Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium

1. ARTICLE
Copyright © 2022 by American Scientific Publishers
All rights reserved.
Printed in the United States of America
Journal of Nanofluids
Vol. 11, pp. 1–12, 2022
(www.aspbs.com/jon)
Hybrid Nanofluid Flow of Non-Newtonian Casson
Fluid for the Analysis of Entropy Through a
Permeable Medium
S. R. Mishra1
, P. Mathur2,∗
, and P. K. Pattnaik3
1
Department of Mathematics, Siksha ‘O’ Anusandhan Deemed to be University, Bhubaneswar 751030, Odisha, India
2
Department of Mathematics, Poornima College of Engineering & Technology, Jaipur, Rajasthan, India
3
Department of Mathematics, College of Engineering Technology, Bhubaneswar 751029, Odisha, India
The magnetohydrodynamic (MHD) flow of an electrically conducting fluid through a rotating channel is used for
the enhancement of heat transfer properties. The entropy analysis due to the irreversibility of the system for the
non-Newtonian Casson hybrid nanofluid is proposed in this investigation. Ethylene glycol (EG) is considered
as the base liquid in which the metal nanoparticle i.e., copper (Cu) and oxide like aluminium oxide (Al2O3)
nanoparticles are submerged into it for the preparation of nanofluid. In addition, the implementation of radiative
heat, viscous and Joule dissipation in the energy equation enhance the profile as well. The dimensionless for
of the governing equations are obtained by using suitable similarity transformations. The solution is obtained
numerically for both the primary and the secondary velocity distributions along with the temperature profiles
and the physical significance of the pertinent parameters are presented via graphs. The simulated results for
the rate coefficients at both the walls are presented in tabular form. Further, the irreversibility process for the
thermal system i.e., the entropy analysis is carried out for the several parameters along with the Bejan number
and deliberated clearly. Further, the important findings are; an increasing rotation retards the thickness of the
bounding surface of the primary velocity near the lower wall however, significant augmentation in the shear rate
as well as heat transfer rate is revealed for the augmenting magnetic and the rotation parameter.
KEYWORDS: Magnetohydrodynamic, Hybrid Nanofluid, Ethylene Glycol Base Fluid, Permeable Channel, Entropy Analysis.
1. INTRODUCTION
Microgravity condition create important physical phe-
nomenon called Marangoni convection. It is induced due
to the gradient of surface tension. In the beginning of
1958 Pearson1
studied about convection cell encouraged
by the surface tension. Further, the heating of the bottom
layer of the fluid causes the less variation in the tem-
perature gradient that leads to the variation in the sur-
face temperature. This contributes a significant flow of
fluid and therefore the fluid temperature maintain a uni-
form distribution of temperature throughout the domain.
In the year 1960, Marangoni discovered the situation in
which the gravity of the liquid dominates the free convec-
tion and it disappeared slowly in the presence of micro
gravity. Therefore, as a conclusive remark it is found
that the cellular motion is obtained because of the forces
of the surface tension in which this criterion is given
∗
Author to whom correspondence should be addressed.
Email: priya.mathur@poornima.org
Received: xx Xxxx xxxx
Accepted: xx Xxxx xxxx
by means of buoyant driven forces. Many researchers2–10
given attention on heat transfer properties and Marangoni
convection. Ganesh with his colleagues11
worked on the
radiative heat presented in the nonlinear form on the
boundary layer flow and it’s Influence. However, they
have presented Al2O3 nanofluid convection and the heat
transport phenomenon over an expanding sheet in asso-
ciation to Marangoni convection. Authors investigated,
the two-dimensional Marangoni boundary layer flow of
Al2O3∼H2O/C2H6O2 in the presence of non-linear thermal
radiation nanofluids. Coupled nonlinear equations solved
by Runge-Kutta method using shooting technique.
To maximize the thermal properties of a nanofluid, it
must first be prepared as a stable and durable nanofluid.
Therefore, for specific purposes, various material combina-
tions could be used, namely: metal nanoparticles, carbides,
oxides, nitrides, and other non-metals dispersed into fluids
such as water, ethylene glycol, or oils.12
The making of
nanofluids requires few simple processes of consistent and
effective approach and sedimentation and fewer aggrega-
tion problems. Two-step and a single-step processes are the
J. Nanofluids 2022, Vol. 11, No. xx 2169-432X/2022/11/001/012 doi:10.1166/jon.2022.1846 1

2. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium Mishra et al.
