The improved thermal efficiency of Prandtl–Eyring hybrid nanofluid via classical Keller box technique

Abstract

Prandtl–Eyring hybrid nanofluid (P-EHNF) heat transfer and entropy generation were studied in this article. A slippery heated surface is used to test the flow and thermal transport properties of P-EHNF nanofluid. This investigation will also examine the effects of nano solid tubes morphologies, porosity materials, Cattaneo–Christov heat flow, and radiative flux. Predominant flow equations are written as partial differential equations (PDE). To find the solution, the PDEs were transformed into ordinary differential equations (ODEs), then the Keller box numerical approach was used to solve the ODEs. Single-walled carbon nanotubes (SWCNT) and multi-walled carbon nanotubes (MWCNT) using Engine Oil (EO) as a base fluid are studied in this work. The flow, temperature, drag force, Nusselt amount, and entropy measurement visually show significant findings for various variables. Notably, the comparison of P-EHNF’s (MWCNT-SWCNT/EO) heat transfer rate with conventional nanofluid (SWCNT-EO) results in ever more significant upsurges. Spherical-shaped nano solid particles have the highest heat transport, whereas lamina-shaped nano solid particles exhibit the lowest heat transport. The model’s entropy increases as the size of the nanoparticles get larger. A similar effect is seen when the radiative flow and the Prandtl–Eyring variable-II are improved.

Full text

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The improved thermal eciency
of Prandtl–Eyring hybrid nanouid
via classical Keller box technique
Wasim Jamshed1*, Dumitru Baleanu2,3,9*, Nor Ain Azeany Moh Nasir4, Faisal Shahzad1,
Kottakkaran Sooppy Nisar5, Muhammad Shoaib6, Sohail Ahmad7 & Khadiga Ahmed Ismail8
Prandtl–Eyring hybrid nanouid (P-EHNF) heat transfer and entropy generation were studied in this
article. A slippery heated surface is used to test the ow and thermal transport properties of P-EHNF
nanouid. This investigation will also examine the eects of nano solid tubes morphologies, porosity
materials, Cattaneo–Christov heat ow, and radiative ux. Predominant ow equations are written
as partial dierential equations (PDE). To nd the solution, the PDEs were transformed into ordinary
dierential equations (ODEs), then the Keller box numerical approach was used to solve the ODEs.
Single-walled carbon nanotubes (SWCNT) and multi-walled carbon nanotubes (MWCNT) using
Engine Oil (EO) as a base uid are studied in this work. The ow, temperature, drag force, Nusselt
amount, and entropy measurement visually show signicant ndings for various variables. Notably,
the comparison of P-EHNF’s (MWCNT-SWCNT/EO) heat transfer rate with conventional nanouid
(SWCNT-EO) results in ever more signicant upsurges. Spherical-shaped nano solid particles have the
highest heat transport, whereas lamina-shaped nano solid particles exhibit the lowest heat transport.
The model’s entropy increases as the size of the nanoparticles get larger. A similar eect is seen when
the radiative ow and the Prandtl–Eyring variable-II are improved.
List of symbols
A∗
1
Prandtl–Eyring parameter-I
A∗
2
Prandtl–Eyring parameter-II
Bς
Brinkman number
b
Initial stretching rate
Cf
Drag force
Cp
Specic-heat
(J kg−1K−1)
Eς
Eckert number
EO Engine Oil
EG
Dimensional entropy
(JK
−1)
Hς
Biot number
hς
Heat transfer coecient
k
Porosity of uid
κ
ermal conductivity

Wm
−1
K
−1
kς
ermal conductivity of the surface
k∗
Absorption coecient
Kς
Porous media parameter
Nς
Radiation parameter
OPEN
1Department of Mathematics, Capital University of Science and Technology (CUST), Islamabad 44000,
Pakistan. 2Institute of Space Sciences, 077125 Magurele-Bucharest, Romania. 3Department of Medical
Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan. 4Department of
Mathematics, Centre for Defence Foundation Studies, Universiti Pertahanan Nasional Malaysia, Kem Sungai Besi,
57000 Kuala Lumpur, Malaysia. 5Department of Mathematics, College of Arts and Sciences, Prince Sattam Bin
Abdulaziz University, Wadi Aldawaser 11991, Saudi Arabia. 6Department of Mathematics, COMSATS University
Islamabad, Attock Campus, Attock, Pakistan. 7Centre for Advanced Studies in Pure and Applied Mathematics
(CASPAM), Bahauddin Zakariya University, Multan 60800, Pakistan. 8Department of Clinical Laboratory Sciences,
College of Applied Medical Sciences, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia. 9 Department of
Mathematics, Cankaya University, Balgat 06530, Turkey. *email: wasiktk@hotmail.com; dumitru@cankaya.edu.tr
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NG
Dimensionless entropy generation
Nux
Local Nusselt number
Pr
Prandtl number
(ν/α)
p
Column vectors of order
J×1
qr
Radiative heat ux
qw
Wall heat ux
Re
Reynolds number
S
Suction/injection parameter
B1,B2
Velocity component in
x
,
y
direction
(ms
−1)
Uw
e velocity of the stretching sheet
Vς
Vertical velocity
x,y
Dimensional space coordinates
(m)
Greek symbols
Y
=
Fluid temperature
Y
=w
e uid temperature of the surface
Y
=∞
Ambient temperature
φ
e volume fraction of the nanoparticles
ρ
Density (
kg m−3
)
σ∗
Stefan Boltzmann constant
ψ
Stream function

