Numerical Examination of a Squeezing Casson Hybrid Nanofluid Flow
Considering Thermophoretic and Internal Heating Mechanisms
AMINE EL HARFOUF1, *, RACHID HERBAZI2,3,4, WALID ABOULOIFA1,
SANAA HAYANI MOUNIR1, HASSANE MES-ADI5, ABDERRAHIM WAKIF6,
MOHAMED MEJDAL 1, MOHAMED NFAOUI 1
1Multidisciplinary Laboratory of Research and Innovation (LaMRI),
Energy, Materials, Atomic and Information Fusion (EMAFI) Team,
Polydisciplinary Faculty of Khouribga,
Sultan Moulay Slimane University,
MOROCCO
2Intelligent Systems and Applications Laboratory (LSIA), EMSI,
Tangier,
MOROCCO
3ENSAT, Abdelmalek Essaâdi University,
Tangier,
MOROCCO
4ERCMN, FSTT, Abdelmalek Essaâdi University,
Tangier,
MOROCCO
5Laboratory of Process Engineering, Computer Science and Mathematics,
National School of Applied Sciences of the Khouribga University of Sultan Moulay Slimane,
MOROCCO
6Faculty of Sciences Aïn Chock, Laboratory of Mechanics,
Hassan II University,
Casablanca,
MOROCCO
*Corresponding Author
Abstract: - One of the main areas of study in the field is increasingly the flow of non-Newtonian fluids. These
liquids find extensive use in nuclear reactors, food processing, paint and adhesives, drilling rigs, and cooling
systems, among other industrial and engineering domains. However, hybrid nanofluids are crucial to the
process of heat transfer. Considering this, this study investigates the motion of a Casson hybrid nanofluid
squeezing flow between two parallel plates under the influence of a heat source and thermophoretic particle
deposition. The Runge–Kutta–Fehlberg fourth–fifth-order approach is utilized to numerically solve the ordinary
differential equations derived from the partial differential equations governing fluid flow, by utilizing suitable
similarity variables. The diagrams show how several important parameters affect fluid profiles both with and
without the Casson parameter. These figures demonstrate how fluid velocity increases as the local porosity
parameter increases. When the heat source/sink parameter is increased, thermal dispersal increases, and when
the thermophoretic parameter is increased, the concentration profile increases.
Key-Words: - Casson fluid, heat source/sink, hybrid nanofluid, parallel plates, thermophoretic particle
deposition, The Runge–Kutta, non-Newtonian fluids.
Received: May 9, 2023. Revised: February 8, 2024. Accepted: March 6, 2024. Published: April 22, 2024.
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Amine El Harfouf, Rachid Herbazi,
Walid Abouloifa, Sanaa Hayani Mounir,
Hassane Mes-Adi, Abderrahim Wakif,
Mohamed Mejdal, Mohamed Nfaoui
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1 Introduction
The nanofluid, which is a novel sort of heat transfer
fluid, is created by suspending a single type of
oxide and metallic nanoparticles, as well as
nonmetallic carbon nanotubes, in carrier liquids
including water, ethylene glycol, and oil, with a
size of less than (). In comparison
to heat transfer liquids, the nanoliquid created by
the solid nanoparticles suspended in the base
liquids would have a higher thermal conductivity.
These solid nanoparticles also possess favorable
thermophysical properties. Additionally, nanofluids
are widely used in a wide range of industrial and
biomedical processes, including the production of
glass fiber, metal spinning, the removal of tumors
that cause hyperthermia, lubricant, the treatment of
asthma, cable drawing, electronic devices, nuclear
reactors, immunological synergy, chilling process,
transportation, and power generation. Because of
this, many researchers are interested in studying the
flow of nanofluids over various geometries when
various physical and chemical processes are
present. The notion of nanofluid was first
introduced in [1], who also illustrated the physical
characteristics of nanoparticles. The stagnation
point stream of nanoliquid across a stretching
cylinder was investigated by [2]. A study on the
effect of activation energy in the presence of a fluid
stream and the suspension of nanoparticles [3].
However, nanofluids were unable to provide the
high heat transfer rate that large-scale
manufacturing companies demanded, and this is
where the heat transfer process became extremely
complex.
