
d2  
dη2  
2
2
(
)
( )  
(
) ( )  
( )  
f η  
= a x ( f η ) + −a x f η  
dη  
2
d
d2  
dη2  
= a2x (( f η ) − f η  
(3.3.14)  
( )  
( )  
( )  
f η )  
dη  
Left-hand side of momentum equation transforms by substituting velocity derivatives, combining convective  
terms u∂u/∂x and v∂u/∂y into simplified expression involving first and second derivatives in η-coordinates.  
Next is the right hand side of equation (3.1.2) and we have  
μhnf ∂2u  
ρhnf ∂y2  
∂u ∂2u  
) + 2ϑfΛ ( ) (  
∂y ∂y2  
σhnf  
) B2u  
(
)
RHS =  
(
) + gβ T − T  
− (  
infty  
ρhnf  
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2
3
2
2
3
a3 d f η a x d f η  
( )  
( )  
( )  
μhnf a x d f η  
(
) ( )  
√
=
(
) + 2ϑfΛx (  
)
+ gβ Tw − T∞ Θ η  
ρhnf ϑf dη3  
ϑf  
dη2 ϑf dη3  
σhnf  
d
B2 (ax  
( )  
f η ),  
−
ρhnf  
dη  
3
2
3
( )  
( ) ( )  
(
)
μhnf d f η  
a d f η d f η  
gβ Tw − T∞  
= a2x (  
+ 2Λax  
+
( )  
Θ η  
√
ρhnfϑf dη3  
ϑf dη2 dη3  
a2x  
σhnf  
d
B2  
(3.3.15)  
( )  
f η ).  
−
aρhnf dη  
Right-hand side of momentum equation transforms by substituting derivatives, incorporating viscous diffusion,  
Darcy-Forchheimer porous terms, thermal buoyancy, and magnetic field effects into simplified η-coordinate  
expression with nanofluid properties.  
Combining the left and right hand sides and equation (3.1.2) becomes  
2
d
d2  
dη2  
a2x (( f η ) − f η  
( )  
( )  
( )  
f η )  
dη  
3
2
3
( )  
( ) ( )  
(
)
μhnf d f η  
a d f η d f η  
gβ Tw − T∞  
= a2x (  
+ 2Λax  
+
( )  
Θ η  
√
ρhnfϑf dη3  
ϑf dη2 dη3  
a2x  
σhnf  
d
B2  
(3.3.16)  
( )  
f η )  
−
aρhnf dη  
2
d
d2  
dη2  
( )  
( )  
( )  
f η  
(
f η ) − f η  
dη  
3
2
3
( )  
( ) ( )  
(
)
μhnf d f η  
a d f η d f η  
gβ Tw − T∞  
( )  
Θ η  
=
−
+ 2Λax  
+
√
ρhnfϑf dη3  
ϑf dη2 dη3  
a2x  
σhnf  
d
B2  
aρhnf dη  
(3.3.17)  
( )  
f η  
Combining left and right sides yields transformed momentum equation in η-coordinates, balancing convective  
acceleration with viscous diffusion, porous medium resistance, thermal buoyancy, and magnetic damping  
effects.  
Simplified momentum equation balances inertial terms with viscous diffusion, Darcy-Forchheimer porous  
resistance, thermal buoyancy force, and Lorentz magnetic force in dimensionless similarity coordinates.  
