Unsteady Compressed Williamson Fluid Flow Behavior under the
Influence of a Fixed Magnetic Field (Numerical Study)
AMINE EL HARFOUF1, *, RACHID HERBAZI2,3,4, SANAA HAYANI MOUNIR1,
HASSANE MES-ADI5, ABDERRAHIM WAKIF6
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: - A numerical investigation is conducted into a two-dimensional mathematical model of magnetized
unsteady incompressible Williamson fluid flow over a sensor surface with fixed thermal conductivity and
external squeezing accompanied by viscous dissipation effect. Based on the flow geometry under consideration,
the current flow model was created. The momentum equation takes into consideration the magnetic field when
describing the impact of Lorentz forces on flow behavior. The energy equation takes varying thermal
conductivity into account while calculating heat transmission. The extremely complex nonlinear, unstable
governing flow equations for the now under investigation are coupled in nature. Due to the inability of
analytical or direct methods, the Runge-Kutta scheme (RK-4) via similarity transformations approach is used to
tackle the physical problem under consideration. The physical behavior of various control factors on the flow
phenomena is described using graphs and tables. For increasing values of the Weissenberg parameter and the
permeable velocity parameter, the temperature boundary layer thickens. As the permeable velocity parameter
and squeezed flow index increased, the velocity profile shrank. The velocity profile grows as the magnetic
number rises. Squeezed flow magnifying increases the Nusselt number's magnitude. Furthermore, the
extremely complex nonlinear complex equations that arise in fluid flow issues are quickly solved by RK-4. The
current findings in this article closely align with the findings that have been reported in the literature.
Key-Words: - thermal conductivity, Williamson fluid, sensor surface, magnetic field, Weissenberg number.
Received: February 13, 2023. Revised: November 26, 2023. Accepted: December 26, 2023. Published: February 29, 2024.
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.8
Amine El Harfouf, Rachid Herbazi,
Sanaa Hayani Mounir, Hassane Mes-Adi,
Abderrahim Wakif
E-ISSN: 2224-347X
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1 Introduction
Owing to the significant advancements in
contemporary technology, careful consideration has
been paid to examining the heat transfer properties
of squeezing flows in a variety of shapes. In
numerous scientific and engineering domains,
including polymer processing, food engineering,
injection molding, lubrication systems, foam
formation, blood flow inside vessels, cooling
towers, bi-axial expansion of bubble boundaries,
hydrodynamical machines, compression, moisture
migration, chemical engineering, dampers,
heating/cooling processes, and many more,
squeezed flows have many important practical and
industrial applications. Nonetheless, the squeezing
flow is caused by the typical stresses that are
applied to the moving surfaces or plates.
[1], [2], has a thorough overview of the
literature and applications related to squeezing
flows. The movement of the human body's
diarthrodial joints and valves, which is related to
the fields of mathematical bioengineering and
biomedicine, is another noteworthy illustration of
squeezing flow, [3]. In today's biological and
chemical technologies, sensors that use stretching
surfaces as sensing elements are crucial for
identifying a wide range of illnesses, dangerous
substances, and biological warfare elements. The
issues are addressed in practice by employing a
micro-cantilever that bends when target molecules
bind to one of its surfaces with the receptor coating.
It is evident that in practice, the micro-cantilever is
typically positioned in a film of thin fluidic cells
with an external squeezing disturbance; this
physical scenario of fluid motion over a micro-
cantilever is modeled as flow about a sensor
surface. Literature, [4], [5] provides a thorough
analysis of micro-cantilever, electrochemicals,
biosensors, and their applications in diverse
biomedical domains. Heat transfer problems,
however, have many scientific applications in the
field of engineering sciences, including conduction
of heat in tissues, thermal energy storage, laser
cooling, magnet, and radiative cooling, cooling of
nuclear reactors, metallurgical processes, space
cooling, and petroleum industries. By creating a
mathematical model, [6], significantly advanced the
field of squeezing flows in this approach. Later, a
lot of researchers carried on with Stefan's problem
by considering various geometries with appropriate
adjustments. The authors [7], assumed that the
length between the plates changed as the inverse
square root of time to study the thermodynamic
behavior of squeezed flow between two elliptic
parallel plates. Additionally, the two-point
boundary value problem was modeled in the
literature, [8] and is currently being solved utilizing
appropriate mathematical techniques such as the
homotopy analysis method (HAM) and
perturbation scheme. Their research demonstrates
that a boundary layer with very little viscosity
forms on the plates at higher squeezed number
values. The magnetized squeezing flow of a
viscous incompressible electrically conducting
fluid film created between two parallel discs was
investigated, [9]. Additionally, their research
assumes that the lower disc will rotate at a
temporary, arbitrary angular velocity. Additionally,
the typical Hermitian finite difference scheme is