Independent similarity variable
θ
Dimensionless temperature
ες
Relaxation time
�ς
Velocity slip parameter
µ
Dynamic viscosity of the uid (
kg m−1s−1
)
ν
Kinematic viscosity of the uid (
m2s−1
)
α
ermal diusivity
(m2s−1)
Π Dimensionless temperature gradient
Subscripts
f,gf
Base uid
nf
Nanouid
hnf
Hybrid nanouid
p,p1,p2
Nanoparticles
s
Particles
SWCNT Single-walled carbon nanotubes
MWCNT Multi-walled carbon nanotubes
Liquid mechanics’ limits are dened by the thin uid or liquid layer in contact with the pipe’s or an aircra wing’s
surface. In the boundary layer, shear forces can damage the liquid. Given that the uid is in touch with the surface,
a range of speeds exists between the maximum and zero boundary layer speeds. Limits on the trailing edge of an
aeroplane wing, for example, are smaller and thicker. A thickening of the ow occurs at the front or upstream end
of these boundaries. In 1904, Prandtl proposed the concept of boundary layers to describe the ow behaviour
of viscous uid near a solid barrier (see Aziz etal.1). Using the Navier Stoke equations, Prandtl constructed and
inferred boundary layer equations for large Reynolds number ows. As a necessary simplication of the original
Navier–Stokes equation, the boundary layer theory equations were critical. Studying wall jets, free jets, uid jets,
ow over a stretched platform/surface, and inductive ow from a shrinking plate helps develop the equations
for these phenomena. Boundary layer equations are oen solved using a variety of boundary conditions that are
specic to a given physical model. For a magnetohydrodynamics (MHD) uid ow with gyrotactic microorgan-
isms, Sankad etal.2 found that the magnetic and Peclet numbers may be utilised to reduce the thermal boundary
layer thickness. Aer that, Hussain etal.3 discovered that the thickness of the thermal boundary layer increases
as a Casson liquid ows towards the growing porous wedge due to convective heat transfer. e literature has
several experiments with various physical parameter impacts on boundary layer ow4–6 and multiple liquids7,8.
A hybrid nanouid is now attracting the attention of many researchers. Hybrid nanouids are cutting-edge
nanouids that combine two dierent types of nanoparticles in a single uid. e thermal properties of the
hybrid nanouid are better than those of the primary liquid and nanouids. In machining and manufacturing,
hybrid nanouids are commonly utilised in solar collectors, refrigeration, and coolants. According to Suresh
etal.9, copper nanoparticles in the alumina matrix mixed at most modest and sucient levels may preserve
the hybrid nanouid’s strength, rst introduced in9. Despite having a lower thermal conductivity than copper
nanoparticles, alumina nanoparticles have excellent chemical inactivity and stability. Yildiz etal.10 developed an
equivalence between theoretical and experimental thermal conductivity models for heat transfer performance
in hybrid-nanouid. In comparison to a mono nanouid, the hybridisation of nanoparticles improved heat
transfer at a lower particle percentage (Al2O3). Waini etal.11 investigated a hybrid nanouid’s unsteady ow and
heat transfer using a curved surface. As the surface curvature changed, the presence of dual solutions resulted in
intensication in the volume percentage of copper nanoparticles. Many years later, Qureshi etal.12 investigated
the hybrid mixed convection nanouid’s characteristics in a straight obstacle channel. ey’ve discovered that
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increasing the barrier’s radius improves heat transfer by as much as 119%. In addition, the horizontal orienta-
tion of the cylinder only supports a heat transfer eciency of 2.54%. Mabood and Akinshilo13 investigated the
inuence of uniform magnetic and radiation on the heat transfer ow of Cu-Al2O3/H2O hybrid nanouid ow-
ing over the stretched surface. Discoveries such as those made at the science fair show how radiation speeds up
heat transport while magnetic forces slow it down. Further hybrid nanouid studies and experiments have been
conducted by these researchers14–24.
e Cattaneo–Christov heat ux model describes the heat transfer in viscoelastic ows caused by an expo-
nentially expanding sheet. ere may be a relationship between thermal relaxation time and the boundaries of
this model. Dogonchi and Ganji25 researched unstable squeezing MHD nanouid ow across parallel plates
using a Cattaneo–Cristov heat ux model some years ago. e thermal relaxation parameter, they found, slowed
heat transfer. Additionally, Muhammad etal.26 discovered that when thermal relaxation increased, the uid
temperature decreased. Other researchers have used the Cattaneo–Christov heat ux model to examine uid
ow and determine the physical features that thermal relaxation aects. Scholars like27–32 may be found in the
literature as examples of this group. Even the temperature of a nanouid may be reduced by the thermal relaxation
parameter, according to Ali etal.33. is nding is critical to the contemporary food, medicinal, and aerospace
industries. Waqas etal.34 introduced mathematical modelling using the Cattaneo–Christov model for hybrid
nanouid ow in a rocket engine. e nding exposed that the temperature is reduced when thermal relaxation
and melting parameters vary, but the Biot number increases. Other types of hybrids nanouid characteristics
using the Cattaneo–Christov model have been discussed by Haneef etal.35. e vital discovery uncovered an
escalation causes shrinkage in wall shear stress in momentum relaxation time. Dierent encounters were found
by Reddy etal.36 in the Cattaneo–Christov model problem for hybrid dusty nanouid ow. It reveals that dusty
hybrid nanouid has a better heat transfer method than hybrid nanouid.
Nevertheless, a few years back, a new type of uid was found called hybrid nanouid, and many researchers
have been eager to search for the characteristics of this type of uid since then. e research for nding the aspect
of non-Newtonian hybrid nanouid also needed to be done. Latterly, Yan etal.37 have conducted an investigation
towards the rheological behaviour of non-Newtonian hybrid nanouid for a powered pump. ey reported at the
highest volume fraction hybrid nanouid, the viscosity reduced at most 21%. Nabwey and Mahdy38 are doing an
inclusive exploration of micropolar dusty hybrid nanouid. e nding indicates that the temperature uctua-
tion in both the micropolar hybrid nanouid and dust phases is strengthened by increased thermal relaxation.
Several investigations have been carried out for the dierent types of non-Newtonian hybrid nanouid, such
as aluminium alloy nanoparticles by Madhukesh etal.39, MWCNT-Al2O3/5W50 by Esfe etal.40 and ZnO–Ag/
H2O by He etal.41 in the literature. Despite that, only a few research available in the literature investigating the
viscoelastic hybrid nanouid behaviour. Several models can be used to examine the physical properties of the
viscoelastic uid, including the power-law model, the Prandtl uid model, and the Prandtl–Eyring model. e
power-law model predicts the non-linear relationship between deformation rate and shear stress. It has been
hypothesised that shear stress is connected to the sine inverse function of deformation rate by the Prandtl model
and that it is related to the hyperbolic sine function of deformation rate by the Prandtl–Eyring model. Hussain
etal.42 have investigated the physical aspect of MHD Prandtl–Eyring uid ow and reported that at all positions
in the ow domain, a substantial rise in momentum transportation had been seen against an increase in the
uid parameter. Rehman etal.43 added in the ndings that Prandtl–Eyring liquid particles are subjected to drag
forces in a ow when their skin friction coecients are high (or low). A similar discovery has been conveyed
by Khan etal.44 which the skin friction improves for the Prandtl–Eyring nanouid. Later, Akram etal.45 model
a MHD Prandtl–Eyring nanouid peristaltic pumping in an inclined channel. is study demonstrates that the
wall tension and mass parameters have a rising inuence on axial velocity, whereas the wall damping param-
eter has a decreasing impact. Li etal.46 have explored the entropy of the Prandtl–Eyring uid ow model over
a rotating cone. e result demonstration the velocity and temperature have been shown to behave dierently
when the viscosity parameter increases in magnitude. Latest study for the Prandtl–Eyring hybrid nanouid
model being carried out by Jamshed etal.47. e outcome was mentioning the entropy upsurged with radiative
ux and Prandtl–Eyring parameter.
e famous numerical technique for solving non-linear boundary layer equations in uid mechanics is derived
by Keller and Cebeci48 called Keller Box Method (KBM). It is being popularised by Cebeci and Bradshaw49. e
technique is known for highly accurate and time computation in solving non-linear problems. A lot of inves-
tigations of uid dynamics have been solved using KBM in the literature. Bilal etal.50 implemented the KBM
for solving Williamson uid ow towards a cylindrical surface and found the results are comparable with other
published results. Similar numerical computation was reported by Swalmeh etal.51 in solving the micropolar
nanouid over a solid sphere using KBM. e computed solution being reported as having a good agreement with
the solution computed by bvp4c (MATLAB). e KBM is a universal solver since it is proven can solve another
type of mathematical modelling, for instance, Carreau uid model (Salahuddin52), micropolar uid (Singh
etal.53), viscous uid model (Bhat and Katagi54), Prandtl nanouid (Habib etal.55), MHD nanouid (Zeeshan
etal.56) and third-grade nanouid (Abbasi etal.57).
Size and distribution descriptors should be chosen to oer the most signicant discrimination for particulate
quality concerning specic attributes or characterisation of a manufacturing process, depending on their use.
If particle form aects these attributes, the shape and distribution of the particles should be studied in addition
to their size. Qualitative terminology like bres or akes can be used, or quantitative terms like elongation,
roundness, and angularity can also be used. Other quantitative terms include percentages of certain model
forms and fractal dimensions. Despite the importance of the particle shape, only a few research can be found
in the literature, such as58–60. e latest research has been done by Sahoo61, which claimed that the particle
shapes heavily inuence the thermo-hydraulic performance of a ternary hybrid nanouid. Similar ndings have
been illustrated by Elnaqeeb etal.62 in hybrid nanouid ow with the impact of suction and stretching surface.
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Meanwhile, Rashid etal.63 suggested that the temperature and Nusselt number proles demonstrate the sphere
shape nanoparticles have superior temperature disturbance and heat transmission on hybrid nanouid ow with
the inuence of relevant factors.
A few publications have examined the impact of the porosity material, viscid dissipative ow, Cattaneo–Chris-
tov heat ow and thermal radiative ow shape-factor along the elongated surface using nanouid Tiwari-Das
type on P-EHNF entropy generation. However, none of these papers has addressed these issues. In the Tiwari-
Das (monotonic model), the uid, speed, and temperature are all the same. As a result, the model is simpler and
easier to solve when using the single-phase technique numerically. However, this technique has the drawback
of resulting in numerical eects that dier from experimental results in some cases. Nanoparticle concentra-
tions in this model volume range from 3 to 20%. Numerical results could only mimic the eects of SWCNT-EO,
MWCNT-EO hybrids, and conventional nanouids in this study. us, in order to bridge the gap, the current
research focuses on the solid–uid characteristics impacts and the level of chaos in the boundary layer using
the Keller-box technique of P-EHNF.
Flow model formulations
e mathematical ow equations shows the moved horizontal plate with the irregular expanding velocity64:
where
b
is an original expanding ratio. Sequestered surface heat is
Y
=w(x,t)=Y
=
∞+b∗x
and for the suitability,
it is presumed to stand at
x=0
, where
b∗,
Y
=w
, where
Y
=∞
signify the temperature variation amount, heat of
surface, and surrounds correspondingly. e plate is supposed to be slippery, and the surface is subjected to a
temperature variation.
Primary addition SWCNT nano solid-particles synthesise the hybrid nanouid in the EO-based liquid at an
interaction volume fraction (
φST
) and it is xed at 0.09 during the examination. MWCNT nano molecules have
been extended in combination to obtain a hybrid nanouid at the concentrated size (
φMT
).
Prandtl–Eyring uid stress tensor. Prandtl–Eyring uid stress tensor is given in the following math-
ematical form (for example, Mekheimer and Ramadan65).
Here the curving velocity indicates the mechanisms
←
B
=