To overcome this limitation, hybrid nanoliquids
are used instead of fluid suspended with single-kind
nanoparticles. Different forms of liquids are
produced when numerous kinds of minute
nanoparticles come together to form hybrid
nanoliquids. These fluids are used in solar energy
storage applications, the automotive industry, brake
fluids for vehicles, and tubular heat exchangers.
The numerical simulation of a water-based hybrid
nanofluid flow over a curved stretching sheet,
considering the Newtonian heating effect, was
recently explained in [4]. Through an annulus, [5]
examined the effects of radiation on the hybrid
nanofluid stream. Information on the hybrid
nanoliquid stream passing through a cylinder [6].
[7] discussed the impact of slippage on a water-
based hybrid nanoliquid stream flowing across a
curved surface. Recently, examination of how
hybrid nanoliquid flow behaves when it passes over
an elastic sheet [8].
One important component of a liquid’s flow
across a media is its fluid rheology, which may be
divided into two primary classes: Newtonian and
non-Newtonian. The essential qualification
between the non-Newtonian and Newtonian fluid
models is the utilitarian relationship between shear
push and shear rate. In differentiating to non-
Newtonian fluids, Newtonian fluids don’t show
abdicate push. These fluids are widely used in
nuclear reactors, food processing, paint and
adhesives, drilling rigs, and cooling systems,
among other industrial and engineering domains.
One of the most significant rheological non-
Newtonian fluid models used in the production of
biological fluids, paints, and pharmaceuticals is the
Casson fluid. Casson fluid is widely used in the
drilling, food processing, and metallurgy industries,
which makes rheological research on it crucial.
The authors in [9] investigated the MHD flow
of nanofluids between parallel plates. The
investigation of the incompressible flow of a
nanofluid between parallel plates using ohmic
heating settings was also explored [10].
Thermophoresis is the transfer of tiny particles
from a high-temperature to a low-temperature
environment. This phenomenon has various
practical applications, such as following the
trajectories of exhaust gas particles from burning
devices, collecting microscopic particles from gas
flows, and studying the deposition of particulate
matter on turbine blades. Several scholars have
investigated this phenomenon over the previous
few decades, taking into consideration various
biological repercussions. The authors [11]
examined the thermophoretic deposition of aerosol
particles on a liquid stream passing over a cylinder.
[12] wrote about particle deposition on the axial
stream of liquid over a cylinder. In literature [13].
Investigated the key aspects of thermophoretic
particle deposition and the Soret-Dufour impact on
the fluid stream over a revolving disk.
The research cited above leads one to the
conclusion that there hasn’t been much discussion
of Casson hybrid nanoliquid flow between two
parallel plates in the presence of particle deposition
and a heat source or sink. Therefore, under the
influence of thermophoretic particle deposition and
a heat source or sink, the stream of non-Newtonian
Casson liquid containing hybrid ferrite
nanoparticles between
two parallel plates is examined in this work.
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2 Problem Description and
Formulation Mathematics
Here, we study the time-dependent flow of a
Casson hybrid nanofluid between two parallel
horizontal plates. Two ferrite nanoparticles,
, are added to the base
liquid, , to create the hybrid nanofluid.
Together, the plates and liquid rotate at an angular
velocity along the y-axis, which is normal to the
plates (Figure 1). The upper plate is located at
and the subordinate plate is
positioned at , where it is stretched by two
opposing and equal forces. Where the initial
location (at time ) is indicated by . When
, the plates are compressed until ,
and when α < 0, the two plates are separated. As a
result, the point (0,0,0) stays in the same place. The
governing equations are given below with these
observations:
(1)
(2)
(3)
(4)
(5)
The following are the related boundary conditions:
(6)
Thermophoretic velocity is given by
,
here, is reference temperature and is
thermophoretic coefficient.
The similarity variables are mentioned below:
(7)
Equation (7) can be substituted into Equations (2)
and (3), and the pressure gradient can then be
removed from the resultant equations to get:
(8)
Equation (7) helps to reduce the equations from (4)
to (5). The following are the converted ordinary
differential equations (ODEs):
(9)
(10)
Here are the diminished boundary conditions:
(11)
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Here
is the squeeze number,
is
the Prandtl number,
is the heat source
parameter
is Schmidt number,
is thermophoretic parameter.
