Entropy Generation in a Magnetohydrodynamic Hybrid Nanofluid
Flow over a Nonlinear Permeable Surface with Velocity Slip Effect
S. O. SALAWU1, H. A. OGUNSEYE2, T. A. YUSUF3, R. S. LEBELO4, R. A. MUSTAPHA5
Department of Mathematics,
Bowen University,
Iwo,
NIGERIA
West Virginia Academy,
Al Wukair, Doha,
QATAR
Department of Mathematics,
Adeleke University,
Ede,
NIGERIA
4Education Department,
Vaal University of Technology,
Vanderbijlpark,
SOUTH AFRICA
5Department of Mathematics,
Lagos State University,
Lagos,
NIGERIA
Abstract: - The current study is designed to model the hydrothermal feature of a hybrid nano liquid slip
flows over a permeable expanding/contracting surface with entropy generation. The model
incorporates Cu-Al2O3 nanoparticles with water as the host liquid to simulate the flow. Additional
impacts incorporated into the novelty of the model are viscous dissipation and Joule heating. The
model is transformed appropriately to its dimensionless form using similarity quantities and the
solution is numerically obtained using the spectral quasi-linearization method (SQLM). The impact of
pertinent factors on the flow characteristics is communicated through graphs for the hybrid nano-
suspension to discuss the hydrothermal variations. The friction factor and the rate of heat transport are
also discussed with sensible judgment through tables. To ensure the code validity, a comparison with
earlier studies is conducted and excellent consensus is accomplished. The result explored that
diminution in the irreversibility ratio is witnessed for rising magnetic field strength along the free
stream, distance away from the permeable surface as the heat dissipation to the surrounding
decelerates. Also, the augmented nonlinearity parameter intensified the heat transfer rate for about
of the hybrid nano-suspension.
Key-Words: - Boundary slip; entropy generation; Magnetohydrodynamic; hybrid nanofluid; nonlinear
permeable surface
Received: October 15, 2022. Revised: May 18, 2023. Accepted: June 19, 2023. Published: July 27, 2023.
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1 Introduction
Fluid flows over stretching or shrinking surfaces
are extremely noteworthy owing to their wide range
of applications. Different aspects of fluid flow
overstretched surfaces under various flow
conditions can be traced out through the open
pieces of literature, [1], [2], [3], [4]. Since the
innovative research articles have devoted interest to
the field of flow through a stretching surface.
However, the blowing (suction) or withdrawal
(injection) phenomena associated with fluid flow
over a stretched surface have a tremendous impact
on the flow field. It has useful engineering
applications in the design of thrust bearings and
radial diffusers to prevent corrosion and thermal oil
recovery. In, [5], the authors discussed the
suction/injection effect on a boundary layer flow
due to stretching the surface with thermal radiation.
The author claimed that the injection tends to
eclipse skin friction, while the suction acts
otherwise. Convection flows over a stretching or
shrinking surface with suction or injection is
numerically investigated by, [6]. Some recent
related articles are, [7], [8], [9]. In, [10], the author
discussed the Darcy-Forchheimer flow of nanofluid
through a stretchy, permeable wall. It was found
that the flow rate decreases with rising Darcy-
Forchheimer numbers.
Some common liquids like water, mineral oils,
and ethylene glycol, are found to have low thermal
characteristics when compared with metals, non-
metals, and their oxides. Researchers have in the
past years come up with the notion of the
suspension of millimeter-sized solid material in a
liquid, [11]. However, when these suspended
particles are larger, a lot of technical issues are
encountered, including an increasing drop in
pressure, erosion of pipelines, etc. Given these
challenges, the idea of improving thermal
conductivity by employing nano-sized materials
was later coined by, [12]. The author
experimentally analyzed the enhancement of
thermal conductivity through the dispersion of
nanoparticles in conventional-based liquids named
nanofluids. Other than improving thermal
conductivity, the choice of adding nano-sized
materials over micro-sized particles inside a
conventional base fluid has been a promising
candidate due to several valid scientific reasons,
such as shape, size, concentration, wide suspension
time (more stability), and significant energy
savings, [13]. In, [14], carried out a comparative
analysis on the viscous fluid of different
nanoparticles in a rotating slip disk and Joule
heating. A high-volume fraction of nanoparticles
dampened the axial velocity. In terms of provision
of efficient heat transfer and coolant issues, a
modified version of nanofluid i.e. hybrid nanofluid,
has recently been introduced and has significant
uses in pharmaceutical medicine, electronics,
chemical industry, agriculture, and so on, [15]. In
the hybrid nanofluid, two or more metallic
nanoparticles are dispersed within the base fluid.