ARTICLE
two most general approaches for preparing nanofluids.13
These may be done using either a chemical or mechan-
ical process. Shadlaghani et al.14
investigated the impact
of natural convection on the transport phenomenon within
the annulus that is equivalent to either circular or square
or may be of triangular cross sections. They have pro-
posed control volume method, a numerical technique for
the solution. Goodarzi et al.15
done experimental study
the effect of concentration and temperature of nanoparti-
cle in engine oil. However, the experimentation is con-
ducted with the contribution of volume fractions by
considering 0.05%, 0.1%, 0.2%, 0.4%, 0.6%, and 0.8%,
whereas the range of the temperature 5–55
C, and the
rate of shear stress within the range 666.5 to 13,330
s-1. Further, “Brookfield digital viscometer (CAP2000)”
is used to measure the viscosity of hybrid nanolubri-
cant. The inclusion of augmented particle concentration
enhances the conductivity of the nanoliquid and that leads
to increase the side effects of viscosity. Looking into
the matter, researchers16–18
have presented their report on
the rheological behaviour of nanofluids. Mishra et al.19
and his team recently proposed their investigation by
considering ethylene–glycol-based liquid for the thermal
enhancement in the nanofluid flow past a semi-infinite
vertical plate imposed with porous matrix. For the bet-
ter enhanced properties of the thermophysical properties
they have used Cu, and oxide, Al2O3, as the nanoparti-
cles. ADM was used to solve the non-linear differential
equations and presented graphical analysis. Authors con-
cluded that, the velocity profiles retards with the enhanced
particle concentration since the density of the Cu nanopar-
ticle is likely to be greater, but the impact is opposite
for Al2O3 nanoparticles. Nazari et al.2021
also devoted
their recent work to understand thermophysical properties
of nanofluids. Williamson nanofluid flow through porous
medium was studied under heat transfer boundary condi-
tions, and in other paper they discussed about micropo-
lar nanofluids. Naimi et al.23
in the year 2002 proposed
their investigation by imposing both analytically as well
as numerically. Sheikholeslami and Chamka24
investigated
the influence of Lorentz forces on nanofluid forced con-
vection with Marangoni convection effects for two-phase
nanofluid.
Further, in recent studies several experimental inves-
tigations have been proposed for the thermal enhance-
ment treatment considering both nano and hybrid nanofluid
in various environments. An exhaustive review on the
free convection of nanofluid in various enclosures have
been presented by Sadeghi et al.25
In different geome-
try i.e., within a gamma-shaped cavity, Chamkha et al.26
presented their investigation on the mixed convection
of an electrically conducting nanofluid. They have also
analyzed the entropy generation within the system due
to the heat transport phenomenon. Dogonchi et al.2728
illustrate the natural convection of nanofluid within a
square cavity as well as wavy channel for the impact of
magnetic field. Their focus goes to the shape factor of
the nanoparticles impacts on the flow behaviour. Several
authors including Chamkha and their co-workers29–35
have
analyzed the influence of various characterizing parame-
ters on the flow of nanofluids in different physical sit-
uation. Biswas et al.36
proposed the hybrid nanoliquid
composed of Cu and Al2O3 nanoparticles considering
water as a base liquid for the impact of half-sinusoidal
non-uniform heating. They have imposed the thermal con-
vection boundary approach in their investigation. Further,
Biswas et al.37
convey magnetohydrodynamic thermal con-
vection in the same hybrid nanofluid past a saturated
porous medium. Recently, Manna et al.38
illustrates the
multi-banding application of the proposed magnetic field
within the porous medium. However, Biswas et al.3940
analyzed the heat transport phenomenon of several
nanofluids within a porous cavity embedding with porous
matrix.
Following aforementioned reference, the present inves-
tigation performs the heat transfer features of a conduct-
ing Casson hybrid nanofluid through a rotating permeable
channel. Further, the thermal properties of nanofluid
enriches with the consideration of radiating heat and
dissipative heat. Numerical treatment is employed for
the solution of the nonlinear problem and the analysis
is carried out through graphs for the numerous param-
eters affecting the flow profiles. Further, the numeri-
cal results for the rate coefficients are deployed and
discussed.
2. MATHEMATICAL FORMULATION
A non-Newtonian Casson hybrid nanofluid is presented for
the addition of both Copper and Aluminium is used in the
base liquid Ethylene glycol (Cu/Al2O3∼C2H6O2) through
infinite permeable parallel plates. Consider MHD flow of
an electrically conducting through parallel plates placed at
a distance ‘h’ apart. The flow is along x-axis and y-axis
is normal to it. A uniform magnetic field is applied and
thermal radiation with dissipative heat in a rotation system
also incorporated for the development in the flow phenom-
ena. The plate temperature at the lower part of the channel
is considered to be T0 (“injection takes place”) whereas at
the upper plate it is Th (“suction occurs”) such that T0
Th. A uniform angular velocity through which the body is
rotating about the y-axis is displayed in Figure 1. Because
of the small value of assumed magnetic Reynolds number,
it is wise to omit the impact of induced magnetic field.