Now, by using equations (3.2.7) and (3.2.9), we have  
σhnf  
A2σf μhnf  
A5μf  
A5  
=
,
=
=
(3.3.18)  
aρhnf  
aA4ρf ρhnfϑf  
A4ρfϑf A4  
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and the momentum equation becomes  
2
df  
d2f A5 d3f  
=
a d3f d2f gβ Tw − T∞  
A2σf  
df  
(
)
(
) − f  
+ 2Λax  
+
Θ −  
B2 (  
)
√
dη  
dη2 A4 dη3  
ϑf dη3 dη2  
a2x  
aA4ρf  
dη  
which is the same as  
2
A5 d3f  
A4 dη3  
a d3f d2f  
ϑf dη3 dη2  
df  
d2f gβ Tw − T∞  
A2σf  
df  
(
)
+ 2Λax  
− ( ) + f  
+
Θ −  
B2 ( ) = 0  
√
dη  
dη2  
a2x  
aA4ρf  
dη  
2
a d2f d3f  
)
df  
d2f gβ Tw − T∞  
A2σf  
df  
(
)
A5  
(
+ 2Λax  
− ( ) + f  
+
Θ −  
B2 ( ) = 0  
√
A4  
ϑf dη2 dη3  
dη  
dη2  
a2x  
aA4ρf  
dη  
2
A5  
A4  
d2f d3f  
)
df  
− ( ) + f  
dη  
d2f  
dη2  
A2 df  
M
(
+ 2We  
+ GrΘ −  
= 0  
(3.3.19)  
dη2 dη3  
A4 dη  
Using enhancement factor ratios simplifies momentum equation into dimensionless form with coefficients  
A₅/A₄, introducing Weissenberg number (We), Grashof number (Gr), and magnetic parameter (M) for compact  
representation.  
where  
(
)
a
gβ Tw − T∞  
σfB2  
We = Λax  
, Gr =  
, M =  
.
√
ϑf  
a2x  
aρf  
Next stage is to consider the energy equation (3.1.3)  
∂T  
∂T  
1
∂2T 16σ∗T∞3 ∂2T  
(
)
u
+ v  
−
(κhnf  
+
− Q0 T − Tw ) = 0  
(3.3.20)  
∂x  
∂y  
∂y2  
3k∗ ∂y2  
(
)
ρcp  
hnf  
and on substitutions, we have  
1
2
1
2
1
2
1
2
(
)
d
)
dη  
1
16σ∗T∞3 a Tw − T∞ d2  
−
( )  
(
( )  
Θ η ) −  
( )  
Θ η  
0 − (a ϑf f η ) (a ϑf Tw − T∞  
(κhnf  
+
)
3k∗  
ϑf  
dη2  
(
)
ρcp  
hnf  
(
)
Q0 Tw − T∞  
(
)
+
Θ − 1 = 0,  
(
)
ρcp  
hnf  
(
)
d
κhnf  
16σ∗T∞3  
1 d2  
ϑf dη2  
Q0 Θ − 1  
( )  
− (f η  
( )  
Θ η ) − (  
( )  
Θ η +  
+
)
= 0.  
∗
dη  
(
)
(
)
(
)
ρcp  
3k ρcp  
a ρcp  
hnf  
hnf  
hnf  
Weissenberg number (We) measures viscoelastic effects, Grashof number (Gr) represents buoyancy-driven  
flow, and magnetic parameter (M) quantifies electromagnetic damping strength in the system.  
Energy equation (3.3.20): Energy equation balances convective heat transfer with thermal conduction  
(including radiation effects) and heat generation/absorption, transformed into similarity coordinates using  
temperature and velocity derivatives.  
Substituting equations (3.2.6) and (3.2.8)  
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(
)
d
A1κf  
16σ∗T∞3  
d2  
dη2  
Q0 Θ − 1  
( )  
− (f η  
( )  
Θ η ) − (  
( )  
Θ η +  
+
)
= 0. (3.3.21)  
A3 ρcp ϑf 3k∗A3 ρcp ϑf  
aA3 ρcp  
dη  
(
)
(
)
(
)
f
f
f
κf  
Recall that the thermal diffusivity αf of the base fluid is defined as αf =  
and so the equation becomes  
(ρcp)f  
(
)
d
A1αf  
A3ϑf  
16σ∗T∞3  
d2  
dη2  
Q0 Θ − 1  
( )  
− (f η  
( )  
Θ η ) − (  
( )  
Θ η +  
+
)
= 0.  
(3.3.22)  
3k∗A3 ρcp ϑf  
aA3 ρcp  
dη  
(
)
(
)
f
f
Substituting enhancement factors and thermal diffusivity definition (αf = κf/(ρcp)f) simplifies energy equation  
into compact dimensionless form with ratios A₁/A₃, incorporating radiation and heat generation effects.  