In, [16], the authors examined the influence of heat
transfer enhancement and thermal conductivity in
hybrid nanofluids of water-based suspended Cu-
A over a stretched surface using a numerical
approach. In, [17], the authors used the concept of a
hybrid nanofluid of Ag-CuO water-based to
communicate the analysis of 3D flow, heat, and
mass transfer of a rotating liquid past a stretching
sheet. They reported through their analysis that
hybridity is boosted by enhancing the heat transfer
rate at the surface. Recently, a semi-analytical
method was employed to examine the 3-D flow of a
water-based hybrid nanofluid over a stretched
surface with Darcy-Forchheimer porous medium
and nonlinear thermal radiation by, [18], [19],
discussed the inclined magnetic field effect on
hybrid nanofluid flow over a slip surface. They
explore Silver and copper oxide nanoparticles as
hybrid nanofluids, while copper oxide is the usual
nanofluid with base fluid water. They claim that
maximum heat transfer capability is noticed for the
hybrid nanofluid when compared with the ordinary
nanofluid. In, [20], the authors numerically studied
a 2-D steady flow with the analysis of heat transfer
of water-based Cu-A nanoparticles over a
permeable stretching sheet with thermal radiation.
Entropy generation, a noteworthy function in
thermal engineering, is often affected mainly due to
conduction, and convection within the system. The
analysis of entropy generation has snatched the
attention of so many researchers globally as it helps
to control the irreversibility losses in a thermal
system that may affect system efficiency, [21]. It is
a pertinent factor, particularly in the industries and
engineering processes where cooling and heating
procedures are fostered. The result of an
irreversible process that takes place in
homogeneous thermodynamic systems can be
linked to thermal dissipation and Joule heating.
These are significant in examining the level of
entropy generated in a thermal system. In an
attempt to obtain an optimum design criterion,
investigation in this direction by several authors
over diverse flow geometries can be found in
references, [22], [23]. In, [24], the authors
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examined the Arrhenius kinetics of thermal
diffusion of nano liquid squeezing second-grade
fluid flow along an infinite plate and entropy
generation. As seen, the Brinkman number
discouraged the rate of generation of entropy.
Meanwhile, great improvements have been reported
for hybrid nanofluids over conventional fluids in
the analysis of entropy generation. For instance, In,
[25], the authors studied the entropy generation
analysis of water-FeO/CNT hybrid nanofluid
flow inside a concentric horizontal annulus using a
numerical approach. Furthermore, In, [22], the
authors examine a steady flow of hybrid nanofluid
with entropy generation by including melting heat
transfer to address the heat transfer analysis. They
concluded that the irreversibility ratio is lower for
hybrids as compared to the usual nanofluid. In,
[26], the authors numerically examine the entropy
generation characteristics of a water-based hybrid
nanofluid flow over a slippery wedge surface with
variable viscosity. They reported that increasing the
volume fraction of nanoparticles eclipse the entropy
production. For further results on the entropy
generation in hybrid nanofluid, interested readers
can see the following references, [27], [28], [29].
Inspired by the above research, we have
presented in this communication the impacts of the
water-based Cu-AlO hybrid nano liquid flowing
through a permeable stretching surface. The
analysis of entropy generation is included to
explore the flow and heat interaction. With many
studies on hybrid nanofluids, no mathematical post
of the present study has been done subject to the
boundary conditions. Despite several solution
techniques, [2], [20], in handling nonlinear ODEs,
resulting equations from this present study were
solved using the spectral quasi-linearization method
(SQLM) owning to its rapid convergence, [30],
[31]. The outcomes convey the impacts for both
hybrid and usual nanofluids. The study objectives
are to examine the: velocity, temperature, and rate
of entropy generation, and the Bejan figures are
captured to have requisite information on the flow
and heat transfer. Skin friction and Nusselt number
are also reported via Table. To the best of our
knowledge, no investigation on the mentioned
issues has so far been reported. Thus, we hope that
this study will provide a basis to gather the
indispensable information of such flow which in
turn helps various technological issues.
2 Problem Formulation
We consider a steady 2-D flow of a two-
dimensional non-linearly stretching permeable
sheet with velocity slip and prescribed surface heat
flux with the working fluid being an electrically
conducting viscous hybrid nanofluid. The
permeable sheet is stretching with assumed velocity
also, the external velocity takes
, where and are positive constants.
Here, is the nonlinearity parameter. A non-
uniform magnetic field acts normally to the flow
direction with strength , where
is the applied magnetic field strength as
geometrically modeled in Figure 1.
Fig. 1: The theoretical physical model
Using the boundary layer approximations and
accounting for the impacts of Joule heating and
viscous dissipation, the hybrid nanofluid model
equations are [2], [20]:
(1)
(2)
(3)
and at the boundary, the model is set as:
, (4)
(5)
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Here, the hybrid nanofluid dynamic viscosity,
density, electrical conductivity, thermal
conductivity, and heat capacity represent ,
, , and respectively.
represent the coordinate along the surface and the
vertical coordinate takes , and respectively
represent a component of the velocity along and
directions, is the hybrid nanofluid temperature,
is the tangential momentum accommodation
coefficient, is the mean free
path, is the suction (injection) velocity, is the
wall temperature, and is the surface temperature
parameter. In addition, the expression to evaluate
these thermo-physical properties for both nanofluid
and hybrid nanofluid is presented in Table 1,
where, the subscript denotes the nanofluid,
indicates the base fluid, indicates the hybrid
nanofluid, while and represents the hybrid
nanoparticles. In this model, water is considered as
a base fluid and Cu-Al2O3 is considered due to the
thermal conductivity and heat convective strength
of the nanoparticles and the base fluid. This is
synthesized by dissipating a fraction of Al2O3
nanoparticle to water (base fluid).