The upper wall is moving in a constant velocity u0 towards
y-direction whereas lower one is at rest. The flow charac-
teristic suggests that except pressure, all the physical quan-
tities depend on ‘y’ only since the channel is long enough.
Though the channel wall is permeable it is assumed that
2 J. Nanofluids, 11, 1–12, 2022

3. Mishra et al. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium
ARTICLE
Fig. 1. Schematic diagram.
the suction velocity is v = −v0. Following,41
the rheolog-
ical equation for the assumed flow phenomena is
ij =
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
2
b +
py
√
2
eij c
2
b +
p
2c
eij c
(1)
where b, dynamic viscosity, py, the yield stress, , defor-
mation rate in the product form namely, = eij eij , eij , the
deformation rate for the i jth
component, and basing
on the model the critical value of is considered as c.
Therefore, Eq. (1) gives rise to c,
ij = b
1+
1
2eij (2)
where = b 2c/py, the Casson fluid parameter.
For the large value of the non-Newtonian parameter i.e.,
→ the problem became Newtonian. Following the
Ohm’s law a resistive force i.e., “Lorentz force”
J ×
B
is formed because of the conjunction of applied magnetic
field where the electric current density
J expressed as
J = hnf
E +
q ×
B (3)
Where the vector quantities present their usual meanings
i.e., hnf the conductivity of the electrical current,
q the
momentum,
E, the electric field, and
B stands for the
magnetic field. The appearance of constant magnetic is
obtained from
B = 0 where
B = 0B00. With respect
to the mentioned quantities and conditions the momentum
and the energy along x- and z-directions take the following
form,2627
− 0hnf
du
dy
+2w
=
p
x
+hnf
1+
1
d2
u
dy2
−B0Jz
−
hnf
k∗
p
u (4)
− 0hnf
dw
dy
−2u
=
p
z
+hnf
1+
1
d2
w
dy2
+B0Jx
+
hnf
k∗
p
w (5)
− 0cphnf
dT
dy
= khnf
d2
T
dy2
−
qr
y
+hnf
1+
1
×
u
y
2
+
w
y
2
+ hnf J2
x +J2
y (6)
The associated boundary conditions are
u = w = 0 T = T0 at y = 0
u = u0 w = 0 T = Th at y = h
(7)
where the velocity components u and w are presented
along the flow directions respectively, T, the fluid tempera-
ture, the current density J with components JxJyJz, p,
the pressure, qr , radiative heat flux, hnf , the density, hnf ,
the dynamic viscosity, cphnf , specific heat, and khnf , ther-
mal conductivity of hybrid nanofluid. The Tables I and II
presents the details on the physical properties of the hybrid
as well as the nanoparticles.
Here, 1 ≈ Cu and 2 ≈ Al2O3
are the volume fractions
of Cu and Al2O3 nanoparticles, respectively. Table II dis-
plays the physical properties of Cu and Al2O3 along with
the base liquid EG. The subscripts s1, s2, denotes the alu-
mina and copper particle correspondingly, f and hnf, used
for standard liquid and hybrid nanoliquid, respectively.
In case of steady state,
×
E = 0 that leads to Ex/y = 0 and
Ez/y = 0 and therefore, Ex and Ez are both
constants.
Hence Eq. (3) becomes
Jx = hnf Ex −B0w Jz = hnf Ez +B0w (8)
For non-conducting channel plate, Jx = 0, Jz = 0 at y = h.
Using the boundary condition at y = h, one can easily find
that Ex = 0 and Ez = −B0u0.
Which in turn yields from Eq. (8),
Jx = − hnf B0w Jz = hnf B0u−u0 (9)
Introducing (9), (4)–(6) gives rise to,
− 0hnf
du
dy
+2w
=hnf
1+
1
d2
u
dy2
−
B2
0 hnf +
hnf
k∗
p
u−u0 (10)
− 0hnf
dw
dy
−2u−u0
=hnf
1+
1
d2
w
dy2
+
B2
0 hnf +
hnf
k∗
p
w (11)
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4. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium Mishra et al.