Also setting  
1
αf  
4σ∗T∞3  
Q0  
=
, R =  
and Q =  
∗
Pr ϑf  
(
)
(
)
k ϑf ρcp  
a ρcp  
f
f
we have  
d
A1  
+
4
3
d2  
dη2  
Q
( )  
− (f η  
( )  
Θ η ) − (  
( )  
Θ η +  
(
)
R)  
Θ − 1 = 0,  
dη  
A3Pr  
A3  
A1  
4
d2  
dη2  
d
Q
( )  
( )  
( )  
Θ η ) −  
(
)
(
+
R)  
Θ η + (f η  
Θ − 1 = 0  
(3.3.23)  
A3Pr  
3
dη  
A3  
Prandtl number (1/Pr = αf/ϑf), radiation parameter (R), and heat generation parameter (Q) transforms energy  
equation into simplified form with A₁/A₃ ratios.  
1
1
−
2
2
Finally, we consider the boundary conditions. Based on the choice of the similarity variables as η = ya ϑf ,  
we have  
at y = 0, η = 0  
and as y → ∞, η → ∞.  
Starting at the wall where y = 0,  
df  
(
)
u 0, x = ax  
⇒
= 1 at η = 0  
dη  
(
)
)
v 0, x = 0  
⇒
⇒
f = 1 at η = 0  
Θ = 1 at η = 0  
(
T 0, x = Tw  
At the free stream,  
f′ → 0 as η → ∞  
(
)
u ∞, x → 0  
⇒
(
)
T ∞, x → T∞  
⇒
Θ → 0 as η → ∞  
Boundary conditions transform using similarity variable η. At wall (y=0, η=0): velocity matches stretching,  
no-penetration, and isothermal conditions. At free stream (η→∞): velocity and temperature gradients vanish.  
The dimensionless equations are therefore  
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2
A5  
A4  
A1  
d2f d3f  
)
df  
d2f  
dη2  
A2 df  
M
(
+ 2We  
− ( ) + f  
dη  
+ GrΘ −  
Q
= 0  
(3.3.24)  
dη2 dη3  
A4 dη  
4
3
d2  
d
( ) ( )  
Θ η + f η  
( )  
Θ η −  
(
)
(
+
R)  
Θ − 1 = 0  
(3.3.25)  
A3Pr  
dη2  
dη  
A3  
with the condition  
df  
f = 0,  
= 1, Θ = 1 at η = 0  
(3.3.26)  
dη  
f′ → 0,  
Θ → 0 as η → ∞  
(3.3.27)  
where  
(
)
a
gβ Tw − T∞  
σfB2  
We = Λax  
, Gr =  
, M =  
(3.3.28)  
√
ϑf  
a2x  
aρf  
κf  
1
αf  
4σ∗T∞3  
Q0  
αf =  
,
=
, R =  
, Q =  
(3.3.29)  
∗
Pr ϑf  
(
)
(
)
(
)
ρcp  
k ϑf ρcp  
a ρcp  
f
f
f
Dimensionless governing equations (3.3.24-3.3.25) couple momentum and energy with boundary conditions  
(3.3.26-3.3.27). Parameters We, Gr, M, Pr, R, Q characterize viscoelasticity, buoyancy, magnetism, thermal  
diffusion, radiation, and heat generation respectively.  
Thermophysical Properties  
The nanoparticles under study are that made from Al₂O₃ and Cu and the choice of base-fluid is molten  
polyethylene (a Maxwell fluid). The thermophysical properties of the base-fluid and that of the nanoparticles  
are obtained from literature and are recorded in table (3.1).  