Table 1. Thermo-physical properties of nanofluid
and hybrid nanofluid
Property
Nanofluid
Hybrid Nanofluid
Dynamic
viscosity
Density
Electrical
conductivi
ty
where
Thermal
conductivi
ty
where
Heat
capacity
Source: [16], [17], [20]
Different volume fractions of Cu are
subsequently added. It is interesting to note that, for
, a working fluid (water) is
retrieved. More so, when and , a
case of Cuwater nanofluid can be obtained. The
thermo-physical features of water and the hybrid
nanoparticles are presented in Table 2.
Table 2. Thermo-physical properties of ethylene
glycol and hybrid nanoparticles
Water
997.1
4 179
0.613
0.05
Cu
8933
385
400
5.96
Al2O3
3970
765
40
3.69
Source: [20]
With aid of the below transformation ([4], [20]):
(5)
here, and , where
represents the stream function, also represent the
kinematic viscosity of the base fluid. Also, to
obtain a similar solution, and are
defined according to [1], as:
and (6)
denotes the constant mass flux parameter
with signifying fluid suction and
depicts fluid injection. Substituting equations (5,6)
into equations (1-4) yields:
(7)
(8)
subjected to:
(9)
(10)
where the magnetic parameter M, Prandtl number
Pr, Eckert number Ec, velocity slip parameter, and
the velocity ratio parameter are defined as:
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(11)
The relevant quantities of engineering interest in
this work and their corresponding mathematical
descriptions of the skin friction coefficient and
the Nusselt number are portrayed as:
(12)
where denotes the
surface shear stress and
represents the wall heat flux. Using
equation (5), equation (12) is gotten as follows:
, (13)
where the local Reynolds number
.
2.1 Entropy Generation
Based on the second law of thermodynamics, the
analysis of entropy generation for the present study
is portrayed as, [9],
(14)
Using the dimensionless variable defined above in
equation (5), the entropy generation number can be
rewritten as:
(15)
where
is the temperature difference
parameter.
The Bejan number (Be) is defined as
(16)
3 Numerical Solution
The analytical solutions to the boundary value
problem given by equations (7-10) are nearly
impossible to obtain, for these equations are highly
nonlinear, hence, we resolve a numerical method.
The solution is therefore obtained via the spectral
quasi-linearization method (SQLM). The employed
numerical technique was adopted for this study
because it is found to be stable, consistent, and
convergence. This method is employed to
numerically integrate the coupled nonlinear
differential equations (7-10). Recently, SQLM has
been shown to be efficient in solving nonlinear
boundary value problems emerging from boundary
layer flow, in terms of accuracy and rapid
convergence, [30], [31].
To apply the SQLM, the following non-linear
differential operators are considered:
(17)
(18)
With respect to SQLM, equations (7,8) are
linearized using the Newton-Raphson algorithm as
described by, [32], to give the following iterative
scheme:
(19)
(20)
subject to the corresponding boundary conditions:
(21)
where the coefficients , are
defined as:
(22)
and
(23)
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The SQLM iterative scheme which is made up
of equations (19-21) is solved numerically with the
aid of the Chebyshev pseudo-spectral scheme.
Equations (19, 20) are discretized with the help of
this scheme. Furthermore, using the transformation
, to transform the interval
. Then, the semi-infinite domain,
is truncated by replacing it with the
domain , where .
The computation of and (i.e the
derivatives of the unknown variables) is done using
the Chebyshev differentiation matrix , [33], at the
collocation points as a matrix-vector product;
(24)
(25)
where
denotes the number of collocation points,
,
and
are vector
functions at the collocation point.
The Gauss-Lobatto points are chosen to define the
nodes in as:
(26)
Higher order derivatives of and are evaluated
as powers of , that is
(27)
Substituting Eqs. (24) – (27) into Eqs. (19) – (21),
we obtain the following matrix form:
(28)
where are
matrices and
and
are
vectors,
such that:
(29)
subject to the boundary conditions
(30)
The SQLM scheme is initialized with the following
initial approximation;
(31)
3.1 Numerical Validation
Using the Maple 18 symbolic package, we
implement the SQLM iterative scheme. To express
the accuracy of our code validity, we extract the
skin friction coefficient values () using the
SQLM iterative scheme for limiting cases available
in the literature. Table 3 compares the value of
when , , and
for varying values of velocity ratio
parameter, and nanoparticle volume fraction,
with the results of [3], [20], and our results agree
with theirs, in Table 3. This establishes the
correctness of the method and validates the results
presented.