ARTICLE
Table I. Physicothermal properties of nano and hybrid nanofluids.50
Attributes Nano fluid Hybrid nano fluid
Density nf = 1−1f +1s1 hnf = 1−2nf +2s2
Dynamic viscosity nf = f 1−1−2 5
hnf = nf 1−2−2 5
Thermal capacity cpnf = 1−1cpf +1cps1 cphnf = 1−2cpnf +2cps2
Thermal conductivity
knf
kf
=
ks +2kf −21kf −ks
ks +2kf +1kf −ks
khnf
knf
=
ks +2knf −22knf −ks
ks +2knf +2knf −ks
Electrical conductivity nf
f
=
s1 +2 f −21 f − s1
s1 +2 f +1 f − s1
hnf
nf
= s2 +2 nf −22 nf − s2
s2 +2 nf +2 nf − s2
− 0cphnf
dT
dy
= khnf
d2
T
dy2
−
qr
y
+hnf
1+
1
×
u
y
2
+
w
y
2
+ hnf B2
0
×w2
+u−u02
(12)
Imposing the Cogley radiation following,42
the radiative
heat flux can be expressed as
qr
y
= 4T −T0
0
K0
e0
T
0
d (13)
where K0
, absorption coefficient, 0, length of the
wave and e0
, Planck’s function, and T0, the reference
temperature.
So Eq. (12) can be written as,
− 0cphnf
dT
dy
= khnf
d2
T
dy2
−4T −T0I +hnf
1+
1
×
u
y
2
+
w
y
2
+ hnf B2
0
×w2
+u−u02
(14)
Where,
I =
0
K
e
T
d (15)
Following variables are considered to get non-dimensional
form:
=
y
h
u1 =
u
u0
w1 =
w
u0
=
T −T0
Th −T0
(16)
Table II. Thermophysical properties of regular fluid, nanoparticle and
hybrid nanoparticle.50
Physical properties Cu Al2O3 C2H6O2
Cp (J/Kg K) 385 765 1115
(Kg/m3
) 8933 3970 2430
(W/mK) 400 40 0.253
5.96×107
35×106
1.07×10−4
Involvement of (16) in (10), (11) and (14) lead to,
−A1Re
du1
d
= A2
1+
1
d2
u1
d2
−A3M +Dau1 −1
−2A1Kw1 (17)
−A1Re
dw1
d
= A2
1+
1
d2
w1
d2
−A3M +Daw1
−2A1Ku1 −1 (18)
−A4Pe
d
d
= A5
d2
d2
−Ra +A2 Pr Ec
1+
1
×
du1
d
2
+
dw1
d
2
+A3M
×u1 −12
+w1
2
(19)
where all the physicothermal parameters
A1A2A3A4A5 and physical parameters such as, Re,
Reynolds number (when Re 0 the upper plate affected
by suction whereas Re 0 injection occurs at the lower
plate), M magnetic parameter, Da, Darcy number, K,
rotation parameter, Pe, Peclet number, Ra, radiation
parameter, Pr, Prandtl number, and Ec, Eckert number are
defined as,
A1 =
hnf
f
A2 =
hnf
f
A3 = hnf
f
A4 =
cphnf
cpf
A5 =
khnf
kf
Re = 0h
f
M =
f B2
0h
2
f f
Da =
h2
k∗
p
K =
h2
f
Pe =
0hcpf
kf
Ra =
4lh2
kf
Pr =
f cpf
kf
Ec =
u2
0
cpf
Th −T0
The non-dimensional boundary conditions are,
u1 = w1 = 0 = 0 at = 0
u1 = 1 w1 = 0 = 1 at = 1
(20)
4 J. Nanofluids, 11, 1–12, 2022

5. Mishra et al. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium
ARTICLE
Taking, F = u1 −1+iw1i =
√
−1, (17) and (18) can
be combined as:
A2
1+
1
d2
F
d2
+A1Re
dF
d
−A3M +Da−2iA1KF = 0
(21)
A5
d2
d2
+A4Pe
d
d
−Ra +A2 Pr Ec
1+
1
dF
d
2
+A3M F 2
= 0 (22)
where : stands for absolute value.
The boundary conditions (20) become,
F = −1 = 0 at = 0
F = 0 = 1 at = 1
(23)
2.1. Shear Stresses
The interference of both the velocities suggests the shear
stresses at both the plates i.e., = 0 and = 1 respec-
tively can be expressed as,
R0 =
du1
d
2
+
dw1
d
2 1/2
=0
=
dF
d
=0
R1 =
du1
d
2
+
dw1
d
2 1/2
=1
=
dF
d
=1
(24)
The simulations for the shear stresses coefficients i.e.,
R0, R1 are tabulated in Table III for several contributing
parameters governing the flow of Casson nanofluid as well
as hybrid nanofluid.
2.2. Rate of Heat Transfer
In thermo-physical system the rate of heat transfer is a
major component to be carried out for several parameters.
The non-dimensional form at the plates = 0 and = 1
can be obtain and presented as
0 =
=0
1 =
=1
(25)
The simulated results for the coefficients − 0 and
− 1 are also tabulated in Table III for the various
parameters affecting the flow phenomena of Casson hybrid
nanofluid.