Table 3.1: Thermophysical properties  
cp  
765  
ρcp  
κ
ρ
3970  
8933  
μ
40  
400  
3037050  
3439205  
2709450  
-
-
Al2O3  
Cu  
385  
molten polyethylene  
0.253 1115  
2430  
18.376  
By substituting these values into equations (3.2.6 - 3.2.9), we have the following  
)(  
(
)
(
)
0.506 1 − ϕ ϕ + 1 + 2ϕ 34.5ϕ1 + 1.2ϕ2  
A1 =  
,
(
)
(
)( )  
0.253 2 + ϕ ϕ + 1 − ϕ 34.5ϕ1 + 1.2ϕ2  
−5  
7
(
)
(
)(  
)
21.4 1 − ϕ ϕ × 10 + 1 + 2ϕ 6.3ϕ1 + 4.25ϕ2 × 10  
A2 =  
,
−5  
7
(
)
(
)(  
)
10.7 2 + ϕ ϕ × 10 + 1 − ϕ 6.3ϕ1 + 4.25ϕ2 × 10  
(
(
)
2011147ϕ1 + 1546600ϕ2  
A3 = 1 − ϕ +  
,
2709450  
5060ϕ1 + 2200ϕ2  
)
A4 = 1 − ϕ +  
.
1115  
Substituting nanoparticle properties into equations (3.2.6-3.2.9) yields explicit enhancement factors A₁ through  
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A₄, representing thermal conductivity, electrical conductivity, heat capacity, and density modifications from  
nanoparticle volume fractions ϕ₁ and ϕ₂.  
Numerical Procedure  
The coupled PDEs governing the MHD flow of hybrid nanofluid is reformulated as a coupled system of ODEs  
through the use of similarity transformation. The resulting the ODEs comes with some boundary conditions at  
the free stream and some initial conditions at the boundary layer. This kind of problem cannot be solved by  
simply adopting a numerical procedure due to the inclusion of boundary conditions. The problem concerning  
the boundary conditions is solved by bringing in the method of Shooting Technique; which seeks the initial  
condition that best satisfies the boundary condition. The Runge Kutta method is used to solve the coupled  
ODEs with the initial conditions. The numerical results are graphed and the results are discussed herewith.  
Analysis of Results  
Introduction  
The parameters that emerged from the nondimensionalisation process are varied to simulate the flow. By  
varying a parameter while fixing other parameters, the profiles for velocity and temperature of the flow are  
plotted against the dimensionless distance η. In any case a parameter is fixed, the following values are chosen  
as the default for the parameters;  
We = 0.1, Gr = 1, M = 2, Pr = 7, Q = 0.21, R = 0.1, ϕ1 = ϕ2 = 0.1.  
Recall that the flow velocity is similar to f′ and temperature is similar to Θ, hence, in the following discussion,  
the same notations shall be retained for these flow properties.  
RESULTS AND DISCUSSION  
The parameters that emerged from the nondimensionalisation process are varied to simulate the flow. By  
varying a parameter while fixing other parameters, the profiles for velocity and temperature of the flow are  
plotted against the dimensionless distance η. In any case a parameter is fixed, the following values are chosen  
as the default for the parameters;  
We = 0.1, Gr = 1, M = 2, Pr = 7, Q = 0.21, R = 0.1, ϕ1 = ϕ2 = 0.1.  
Recall that the flow velocity is similar to f′ and temperature is similar to Θ, hence, in the following discussion,  
the same notations shall be retained for these flow properties.  
Effect of Grashof Number  
The Grashof number Gr measures the ratio of buoyancy to viscous forces, often arising in natural convection  
and defined in this study as  
(
)
gβ Tw − T∞  
Gr =  
.
a2x  
The behaviour of temperature with increasing Grashof number is shown in figure (4.1) while the behaviour of  
velocity is illustrated in figure (4.2). The figures illustrated a decrease in temperature and an increase in  
velocity with Grashof number. A rise in Grashof number is a consequence of increasing buoyancy force which  
enhances flow velocity but reduces thermal boundary layer. Hence, as Grashof number rises, temperature goes  
down and velocity goes up.  