Table 3. Comparison of the SQLM results for
with [3], [20], for the following values:
, , and for
varying values of and In the case of Cu/water
nanofluid )
[3]
[20]
Present
-0.5
0.1
2.2865
2.286512
2.28651169
-0.5
0.2
3.1826
3.182538
3.18253843
0
0.1
1.8843
1.884324
1.88432376
0
0.2
2.6226
2.622743
2.62274312
0.5
0.1
1.0904
1.090453
1.09045278
0.5
0.2
1.5177
1.517774
1.51777395
4 Discussion of Results
This segment is designed to address the behavior of
the fluid parameters on the velocity profile (),
temperature distribution (), entropy generation
number, Bejan number, the skin friction, and
Nusselt number through plots and tables for the
hybrid nanosuspension. Following, [20], here, the
Prandtl number is taken as during our
discussion.
In Table 4, a comparative analysis of the
Nusselt number in both Cu/water nanofluid and Cu-
Al2O3water hybrid nanofluid is exposed to
demonstrate the effective thermal features of the
hybrid nano-liquid. We observed from the result
presented in Table 4 that an improvement in the
heat transfer coefficient is captured for hybrid
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nanofluids. In addition, Table 5 (Appendix)
presents the values of the skin friction coefficient
() and Nusselt number for
varying the nanofluid flow parameters.
and are enhanced with increasing
values of , , , and . However, with
increasing value of , decreases and
increases. Similar behavior is noticed
for the case when . Also, is an
increasing function of Ec.
Table 4. Nusselt number for Cu/ethylene gylcol and
Cu-Al2O3water when
, and , for various values of .
Cu/water
()
Cu-
Al2O3water
()
difference
0.001
3.10125361
3.32810096
7.31
0.005
3.11135376
3.34032291
7.36
0.01
3.12405619
3.35565937
7.41
0.03
3.17567132
3.41765319
7.62
0.1
3.36535206
3.64289688
8.25
Fig. 2: Effect of varying the nanoparticle volume
fraction () on the hybrid nanofluid on (A)
velocity profiles, (B) temperature profiles, (C)
entropy generation rate, and (D) Bejan number.
Figure 2 shows the flow characteristics for
rising values of nanoparticle volume fraction
for the hybrid nano-liquid. Figure 2(A-B) depicts
the flow rate, heat transfer, entropy generation, and
irreversibility ratio profiles, respectively for various
values . The volume fraction represents the
ratio of the constituent volume of Cu-Al2O3/water
to all constituent volume mixtures. As observed,
the velocity field (Figure 2A) drops as the
nanomaterial volume fraction is enhanced. This is
because the overall constituent volume mixtures
control the reaction as the hybrid nanofluid reduces
in the stretching boundless domain. Also, we
observed that the momentum boundary layer
encourages convective heat transfer that leads to
strengthening in the liquid bonding force; hence,
the flow is dragged thereby causing a reduction in
the velocity profile. Meanwhile, the heat
distribution is enhanced as the nanomaterial volume
fraction is raised as presented in Figure 2B.
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The rise in the temperature field is due to an
increase in the thermal conducting strength of the
nanoparticle, which encourages conduction and
convective heat transfer within the reactive
mixture. As such, the thermal boundary layer is
enhanced to reduce heat diffusion, thereby causing
a rise in the temperature magnitude of the system.
Hence, the temperature distribution is upsurged, as
observed in the profiles. In Figures 2C and 2D, the
magnitude of the irreversibility and Bejan number
increases as the parameter varies. Close to the
permeable stretching sheet, significant rises in both
entropy generation rate and Bejan fields is noticed
due to the respective dominance of fluid viscosity,
Joule heating, and viscous dissipation as well as
heat transfer. However, a little distance away from
the plate, the irreversibility and irreversibility ratio
reduce gradually towards the far stream as the
dissipation of energy decreases. Therefore, entropy
generation and Bejan number are minimized for the
hybrid nano-liquid reactive mixture as energy loss
diminishes. Hence, the nanoparticle volume
fraction reduces the energy loss far away from
the porous moving plate.
Fig. 3: Effect of varying the magnetic parameter
() on the hybrid nanofluid on (A) velocity
profiles, (B) temperature profiles, (C) entropy
generation rate, (D) Bejan number
The reaction of hybrid nanofluid to rising
values of a magnetic field for the flow thermo-
physics properties is displayed in Figure 3. For the
value of or , a respective
momentous increase or decrease in the flow
velocity is seen in Figure 3A. The respective flow
behaviour is a result of the boost or diminishment
in the heat source terms and nanoparticle thermal
conductivity that leads to the breaking or
strengthening of the fluid bonding force. Figure 3B
shows the response of the temperature field to
changes in the parameter for different values of
the velocity ratio parameter . The heat distribution
for a stagnation hybrid nano liquid decreases
significantly for but gradually increases for
for an increasing value of . The internal
heat generation is discouraged for velocity ratio
which leads to a reduction in the heat
distribution. Meanwhile, internal heat is raised for
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velocity ratio though with little effect on the
hybrid nanoparticle reactive mixtures. The heat
transfer due to viscous heating is boosted all over
the flow region as the magnetic field parameter
is enhanced in Figure 3C. The magnitude of
entropy generated due to irreversibility is increased
as a result of energy lost to the surrounding
environment close to the permeable surface.