B
1
x,y,0

,B
2
x,y,0

,0

.
Ad
and
C
is fluid
parameters.
Suppositions and terms of system. e following principles, as well as the constraints, apply to the ow
system:
2-D laminar time-dependent curving Domenating-layer approximations
Single phase (Tiwari-Das) scheme Non-Newtonian P-EHNF
Porous medium Cattaneo–Christov heat ux
ermal radiative ow Viscid dissipative owing
Nano solid-particles shape-factor Porousness elongated surface
Slippery boundary constraints ermal jump boundary con-
straints
Formal model. e formal (geometric) owing model is displayed as (Fig.1):
Model equations. e constitutive ow formulas66 of the viscous Prandtl–Eyring hybrid nanouid, in com-
bination with a porous material, Cattaneo–Christov heat ux and thermal radiative ow utilising the approxi-
mate boundary-layer are
(1)
Uw(x,t)=bx,
τ
=
AdSin−11
C∂B1
∂y2
+∂B1
∂y2
1
2

∂B1
∂y

2
+

∂B1
∂y

2
1
2

∂B1
∂y
.
(2)
∂B
1
∂x
+
∂B
2
∂y
=
0,
(3)
B
1
∂B1
∂x+B2
∂B1
∂y=Ad
Cρhnf

∂2B1
∂y2

−Ad
2C3ρhnf
∂2B1
∂y2

∂B1
∂y

2

−µhnf
ρhnf kB1
,
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the appropriate connection conditions are as follows, which can be located in Aziz etal.67:
We formulate the
Y
=
as a uid heat. Other vital parameters are surface permeability
Vς
, heat transfer coe-
cient
hς
, porosity
(k)
and heat conductivity of rm
kς
. Physical features identical, Convectional animated surface
experienced its heat loss through conductive (Newtonian thermal) and owing swiness close to the sheet is
comparative to the cut stress exerts in it (slippy form) are deliberate.
Heat-physical possessions of P-ENF. Nano solid particles dispersed in EO induce improved thermo-
physical characteristics. e next Table1 equations summarize P-ENF substance variables68,69.
φ
is the nano solid-particle size coecient.
µf
,
ρf
,
(Cp)f
and
κf
are dynamical viscidness, intensity, function-
ing thermal capacity, and thermal conductivity of the standard uid, respectively. e additional characteristics
ρs
,
(Cp)s
and
κs
are the concentration, eective heat capacitance, and heat conductance of the nano molecules,
correspondingly.
Thermo-physical properties of P-EHNF. e primary assumption of hybrid nanouids is the suspen-
sion of two distinct forms of nano solid particles inside the basis uid70. is assumption improves the capacity
for heat transmission of common liquids and is a higher heat interpreter than nanouids. P-EHNF variables
content is summarised in Table271,72.
In Table2,
µhnf
,
ρhnf
,
ρ(Cp)hnf
and
κhnf
are mixture nanouid functional viscidness, concentration, exact
thermal capacitance, and thermal conductance.
φ
is the volume of solid nano molecules coecient for mono
(4)
B
1
∂Y
=
∂x+B2
∂Y
=
∂y=
1
ρCpκhnf

khnf ∂2Y
=
∂y2+µhnf ∂B1
∂y2
−∂qr
∂y

,
−δ∗

B1
∂B1
∂
x
∂Y
=
∂
x
+B2
∂B2
∂
y
∂Y
=
∂
y
+B1
∂B2
∂
x
∂Y
=
∂
y
+B2
∂B1
∂
y
∂Y
=
∂
x
+B2
1
∂2Y
=
∂
x
2+B2
2
∂2Y
=
∂
y
2+2B1B2
∂2Y
=
∂
x
∂
y.
(5)
B
1(x,0
)=Uw+Nς