Figure 4.1: Effects of temperature to Grashof number  
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Figure 4.2: Effects of Velocity to Grashof number  
Effect of Magnetic field  
A magnetised hybrid nanofluid flow experiences an opposing force called Lorenz force. This implies that  
Lorenz force becomes stronger as magnetism increases and therefore, the flow experiences more opposition. In  
this study, the magnitude of magnetism is obtained as  
σfB2  
M =  
.
aρf  
We increase M, consequently increasing Lorenz force, and study the response of temperature and velocity to  
increasing magnetism. Figures (4.3) and (4.4) display the behaviours. The stronger the magnetism, the higher  
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the temperature becomes but the lower the velocity. Fluid motion get impeded by the Lorenz force and thereby  
slow down the fluid particles, causing a reduction in velocity. The drag in the flow produce thermal energy  
which increases the temperature if the flow, hence an increase in flow temperature.  
Figure 4.3: Effects of Temperature to Magnetism M on Temperature Profile Θ(η)  
Higher magnetic parameters induce Joule heating through electrical resistance, elevating fluid temperature.  
Electromagnetic work dissipates as thermal energy, increasing Θ throughout. Arrows indicate temperature  
enhancement with stronger fields.  
Figure 4.4: Effects of Temperature to Magnetism M on Velocity Profile f'(η)  
Increasing M strengthens Lorentz force, opposing fluid motion and retarding momentum diffusion. This  
creates steeper velocity gradients near the wall and thinner momentum boundary layers, as arrows indicate.  
Effect of Volume Fraction  
The volume fractions for the nanoparticles are ϕ1 and ϕ2, where 1 represents alumina (Al₂O₃) nanoparticles and  
2 represents copper (Cu) nanoparticle. In this study, we have considered only the case where the two volume  
fractions are equal ϕ1 = ϕ2. Figure (4.5) shows the behaviour of the temperature of flow as ϕ1 and ϕ2 increase.  
The temperature increases as ϕ1 and ϕ2 get bigger. Increasing ϕ1 and ϕ2 increases the surface area of the solid  
particles and thereby allowing quick exchange of heat energy. This is responsible for the rise in temperature as  
ϕ1 and ϕ2 increase. However, figure (4.6) shows that velocity decreases with increasing ϕ1 and ϕ2. Due to the  
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increase in ϕ1 and ϕ2, nanoparticles agglomeration tends to cause a retardation in the flow and thereby reducing  
velocity as ϕ1 and ϕ2 gets larger.  
Figure 4.5: Effects of Temperature to Volume fractions  
Figure 4.6: Effects of Velocity to Volume fractions  
CONCLUSION AND RECOMMENDATION  
Conclusion  
This study considers the flow of fluid obtained by releasing two nanoparticles of different solid materials in the  
molten polyethylene. The flow is represented in mathematical form as a coupled PDE with some initial-  
boundary condition, which is reformulated as a coupled ODEs by the use of similarity transformation. We  
employed the shooting technique to find an analogous the initial value problem to the initial-boundary  
condition problem. By varying a parameter while keeping other parameter fixed, the flow is simulated and the  
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following results were obtained;  
Decrease in temperature and increase in velocity with Grashof number: As the Grashof number  
increases, indicating stronger buoyancy forces relative to viscous forces, the temperature decreases. This  
is because the fluid experiences less resistance from buoyancy, allowing it to flow more freely and  
increase in velocity.  
Magnetism raises temperature but lowers velocity: The introduction of magnetism increases the fluid  
temperature due to heat generation from magnetic nanoparticles. However, it also decreases fluid  
velocity, possibly due to altered flow patterns or increased frictional forces.  
Temperature increases but velocity decreases as volume fraction increases: Higher nanoparticle  
concentration leads to increased fluid temperature due to enhanced thermal conductivity or heat  
generation. However, the fluid velocity decreases as nanoparticles impede flow, causing greater  
resistance.  