Irreversibility decreases momentarily for rising
magnetic fields along the free stream as distance
away from the permeable surface decreases and
heat dissipation to the surrounding area decreases.
However, in Figure 3D, the irreversibility ratio
(Bejan number) due to heat transfer diminishes,
which enhances the dominance of fluid viscosity
and viscous heating. Therefore, the Bejan number
profile reduces as the thermal boundary layer
becomes thinner, which causes more heat
dissipation into the environment from the hybrid
nanofluid mixture. Hence, the irreversibility ratio
field is discouraged, as presented in Figure 3D, as
heat diffuses out of the mixtures.
Fig. 4: Effect of varying the nonlinearity parameter
() on the hybrid nanofluid on (A) velocity
profiles, (B) temperature profiles, (C) entropy
generation rate, (D) Bejan number.
Figure 4 represents the reaction of the hybrid
nano liquid flow system behaviour to rising values
of the non-linearity term . The overall flow
characteristics are significantly influenced by the
non-linearity term, as noticed in Figure 4. In
Figures 4A and 4B, the magnitude of the flow
velocity and heat distribution reduce momentously
for the considered Cu-Al2O3/water mixture. This
behaviour is due to the low internal heat generation
and heat conductivity of the nanoparticle, which
leads to damping in the flow rate and heat transfer
of the reacting mixture. Hence, the non-linearity
term declines the stagnation of the hybrid nanofluid
mixture. Meanwhile, in Figures 4C and 4D, the
non-linearity term causes an early increase in the
entropy generation and Bejan number near the
permeable moving sheet. However, at a few
distances from the sheet, the entropy generation and
Bejan number diminish due to low heat dissipation
and energy loss in the reacting nano-liquid mixture.
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The continuous minimization in the irreversibility
and Bejan numbers is observed in the free flow
until no energy or heat is lost to the surroundings.
The drastic reduction in the irreversible process
enhances the optimal efficiency of the hybrid nano-
liquid. This is because the thermal conductivity
strength of the nanoparticle is boosted. Therefore,
the irreversibility profile and irreversibility ratio
decrease.
Fig. 5: Effect of varying the suction/injection
parameter () on the hybrid nanofluid on (A)
velocity profiles, (B) temperature profiles, (C)
entropy generation rate, (D) Bejan number.
The impact of the suction or injection term ()
on the hybrid nanomaterial properties is
demonstrated in Figure 5. As seen in Figures 5A
and 5B, the velocity and temperature fields are
discouraged by rising values of the parameter ().
The noteworthy declination (S) is a result of thinner
momentum and heat boundary layers that dampen
heat source terms and allow more heat to leave the
reactive mixture. A rise in heat diffusion
strengthens the hybrid nanoparticle bonding force
that thereby drags the hybrid Cu-Al2O3/mixture.
Thus, the reactive mixture flow velocity reduces in
the system. More also, increasing heat lost to the
ambient reduces the amount of heat within the
vertical moving plate; as such, the temperature
profile declines as depicted in Figure 5B. The
suction effect on the hybrid nanofluid irreversibility
and its ratio is examined in Figures 5C and 5D. An
early rise in irreversibility is noticed due to the
dominance of the nanofluid viscosity and viscous
dissipation. However, after a while, the
nanoparticle heat transfer rate reduces as a result of
declining energy loss, which in turn decreases the
entropy generation rate and irreversibility ratio. The
hybrid nanofluid entropy generation and Bejan field
are discouraged as the suction/injection term is
raised. Therefore, the nanomaterial thermal
conductivity is enhanced, which leads to increasing
performance of the hybrid Cu- Al2O3/material
mixture.
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Fig. 6: Effect of varying the velocity slip parameter
() on the hybrid nanofluid on (A) velocity profiles,
(B) Bejan number.
Figure 6 represents the reaction of the flow
velocity and Bejan number to variation in the slip
velocity term (). In Figure 6A, the flow rate
decreases due to very low internal heat production
in the Cu-Al2O3 reactive mixture, which leads to an
enhancement in the particle bonding force. Hence,
the nanofluid velocity distribution is dragged as the
fluid particles are restricted from free collision.
Meanwhile, the Bejan number profile is raised for
the reactive mixture because irreversibility due to
heat transfer controls the hybrid nano-liquid
mixture. However, the colloidal suspension
irreversibility mixture decreases towards the free
flow until the irreversibility vanishes away in the
system, as depicted in Figure 6B. Hence, the Bejan
number reduces as slip velocity is increased.
Fig. 7: Effect of varying the Eckert number (Ec) on
the hybrid nanofluid on (A) temperature profiles,
(B) entropy generation rate, (C) Bejan number.
