Analysis of Thermal Radiation and Ohmic Heating Effects on the
Entropy Generation of MHD Williamson Fluid through an Inclined
Channel
ABIODUN OPANUGA1, GBEMINIYI SOBAMOWO2, HILARY OKAGBUE1,
PETER OGUNNIYI1
1Department of Mathematics, Covenant University, Ota, NIGERIA
2Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, NIGERIA
Abstract: -In this present work, the heat irreversibility analysis of thermal radiation, Ohmic heating, and angle
of inclination on Williamson fluid is presented. The developed equations are converted to dimensionless forms,
and Homotopy perturbation method (HPM) is used to solve the resulting coupled ordinary differential
equations. The heat irreversibility analysis is achieved by substituting the obtained results into entropy
generation and Bejan number expressions. The HPM solution for the velocity profile is validated by comparing
it with a previously published study in some limited cases, and an excellent agreement is established. Fluid
motion is accelerated by the increasing values of thermal radiation parameter, whereas the magnetic parameter
and Reynolds number reduce it. Furthermore, except for the Weissenberg and Prandtl numbers, all of the flow
parameters examined enhance fluid temperature. In addition, entropy generation is enhanced at the channel's
upper wall for all parameters except magnetic field parameter.
Key-Words: -Williamson fluid, heat irreversibility, thermal radiation, inclined channel, Homotopy perturbation
method.
Received: July 28, 2021. Revised: October 11, 2022. Accepted: November 15, 2022. Published: December 31, 2022.
1 Introduction
Many researchers have been fascinated by the study
of viscous incompressible non-Newtonian fluids in
recent decades because of their wide applications in
engineering and industry. Non-Newtonian fluids do
not follow Newton's law of viscosity (the shear
stress to shear rate ratio is always the same). Since
the Navier-Stokes equations alone are insufficient to
represent the rheological features of these fluids,
therefore to address this flaw, several rheological
models have been proposed, including the Ellis,
Power-law, Carreaus, Cross, and pseudoplastic
fluids. Pseudoplastic fluids are more significant
among these because of their extensive applications
in industries such as melting of high molecular
weight polymers, photographic films, and polymer
sheet extrusion. Williamson, [1], investigated the
flow of pseudoplastic materials and developed a
model to characterize their behaviour, and further
explained that the fluid captures the thinning
properties of non-Newtonian fluids. Thereafter,
several investigators have been involved in the
analysis of this fluid. Khan and Alzahrani, [2],
discovered that the Weissenberg number (We) has
an inverse relationship with velocity. However, it is
enhanced as the mixed convection parameter
increases. The stretching sheet Williamson flow was
studied by Nadeem et al., [3]. As demonstrated by
Hayat et al., [4], magnetic and electric fields, as well
as thermal radiation, have an effect on the flow
pattern of a two-dimensional flow of Williamson
fluid with porosity. Krishnamurthy et al., [5],
investigated a steady flow of Williamson fluid in a
horizontal linearly stretched sheet with simultaneous
chemical reaction, melting heat transfer, and
nanoparticles. Dapra and Scarpi,[6],presented the
perturbation solution for pulsatile motion of
Williamson fluid. In view of the significance of
peristaltic motion of the non-Newtonian fluid
through asymmetric channels along porous walls,
Vajravelu et al., [7], explored this using varied
phase and amplitude, as well as the manipulation of
different wave patterns on the fluid flow model.
Raza et al., [8], examined hydromagnetic
Williamson nanofluid with slip conditions. It was
submitted that Williamson fluid parameter and the
temperature profile had a direct relationship.
Vasudev et al., [9], investigated the peristaltic
Williamson fluid in a planar channel, and the flow
was explored in a wave framework that flowed with
the speed of the wave. Nadeem et al., [10], proposed
a model for the motion of Williamson fluid in an
annular zone. Using the Keller box approach, Malik
et al., [11], studied the Williamson fluid model
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across a stretched cylinder Vittal et al., [12],reported
hydromagnetic stagnation point Williamson fluid
flow on exponential stretchable surface. Monica et
al., [13], proposed an analysis for non-Newtonian
fluids stagnation point flow through a stretching
sheet. The modelling of two-dimensional
Williamson fluids through a circular cylinder was
presented by Nagaraja and Reddy, [14]. Using the
Adomian decomposition method, Siddiqui et al.,
[15], discovered an analytical approach of
Williamson fluid Blade coating analysis.