Table III. Validation of shear rate for Da = 0.
R0 (2 = 0) R1 (2 = 0 05)
M K Re Das et al.50
Present Das et al.50
Present
5 4 −1 0.5 1.71944 1.70327 1.78261 1.76932
0 1.89867 1.86493 1.98017 1.96275
2.3. Entropy Generation
The contribution of entropy generation is an important
aspect in the current investigation within a flow system. It
is wise to note that to preserve the quality of energy, the
entropy generation is to minimize in a thermal system. Fol-
lowing Das et al.43
and Cogley et al.,44–46
the entropy gen-
eration, the local volumetric rate in the flow of a viscous
electrically conducting hybrid nanofluid through a parallel
plate channel under the action of radiation is proposed as,
EG =
khnf
T 2
0
dT
dy
2
Thermal irreversibility
+
hnf
T0
1+
1
u
y
2
+
w
y
2
Fluid friction irreversibility
+ hnf B2
0
T0
u−u02
+w2
Joule dissipation irreversibility
(26)
The beginning two terms are organized for the irreversibil-
ity due to the heat transfer and the frictional force of the
fluid, however, the remaining suggests the irreversibility
occurs because of the interaction of applied magnetic field.
The entropy generation is presented as,4748
Ns =
T 2
0 h2
EG
kf Th −T02
(27)
where Br = PrEc = f v2
0/kf Th −T0, the Brinkmann
number corresponds to the relation between the heat con-
duction obtained at the surface to the heat caused by
the shear stress within the bounding surface and p =
Th −T0/T0, the difference in temperature.
On the use of (16) and (27), Eq. (26) reduces to,
Ns = Nh+Nf (28)
Nh = A5
d
dy
2
, entropy caused by heat transfer
Nf =
Br
p
A2
1+
1
u1
y
2
+
w1
y
2
+A3u1 −12
+w2
1
=
Br
p
A2
1+
1
dF
d
2
+A3 F 2
entropy caused by the frictional force for the conjunction
of magnetic field.
2.4. Analysis of Irreversibility
The process of the irreversibility ratio between the contri-
bution of entropy due to hea and the frictional force of the
fluid is termed as
=
Nh
Nf
=
heat transfer irreversibility
fluid friction irreversibility
(29)
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6. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium Mishra et al.
ARTICLE
For 0 1 the dominance of the heat transfer irre-
versibility is rendered and the fluid friction with magnetic
field dominates for 1. When = 1, the contribu-
tion of the frictional force conducted by the magnetic field
and permeability of the medium is equivalent to the heat
transfer in the fluid flow therefore, Paoletti et al.51
pre-
sented a different irreversibility, known as Bejan number
and expressed as:
Be =
Nh
Ns
=
Nh
Nh +Nf
=
1
1+
(30)
The range of the Bejan number is described as 0 Be 1.
Here, Be = 1 indicates the case of no irreversibility for
the heat transfer therefore only dissipation is accountable
for the process of irreversibility. However, Be 0 5 means
the fluid friction irreversibility dominants over the heat
transfer irreversibility and Be = 0 5 indicates that the heat
transfer and fluid friction for the consideration of magnetic
field the production of entropy rates are identical.
3. RESULTS AND DISCUSSION
The conducting Casson hybrid nanofluid flow through a
rotating channel subjected to porous medium is presented
in this analysis. For enriching the heat transfer criterion,
ethylene glycol (EG) is taken care of as a base fluid
and as a best conductor Cu is considered as the metal
nanoparticle and for oxide Al2O3 is used. Radiative heat
along with dissipative heat augments the temperature pro-
files. The physical properties relating to the nanofluid as
well as the hybrid nanofluid such as density, conductiv-
ities associated to both thermal and electrical are elabo-
rated in Table I. Further, Table II deliberates the physical
parameters of the base liquid EG and nanoparticles. The
transformed governing equations are handled numerically
employing Runge-Kutta technique. The profiles of both
the primary and secondary velocity and the temperature
profile are obtained for the several values of the param-
eters involved in the flow phenomena and illustrated via
Figures 2–21. The corroboration of the existing outcomes
for the shear rate with the nonappearance of the porous
matrix is presented in Table III. It displays the results are
good correlation with the work of Das et al.50
Further,
the shear stress and the Nusselt number at both the walls
are presented after getting the simulated results that are
displayed in Table IV. The main attraction is the entropy
analysis because of the irreversibility of the process. The
comparative study reveals that the earlier analytical and
the current numerical treatment correlates each other with
a good agreement and suggests achieving our goal.
Figure 2 deliberates the primary velocity profiles for
the non-Newtonian hybrid nanofluid with an inclusion of
magnetic parameter. The result is obtained for both the
nanofluids comprised of Cu/Al2O3∼EG hybrid nanofluid.