The influence of Eckert number on the
hybrid nanofluid temperature field, entropy
generation, and Bejan number is demonstrated in
Figure 7. Eckert number is characterized by
dissipative heat transport in a system; it denotes the
correlation between the enthalpy boundary layer
difference and the considered nanofluid kinetic
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.4
S. O. Salawu, H. A. Ogunseye,
T. A. Yusuf, R. S. Lebelo, R. A. Mustapha
E-ISSN: 2224-347X
44
Volume 18, 2023
energy. Enhancing the term in the chemical
reaction of hybrid nano liquid steadily increases the
nanoparticles' kinetic energy, which raised the
particles' collision rate and then encourages
temperature distribution. Therefore, the ratio of
advective transport to dissipative heat potential
rises, thereby causing the magnitude of heat
transfer to increase, hence the temperature profile
rises as denoted in Figure 7A. The entropy
generation due to irreversibility and its ratio is
illustrated in Figures 7B and 7C. In Figure 7B, the
dissipated kinetic energy ratio to the conducted
thermal energy increases as the Eckert number is
boosted, this leads to a high irreversible rate in the
Cu- Al2O3/mixture. Hence, the entropy generation
is increased. Irreversibility due to chemical reaction
controls the reactive mixture, as seen in Figure 7C.
A rise in the dissipative term decreases the
Bejan number field due to strengthening in the
hybrid nano liquid reaction mixture and a decrease
in irreversibility due to friction and thermal energy.
Therefore, the heat transfer irreversibility reduces,
thereby causing a decrease in the Bejan number
profile.
5 Conclusion
The study described the entropy generation analysis
on a steady incompressible flow featuring hybrid
nanofluids over a stretched permeable surface using
a stable and consistent numerical scheme. The
impact of pertinent factors on the flow field,
thermal field, rate of entropy generation, and Bejan
number is communicated through graphs for the
hybrid nanosuspension to discuss the hydrothermal
variations. The key outcomes of the present
analysis are highlighted as follows:
• Enhanced values of the nonlinearity parameter
cause a diminution of the flow and thermal
distributions.
• The hybrid nanofluid possesses low skin
friction for the nonlinearity parameter, whereas
it augments for increasing values of the slip
parameter.
• As the values of the nanoparticle volume
fraction increase, the heat transfer rate is
augmented to about at the plate surface.
• The entropy generation rate upsurges with a
rise in the values of the magnetic field
parameter and the volume fraction parameter,
while it peters out for the nonlinearity
parameter.
• The employed method demonstrates excellent
potential with respect to accuracy and
convergence for simulating flow over stretched
surfaces.
Due to the significance of the study in thermal
engineering, an extension of the investigation is
encouraged. As such, this work can be extended to
the flow through an annular cylinder in the
presence of Arrhenius kinetic and nonlinear
radiation.
References:
[1] M. Bilal, A. Saeed, T. Gul, I. Ali, W.
Kumam, P. Kumam, Numerical
approximation of microorganisms hybrid
nanofluid flow induced by a wavy fluctuating
spinning disc, Coatings, 9(11) (2021), 1032.
[2] S. O. Salawu, A. M. Obalalu, E. O.
Fatunmbi, MD Shamshuddin, Elastic
deformation of thermal radiative and
convective hybrid SWCNTAg and MWCNT-
MoS4 magneto-nanofluids flow in a cylinder,
Results in Materials, 17 (2023), 100380.
[3] A. Ishak, N. Bachok, I. Pop, Stagnation-point
flow over a stretching/shrinking sheet in a
nanofluid, Nanoscale Research Letters,
(2011), 623.
[4] T. Gul, M.A. Khan, W. Noman, I. Khan, T.A.
Alkanhal, I. Tlili, Fractional order forced
convection carbon nanotube nanofluid flow
passing over a thin needle, Symmetry, 11(3)
(2019), 312.
[5] S.O. Salawu, A.M. Obalalu, S.S. Okoya,
Thermal convection and solar radiation of
electromagnetic actuator Cu-Al2O3/C3H8O2
and Cu-C3H8O2 hybrid nanofluids for solar
collector optimization, Materials Today
Communications 33 (2022), 104763.
[6] B. Jalili, S. Sadighi, P. Jalili, D.D. Ganji,
Characteristics of ferrofluid flow over a
stretching sheet with suction and injection,
Case Studies in Thermal Engineering, 14
(2019), 100470
[7] S.O. Salawu, R.A. Oderinu, A.D. Ohaegbue,
Thermal runaway and thermodynamic second
law of a reactive couple stress hydromagnetic
fluid with variable properties and navier
slips, Scientific African, 7 (2020), e00261.
[8] H.A. Ogunseye, S.O. Salawu, S.D. Oloniiju,
M.T. Akolade, Y.O. Tijani, R. Mustapha, P.
Sibanda, MHD Powell-Eyring nanofluid
motion with convective surface condition and
Dufour-Soret impact past a vertical plate: Lie
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.4
S. O. Salawu, H. A. Ogunseye,
T. A. Yusuf, R. S. Lebelo, R. A. Mustapha
E-ISSN: 2224-347X
45
Volume 18, 2023
group analysis, Partial Differential Equations
in Applied Mathematics, 6 (2022), 100459.