Shashikumar et al., [16], analysed the steady flow of
Williamson fluid in a micro-channel caused by
viscous dissipation, magnetic effect, and Joule
dissipation. For further details on flow accounting
for the viscoelastic shear thinning characteristics of
non-Newtonian fluids, interested readers can consult
the extensive research in [17-18].
Ohmic heating is a type of heating technique in
which electrical current is used to generate heat in
fluid materials. It is produced by the applied electric
field and fluid electrical resistance, which is the
conversion of electric energy to thermal energy.
Numerous researchers have investigated Newtonian
and non-Newtonian fluid flow problems in the
context of Ohmic heating and heat transfer. The
viscosity effect on the Joule heating rate of solid-
liquid mixtures was investigated by Khalaf and
Sastry, [19]. It was submitted that fluid mixture with
higher viscosity has higher rate of heating than the
mixture with lower viscosity fluid. The study of
hydromagnetic heat transfer and boundary layer
flow with Ohmic heating and chemical reaction was
conducted by Rao et al., [20]. It was submitted that
increasing values of Joule heating parameter
improves temperature and concentration
distributions of nanofluid. Prakash et al., [21],
investigated an electrically conducting nanofluid's
mixed convective flow in a porous medium: the
effects of a variable magnetic field. Tsai et al.,
[22],studied the effects of Ohmic heating and heat
transfer on electrically conducting flow with
variable viscosity. Awasthi, [23], analysed the
significance of Ohmic heating and thermal radiation
effects on MHD convective flow using perturbation
technique. Furthermore, Muhammad et al., [24],
presented an analyses on chemical reaction and
viscous dissipation influence on electrically
conducting flow of Newtonian fluid past an
exponentially stretching sheet with the Ohmic
heating. Adegbie et al., [25], conducted an analysis
on free convection flow over a moving porous
surface under the influence of Joule heating and
magnetic field. Osalusi et al., [26], considered Joule
heating and viscous dissipation effects on transient
hydromagnetic and slip flow over a permeable
rotating disk.
Ohmic heating effect on non-Newtonian fluids
has also been investigated by several researchers.
Goud and Nandeppanavar, [27], conducted a study
on Ohmic heating and chemical reaction effects on
hydromagnetic flow of micropolar fluid. Hasan et
al., [28], considered Peristaltic wave-induced Hall
current and Ohmic heating in a non-Newtonian
channel flow. Gireesha et al., [29],investigated the
influence of Joule heating on Casson fluid
hydromagnetic mixed convection flow by taking
cross diffusion into account. Samuel and Olajuwon,
[30],studied theeffects of thermal radiation,Joule
heating on Maxwell fluid with Lorentz and
buoyancy forces.
Irreversibility analysis in gravity-driven flows
has applications in nature, for example, in the
printing field, during paper manufacturing
processes, wire drawing, spaying,fibreglass in
metallurgical technology. Bejan,[31,32], applied the
second law of thermodynamics to aid the
understanding of fluid entropy generation rate and
minimization of irreversibilities processes.
Furthermore, Bejan, [33, 34], analysed the
volumetric entropy generation rate in fluid flow
processes, and this has been adopted by several
scholars. To provide a brief overview of the
application of the second law analysis approach to
monitoring entropy buildup on inclined walls. The
flow of a Newtonian film along a heated inclined
plate has been reportedby Saouli and Aboud-Saouli,
[35], with the goal of increasing the available
energy for work. The notion of a variable viscosity
fluid flowing down the channel was developed by
Havzali et al., [36]. Tshela, [37], used aspect ratio
approximations in the lubrication theory for a
temperature-dependent viscous flow in the boundary
layer with Newtonian heating. Furthermore, Al-
Ahmed et al., [38],studied energy reduction in free
and constrained gravity-driven film flows of varied
viscosity through heated plates.The steady, reactive
flow of a couple stress fluid through a porous
medium was studied by Adesanya et al., [39].
Adesanya and Makinde, [40], used the Adomian
decomposition method to investigate entropy
generation of third-grade fluid flow along a vertical
channel and the impact of internal heat generation.