∂
B
1
∂
y
,B2(x,0
)=Vς,−kς

∂
Y
=
∂
y
=hς(Y
=w−Y
=)
,
(6)
B1→0, Y
=→Y
=∞as y →∞.
Figure1. Diagram of the ow model.
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nanouid and
φhnf =φ
ST
+φMT
is the nano solid particles magnitude measurement for the combination nano-
uid.
µf
,
ρf
,
(C
p
)f
,
κf
and
σf
are functional viscidness, density, exact thermal capacity, and heat conductivity of
the base uid.
ρp1
,
ρp2
,
(Cp)p1
,
(Cp)p2
,
κp1
and
κp2
are the density, specic heat capacity, and thermal conductivity
of the nano-molecules.
Nano solid-particle shape-factor m. e scale of the multiple nano solid-particles is dened as the
shaped-nanoparticles factor. Table3 shows the importance of the experiential form factor for dierent particle
forms (for instance, see Xu and Chen73).
Nano solid-particles and baseuid lineaments. In this analysis, the material characteristics of the
primary oil-based liquid of the engine are specied in Table474,75.
Rosseland approximation. Radiative ow only passes a shortened distance because its non-Newtonian
P-EHNF is thicker. Because of this, the approximation for radiative uxing from Rosseland76 is utilised in for-
mula (4).
herein,
σ∗
signies the constant worth of Stefan–Boltzmann and
k∗
symbols the rate.
Dimensionless formulations model
Given the similarity technology that transforms the governing PDEs into ODEs, the BVP formulas (2)–(6) are
modied. Familiarising stream function
ψ
in the formula75
e specied similarity quantities are
into Eqs. (2)–(4). We get
with
(7)
q
r=−
4σ
∗
3k
∗
∂Y
=
4
∂
y,
(8)
B
1=
∂ψ
∂
y
,B2=−
∂ψ
∂
x.
(9)
�
x,y

=

b
νf
y,ψ

x,y

=

νfbxf (�),θ(�)=Y
=−Y
=∞
Y
=w−Y
=∞
.
(10)
A
∗
1f′′′

1−A∗
2f′′2

+φb

ff′′
−f′2

−
1
φa
Kςf′
=
0,
(11)
θ
′′

1+
1
φ
d
PςNς

+Pς
φc
φ
d
fθ′−f′θ+
E
ς
φ
a
φ
c
f′′2−ες

ff′θ′+f2θ′′

=
0.
Table 1. ermo-physical features for nano liquids.
Features Nano liquid
Dynamical viscidness
(µ)
µnf
=µ
f
(1−φ)
−2.5
Density
(ρ)
ρnf =(1−φ)ρf−φρs
Heat capacity
(ρCp)
(ρCp)nf =(1−φ)(ρ Cp)f−φ(ρ Cp)s
ermal conductivity
(κ)
κ
nf
κf=

κs+(m−1)κf−(m−1)φκf−κs

κs+(m−1)κf

+φ

κf−κs

Table 2. ermo-physical features of hybrid nanouids.
Features Hybrid nanouid
Viscidness
(µ)
µhnf
=µ
f
(1−φ
ST
)
−2.5
(1−φ
MT
)
−2.5
Density
(ρ)
ρhnf =[(1−φMT ){(1−φST )ρf+φST ρp1}] + φGoρp2
Heat capacity

ρC
p
(ρCp)hnf =[
(1−φMT ){(1−φST )(ρCp)f+φST (ρ Cp)p1}] + φMT (ρCp)p2
ermal conductivity
(κ)
κ
hnf
κgf =

κp2+(m−1)κgf −(m−1)φST κgf −κp2

κp2+(m−1)κgf

+φMT

κgf −κp2

;
κ
gf
κf=

κp1+(m−1)κf−(m−1)φST κf−κp1

κp1+(m−1)κf

+φST κf−κp1)

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where
φ′
is
is
a≤i≤d
in formulas (10) and (11) signify the subsequent thermo-physical structures for P-HNF
(12)
f(0)=S,f
′
(0)=1+ςf
′′
(0),θ
′
(0)=−Hς(1−θ(0))
f′()→0, θ()→0, as →∞.

(13)
φ
a=(1−φST )2.5(1−φMT )2.5 ,φb=(1−φMT )

(1−φST )+φST
ρp1
ρ
f
+φMT
ρp2
ρ
f
,
(14)
φ
c=(1−φMT )

(1−φST )+φST
(ρ
C
p)p1
(ρC
p
)
f
+φMT
(ρ
C
p)p2
(ρC
p
)
f
,
Table 3. Shape-factor worth for dierent molecules shape.
Nanoparticles type Shape Size (
m
) Sphericity
Sphere
3 1.0
Hexahedron
3.7221 0.87
Tetrahedron
4.0613 0.82
Column
6.3698 0.61
Lamina
16.1576 0.33
Table 4. Fabricated materials thermo-physical attributes.
ermophysical
ρ
(kg m−3)
cp
(J kg−1K−1)
k
(W mK−1)
SWCNTs 2600 425 6000
MWCNTs 1600 796 3000
Engine Oil (EO) 884 1910 0.144
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Explanation of the entrenched control constraints. Equation(2) is accurately conrmed. Previously,
the representation
′
existed for demonstrating the derivatives regarding

.
Symboles Name Formule Default
value
A∗
1
Prandtl–Eyring parameter-I
A
∗
1=
A
d
µ
fC
1.0
A∗
2
Prandtl–Eyring parameter-II
A
∗
2=b
∗
x
2
2
C
2ν
f
0.4
ες
Relaxation timeparameter
ες=b0
0.2
Pς
Prandtl number
Pς
=
νf
αf
6450
φ
Volume fraction 0.18
Kς
Porosity parameter
Kς
=
νf
bk
0.2
S
Suction/injection parameter
S
=−Vς

1
νfb0.4
Nς
ermal radiation parameter
N
ς=16
3
σ
∗
Y
=3
∞
κ∗ν
f
(ρCp)
f
0.3
Eς
Eckert number
E
ς=
U2
w
(
Cp
)
f
(
Y
=
w
−
Y
=
∞)
0.3
Hς
Biot number
H
ς=hς
kς
νf
b
0.3
m
Shape parameter (spherical)
m
3
�ς
Velocity slip
�
ς=

b
ν
f
N
ς
0.3
Drag-force and Nusselt number. e drag-force

C
f
combined with the Nusselt amount
(Nux)
are the
interesting physical amounts that controlled the owing and specied as66
where
τw
and
qw
determine as
e dimensionless transmutations (9) are implemented to obtain
where
Nux
means Nusselt aggregate and
Cf
states drag force constant.
Re
x
=uwx
ν
f
is local
Re
built in the extended
swiness
uw(x)
.
Classical Keller box technique
Because of its rapid convergence, the Keller-box approach (KBM)77 is used to nd solutions for model formulas
(Fig.2). KBM is used to nd the localised solve of (10) and (11) with constraints (12). e policy of KBM is
specied as next:
Stage 1: ODEs adaptation. In the early stage, all of the ODEs must be changed into 1st-order ODEs
(10)–(12)
(15)
φ
d=