[9] H.A. Ogunseye, Y.O. Tijani, P. Sibanda,
Entropy generation in an unsteady eyring-
powell hybrid nanofluid flow over a
permeable surface: A lie group analysis, Heat
Transfer, 10 (2020), 21778.
[10] B. Jalili, A.D. Ganji, P. Jalili, S.S. Nourazzar,
D.D. Ganji, Thermal analysis of Williamson
fluid flow with Lorentz force on the
stretching plate, Case Studies in Thermal
Engineering, 39 (2022), 102374
[11] D. Lu, M. Ramzan, M. Mohammad, F.
Howari, J.D. Chung, A thin film flow of
nanofluid comprising carbon nanotubes
influenced by cattaneo-christov heat flux and
entropy generation, Coatings, 9(5), (2019),
9050296.
[12] S.U.S. Choi, J.A. Eastman, Enhancing
thermal conductivity of fluids with
nanoparticles, Proceedings of the ASME
International Mechanical Engineering
Congress and Exposition, 66, (1995) p.687-
694.
[13] S.U.S. Choi, W. Yu, S.K. Das, T. Pradeep,
Nanofluids, Science and technology, John
Wiley & Sons, Inc, (2008).
[14] S. Qayyum, I.M. Khan, T. Hayat, A. Ahmed,
Comparative investigation of five
nanoparticles in flow of viscous fluid with
joule heating and slip due to rotating disk,
Physica B: Condensed Matter, 534, (2018),
p.173–183.
[15] S.O. Salawu, R.A. Kareem, J.O. Ajilore,
Eyring-Powell MHD nanoliquid and entropy
generation in a porous device with thermal
radiation and convective cooling, J. of the
Nigeria Society of Physical Sciences, 4,
(2022), p.924.
[16] S.U. Devi, S.A. Devi, Heat transfer
enhancement of Cu-Al2O3/water hybrid
nanofluid flow over a stretching sheet, J. of
the Nigerian Mathematical Society, 36,
(2017), p.419-433.
[17] T. Hayat, S. Nadeem, Heat transfer
enhancement with Ag-CuO/water hybrid
nanofluid, Results in Physics, 7, (2017),
p.2317–2324.
[18] T. Hayat, F. Haider, T. Muhammad, A.
Alsaedi, Darcy-forchheimer three
dimensional flow of carbon nanotubes with
nonlinear thermal radiation, J. of Thermal
Analysis and Calorimetry, 140, (2020)
s10973.
[19] P. Jalili, A.A. Azar, B. Jalili, Z. Asadi, Heat
transfer analysis in cylindrical polar system
with magnetic field: A novel hybrid analtical
and numerical technique, Case Study in
Thermal Engin., 40 (2022), 102524.
[20] A. Ishak, I. Waini, I. Pop, MHD flow and
heat transfer of a hybrid nanofluid past a
permeable stretching/shrinking wedge,
Applied Mathematics and Mechanics, 41
(2020), p.507-520.
[21] M. Ramzan, S. Riasat, C.J. Dong, Y.M. Chu,
M. Sheikholeslami, S. Kadry, F. Howari,
Upshot of heterogeneous catalysis in a
nanofluid flow over a rotating disk with slip
effects and entropy optimization analysis,
Scientific Reports, 11 (2021), s41598.
[22] B. Jalili, N. Aghaee, P. Jalili, D.D. Ganji,
Novel usage of the curved rectanglar fin on
the heat transfer of a double-pipe heat
exchanger with a nanofluid, Case Study in
Thermal Engin., 35 (2022), 102086.
[23] S.O. Salawu, Two-step exothermic reaction-
diffusion of hydromagnetic Prandtl-Eyring
viscous heating fluid in a channel, Int. J. of
Thermofluids, 17 (2023), 100300.
[24] M.I. Khan, S. Qayyum, S. Kadry, W.A.
Khan, S.Z. Abbas, Irreversibility analysis and
heat transport in squeezing nanoliquid flow
of non-newtonian (second-grade) fluid
between infinite plates with activation
energy, Arabian Journal for Science and
Engineering, 45 (2020), p.4939–4947.
[25] A. Shahsavar, P.T. Sardari, D. Toghraie, Free
convection heat transfer and entropy
generation analysis of water-fe3o4/cnt hybrid
nanofuid in a concentric annulus, Int. J. of
Numerical Methods for Heat & Fluid Flow,
29(4) (2019), p.2018-2024.
[26] S. Ahmad, S. Nadeem, N. Ullah, Entropy
generation and temperature dependent
viscosity in the study of swcntâĂŞmwcnt
hybrid nanofuid, Applied Nanoscience, 13
(2020), s13204.