See Refs [41-46] for further reading.
He [47] proposed the Homotopy Perturbation
Method (HPM), which is a combination of
topology's homotopy and traditional perturbation
techniques. This enables us to obtain analytic or
approximate solutions tonumerous problems
occurring in a number of scientific fields. The
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technique demonstrates rapid convergence of the
series solution and provides a wide range of
benefits. It also offers an analytical approximate
solution for non-linear equations in the form of an
infinite series, from which each term can be
deduced one by one. In recent years, researchers
have concentrated their efforts on solving both
linear and nonlinear ordinary and partial differential
equations. The approach has successfully handled
several types of linear and nonlinear differential
equationsincluding the Lighthill equation [48],
Duffing equation [49], Blasius equation [50], wave
equations [51], and boundary value problems [52].
Numerous authors have lately conducted extensive
research on the homotopy perturbation technique
and used it to tackle nonlinear problems. This
approach has been improved [53–56] to provide
more accurate results, help accelerate the series
solution's rapid convergence, and minimize the
computational effort.
In light of the present literature survey, it is
obvious that entropy generation in Williamson fluid
in the presence of Rosseland thermal radiation and
Joule heating in an inclined channel has not been
accorded the required attention in view of the
important applications of flow powered by
gravitational force in many food industries,
polymerization operations, heat exchangers, film
evaporators,wire and glass production, and drying
processes. In addition, the outcome of this
investigation can support research work on emulsion
coated sheets like photographic films and blood
flow. Consequently, this current study's goal is to
apply the energy evaluation technique as a useful
tool for identifying variables related to the loss of
available energy for work and as a measure of
thermal equipment efficiency in order to reduce
waste. Furthermore, the HPM approach applied in
this study is presented in a more appealing and
systematic manner. The technique is easy to apply
because it does not require discretization,
transformation, formulation of Adomian and
homotopy polynomials.
2 Problem Formulation
A fully developed steady and incompressible
hydromagnetic laminar flow of Williamson fluid is
considered as depicted in Figure 1. The fluid is
flowing between two infinite parallel plates with
distance
h
apart, inclined at an angle
. The
Cartesian coordinates system approach is such that
the flow is along
x axis
direction while the
y axis
is perpendicular to the flow. A magnetic
field of strength
0
B
is applied transversely to the
flow. It is assumed that the physical quantities
depend on
y
only and the magnetic Reynolds
number is considered to be low, resulting in a
negligible induced magnetic field in comparison to
the applied magnetic field. It is furthermore
assumed that the imparted electric field is zero, and
that the Hall effect is disregarded. The fluid motion
is given by the following set of equations using
Boussinesq's approximation.
Fig. 1: Schematic Diagram of the Problem
For an electrically conducting fluid, the Ohm's law
when Hall currents, ion-slip and thermoelectric
effects, as well as the electron pressure gradient are
ignored is:
J E q B
(1)
where
E
denotes the electric field vector,
J
is the
current density vector,
q
is the velocity vector, and
represents the fluid's electrical conductivity and
B
is the magnetic field vector. The total magnetic
field
0
B B b
. The applied and induced
magnetic fields are denoted by
0
B
and
b
,
respectively. The induced magnetic field is not
taken into account for small magnetic Reynolds
numbers.
Following the aforementioned assumptions, the
momentum and energy equations take the form [16]
2
02
0
12
sin
1z
du dp du d u
vdy dx dy dy
g T T B J
(2)
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2
2
02
2
12
p
r
dT d T du du
c v k
dy dy dy dy
dq J
dy
(3)
The index
is set at 1 for the inclusion of Joule
dissipation. Taking
0
0, 0, 0,
x z x x Bu
E E J J
(4)
Equations (2) and (3) are transformed by the use of
(4)
22
00
2
1
12
sin
du dp du d u
v B u
dy dx dy dy
g T T
(5)
2
2
02
22
0
12
p
r
dT d T du du
c v k
dy dy dy dy
dq
Bu dy
(6)
The boundary conditions are as follows:
21
0 0, 0, 0 ,u u h T T T h T
(7)
The net radiative heat flux
r
q
for optically thick
material is obtained using the Rosseland
approximation.
4
4
3
c
rc
T
qky
(8)
4
T
is being expanded in a Taylor series about
0
T
as.