used in the literature, [10], to produce numerical
solutions. Their analysis did reveal, however, that
the torque on the bottom disc is amplified by
increasing angular velocity and magnetic number
as well as by lengthening the distance between the
plates, which increases the load. In this work, [11],
investigated the problem of incompressible
rectilinear time-dependent, two-dimensional
magnetized viscous squeezed flow via an infinite
channel using a homotopy analysis approach. They
find that the viscous behavior of the fluid under
consideration can be explained by a diminishing
magnetic field. The analytical solution of the
squeezing flow between circular plates was
addressed in [12], using semi-numerical techniques
as the homotopy analysis method. Furthermore, the
crossflow behavior on the axial velocity profile is
shown by the improved Reynolds number. In this
work, [13], the authors used HAM to study the
issue of incompressible transient viscous squeezed
flow of two-dimensional fluid between two parallel
plates under the influence of chemical reaction and
viscous dissipation. According to the literature,
[13], the magnifying squeezing number raises the
momentum transmission coefficient and decreases
the concentration field. The magnetized time-
dependent, two-dimensional, incompressible, pair
stress microfluid flow between two parallel plates
with chemical reaction effects was established in
[14]. According to their research, the heat field
decreases as the squeezing flow parameter
increases. Additionally, in the solution regime, the
axial velocity profile displays the crossflow
behavior with greater magnetic parameter values.
Owing to the enormous advancements in
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DOI: 10.37394/232013.2024.19.8
Amine El Harfouf, Rachid Herbazi,
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Abderrahim Wakif
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bioengineering technology, numerous researchers
were interested in magnetic fluxes due to their
numerous applications. Additionally, by providing a
magnetic field, the flow and heat transfer properties
are adjusted by the requirements. The concepts of
magneto-hydrodynamics, or MHD, are widely used
in current biomedicine to treat tiresome disease
conditions. These explanations explain why the
MHD concept has good practical benefits today.
The author [15], conducted a theoretical and
experimental examination to examine the impact of
magnetic numbers on lubricating flows between
two parallel plates. This study [16], used an HPM
technique to examine how a magnetic field affected
the incompressible two-dimensional Casson fluid
flow between two parallel plates. Their analysis
demonstrates that the growing magnetic parameter
and the decreasing velocity field are related. The
influence of the Cattaneo-Christov heat flux model
for single- and multi-wall carbon nanotubes on the
stagnation point flow of micropolar nanofluid
across a stretching surface with slip effects was
investigated in [17]. Their study shows that, in the
flow zone, the Bejan number is a decreasing
function of the Brinkman parameter. The effects of
mass flux and Cattaneo-Christov double diffusion
heat models on the transient nanofluid flow
between two parallel plates with Joule heating and
chemical reaction were numerically examined.
Their study has noted that the concentration profile
is an increasing function of the Brownian motion
parameter and a decreasing function of the
thermophoresis parameter. In their inquiry showed
that the power-law fluid exhibits a dual behavior in
the presence of an applied magnetic field, and they
also proved the effect of non-uniform heat source
on magnetized power-law fluid flow across a
stretching sheet with non-Darcian porous medium.
The literature review and the benefits of
optimizing flow in multiple scientific and
engineering domains, such as biology and
biomedicine, served as the authors' driving forces.
Because of this, researchers have attempted to
examine the behavior of unstable Williamson fluid
in terms of flow and heat transfer over a horizontal
sensor surface while subjected to external
compression and a transverse magnetic field. The
current problem is of great interest to engineers, as
the literature review has shown.
2 Mathematical Formulations
An investigation is conducted using numerical
methods on a two-dimensional mathematical model
of magnetized time-dependent, viscous
incompressible, electrically conducting Wilson
fluid flow around a sensor surface with changing
thermal conductivity and external squeezing with
viscous dissipation effect. The flow configuration
of the current issue (closed compressed channel) is
shown in Figure 1 together with all required
parameters. Furthermore, let be the time-
dependent height of the closed channel, measured
from to , and be substantially greater than the
thickness of the boundary layer. Additionally, the
channel encloses the microcantilever sensor of
length , with the lower surface fixed and the upper
surface squeezed. It is evident, therefore, that the
squeezing action is thought to begin at the tip of the
sensor surface and work its way down to the free
stream fluid. In addition, the thermodynamic
behavior of the current physical situation is
addressed with the aid of a well-established
rectangular coordinate system in which
is taken along axial direction and y-
axis is chosen normal to . The fluid
flow is driven by the external free stream velocity
and the magnetic field of strength is
applied normal to the channel.