Here, M = 0 validates for the earlier result without mag-
netic field. Further, the enhanced values of the magnetic
parameter showing the increasing behaviour that’s resulted
in the thickness of the bounding surface decelerates. The
involvement of the “Lorentz force” due to the inclusion
of magnetic field offers a resistive force that causes a
strong retardation throughout the domain and further meets
the boundary condition smoothly. The comparative results
reveals that Cu∼EG nanofluid has stronger retarding effect
as that of the hybrid nanoliquid. Since, Cu has higher
density than Al2O3 so that the profiles of primary veloc-
ity decrease in case of Cu∼EG nanofluid. The impact
of magnetic parameter on the secondary velocity is pre-
sented in Figure 3. From the observation it is found that
the profile gets enhanced near the lower wall and further
the profile retards with the increasing values of magnetic
parameter to meet the upper wall. Comparison shows that
Cu∼EG nanofluid has greater retarding effect than that of
Cu/Al2O3∼EG hybrid nanofluid. The behaviour of thermo-
physical properties associated to nanofluid is vital for the
enrichment of energy profile. Figure 4 reveals the magne-
tization properties on the temperature profile for both the
Fig. 2. Variation of M on primary velocity profile.
Fig. 3. Variation of M on secondary velocity profile.
6 J. Nanofluids, 11, 1–12, 2022

7. Mishra et al. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium
ARTICLE
Fig. 4. Variation of M on temperature profile.
Fig. 5. Variation of Da on primary velocity profile.
Fig. 6. Variation of Da on secondary velocity profile.
Fig. 7. Variation of Da on temperature profile.
Fig. 8. Variation of K on primary velocity profile.
Fig. 9. Variation of K on secondary velocity profile.
J. Nanofluids, 11, 1–12, 2022 7

8. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium Mishra et al.
ARTICLE
Fig. 10. Variation of K on temperature profile.
Fig. 11. Variation of Re on primary velocity profile.
Fig. 12. Variation of Re on secondary velocity profile.
Fig. 13. Variation of Re on temperature profile.
Fig. 14. Variation of on primary velocity profile.
Fig. 15. Variation of on secondary velocity profile.
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9. Mishra et al. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium
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Fig. 16. Variation of on temperature profile.
Fig. 17. Variation of Ra on temperature profile.
Fig. 18. Variation of Pe on temperature profile.
Fig. 19. Variation of M and Pe on Bejan number and entropy variation.
Fig. 20. Variation of Ra and on Bejan number and entropy variation.
Fig. 21. Variation of Brp−1
on Bejan number and entropy variation.
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10. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium Mishra et al.
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Cu∼EG nanofluid and Cu/Al2O3∼EG hybrid nanofluid.
The thermal bounding surface near the lower wall decel-
erates causes the fluid temperature increases significantly.
The fact is that heat conductivity of Cu is good for which
the profile enhances within the domain 0 2 and fur-
ther the profile retards. However, in the second region the
impact is also opposite to the behaviour shown in the first
region. As described earlier, similar to the earlier discus-
sion, porosity is also a resistive force and the impact is
deliberated due to the flow through the permeable medium.
Here, Da = 0 suggests the flow through a clear fluid region
and Da = 0 indicates the behaviour through the permeable
region. Figure 5 illustrates the characteristic of Darcy num-
ber on the primary velocity profile. Increasing Darcy num-
ber produces a force that has a tendency to resists the fluid
motion so that the lower wall thickness retards and further
it became smooth up to the upper wall of the channel. The
behaviour of the Darcy number on the secondary velocity
is rendered in Figure 6. The impact is similar to that of
the magnetic parameter as described earlier in Figure 3.