κp2+(m−1)κnf

−(m−1)φMT

κnf −κp2


κp2+(m−1)κnf

+φMT

κnf −κp2
 
κp1+(m−1)κf

+φST

κf−κp1


κp1+(m−1)κf

−(m−1)φST

κf−κp1
.
(16)
C
f=
τw
1
2
ρ
f
U2
w
,Nux=
xqw
kf(Y
=w−Y
=∞
)
(17)
τ
w=

Ad
C
∂B1
∂y+Ad
6C3

∂B1
∂y
3y
=
0
,qw=−khnf

1+
16
3
σ∗Y
=3
∞
κ∗νf(ρCp)f

∂Y
=
∂y

y=
0
(18)
C
fRe
1
2
x=A∗
1f′′(0)−
1
3A∗
1A∗
2

f′′(0)

3,NuxRe−1
2
x=−
k
hnf
k
f
1+Nς

θ′(0)
,
(19)
z1
=
f′,
(20)
z2
=
z
′
1,
(21)
z3=θ′,
(22)
A
∗
1z′
2

1−A∗
2z2
2

+φb

fz2−z2
1

−
1
φa
Kςz1=
0,
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Stage 2: separation of domains. Discretisation plays a very important in the eld of awareness. Dis-
cretising is usually conducted by making the area separated into equivalent-sized grids. Relatively lesser grids
results are chosen in obtaining a higher precision for the calculation outcomes.
where
j
is used for the spacing in
h
in a horizontal direction to show the position of the coordinates. e solu-
tion to the problem is to be found without any initial approximation. It is very crucial for nding velocity, tem-
peratures, temperature variations, and entropy to make a preliminary assumption between
=0
and
=∞
.
e frameworks from the result have been approximatedsolutions provided as they can happen the boundary
conditions of the problem. It is imperative to remark that the results must be equalled with dierent preliminary
estimations are chosen, but the replication computation and time are varied which have been taken for conduct-
ing the calculations (see Fig.3):
By implementing signicant dierences, dierence equivalences are gured, and functions are used to replace
the mean values. e 1st-order ODEs (19)–(23) have been modied to algebraic formulas which are non-linear.
(23)
z
′
3

1+
1
φd
PςNς

+Pς
φc
φd
fz3−z1θ+Eς
φaφc
z2
2−ες

fz1z3+f2z′
3

=
0.
(24)
f(0)=S,z1(0)=1+�ςz2(0),z3(0)=−Hς(1−θ(0)),z1(∞)→0, θ(∞)→0.
�0=0, �j=�j−1+h,j=1, 2, 3, ...,J−1, �J=�∞.
(25)
(z
1)j+(
z
1)j−1
2
=
f
j−
f
j−1
h,
(26)
(z
2
)
j
+(z
2
)
j−1
2
=
(z
1
)
j
−(z
1
)
j−1
h,
(27)
(z
3
)
j
+(z
3
)
j−1
2
=
θ
j
−θ
j−1
h,
Figure2. Chart of KBM steps.
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Stage 3: linearisation based on Newton’s method. e resulting formulas have been completed lin-
early by using Newton’s process.
(i
+
1)th
iterations can be found in the earlier equations
By replacing this in (25) to (29) and aer overlooking the higher-elevated bounds of
¨
θi
j
a linear tri-diagonal
equation scheme has been resulting as follows:
where
(28)
A∗
1(z2)j−(z2)j−1
h

1−A∗
2(z2)j+(z2)j−1
22

+

φb

fj+fj−1
2

(z2)j+(z2)j−1
2

−

(z1)j+(z1)j−1
2

2

−Kς
1
φa

(z1)j+(z1)j−1
2
,
(29)
(z
3
)
j
−(z
3
)
j−1
h

1+
1
φd
PςNς

+Pς
φc
φd
f
j
+f
j−1
2
(z
3
)
j
+(z
3
)
j−1
2

+Pr φc
φdEς
φaφc(z2)j+(z2)j−1
22
−(z1)j+(z1)j−1
2θj+θj−1
2
−Pr φc
φd

ες

fj+fj−1
2

(z1)j+(z1)j−1
2

(z3)j+(z3)j−1
2

+

fj+fj−1
2

2

(z3)j−(z3)j−1
h

=
0.
(30)
()(i+1)
j
=()
(i)
j
+¨
θ()
(i)
j.
(31)
¨
θ
fj−¨
θfj−1−
1
2
h(¨
θ(z1)j+¨
θ(z1)j−1)=(r1)j−1
2
,
(32)
¨
θ
(z1)j−¨
θ(z1)j−1−
1
2
h(¨
θ(z2)j+¨
θ(z2)j−1)=(r2)j−1
2
,
(33)
¨
θ
θj−¨
θθj−1−
1
2
h(¨
θ(z3)j+¨
θ(z3)j−1)=(r3)j−1
2
,
(34)
(
a1)j
¨
θfj+(a2)j
¨
θfj−1+(a3)j
¨
θz1j+(a4)j
¨
θz1j−1+(a5)j
¨
θz2j+(a6)j
¨
θz2j−
1
+(a7)j¨
θθj+(a8)j¨
θθj−1+(a9)j¨
θ(z3)j+(a10)j¨
θ(z3)j−1=(r4)j−1
2
,
(35)
(
b1)j
¨
θfj+(b2)j
¨
θfj−1+(b3)j
¨
θz1j+(b4)j
¨
θz1j−1+(b5)j
¨
θz2j+(b6)j
¨
θz2j−
1
+(b7)j¨
θθj+(b8)j¨
θθj−1+(b9)j¨
θ(z3)j+(b10)j¨
θ(z3)j−1=(r5)j−1
2
.
(36)
(
r1)j−1
2
=−fj+fj−1+
h
2
(z1)j+((z1)j−1)
,
Figure3. Net rectangle for showing dierence approximations.
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Converted boundary conditions aer the similarity process were given below
Stage 4: the bulk scheme and eliminating. At the nal, bulk tridiagonal matrix has been reached from
the formulations in (30)–(35) as follows,
where
where
5×5
block-sized matrix is denoted by
F
that corresponds to the size of
J×J
. However, the vector of order
J×1
is represented by
¨
θ
and
p
. An esteemed LU factorising method is used for solving
¨
θ
later. e equation
F¨
θ=
p
denotes that F with an array
¨
θ
is used to yield a production array marked by
p
. Further, F is splinted into
lower and upper trigonal matrices, i.e.,
F=LU
can be written as
LU ¨
θ=
p
. Let
U¨
θ=
y
tends to
Ly =p,
which
is used to provide the solution of
y
. Further, the values of
y
computed are replaced into the equation
U¨
θ=
y
for
solving
¨
θ
. e technique of back-substitution has been implemented as this is the easy method to nd a solution.
Code verication
On the other, by measuring the heat transmission rate outcomes from the current technique against the recent
results available in the literature78,79, the validity of the method was evaluated. Table5 summarises the compar-
ing of reliabilities current during the researches. Nevertheless, the outcomes of the current examination are
exceedingly accurate.
Second law of thermodynamics
Porous media generally increase the entropy of the system. Jamshed etal.80 and Jamshed81 described the nanouid
entropy production by:
e non-dimensional formulation of entropy analysis is as follows82–84,
(37)
(
r2)j−1
2
=−(z1)j+(z1)j−1+
h
2
((z2)j+(z2)j−1)
,
(38)
(
r3)j−1
2
=−θj+θj−1+
h
2
((z3)j+(z3)j−1)
,
(39)
(
r4)j−1
2
=−h