2
4 4 3 2
0 0 0 0
3
00
0
46
4
T T T T T T T T
T T T
(9)
and ignoring the higher order components, yields
4 3 3
00
43T T T T
(10)
In light of equations (4) and (6), equation (3) yields
2
2
02
32
22
02
12
16
3
p
c
c
dT d T du du
c v k
dy dy dy dy
T d T
Bu k dy
(11)
To obtain similarity equations, the following
similarity transformations are utilized.
1
21
,, TT
y
uf
h h T T
(12)
which yield
2
1 Re
sin 0
Wef f f M f
Gr A
(13)
2
22
4
1 RePr
3
Pr 1 2
0
Ra
We
Ec f f
Mf
(14)
with the relevant boundary conditions given as
0 0, 1 0, 0 1, 1 0ff
(15)
where
0
Re vh
,
3
2
h dp
Adx
,
2
2
We h
,
22
20
Bh
M
,
23 21
2
h g T T
Gr
, (16)
3
4c
c
T
Ra kk
3 Analysis of Homotopy Perturbation
Method(HPM)
Given a nonlinear differential equation
0, ,u f r r
(17)
And the boundary conditions expressed as
, 0, ,
u
ur
(18)
Where
is a general differential operator,
is a
boundary operator,
fr
is a known analytical
function, and
is the boundary of the domain
.
The operator
is written as linear operator,
L
and
nonlinear operator
N
. Then, equation(1)can be
expressed as:
0, .L u N u f r r
(19)
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Using He’s Homotopy technique to construct a
homotopy
, : 0,1 ,r p R
and this satisfies
0
,1
0
H p p L L u
p f r
(20)
or
0
0
,
0,
H p L L u
p L u N f r
(21)
where
0,1p
is an embedding parameter,
0
u
is an
initial approximation of Eqn. (1).
It is obvious that equations (20) and (21) yield
0
,0 0,H L L u
,1 0,H f r
As p varies from zero to unity,
,rp
from
0
ur
to
ur
. In topology, this is regarded to as
homotopy.The embedding parameter p can be
utilized by HPM as a small parameter and assuming
that the solutions of equations (20) and (21) can be
written as a power series in p.
0 1 2 3
0 1 2 3
p p p p
. (22)
Setting
1p
, yields the approximate solution of
equation (17)
0 1 2 3
lim1p
. (23)
4 Method of Solution by HPM
Homotopy perturbation method is used to solve the
dimensionless nonlinear Equations (13)-(14). As a
result, a homotopy construction for Equations (13)-
(14) is developed for the current problem:
,1
Re
2sin
H f p p f
f Wef f f
pM f Gr A
(24)
and
4
, 1 1 3
4
1 RePr
3
2
Pr 1 2
H f p p Ra
Ra
pWe
Ec f f M f
(25)
and
0,1p
is an embedding parameter, then for
0p
and
1p
01
01
,0 , ,1 ;
,0 , ,1
f f f f
(26)
It is noteworthy that as
p
rises from zero to one,
,fp
shifts from
0
f
to
1
f
and
,0
shifts from
0
to
1
. Assume that the solution
to Equations (13)-(14) is represented as a
p
-series:
2
0 1 2
330
nii
i
f f pf p f
p f p f
(27)
and
2
0 1 2
330
nii
i
pp
pp
(28)
After substituting Equations (27) and (28) into
Equations (13) and (14) respectively, the subsequent
expression will appear in the form of a polynomial
in
p
and expanding such that terms of the same
order of
p
are categorized together. Furthermore, a
set of differential equations is derived together with
their associated boundary conditions by setting the
polynomial coefficients in
p
to one, yields
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00 0 0
12
1 0 0 0 0
0 1 1
22 0 1 1 0 1
21 1 2 2
33 0 2 2 0 1 1
2
2 2 2
33
: 0; 0 0, 1 0,
: Re
sin 0 0, 1 0,
: Re
sin 0 0, 1 0,
:
Re sin
0 0, 1
;
;
;
p f A f f
p f We f f f M f
Gr f f
p f We f f f f f
M f Gr f f
p f We f f f f f f
f M f Gr
ff
0
(29)
And
00 0 0
2
11 0 0
32
2
0 0 1
1
221
0 1 0 0 1
20 1 2 2
3
4
: 1 0, 0 1; 1 0,
3
4
: 1 RePr Pr
3
Pr 0; 0 0,
2
1 0,
4
: 1 RePr
3
Pr 2 Pr 3
2
2 0; 0 0, 1 0
4
:1 3
p Ra
p Ra Ec f
We
Ec f M f
p Ra
We
Ec f f Ec f f f
M f f
p Ra
32
2
0 2 1
0 0 2 0 1 1
2
20 2 3 3
RePr
Pr 2
Pr 3 3
2
2 0; 0 0, 1 0.