The following gives the thermophysical properties:
Equation (7) and (12) are used to reduce Equation
(13)
Fig 1: Geometry of the physical problem.
The definitions of the skin friction coefficient,
Sherwood number, and Nusselt number are as
follows:
3 Outcome and Discussion
This section includes a brief explanation for better
comprehension as well as a graphic representation
of the nature of fluid profiles for several relevant
parameters. For both the Casson hybrid nanoliquid
and the hybrid nanoliquid ,
the key effects of the porosity parameter, squeezing
number, heat source or sink parameter, and
thermophoretic parameter on fluid profiles are
covered in detail. Graphs are also used to assess the
fluctuation in the rate of mass and heat transfer.
Table 1 lists the thermophysical characteristics of
the nanoparticles , , and
. By applying the appropriate similarity
transformations, the flow’s governing equations are
converted into ODEs. Using the Runge–Kutta–
Fehlberg fourth–fifth order (RKF 45) approach, a
transformed collection of ODEs are solved. To
validate the current result , Table 2 is
produced using the published outcomes of [14].
when , and are not present.
The velocity profile is shown in Figure 2
for a range of squeezing number S values. This
graph shows that, for both and ,
dramatically decreases with increased values
of for and rises noticeably for .
Table 1. Thermophysical properties of , , and
Particles
(14)
(12)
(13)
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Table 2. Validation of the problem with existing work of, [14]. for different values of S when
and
Fig. 2: Effect of on
The influence of S on the temperature profile
is explained in Figure 3 for both and
. In this example, has a declining
trend when S is increased, and it rapidly decreases
when S is increased for both the fluid cases where
and . From the physical point of
view, the thickness of the thermal boundary layer
upsurges for the absolute magnitude of the squeeze
number due to the simultaneous movement of two
plates. Physically, an enhancement of has a
significant effect on the velocity profile. When is
less than , the velocity in the squeezing flow
increases because of a rise in the absolute value of
, whereas it decreases when is greater than .
Moreover, the temperature profile is reduced when
’s magnitude increases. This is because the
velocity profile decreases toward the lower plate as
the distance between the dual plates gets smaller as
the positive values increase.
Figure 4 illustrates the significant influence of
heat source or sink parameter on temperature
profile θ (η). For better values of , in both
scenarios and , the
increases. Positive and negative indications of Hs
correspond to the flow’s heat generation and
absorption, respectively. The energy released in the
flow field during heat generation causes the
temperature profile to improve.
Squeezing number
Ref [14]
Current results
4.167389
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Fig. 3: Effect of on
Fig. 4: Effect of on
The behavior of the concentration profile
for increased values of is shown in Figure 5 for
both the scenarios when and . It
reveals that with better values of , drops
quickly. In Figure 6, the effect of on is
shown. This graph shows that for both
and , increases as rises.
Figure 7 illustrates how behaves similarly
for enlarged values of Sc in the scenarios when
and .
Figure 8 shows the change in the solid volume
fraction over . The picture shows that
when , increases for upgrading the
solid volume percentage of , but continues
to decrease when values drop from to .
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Fig. 5: Effect of on
Fig. 6: Effect of on
Fig. 7: Effect of on
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Fig. 8: Effect of on
Fig. 9: Effect of on
As the boundary layer thickness increases, the
hybrid fluid’s velocity increases when and
decreases when , which corresponds to the
Casson hybrid fluid. Figure 9 shows the change in
the solid volume fraction over . The rate of
heat transmission is accelerated by the addition of
. Compared to other hybrid nanoliquids, Casson
hybrid nanoliquid has a higher rate of heat
transmission. Figure 10 shows the fluctuation of the
solid volume fraction over . Because the
Casson parameter has been added, the
concentration in the Casson hybrid nanoliquid is
higher than in the hybrid nanoliquid.
Table 3, Table 4 and Table 5 show the behavior
of mass, heat, and momentum transport rates for
various values of the flow parameters. Table 3
shows that the momentum transport coefficient
() rises when the plate is moving toward
one another and falls when the plate is moving
away from one another. It is also observed that the
skin-friction coefficient near the wall lowers when
the plates separate due to the increasing value of .