Equation of continuity:
(1)
Momentum equation:
Free stream equation:
Energy equation:
The velocity components along the x and y
directions in the Eqs. (1) – (4) are represented by
the variables and , the free stream velocity
(2)
(3)
(4)
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along the axial flow path by , the fluid
temperature by , the time interval by , the
working fluid density by , the time constant by ,
the fixed thermal conductivity by , the kinematic
viscosity of the working liquid by
, and the
magnetic strength by . Eqs. (2) and (4)
additionally meet the flow conditions required
within the boundary layer region taken into
consideration for this investigation. Additionally,
the outer free stream flow , which
is presumptively inviscid and uniform about the
normal coordinate, is governed by Eq. (3).
However, by considering the small sensor length in
line with channel height, the mistake in the forecast
was eliminated. Lastly, by removing the pressure
factor from Eq. (2) and using Eq. (3) as, the
necessary flow equation that follows is achieved.
Therefore, the flow equations (4) and (5) under
consideration are simplified about thermal and
hydromagnetic circumstances as follows.
(6)
Fig. 1: Physical manifestation of the current issue
The symbols utilized in Eq. (6) are as follows:
represents the surface heat flux, and
and
stands for the ambient fluid velocity and
temperature. Furthermore, where is the small
quantity in the current situation of the fixed thermal
conductivity Assume that when the wall is
thought to be permeable, describes the reference
velocity next to the surface if the sensor surface
behaves as a function of the surface heat flow q(x).
However, the set of coupled two-dimensional
transient Williamson fluid flow Eqs. (4) and (5)
with sufficient conditions Eq. (6) are changed into a
set of nonlinear ordinary differential equations by
applying the proper similarity transformations.
Therefore, the following similarity transformations
are applied to accomplish this goal.
(7)
Ultimately, applying Equation (7) to the Eqs. (4) –
(6) results in the nondimensional flow system that
follows considering .
(8)
(9)
The modified boundary conditions concerning
are as follows.
(10)
The derivative concerning eta is shown by the
superscript "prime" in the Equations (8)– (10).
Furthermore, the following definitions apply to the
governing physical factors regulating the fluid
flow.
(Magnetic number),
(Weissenberg number), (squeezed flow index),
(permeable velocity parameter),
(Prandtl number), and
(Eckert
number).
One dimensionless number that is an inherent
characteristic of a fluid is the Prandtl number.
Small Prandtl numbers indicate strong thermal
conductivity and free flow, making them excellent
choices for heat-conducting liquids. Liquid metals
(5)
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are excellent heat transmission media. It's
interesting to note that while common organic
solvents are not good heat transfer liquids, air is.
The momentum transmission takes precedence over
the heat transport as viscosity increases, rendering
these liquids unsuitable for heat conduction. For
non-Newtonian fluids, the Prandtl number is
therefore assumed to be small. The influence of
fluid self-heating because of dissipation effects is
measured using the Eckert number.
In many technological applications, the
engineering numbers of interest, the skin friction
coefficient, and heat transfer rate—contribute
significantly to the results. As a result, in this work,
we have computed computer-generated numerical
values for the skin friction coefficient and heat
transfer rate, which are reported in Figure 13,
Figure14 and Table 1. Nonetheless, the wall shear
stress and heat transfer rate are provided below
with the aid of defined thermodynamic conditions.
(11)
(12)
The values of
and in the Eqs. (11) and
(12) are determined as follows:
(13)
Ultimately, using the similarity variable, the
skin-friction coefficient and Nusselt number
equations are derived by incorporating Equations
(7) and (13) into Equations (11) and (12). This
process yields the following results:
(14)
(15)
Therefore, the necessary equations for the skin-
friction coefficient and Nusselt number are
equations (14) and (15). Additionally, the local
Reynolds number is shown by the formulas (14)
and (15) above, where
.
3 Analysis of the Results
This section examines how different embedding
factors affect the flow of a Williamson fluid over a
sensor surface as it is squeezed, considering the
Nusselt number, momentum transport coefficient,
and temperature and velocity profiles. Additionally,
Figure 2, Figure 3, Figure 4, Figure 5, Figure 6,
Figure 7, Figure 8, Figure 9, Figure 10, Figure 11
and Figure 12 describe the effects of the
Weissenberg number , compressed flow index
, permeability velocity parameter ,
magnetic number, Eckert number , and
Prandtl number on temperature and velocity
profiles. Additionally, Table 1, Figure 13, Figure 14
and Figure 15 display the variations in momentum
and heat transfer rates observed within the study
zone for different values of flow parameters.