Industrial Applications  
These findings directly inform:  
Heat exchanger optimization: Tailoring nanoparticle concentration and magnetic fields maximizes  
heat transfer while minimizing pressure drop  
Polymer extrusion control: Adjusting thermal and flow parameters improves product uniformity and  
reduces defects  
Thermal management systems: Hybrid nanofluids offer superior cooling performance for electronics  
and automotive applications  
Magnetorheological processing: Magnetic field control enables real-time adjustment of flow and  
thermal characteristics  
Recommendation  
Based on the findings of this study, the following are recommended:  
1. There is a crucial need for experimental validation of the simulation results. Conducting rigorous  
experiments would not only validate the accuracy of the model but also provide a real-world context for  
understanding the observed phenomena. By comparing simulation results with experimental data,  
researchers can gain confidence in the predictive capabilities of the model and refine it further.  
2. Exploring the effects of varying nanoparticle properties, such as size, shape, and surface characteristics,  
could yield valuable insights. Understanding how these factors influence fluid behaviour and thermal  
dynamics could lead to the development of optimized nanoparticle designs for specific applications. This  
avenue of research has the potential to unlock new possibilities in areas such as heat transfer  
enhancement and nanofluid-based technologies.  
3. In addition to nanoparticle properties, investigating the impact of external factors on the system is  
essential. Factors such as pressure variations, different fluid compositions, or external fields could  
significantly influence fluid flow and temperature distribution. Exploring these factors could not only  
expand the scope of potential applications but also enhance our understanding of the system's robustness  
and adaptability.  
REFERENCES  
1. Ahmed, F., Reza-E-Rabbi, S., Ali, M. Y., Ali, L. E., Islam, A., Rahman, M. A., ... & Ahmmed, S. F.  
(2024). Numerical modeling of a MHD non-linear radiative Maxwell nano fluid with activation  
energy. Heliyon, 10(2).  
2. Alfvén, H. (1942). Existence of Electromagnetic-Hydrodynamic Waves. Nature, 150(3805):405–406.  
3. Ali, A., Kanwal, T., Awais, M., Shah, Z., Kumam, P., and Thounthong, P. (2021). Impact of thermal  
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radiation and non-uniform heat flux on MHD hybrid nanofluid along a stretching cylinder. Scientific  
Reports, 11(1).  
4. Ali, B., Ahammad, N. A., Awan, A. U., Oke, A. S., Tag-ElDin, E. M., Shah, F. A., & Majeed, S. (2022).  
The dynamics of water-based nanofluid subject to the nanoparticle’s radius with a significant magnetic  
field: The case of rotating micropolar fluid. Sustainability, 14(17), 10474.  
5. Allehiany, F. M., Bilal, M., Alfwzan, W. F., Ali, A., and Eldin, S. M. (2023). Numerical solution for the  
electrically conducting hybrid nanofluid flow between two parallel rotating surfaces subject to thermal  
radiation. AIP Advances, 13(7).  
6. Algehyne, E. A., Alamrani, F. M., Khan, A., Khan, K. A., Lone, S. A., & Saeed, A. (2024). On thermal  
distribution of MHD mixed convective flow of a Casson hybrid nanofluid over an exponentially  
stretching surface with impact of chemical reaction and ohmic heating. Colloid and Polymer  
Science, 302(4), 503-516.  
7. Asghar, A., Lund, L. A., Shah, Z., Vrinceanu, N., Deebani, W., and Shutaywi, M. (2022). Effect of  
thermal radiation on three-dimensional magnetized rotating flow of a hybrid nanofluid. Nanomaterials,  
12(9):1566.  
8. Choi, U. S. and Eastman, J. (1995). Enhancing Thermal Conductivity of Fluids with Nanoparticles.  
ASME International Mechanical Congress and Exposition, San Francisco, (12–17).  
9. Farooq, U., Lu, D., Munir, S., Ramzan, M., Suleman, M., and Hussain, S. (2019). MHD flow of  
maxwell fluid with nanomaterials due to an exponentially stretching surface. Scientific Reports, 9(1).  
10. Fatunmbi, E. O., Badru, J. O., & Oke, A. S. (2020). Magneto-Reactive Jeffrey Nanofluid Flow over a  
Stretching Sheet with Activation Energy, Nonlinear Thermal Radiation and Entropy Analysis.  