Table 4 provides an illustration of how physical
characteristics affect the heat transfer rate ().
Table 4 makes it evident that the Nusselt number
rises as the squeezing number’s magnifying values
do. This results from the fluid in the channel being
less viscous.
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Fig. 10: Effect of on
Table 3. Local skin-friction coefficient for various values of and
Table 4. Local Nusselt number coefficient for various values of and
-1.50
3.00
4.732768497
-0.50
6.167692445
0.20
7.001042863
1.50
8.311880921
2.50
9.175911670
4.00
10.31304423
2.00
2.00
9.632617717
2.50
9.107675851
3.00
8.756605950
3.50
8.505227354
5.00
8.051288352
-0.30
0.10
0.1126336986
-0.10
0.1089894740
0.20
0.1039879097
0.50
0.0994707971
1.50
0.0871236816
2.00
0.20
0.1737913056
0.30
0.2771814053
0.40
0.3953297053
0.50
0.5323996434
0.60
0.6943066029
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Table 5. Local Sherwood number coefficient for various values of and
-0.30
0.50
1.50
1.842767138
-0.10
1.675170790
0.20
1.474330027
1.50
0.962905129
2.50
0.751230420
2.00
0.70
1.023652562
0.80
1.072418601
0.90
1.095900128
0.91
1.096956906
0.92
1.097791699
0.50
-1.40
-0.903792841
-0.50
-0.314212136
1.50
0.845402744
2.00
1.079439671
2.50
1.280753231
In the same way, Table 5 shows how different
control settings affect the mass transfer rate ()
in the flow region. Table 5 demonstrates
unequivocally that the mass transmission rate at the
wall falls as increases. The primary cause of this
is the channel’s construction. Additionally, it is
observed that as Sc levels rise, mass transfer rate
rises as well. Furthermore, because it is
constructive, () decreases for
and grows for .
4 Conclusion
The current study examines the flow behavior of a
Casson hybrid nanofluid between two parallel
plates under the influence of a heat source or sink
and thermophoretic particle deposition. Appropriate
similarity variables are used to convert the
governing equations representing the fluid flow
into nonlinear ODEs, which are then numerically
solved using the RKF 45 method. Consequently,
the effects of physical parameters on the distinct
fluid profiles are depicted via graphs. The
significant conclusions of the present inquiry are as
follows:
For both Casson hybrid nanofluid and
conventional hybrid nanoliquid, fluid velocity
decreases with increased squeezing number
values for η < 0.5 and increases for η > 0.5;
however, a reversal tendency is observed for
varying local porosity parameter values.
The temperature profile decreases as the
squeezing number increases, but it increases for
higher heat source/sink parameter values in
both the Casson hybrid nanofluid and hybrid
nanoliquid situations.
For both Casson hybrid nanofluid and ordinary
hybrid nanoliquid, the concentration profile
exhibits the opposite behavior with increased
values of squeezing number, but it greatly
improves with an increase in thermophoretic
parameter and Schmidt number.
An increase in the thermophoretic parameter
over the Schmidt number causes the rate of
mass transfer to accelerate.
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Hassane Mes-Adi, Abderrahim Wakif,
Mohamed Mejdal, Mohamed Nfaoui
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
A. EL Harfouf: Conceptualization, Formal
analysis, Investigation, Methodology, Project
administration, Resources, Validation, Writing –
original draft, Data curation, Software,
Visualization.
R. Herbazi: Conceptualization, Formal analysis,
Investigation, Methodology, Project administration,
Resources, Validation, Writing – review & editing.
W. Abouloifa: Conceptualization, Investigation,
Writing – review & editing. H. Mes-adi:
Conceptualization, Formal analysis, Investigation,
Methodology, Project administration, Resources,
Validation, Writing – review & editing. A. Wakif:
Conceptualization, Investigation, Project
administration, Supervision, Writing – review &
editing. M. Mejdal: Conceptualization, Formal
analysis, Investigation, Methodology, Project
administration, Resources, Validation, Writing –
review & editing. M. Nfaoui: Conceptualization,
Investigation, Project administration, Writing –
review & editing. S. Hayani Mounir:
Conceptualization, Investigation, Project
administration, Supervision, Writing – review &
editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
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