Weissenberg number 's effect on and
:
Figure 2 and Figure 3 show how the Weissenberg
number affects the temperature and velocity fields.
Figure 2 illustrates how the axial velocity profile
decreased as the Weissenberg number increased.
The ratio of relaxation time to the process time is
known as the physical Weissenberg number. Here,
the Weissenberg number magnifying values
increase the fluid relaxation time, which
strengthens the sensor surface's resistance to the
Williamson fluid flow. As a result, this
circumstance causes the flow region's resistance to
magnify, which lowers velocity as shown in Figure
2. Additionally, Figure 3 shows how the
Weissenberg number affects the thermal profile.
Figure 3 shows that as the Weissenberg number
increases, the temperature field also does,
increasing the thickness of the thermal boundary
layer in the flow area. As a result, the relationship
between temperature and Weissenberg number
increases.
Impact of squeezed flow index on and
:
The effects of on the temperature field and
velocity fields are shown in Figure 4 and Figure 5,
respectively. On the other hand, Figure 4 illustrates
how the velocity field decreases as the compressed
flow index increases. The velocity profile is
decreasing because the motion of the Williamson
fluid molecules in the flow direction is amplified
by an increase in the squeezing process. It is
observed that the strength of the squeeze flow and
the squeezed flow index have the opposite
relationship. As a result, this condition results in a
decreased flow velocity increasing . Additionally,
dual velocity behavior is seen in the channel
because of the fluctuations at the border.
Additionally, Figure 5 shows how affects the
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flow domain's thermal field. It is noteworthy to see
that the thermal profile was subdued by the
magnification . Larger b values physically lessen
the force that squeezes velocity, which in turn
lessens the thermal field. The thickness of the
temperature boundary layer decreases as the
squeezed flow index increases.
Influence of permeable velocity parameter
on and :
The effect of on velocity and temperature fields
is depicted in Figure 6 and Figure 7, respectively.
Figure 6 shows that increases and decreases the
velocity field. This decay in the axial flow field is
caused by the fluid being mostly attached to the
sensor surface , which is a physical
condition that decays the velocity field inside the
channel. Additionally, Figure 7 shows that has
an impact on the temperature field. Figure 7
illustrates how the thermal field is enhanced by
an increase in the permeability velocity
parameter. In this circumstance, the sensor
surface's cooling is improved beneath the
suction enclosure . In addition, the
temperature boundary layer's thickness
decreased as the permeability velocity
parameter increased.
Effect of Magnetic parameter on and
:
The influence of the magnetic parameter on the
velocity and heat profiles is shown in Figure 8 and
Figure 9, respectively. Figure 8 illustrates how the
velocity profile is enhanced by an increase in the
magnetic parameter. As a result of the upper plate
being squeezed, a rising magnetic parameter
physically increases resistance along the axial flow
direction. This eliminates the effect of applied
magnetic field strength on velocity and increases
velocity inside the channel. Additionally, Figure 8
shows that as the magnetic number increases, the
velocity boundary layer thickness increases. The
impact of magnetic numbers on thermal fields is
examined in Figure 9. Figure 9 illustrates how the
thermal profile is increased by the increasing
Lorentz forces. Furthermore, when the magnetic
parameter values increased, so did the thickness of
the thermal boundary layer. As a result, the
temperature field increases as the magnetic
parameter does.
Effect of Prandlt number on :
Figure 10 show how the Prandtl number
behaves thermodynamically on a thermal profile.
Figure 10 illustrates how affects the thermal
field. Figure 10 shows that the thermal profile
decays with increasing . The reason for this
decline in the thermal profile is because a rise in
causes the temperature diffusivity to decrease,
which suppresses the thermal field. Additionally,
when values increased, the temperature
boundary layer's thickness decreased.
Impact of Eckert number on :
The viscous dissipation effect, which is always
positive and indicates a source of heat due to the
frictional forces among the fluid particles, is
represented by the second term on the right-hand
side of Eq. (13). Additionally, the irreversible
process known as viscous dissipation converts the
work that a fluid performs on nearby fluid layers
because of shear forces into heat. In scientific and
engineering fields including aerodynamics,
injection molding, polymer processing, and others,
the viscous dissipation effect has a stronger
influence. In real applications, however, boundary
layer flows with viscous dissipation effect over
sheets/surfaces garner significant attention across a
wide range of engineering systems. Figure 11
shows the effect of Eckert number on
temperature profile. The Eckert number is the
primary unit of calculation for heat dissipation in
the specified physical system. When the Eckert
number is higher than the enthalpy changes in each
physical system or medium, it indicates the
advective transport mechanism, which has a
significant impact on heat transfer phenomena.