11. Govindarajulu, K., & Subramanyam Reddy, A. (2022). Magnetohydrodynamic pulsatile flow of third  
grade hybrid nanofluid in a porous channel with Ohmic heating and thermal radiation effects. Physics of  
Fluids, 34(1).  
12. Hady, F. M., Ibrahim, F. S., Abdel-Gaied, S. M., and Eid, M. R. (2012). Radiation effect on viscous  
flow of a nanofluid and heat transfer over a nonlinearly stretching sheet. Nanoscale Research Letters,  
7(1).  
13. Jaafar, A., Waini, I., Jamaludin, A., Nazaar, R., and Pop, I. (2022). Mhd flow and heat transfer of a  
hybrid nanofluid past a nonlinear surface stretching/shrinking with effects of thermal radiation and  
suction. Chinese Journal of Physics, 79:13–27.  
14. Jayaprakash, J., Govindan, V., Santra, S. S., Askar, S. S., Foul, A., Nandi, S., & Hussain, S. M. (2024).  
Thermal radiation, Soret and Dufour effects on MHD mixed convective Maxwell hybrid nanofluid flow  
under porous medium: a numerical study. International Journal of Numerical Methods for Heat & Fluid  
Flow, 34(10), 3924-3952.  
15. Khan, N., Ali, F., Ahmad, Z., Murtaza, S., Ganie, A. H., Khan, I., & Eldin, S. M. (2023). A time  
fractional model of a Maxwell nanofluid through a channel flow with applications in grease. Scientific  
Reports, 13(1), 4428.  
16. Kigio, J. K., Mutuku, W. N., & Oke, A. S. Analysis of volume fraction and convective heat transfer on  
MHD Casson nanofluid over a vertical plate, Fluid Mechanics, 7 (2021), 1–8.  
17. Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Nature, 7:478–480.  
18. Rajesh, V., Srilatha, M., and Chamkha, A. J. (2020). Hydromagnetic effects on hybrid nanofluid (cu–  
aluminium-water) flow with convective heat transfer due to a stretching sheet. Journal of Nanofluids,  
9(4):293–301.  
19. Sakiadis, B. C. (1961). Boundary-layer behavior on continuous solid surfaces: I. boundary-layer  
equations for two-dimensional and axisymmetric flow. AIChE Journal, 7(1):26–28.  
20. SAMUEL, O. A. (2022). Coriolis Effects on Air, Nanofluid and Casson Fluid Flow Over a Surface with  
Nonuniform Thickness. PhD diss., KENYATTA UNIVERSITY  
21. Suresh, S., Venkitaraj, K., Selvakumar, P., and Chandrasekar, M. (2011). Synthesis of al2o3–cu/water  
hybrid nanofluids using two step method and its thermo physical properties. Multidisciplinary Digital,  
388(1-3):41–48.  
22. Vijay, N., & Sharma, K. (2023). Dynamics of stagnation point flow of Maxwell nanofluid with  
combined heat and mass transfer effects: A numerical investigation. International Communications in  
Page 2133  
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INTERNATIONAL JOURNAL OF RESEARCH AND INNOVATION IN APPLIED SCIENCE (IJRIAS)  
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Heat and Mass Transfer, 141, 106545.  
23. Wang, F., Jamshed, W., Ibrahim, R. W., Abdalla, N. S. E., Abd-Elmonem, A., & Hussain, S. M. (2023).  
Solar radiative and chemical reactive influences on electromagnetic Maxwell nanofluid flow in  
Buongiorno model. Journal of Magnetism and Magnetic Materials, 576, 170748.  
24. Zainal, N. A., Nazar, R., Naganthran, K., and Pop, I. (2022). The impact of thermal radiation on  
maxwell hybrid nanofluids in the stagnation region. Multidisciplinary Digital Publishing Institute.  