A∗
1(z2)j−(z2)j−1
h

1−A∗
2(z2)j+(z2)j−1
22

−h

φb

fj+fj−1
2

(z2)j+(z2)j−1
2

−

(z1)j+(z1)j−1
2

2

−Kς
1
φa

(z1)j+(z1)j−1
2
,
(40)
(
r5)j−1
2
=−h

((z3)j−z3)j−1
h

1+
1
φd
PςNς+φcPr
φd

fj+fj−1((z3)j+z3)j−1
4

+hφcPς
φd

θj+θj−1z1j+z1j−1
4
−Eς
φ1φ3(z2)j+(z2)j−1
22

+hφcPς
φd

ες

fj+fj−1
2

(z1)j+(z1)j−1
2

(z3)j+(z3)j−1
2

−

fj+fj−1
2

2

(z3)j−(z3)j−1
h
.
(41)
¨
θf0
=
0, ¨
θ
(z1)0
=
0, ¨
θ
(z3)0
=
0, ¨
θ
(z1)J
=
0, ¨
θ
θJ
=
0.
(42)
F¨
θ=
p,
(43)
F
=










A
1
C
1
B2A2C2
.........
.........
BJ−1AJ−1CJ−1
BJAJ










,¨
θ=







¨
θ1
¨
θ2
.
.
.
¨
θj−1
¨
θj







,p=








(r1)j−1
2
(r2)j−1
2
.
.
.
(rJ−1)j−1
2
(rJ)j−1
2








.
(44)
E
G=khnf
Y
=2
∞

∂Y
=
∂y

2
+
16
3
σ∗Y
=3
∞
κ∗νf(ρCp)f

∂Y
=
∂y

2

+µhnf
Y
=∞

∂B1
∂y

2
+µhnf B2
1
kY
=∞
.
(45)
N
G=Y
=
2
∞b
2
EG
k
f
(Y
=
w
−Y
=∞)2
.
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By formula (9), the non-dimensional entropy formula is:
Here
Re
is the Reynolds number,
Bς
signies Brinkmann amount and

symbols the non-dimensional vari-
ation of the temperature.
Results and discussion
An adequate discussion is indicated by numerical results that reach the model described before. As a result of
these potential parameters, the values for
A∗
1
,
A∗
2
,
Kς
,
φ
,
�ς
,
S
,
Nς
,
ες
,
Eς
Hς,Re
and
Bς
been illustrated. ese
parameters show the physical performance of the non-dimensional quantities in Figs.4, 5, 6, 7, 8, 9, 10, 11, 12,
(46)
N
G=Re