1
Ec f f f
We
Ec f f f f f f
M f f f
(30)
The following results are obtained upon solving
Equations (29)–(30).
2
0
1
2
f A A
, (31)
1
22
2 2 2 2 2
3 2 3 2 4
2
3
2 Re 2
6 Re 6 2
14 Re 4
24 8 sin 12 sin
4 sin
f
AM A A We
A A We AM
A A We AM
Gr Gr
Gr
(32)
01
, (33)
1
2 2 2
2
2 2 2
3 2 2 3
3 3 2 2 4
2 4 3 4
2 2 5 3 5
2 2 5
1
160 3 4
2 20 Pr
240Pr Re 3 Pr
60 Pr 240Pr Re
15 Pr 80 Pr
30 Pr 10
40 Pr 30 Pr
12 12 Pr
4
Ra
A M A Ec
A Ec We
A Ec
A Ec We A Ec
A Ec We A M
A Ec A Ec We
A M A Ec We
AM
(34)
The expressions for
2
f
and
2
are obtained in
the same way. These expressions, however, not
included in this write-up. The HPM solutions of
Equations (24) and (25) can be written as:
2
0 1 2
2
0 1 2
f f pf p f
pp
(35)
At
1p
, the approximate expressions for
equations (13) and (14) can be written as
01
2
01
2
lim1
lim1
f f f f
p
f
p
(36)
Substituting Equations (31)-(34) into Equation (36),
yields
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2
2
22
2 2 2 3
3 2 3
2 4 2 4
1
2
2 Re
2 6 Re
162
24 4 Re 4
f A A
AM A
A We A
A We AM
A A We
AM AM
(37)
and
2 2 2
3
2 2 2
3 2 2 3
33
1
1160 3 4
2 20 Pr
240Pr Re 3 Pr
60 Pr 240Pr Re
15 Pr 80 Pr
30 Pr
Ra
A M A Ec
A Ec We
A Ec
A Ec We A Ec
A Ec We
(38)
5 Result Validation
The exact solution of the Newtonian case
0We
as considered by Makinde and Eegunjobi,[57],is
compared with the HPM solution and presented in
Table 1, to validate the Homotopy perturbation
method solution used in this work. The exact and
HPM solutions are found to be in perfect agreement.
The exact solution as obtained by Makinde and
Eegunjobi,[57],is
Re
Re
Re
1
Re 1
A e e
fe
(39)
The momentum Equation (13) at
Re 1, 1, 0, 0, 0A We M Gr
reduces to
0, 0 0f f A f
(40)
The HPM solution of Eqn. (40) is expressed as
2 2 3
3
2 3 4
45
32
2 12
10
2
24 720 15 6
AA
f
AA
(41)
The convergence of the solution is also displayed in
Table 2.
Table 1. Comparison of Exact and HPM solutionsof
Equations (13) at
Re 1, 1, 0,A We
0, 0M Gr
.
Exact [57]
RK4 [57]
Current
0.1
0.03879297
0.03879297
0.03879298
0.2
0.07114875
0.07114875
0.07114876
0.3
0.09639032
0.09639032
0.09639033
0.4
0.11376948
0.11376948
0.11376949
0.5
0.12245933
0.12245933
0.12245933
0.6
0.12154600
0.12154600
0.12154601
0.7
0.11001953
0.11001953
0.11001954
0.8
0.08676372
0.08676372
0.08676373
0.9
0.05054498
0.05054498
0.05054499
Table 2. Convergence of HPM result for
2
Re 1, 0A M We Gr
k
2
2k
df
dy
2
2
0
nk
k
df
dy
0
1.000000000
1.000000000
1
0.500000000
1.500000000
2
0.041666667
1.541666667
3
-0.005555556
1.536111111
4
-0.000090939
1.536020172
5
0.0001361332
1.536156305
6
-0.000012261
1.536144044
7
0.000002121
1.536141923
8
-0.0000000002
1.536142450
9
-0.00000001
1.536142450
5.1 Irreversibility Analysis
The irreversibility rate within the flow is calculated
using the obtained temperature and velocity results.