Nevertheless, Figure 11 shows that the thermal
profile increases when the Eckert number increases.
directly affects the mechanism of heat
dissipation, which strengthens the thermal field,
making the increase in temperature evident.
Additionally, as the Eckert number increases, the
thickness of the thermal boundary layer decreases.
Skin-friction coefficient and Nusselt number
behavior
Table 1, Figure 12 and Figure 13 show how
different control settings affect the skin friction
coefficient and Nusselt number. The skin-friction
coefficient numerical values produced by the
computer for a range of flow parameter values are
tabulated in Table 1. On the other hand, Table 1
makes it evident that while improving b reduces the
skin friction coefficient, raising We and M values
increases it. Additionally, the skin-friction
coefficient increases for and decreases for
. Furthermore, because and are not
explicitly included in the momentum equation, they
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Abderrahim Wakif
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have no discernible impact on the skin friction
coefficient.
In a similar vein, Figure 12 and Figure 13 show
how , affect the profile of heat transfer rate.
Because both the temperature boundary condition
(see Eq. (10)) and the heat transfer rate formula
(see Eq. (15) start with a negative sign, Figure 12
and Figure 13 also start with negative values.
Figure 12 clearly shows that as values of grow,
the thermal barrier layer thickness reduces while
the increased squeezed flow index increases the
rate of heat transmission. Physically, by creating
high heat molecular forces and high pressure on the
fluid flow, increasing the squeezed flow index
enhances the rate of heat transmission.
Furthermore, Figure 13 shows that the permeability
velocity parameter decreases as the heat transfer
rate profile increases. Additionally, when the values
of grow, so does the thickness of the thermal
boundary layer.
4 Conclusion
A numerical study is conducted on a two-
dimensional magnetized Williamson fluid flow
surrounding a sensor surface that has variable
thermal conductivity and external squeezing with a
viscous dissipation effect. The time-dependent
coupled nonlinear partial differential equations
resulting from the examined physical problem are
reduced to ordinary differential equations by the
incorporation of appropriate similarity
transformations. Using the shot technique and the
Runge-Kutta fourth-order integration scheme, the
resulting nonlinear flow system is solved. By
performing the parametric analysis relating to the
numerous physical parameters, the physics
underlying the current situation is unearthed. The
temperature behavior and flow sensitivity of the
Williamson fluid around a sensor surface are
depicted in the tables and graphs. The following is
a limited list of the key findings related to the
current numerical study:
As the Weissenberg number increases, the
thickness of the momentum boundary layer
is observed to decrease.
As the Weissenberg number increases, so
does the temperature profile.
Temperature and velocity fields were
suppressed as the compressed flow index
increased.
The velocity profile became less as the
permeability velocity parameter increased.
On the temperature profile, however, the
opposite tendency is seen.
As the Eckert number climbed, so did the
thermal profile.
The momentum transmission coefficient
increased as the magnetic parameter and
Weissenberg number increased. However,
the increasing compressed flow indicator
shows the opposite trend.
Science and engineering-related flow
issues can be effectively resolved by using
the Runge-Kutta scheme and firing
approach.
The Nusselt number's magnitude rises as
values rise.
Furthermore, a wide range of commercial
applications, including solar collectors, fluidic cells
for thermal flow, oil recovery, and so on, are
predicted to benefit from the current numerical
analysis.
Fig. 2: Effect of on
Fig. 3: Effect of on
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Fig. 4: Effect of on
Fig. 5: Effect of on
Fig. 6: Effect of on
Fig. 7: Effect of on
Fig. 8: Effect of on
Fig. 9: Effect of on
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Fig. 10: Effect of on
Fig. 11: Effect of on
Fig. 12: Effect of on
Fig. 13: Effect of on
Table 1. Numerical values of the momentum transmission coefficient produced by a computer for various flow
parameter values
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[17] Mishra SR, Baag S, Dash GC, Acharya MR.
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WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.8
Amine El Harfouf, Rachid Herbazi,
Sanaa Hayani Mounir, Hassane Mes-Adi,
Abderrahim Wakif
E-ISSN: 2224-347X
81
Volume 19, 2024
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. S. Hayani Mounir:
Conceptualization, Investigation, Project
administration, 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.
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
n_US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.8
Amine El Harfouf, Rachid Herbazi,
Sanaa Hayani Mounir, Hassane Mes-Adi,
Abderrahim Wakif
E-ISSN: 2224-347X
82
Volume 19, 2024