Further, the nano as well as hybrid nanofluid temperature
profile for the behaviour of the Darcy number in deployed
in Figure 7. It is seen that the profile augmented with
augment in the Darcy number for the Cu∼EG nanofluid
and the amount of enhancement is more in contrast to the
Cu/Al2O3∼EG hybrid nanofluid. It is quite interesting to
observe that in the second region i.e., for 0 2 the pro-
files for both the nano as well as hybrid nanofluid have
reverse impact due to increasing Darcy number. Both the
primary and the secondary velocity combine to each other
due to the appearance of the rotational parameter that is
presented in Eqs. (17) and (18). Figure 8 portrays the
influence of the rotational parameter on the primary veloc-
ity distribution for the presence of other fixed parameters
involved in the governing flow phenomena. Here, K = 0
suggests the without rotation the primary velocity for the
nanofluid and the hybrid nanofluid coincides each other
whereas the increasing rotation decelerates the thickness
of the bounding surface near the lower wall of the chan-
nel. Figure 9 portrays the secondary velocity distribution
for the variation of the rotation parameter. The pick in the
profiles is rendered near the lower wall of the channel and
thus it again boosts up the profile with the decelerating
nature throughout the domain. The hike in the values of
the rotational parameter on the fluid temperature is pre-
sented in Figure 10. In all the profiles for the variation of
rotation parameter it is observed that the Cu∼EG nanofluid
has a greater impact on the primary velocity than that
of Cu/Al2O3∼EG hybrid nanofluid. Further, the impact is
reversed in case of secondary velocity. Inertial force in
combination with the viscous force leads to the effect of
Reynolds number and the impact of Re is vital for the flow
phenomena. The influence of Re on the primary velocity
is rendered in Figure 11. Decelerating effect of viscous
force enriches the Reynolds number and the result shows
that the primary velocity enhances. This is because of the
dominating effect of the inertial force. Due to this reason
the thickness of the bounding surface decreases. It quan-
tifies the relative importance of these two forces. Further,
the behaviour of the Reynolds number on the secondary
velocity profiles. The dominating nature of inertial force
over the viscous force retards the secondary velocity that is
displayed in Figure 12. In both of these figures it is seen
that the Cu∼EG nanofluid has a tendency to decelerate
the velocity profiles in assessment to the Cu/Al2O3∼EG
hybrid nanofluid. The fact is straight forwards because
of the density of the Cu particle and thus the agglom-
eration of the particle is found near the lower wall sur-
face. Further, it is found that as the domain increases
the profile became sooth to meet the boundary condi-
tion. Figure 13 describes the impact of Reynolds number
on the temperature profile considering both the nano and
hybrid nanofluid. The enhanced value of Re augments the
fluid temperature nearby the lower wall and afterwards the
impact decelerates gradually. Casson parameter signifies
the non-Newtonian features of the magnetized fluid. Large
value of Casson parameter leads to perform the Newtonian
characteristics. Figure 14 illustrates the behaviour of the
Casson parameter on the flow phenomenon of the primary
velocity of hybrid nanofluid. The elasticity of the param-
eter is due to the relationship between the relaxations
with retardation time. An increase in Casson parameter
the flow profile increases showing the lower bounding sur-
face thickness ceases to zero. The fact is, higher Casson
value suggests the Newtonian case for which the primary
velocity rises up. Further, reverse impact is rendered for
the secondary velocity distribution that is presented in
Figure 15. Initially the hike is faster nearby the lower
surface and then the behaviour is opposite. However, the
comparative result exposes that the denser Cu nanoparticle
agglomerated near the lower surface resulted in the decel-
eration is ore in the case of Cu∼EG nanofluid than that of
Cu/Al2O3∼EG hybrid nanofluid. Figure 16 portrays a sig-
nificant deceleration in the fluid temperature due to an aug-
mentation in the Casson parameter. Thus, it is concluded
that the non-Newtonian characteristic of the magnetized
fluid favors to enhance the nanofluid as well as the hybrid
nanofluid temperature profiles at points within the domain.
The release of electromagnetic radiation from the fluid sur-
face encoded as radiation. Figure 17 exhibits the behaviour
of the thermal radiation on the fluid temperature. The
radiative heat energy is the reciprocal of the thermal con-
ductivity of the base fluids. Thermal radiation enhances
with decreasing conductivity. Therefore, increasing radia-
tion the amount of heat radiate from the lower wall surface
radiated greatly and thus the fluid temperature increases
significantly. Figure 18 shows the significance of the Peclet
number on the fluid temperature profiles considering both
the nanofluid and the hybrid nanofluid. Pe is defined as
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11. Mishra et al. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium
ARTICLE
Table IV. The rate coefficients for several values of the parameters.