φd

1+Nς

θ′2+
1
φ
a
B
ς
�

f′′2+Hςf′2
,
Table 5. Comparing of
−θ′(
0
)
values with
Pς
, when
φ
=
0
,
φhnf
=
0
,
ες
=
0
,
�ς
=
0
,
Eς
=
0
Nς
=
0
,
S=0
and
Hς→∞
.
Pr
Ref.78 Ref.79 Present
72 × 10−2 080863135 × 10−8 080876122 × 10−8 080876181 × 10−8
1 × 1001 × 1001 × 1001 × 100
3 × 100192,368,259 × 10−8 192,357,431 × 10−8 192,357,420 × 10−8
7 × 100307,225,021 × 10−8 307,314,679 × 10−8 307,314,651 × 10−8
10 × 100372,067,390 × 10−8 372,055,436 × 10−8 372,055,429 × 10−8
Figure4. Velocity change with
A1∗
.
Figure5. Temperature change with
A1∗
.
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13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 and 26, such as velocity, energy, and entropy production. e results
are obtained for Cu-EO normal P-ENF and MWCNT-SWCNT/EO non-Newtonian P-EHNF. e coecient of
skin friction and temperature variations are shown in Table6. For example, the default values were 1.0 for A
∗
1
and 0.4 for
A∗
2
,
Kς
was set to be equal to 0.1, and
φ
= 0.18,
φMT
was set to 0.09,
�ς
was set to 0.3,
S
was set to 0.4,
and
Nς
was set to 0.3,
ες
was set to 0.1,
Eς
was set to 0.3,
Hς
was set to 0.3, and
Re
and
Bς
was set to 5.
Figure6. Entropy change with
A1∗
.
Figure7. Velocity change with
A2∗
.
Figure8. Temperature change with
A2∗
.
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Inuence of Prandtl–Eyring parameter
A∗
1
. Figures4, 5 and 6 illustrate the inuence of the Prandtl–
Eyring parameter
A∗
1
on the velocity, energy, and entropy distributions of the Prandtl–Eyring hybrid nanouid,
respectively.
A∗
1
′s
velocity uctuation (
f′
) is seen in Fig.4. As the value of
A∗
1
was elevated, so was the velocity
prole for both uids. e physical reason for this occurrence is that it causes the uid’s viscosity to decrease,
reducing resistance while boosting uid velocity. MWCNT-SWCNT nanouid, on the other hand, has faster
Figure9. Entropy change with
A2∗
.
Figure10. Velocity change with
Kς
.
Figure11. Temperature change with
Kς
.
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acceleration than SWCNT nanouid. It can be explained as the hybrid nanouid have an enormous density
impact rather than the nanouid. e temperature curve for the Prandtl–Eyring parameter
A∗
1
is shown in Fig.5.
MWCNT-SWCNT hybrid nanouid had a lower temperature prole since the value of
A∗
1
was raised, while the
Cu nanouid had a higher temperature prole. More heat can be conveyed faster when the caused in this low-
ered manner due to velocity improve and expand. Another important distinction is that the hybrid nanouid
Figure12. Entropy change with
Kς
.
Figure13. Velocity change with
φ/φhnf
.
Figure14. Temperature change with
φ/φhnf
.
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exhibits signicantly reduced thermal conductivity when compared to pure nanouid. Figure6 depicted the
Prandtl–Eyring hybrid nanouid entropy uctuation based on its parameter
A∗
1
. e quantity of entropy pro-
duced decreased as the amount of
A∗
1
enhanced. MWCNT-SWCNT uid exhibited a lower entropy value than
SWCNT hybrid nanouid, even though their values were the same at one point in the graph. is phenomenon
Figure15. Entropy change with
φ/φhnf
.
Figure16. Velocity change with
�ς
.
Figure17. Temperature change with
�ς
.
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occurs due to the low temperatures reducing hybrid nanouid mobility, causing the system’s entropy to prolif-
eration.
Inuence of Prandtl–Eyring parameter A
∗
2
. ere was an inuence of Prandtl–Eyring Parameter
A∗
2
on
the Prandtl–Eyring hybrid nanouid temperature, velocity, and entropy production prole (see Figs.7, 8). Fig-
Figure18. Entropy change with
�ς
.
Figure19. Temperature change with
Nς
.
Figure20. Entropy change with
Nς
.
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ure7 depicts the varying
A∗
2
with velocity . e velocity prole narrows as
A∗
2
rises, with MWCNT-SWCNT/EO
achieving a higher top speed than SWCNT-EO. Hybrid nanouid particles have resistance due to the fact that
they vary inversely with momentum diusivity. As a result, the ow’s velocity will be reduced with A
∗
2
. is phe-
nomenon is because SWCNT-EO has a higher density and hence has a thicker ow than MWCNT-SWCNT/EO,
making the uid challenging to transport. Figure8 shows the temperature change aer
A∗
2
has had its impact. As
Figure21. Temperature change with
ες
.
Figure22. Entropy change with
ες
.
Figure23. Temperature change with
m
.
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the value of
A∗
2
grew, so did the temperature, with SWCNT-EO quickly reaching the desired temperature. e
occurrence happens because the ow velocity dropped, and as a result, the heat transmission from the surface
was degraded. Figure9 shows the change in entropy according to the Prandtl–Eyring parameter
A∗
2
. e entropy
prole grew as the value of
A∗
2
grew, showing a clear connection between the two. It suggested that
A∗
2
amplifying
the impediment in the system, resulting in the entropy of the developing system being elevated.
Figure24. Entropy change with
m
.
Figure25. Entropy change with
Re
.
Figure26. Entropy change with
Bς
.
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Eect of porous media variable
Kς
. Figures 10, 11 and 12 demonstrate that surface porosity aects
several outputs, including ow speed, domain heat, and entropy generation. Improving the variable (
Kς
) in
Fig.10 makes the surface more porous, allowing more uid to ow through it. Due to the other particles, the
hybrid nanouid moves more slowly through the porous surface when compared to MWCNT-SWCNT/EO
Prandtl–Eyring nanouid. is occurrence might be because the added particles delay the hybrid nanouid’s
ow through the porous surface. Figure11 displays the expansion of the porous medium variable (
Kς
) results in
better heat dispersion throughout the domain. When a hole is made in a porous medium, the ow slows down,
allowing more time to collect heat from the surface. is phenomenon improves the thermal distribution around
the area. Since particle motions across porous media are sluggish, the porosity aids in the irreversibility of energy
transfer across the domain during entropy production
(NG)
(Fig.12).
Eect of nanomolecules size
φ
and
φ
hnf. e ecacy of the nanouid and hybrid versions appears
to be determined by the fractional nanoparticle size in the base uid. e more excellent fractional range of
nanoparticles reduces owability because of the additional load it adds. For some reason, the fractional upgrade
prefers the hybrid nanouid over the single-nanouid, which ows lower in Fig.13. is incidence displayed
the primary reason for utilising nano- and hybrid-based uid mixtures because of their exceptional heat trans-
mission properties. is degradation occurs as a result of excessive nanoparticle surface area and higher hybrid
nanouid density. As the fractional volume of both kinds of ow uids improved, so did the resultant thermal
distribution, as shown in Fig.14. Because of the temperature dierence, when the nano molecule size is reduced,
the molecules will disperse in the far-eld ow. e thermal boundary layer’s thickness will rise as a result of this
change. e minimal size of nano molecules can be utilised to create the lowest possible temperature prole, as
determined through experimentation. Figure15 exhibits the leading nanouid varies in the middle and settles
down to the hybrid nanouid at the far end, with energy entropy uctuations also intensifying for fractional
volume. SWCNT-EO has a greater entropy than MWCNT-SWCNT/EO because the hybrid nanouid has a far
higher thermal conductivity than nanouid.
Table 6. Values of
Cf
Re
1/2
x
and
NuxRe
−
1/2
x
for
Pς
=
6450
.
A∗
1
A∗
2
Kς
φ
φMT
�ς
S
Nς
ες
Hς
CfRe
1
2
x
SWCNT-EO
Cf
Re
1
2
x
MWCNT-SWCNT/
EO
NuRe
−1
2
x
SWCNT-EO
NuRe
−
1
2
x
MWCNT-
SWCNT/EO
1.0 0.4 0.1 0.18 0.09 0.3 0.4 0.3 0.2 0.3 4.7980 5.4521 2.5615 3.0496
1.4 4.8262 5.4884 2.5974 3.0727
1.7 4.8561 5.5173 2.6299 3.1092
0.4 4.7980 5.4521 2.5615 3.0496
0.6 4.7629 5.4264 2.5426 3.0138
0.8 4.7487 5.3933 2.5273 3.0095
0.1 4.7980 5.4521 2.5615 3.0496
0.6 4.8113 5.4816 2.5381 3.0230
1.6 4.8335 5.5250 2.5045 3.0045
0.09 4.7372 – 2.5126 –
0.15 4.7543 – 2.5458 –
0.18 4.7980 – 2.5615 –
0.0 – 4.7372 – 2.5126
0.06 – 5.4377 – 3.0123
0.09 – 5.4521 – 3.0496
0.1 4.8626 5.5143 2.6083 3.0911
0.2 4.8397 5.4916 2.5851 3.0648
0.3 4.7980 5.4521 2.5615 3.0496
0.2 4.7647 5.4305 2.5229 3.0092
0.4 4.7980 5.4521 2.5615 3.0496
0.6 4.8131 5.4845 2.5882 3.0675
0.1 4.7980 5.4521 2.5237 3.0282
0.3 4.7980 5.4521 2.5615 3.0496
0.5 4.7980 5.4521 2.5925 3.0755
0.1 4.7980 5.4521 2.5949 3.0754
0.2 4.7980 5.4521 2.5615 3.0496
0.3 4.7980 5.4521 2.5337 3.0042
0.1 4.7980 5.4521 2.5359 3.0148
0.3 4.7980 5.4521 2.5615 3.0496
0.5 4.7980 5.4521 2.5979 3.0636
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Eect of velocity slip variable
�ς
. Figures16, 17 and 18 evaluate the impact of enhanced slip circum-
stances on ow nature, thermal features, and entropy forms. Figure16 illustrates the ow conditions in Prandtl–
Eyring uid mixtures are primarily centred upon the viscous behaviour. Due to this occurrence, slip conditions
become incredibly critical in uids as a whole. For a hybrid suspended Prandtl–Eyring nanouid, the viscous
nature and higher levels of ow slip generate more complex uidity circumstances, with the result that the
uidity of the single nanouid drops even more rapidly. Due to the ow hierarchy, the SWCNT-EO nanouid
maintains a higher temperature state than the MWCNT-SWCNT/EO hybrid nanouid, which is depicted in
Fig.17. e improvement in boundary layer viscosity due to the decline in velocity will have a similar eect. As
a result, it will have skyrocketed the ow’s temperature. Because the hybrid nanouid has less viscosity than the
conventional nanouid, it is predicted MWCNT-SWCNT/EO to have a lower temperature than SWCNT-EO.
A descending trend in entropy formation can be seen for higher slip parameters because the slipped ow acts
against the domain’s entropy formation.
Thermal radiative variable
Nς
and relaxation time parameter (
ες
) inuence. Figures19 and 20
highlights the actual status of thermal diusion and entropy generation under enhanced heat radiative ow limi-
tation