[27] Yusuf, T. A., R. Ukaegbu, J.C., and Ayinde,
A.M., Irreversibility analysis in
thehydrothermal flow of γAl2O3/H2O and
γAl2O3/C2H6O2 over a permeable stretching
surface with effective Prandtl number, Waves
in Random and Complex Media, (2022),
https://doi.org/10.1080/17455030.2022.2155
323
[28] P. Jalili, A.S. Ghahare, B. Jalili, D. D. Ganji,
Analytical and numerical investigation of
thermal distribution for hybrid nanofluid
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.4
S. O. Salawu, H. A. Ogunseye,
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E-ISSN: 2224-347X
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Volume 18, 2023
through an oblique artery with mild stenosis,
SN Applied Sci., 95 (2023), s42452.
[29] S.O. Salawu, A.M. Obalalu, MD.
Shamshuddin, Nonlinear solar thermal
radiation efficiency and energy optimization
for magnetized hybrid Prandtl-Eyring
nanoliquid in aircraft. Arabian J. for Sci. and
engin., 22 (2022), 070801.
[30] S.S. Motsa, A new spectral local linearization
method for nonlinear boundary layer flow
problems, J. of Applied Mathematics, 13
(2013), 423628.
[31] H.A. Ogunseye, E.O. Fatunmbi, P. Sibanda,
Magnetohydrodynamic micropolar fluid flow
in a porous medium with multiple slip
conditions, Int. Commun. in Heat and Mass
Transfer, 115 (2020), 104577.
[32] R.E. Bellman, R.E. Kalaba,
Quasilinearization and nonlinear boundary-
value problems, (1965).
[33] L.N. Trefethen, Spectral methods in
MATLAB, Siam, 10 (2000).
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.4
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Appendix
Table 5. The statistical data for the skin friction coefficient and Nusselt number for Cu-AlOwater (
) hybrid nanofluid
m
M
S
Ec
0.001
0.5
0.5
0.5
2
0.5
0.5
1.204108576
3.315391243
0.005
0.5
0.5
0.5
2
0.5
0.5
1.220313025
3.328393413
0.01
0.5
0.5
0.5
2
0.5
0.5
1.240694846
3.344756145
0.1
0.2
0.5
0.5
2
0.5
0.5
1.642556027
3.659691749
0.1
0.5
0.5
0.5
2
0.5
0.5
1.098739958
1.073055261
0.1
0.8
0.5
0.5
2
0.5
0.5
1.240694846
3.344756145
0.1
1
0.5
0.5
2
0.5
0.5
1.342452049
4.810744382
0.1
0.5
0.5
0.5
2
0.5
0.5
1.397090086
5.609892787
0.1
0.5
0
0.5
2
0.5
0.5
1.17458904
5.007446213
0.1
0.5
0.3
0.5
2
0.5
0.5
1.215708203
3.999445799
0.1
0.5
0.5
0.5
2
0.5
0.5
1.240694846
3.344756145
0.1
0.5
0.8
0.5
2
0.5
0.5
1.275110522
2.384597808
0.1
0.5
0.5
-0.5
2
0.5
0.5
1.066719419
1.648327654
0.1
0.5
0.5
-0.3
2
0.5
0.5
1.101391086
1.906004528
0.1
0.5
0.5
0
2
0.5
0.5
1.153722426
2.369501198
0.1
0.5
0.5
0.3
2
0.5
0.5
1.206039213
2.925282015
0.1
0.5
0.5
0.5
2
0.5
0.5
1.240694846
3.344756145
0.1
0.5
0.5
0.5
-0.5
0.5
0.5
1.662773645
1.612424622
0.1
0.5
0.5
0.5
-0.3
0.5
0.5
1.459327908
2.098984483
0.1
0.5
0.5
0.5
1
0.5
0.5
1.46244E-07
3.664105655
0.1
0.5
0.5
0.5
1.5
0.5
0.5
0.609911794
3.652495598
0.1
0.5
0.5
0.5
2
0.5
0.5
1.240694846
3.344756145
0.1
0.5
0.5
0.5
0.5
0.1
0.5
2.034448487
2.592949542
0.1
0.5
0.5
0.5
0.5
0.3
0.5
1.537704272
3.108090507
0.1
0.5
0.5
0.5
0.5
0.5
0.5
1.240694846
3.344756145
0.1
0.5
0.5
0.5
0.5
1
0.5
0.840924452
3.572167774
0.1
0.5
0.5
0.5
0.5
0.5
0.1
1.240694846
5.038423866
0.1
0.5
0.5
0.5
0.5
0.5
0.3
1.240694844
4.191590005
0.1
0.5
0.5
0.5
0.5
0.5
0.5
1.240694846
3.344756145
0.1
0.5
0.5
0.5
0.5
0.5
0.8
1.240694847
2.074505355
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
All authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Data Availability
Data sharing not applicable to this article as no
datasets were generated or analysed during the
current study
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 conflict 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: 10.37394/232013.2023.18.4
S. O. Salawu, H. A. Ogunseye,
T. A. Yusuf, R. S. Lebelo, R. A. Mustapha
E-ISSN: 2224-347X
48
Volume 18, 2023