R0 ×102
R1 ×102
− 0×102
− 1×102
M K Re 2 = 0 2 = 0 05 2 = 0 2 = 0 05 2 = 0 2 = 0 05 2 = 0 2 = 0 05
1 10 10 0.5 0.0164 0.0157 0.0077 0.0079 −2.3742 −2.2169 0.4023 0.4569
3 0.0219 0.0215 0.0052 0.0053 −2.862 −2.7362 0.2446 0.2834
5 0.0314 0.0314 0.0027 0.0027 −3.9213 −3.821 0.1334 0.1637
1 5 0.0343 0.0338 0.0025 0.0026 −4.3764 −4.1895 0.1319 0.1636
10 0.048 0.0461 0.0016 0.0019 −6.812 −6.3169 0.1247 0.1626
15 0.06 0.0573 0.06 0.0011 −8.9819 −8.3188 0.1194 0.1608
5 5 0.0391 0.0383 0.0013 0.0014 −4.7961 −4.5944 0.0965 0.1274
10 0.0505 0.0484 0.0004 0.0005 −6.1276 −5.7551 0.0747 0.1038
15 0.0663 0.0623 0.0001 0.0002 −7.996 −7.3777 0.0631 0.0901
5 1 0.0389 0.0388 0.0016 0.0016 −3.2366 −3.1548 0.0663 0.0853
5 0.0512 0.051 0.0006 0.0006 −2.5735 −2.5073 0.0299 0.0405
10 0.0537 0.0534 0.0005 0.0005 −2.4767 −2.4126 0.0263 0.0359
the product of Re and Pr. The use of base fluid EG corre-
sponds to the higher values of the Prandtl number there-
fore, increasing Pr retards the fluid temperature for both of
the cases. Entropy generation is the major aspects of the
energy system that is due to various irreversibilities i.e.,
the combination of the thermal, fluid friction and Joule
dissipation irreversibility. Figures 19–21 show the entropy
analysis along with the Bejan number for the impact of
various contributing parameters. The effects of Pe num-
ber on the entropy and the variation of magnetic param-
eter on the Bejan value is deliberated in Figure 19. It is
clear to see that increasing Pe number enriches the entropy
throughout the domain. The profile of Bejan shows its dual
characteristics for the variation of magnetic parameter. It
is observed that the increasing magnetic parameter decel-
erates the Bejan value near the lower wall whereas reverse
impact is rendered far away from it. Figure 20 describes
the behaviour of the non-Newtonian Casson parameter on
the entropy generation and the behaviour of the Ra on
Bejan number. An augmentation in the Casson parame-
ter retards the entropy and the similar behaviour of Ra on
Bejan value is rendered as described in the earlier figure.
The impact of Brinkman number on the entropy analy-
sis and the Bejan number is presented in Figure 21. Br
is the product of Pr and Ec. An augmentation in the Br
enriches the entropy and the reverse impact is detected for
the Bejan value. Table IV shows the shear stress coeffi-
cients and Nusselt number at both of the lower and upper
walls of the channel in two different situations i.e., in case
of nanofluid 2 = 0 and in case of hybrid nanofluid 2 =
0 05. The behaviour of magnetization, rotational as well
as Reynolds number and the Casson parameter is obtained
keeping others as fixed. An inclusion of magnetic, rota-
tion parameter and the Reynolds number enhance the skin
friction rate at the lower surface whereas the influence is
counterproductive in the upper surface for both the case
of nanofluid and hybrid nanofluid. Similar observation is
encountered for the rate of heat transfer. The compara-
tive result reveals that the magnitude is more for Cu∼EG
nanofluid. Further, it is clarified that the non-Newtonian
parameter exhibits greater impact to enhance the skin fric-
tion rate about lower part and decelerates about the upper
wall whereas the impact is reversed in case of rate of heat
transfer for both the nano and hybrid nanofluid.
4. CONCLUSION
Flow of conducting Casson hybrid nanoliquid via a rotat-
ing permeable channel is considered in this study. The
novelty is to minimize the energy flow rate due to the irre-
versibility of the process for the analysis of the entropy.
The thermal enhancement is carried out due to the inclu-
sion of the radiative and dissipative eat energy. However,
the characteristic of the thermophysical properties of sev-
eral nanoparticles i.e., Cu and Al2O3 with the base fluid
EG augments the significance of the flow properties. The
numerical treatment is obtained for the solution of com-
plex nonlinear problem and further, the behaviour of the
physical parameters is presented graphically and then dis-
cussed. However, the following conclusions are presented
as;
• The inclusion of metal and oxide nanoparticle with the
base fluid EG enriches the flow phenomena of the nanoflu-
ids as well as the hybrid nanofluid, Thermophysical prop-
erties therefore the flow profiles boost up.
• The magnetized fluid energizes the primary as well as
secondary velocity that resulted in to decelerates the thick-
ness of the lower bounding surface further, the energy
transport boost up the profiles significantly.
• The augmentation in the inertial force dominating over
viscous force augments the Reynolds number for which
the primary and secondary velocity decelerates the thick-
ness of both the velocity profiles and enhances the thermal
bounding surface.
• Similar tendency is rendered for the variation of the
nono-Newton Casson parameter for the velocity profiles
however the temperature profile also retards.
• Rate of shear stress and heat transfer rate rises up for
the increasing magnetic parameter, rotation parameter, and
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12. Hybrid Nanofluid Flow of Non-Newtonian Casson Fluid for the Analysis of Entropy Through a Permeable Medium Mishra et al.
ARTICLE
Reynolds number near the lower wall surface whereas
impact is reversed near the upper wall.
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