N
ς.
ermally diusing nanouids have a propensity to rise in temperature past the interesting domain,
boosting the heat transmission burden for radiation constrictions on the transient nanouid. is temperature
rise may be explained in a physical sense by supposing that thermal radiation is converted into electromagnetic
energy. As a result, the distance from the surface from which radiation is emitted rises, ultimately superheating
the boundary layer ow. As a result, the thermal radiative variable is critical in determining the system’s tempera-
ture prole. A limit on radiative ow

N
ς
via entropy generation is illustrated in Fig.20 by the overlled disper-
sions. For dierent

N
ς
values, the entropic side-by-side leans toward developing more in MWCNT-SWCNT/
EO than in SWCNT-EO nanouid. A reasonable explanation for this occurrence is the system’s irreparable heat
transfer mechanism is entirely irreversible. According to Fig.21, greater values of the relaxation time parameter
cause a rise in the temperature of the Tangent hyperbolic hybrid nanouids, as seen in the graph. As the tem-
perature drops, the thickness of the thermal boundary layer reduces. Table5 shows that when the rate of heat
output eciency, the eectiveness of the thermal system improves as well. Figure22 shows the impact of engine
oil-based nanouid entropy proles. e velocity prole, on the other hand, shows no change, while the entropy
of the system increases with varying values
ες
.
Eect of the diverse solid particle shape m. It is well-known that NPs have high thermal conductiv-
ity and transfer rates under a variety of physical conditions. In porous medium diculties, such nano-level
particles become an issue, modelled using the shape variable (m) in this study. From spherical (m = 3) to lamina
(m = 16.176), the forms considered here ranged. To improve the thermal state, Fig.23 indicates that nanoparti-
cle shapes impact it. In comparison to SWCNT-EO mono nanouid, the MWCNT-SWCNT/EO hybrid nano-
uid has a more signicant form impact. Hybrid nanouid has a broader thermal layer boundary and a more
excellent thermal distribution than nanouid. Even in the MWCNT-SWCNT/EO hybrid nanouid, the lamina
(m = 16.176) shaped particles remain ahead of the others. e main physical reason for this phenomenon is the
lamina shape particles has the most remarkable viscosity while the sphere has the minimum viscosity. It is also
noted that at a higher temperature, the viscosity of the particles will be diminished. is phenomenon happens
because of the temperature-dependent shear-thinning characteristic. e proles in Fig.24 indicate the form
factors have a more substantial inuence in MWCNT-SWCNT/EO NHF, which has a higher entropy rate than
SWCNT-EO mono nanouid, even though the morphologies of the particles have a much less impact.
Entropy variations for Reynolds number (
Re
) and Brinkman number (
Bς
). Figure25 depicted as
Reynolds number proliferations
(Re)
, the entropy rate
(NG)
improves as well. An aggregate Reynolds number
supports nanoparticle mobility in porous media because of the dominance of inertial over viscous forces in the
system. Consequently, entropy can be generated over the domain. MWCNT-SWCNT/EO HNF generated a
higher entropy rate than MWCNT-SWCNT/EO nanouid because of the combined eciency of the particles.
e Brinkman number

B
ς
was used to describe the heat created by viscous properties because it enhances the
generated heat above and beyond other thermal inputs. e heightened heat-inducing ability of such viscosity
enhancement promotes entropy production in the system as a whole
(NG)
. Figure 26 illustrated the elevated
entropy layers, which improved the Brinkman number

B
ς
values. e primary feature of viscous dislocation
heat produces a decrease in the escalating Brinkman numbers, which theoretically leads to a higher rate of
entropy development.
Final results and future guidance
e entropy production and heat transmission by a Prandtl–Eyring hybrid nanouid (P-EHNF) over a stretched
sheet is studied. By utilising a single model phase, a computational model may be developed. Several physical
characteristics are used to derive the results. ese include changes in velocity, energy, and entropy. Catta-
neo–Christov heat ux
ες
is also discussed in this study, as are the eects of Prandtl–Eyring parameters
A∗
1
and
A∗
2
as well as
Kς
for porous medium, nanomolecular size
φ
and
φhnf
,
�ς
for velocity slip, thermal radiative vari-
able
Nς
and Biot number
Hς
as well as various solid particle shapes
m
,
Re
and
Bς
. e following are the study’s
signicant ndings:
1. In comparison to traditional Prandtl–Eyring nanouid (SWCNT/-EO), hybrid Prandtl–Eyring nanouid
(MWCNT-SWCNT/EO) is shown to be a superior thermal conductor.
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2. Swelling the size of EO’s nano solid particles can increase the rate of heat transmission.
3. Upsurges in the porous media variable
K
, the size parameters
φ
and
φ
hnf , thermal radiative ow (
Nς
), the
Brinkman number (
Bς
) and the Reynolds number (
Re
) growth the system’s entropy, whereas the increase in
the velocity slip parameter (
�ς
) reduces it.
4. Porous media variable increments enhance the velocity magnitude, whereas nano molecule swelling causes
the speed to drop.
e current study’s ndings may help lead future heating system upgrades that evaluate the heating system’s
heat eect using a variety of non-Newtonian hybrid nanouids (i.e., second-grade, Carreau, Casson, Maxwell,
micropolar nanouids, etc.). It’s possible to depict the eects of temperature-dependent viscosity, temperature-
dependent porosity, and magneto-slip ow by signicantly expanding the scheme’s capabilities.
Received: 3 September 2021; Accepted: 22 November 2021
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Acknowledgements
is study was supported by Taif University Researchers Supporting Project Number (TURSP-2020/117), Taif
University, Taif, Saudi Arabia.
Author contributions
W.J. and F.S. framed the issue. W.J., D.B., N.A.A.M.N., F.S., and K.S.N. resolved the problem. W.J., F.S.,
N.A.A.M.N., K.S.N., M.S., S.A. and K.A.I. computed and analysed the results. All the authors equally contrib-
uted to the writing and proofreading of the paper.
Competing interests
e authors declare no competing interests.
Additional information
Correspondence and requests for materials should be addressed to W.J.orD.B.
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