The Matlab Code  
clc; clear all; format compact  
global We Gr M Pr R Q phi mu_f phi kappa sigma rho cp  
mu_f = 18.376; We = 0.1; Gr = 1; M = 2; Pr = 7; Q = 0.21; R = 0.1;  
phi = [0.1, 0.1]; tspan=linspace(0,7,500); x_initial = zeros(1,5); i=1;  
%nanoparticles are Al2O3 and Cu  
%base fluid is molten polyethylene  
kappa = [40, 400, 0.253]; sigma = [6.3e7,4.25e7,10.7e-5];  
rho = [3970, 8933, 1115]; cp = [765, 385, 2430];  
Legend_Entries = []; txt_var = "phi"; Line_Style = ["k-", "g-", "r-", "b-"];  
for Val = [0.1, 0.15, 0.2, 0.25]  
phi = [Val, Val];  
solinit=bvpinit(tspan,x_initial);  
sol = bvp4c(@Fluid,@Bc,solinit);  
t = sol.x; s = sol.y;  
%Legend_Entries = [Legend_Entries,strcat(txt_var," = ", num2str(Val))];  
Legend_Entries = [Legend_Entries,strcat("\phi_1 = \phi_2 = ", num2str(Val))];  
txt = Line_Style(i);  
if i ~= 4  
figure(2), plot(t,s(2,:),txt,'LineWidth',2)  
hold on  
figure(4), plot(t,s(4,:),txt,'LineWidth',2)  
hold on  
elseif i == 4  
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figure(2), plot(t,s(2,:),txt,'LineWidth',2)  
xlabel(" dimensionless distance \eta")  
ylabel("velocity, f^\prime(\eta)")  
legend(Legend_Entries); legend('boxoff')  
txt_vel = strcat(txt_var,"_velocity");  
saveas(gcf,txt_vel,'fig')  
figure(4), plot(t,s(4,:),txt,'LineWidth',2)  
xlabel(" dimensionless distance \eta")  
ylabel("temperature, \Theta(\eta)")  
legend(Legend_Entries); legend('boxoff')  
txt_temp = strcat(txt_var,"_temperature");  
saveas(gcf,txt_temp,'fig')  
end  
i=i+1;  
end  
function res = Fluid(eta,x)  
global We Gr M Pr R Q phi mu_f phi kappa sigma rho cp  
f = x(1); f_p = x(2); f_pp = x(3); theta = x(4); theta_p = x(5);  
PHI = sum(phi);  
A1_num = 2*(1-PHI)*PHI*kappa(3) + (1 + 2*PHI)*sum(phi.*kappa(1:2));  
A1_den = (2+PHI)*PHI*kappa(3) + (1 - PHI)*sum(phi.*kappa(1:2));  
A1 = A1_num/A1_den;  
A2_num = 2*(1-PHI)*PHI*sigma(3) + (1 + 2*PHI)*sum(phi.*sigma(1:2));  
A2_den = ((2+PHI)*PHI*sigma(3) + (1 - PHI)*sum(phi.*sigma(1:2)));  
A2 = A1_num/A1_den;  
% A2 = (1-PHI + sum(phi.*sigma(1:2))/sigma(3));  
A3 = 1 - PHI + sum(phi.*rho(1:2).*cp(1:2))/(rho(3)*cp(3));  
A4 = 1 - PHI + sum(phi.*rho(1:2))/rho(3);  
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A5 = 0.904*exp(0.148*PHI);  
dx1 = x(2); dx2 = x(3);  
dx3 = (f_p^2 - f*f_pp - Gr*theta + (A2/A4)*M*f_p)/((A5/A4) + We*f_pp);  
dx4 = x(5);  
dx5 = (-f*theta_p + Q*(theta - 1)/A3)/(A1/(A3*Pr) + (4*R/3));  
res = [dx1, dx2, dx3, dx4, dx5];  
end  
function res = Bc(y0,yinf)  
global We Gr M Pr R Q phi mu_f phi kappa sigma rho cp  
res = [y0(1)  
y0(2)-1  
y0(4)-1  
yinf(2)  
yinf(4)];  
end  
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