The expression for heat irreversibility in the
presence of magnetic field and thermal radiation is
22
3
2
1
2
22
0
1
16
3
112
c
c
k dT T dT
Eg T dy k dy
HTEG TREG
du du Bu
T dy dy MFEG
FFEG
(42)
Where HTEG = Heat Transfer Entropy Generation,
TREG = Thermal Radiation Entropy Generation,
FFEG = Fluid Friction Entropy Generation and
MFEG = Magnetic Field Entropy Generation.
The non-dimension form of equation (42) is
obtained by applying relations (16) to yield
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2
222
4
13
Pr 1 2
Ns Ra
We
Ec L f f M f
(43)
Denoting
2
4
1
13
N Ra
and
222
2Pr 1 2
We
N Ec L f f M f
(44)
Where
1
21
T
LTT
and
22
1
21
E T h
g
Ns k T T
The Bejan number is then written as
12
1
1,
1
S
NN
Be NN
(45)
The distribution ratio of fluid irreversibility is
denoted by
. From 1 to 0, the Bejan number can
have any value. When
0Be
, irreversibility of fluid
friction dominates entropy generation,when
1Be
,
irreversibility of heat transfer dominates, and when
both enhance entropy generation equally
0.5Be
.
6 Results and Discussion
For velocity, temperature, and entropy generation
the effects of various physical parameters
influencing the flow are analyzed for
0.1 1We
,
0.5 1.5M
,
0.1 4.5Ra
,
0.71 Pr 7
,
1.5 Re 2.5
and
7
6 18
while
0.5Ec
,
A
and
L
, on the other hand, are each fixed as 1.
Fig. 2A:
versus Velocity Profile
Fig. 2B: We versus Velocity Profile
Fig. 2C: M versus Velocity Profile
Fig. 2D: Re versus Velocity Profile
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Fig. 2E: Ra versus Velocity Profile
Fig. 2F: Pr versus Velocity Profile
In Figure 2A, the influence of the angle of
inclination on fluid velocity is depicted. It is
observed that fluid velocity increases as the angle of
inclination increases. This observation is due to
increased forces acting on the fluid flow. In Figure
2B, the Weissenberg number versus velocity profile
is displayed. An increasing trend at the lower wall
of the channel is observed while there is a reduction
in fluid velocity at the upper wall. Fluid velocity
increases with the Weissenberg number (We) at the
lower wall due to the shear-thinning effect, while
the reduction at the upper wall is due to higher
viscosity of the non-Newtonian fluid. The shear-
thinning effect is one of the characteristic features of
the Williamson fluid. It is a non-Newtonian fluid
feature where the fluid's viscosity decreases as the
shear stress rises. In addition, the Williamson fluid
parameter, measures the effect of viscosity to
elasticity. It therefore causes a drop in the velocity
profile due to low resistance to flow. In Figure 2C,
the response of fluid velocity to the external
magnetic field effect is displayed. It is indicated that
fluid velocity experiences a reduced momentum.
This is to be expected, because the Lorentz force
resulting from the applied magnetic field develops a
resistant force within the flow, causing the
Williamson fluid motion to reduce. In addition, it is
also worth noting that the velocity boundary layer's
thickness decreases with a rise in M's value. The
velocity response to variations in Reynolds number
is displayed in Figure 2D. It is shown that fluid
velocity reduces as values of Reynolds number
increase. This is physically correct since the
Reynolds number signifies the significance of the
inertia effect in comparison to the viscous effect.
Hence, fluid velocity is retarded as depicted in the
figure. An enhancement in fluid velocity with an
increase in radiation parameter is observed in Figure
2E, while a falling trend is noticed in Figure 2F for
increasing values of Prandtl number. The boundary
layer thickness increases, causing more fluid flow
and thereby increasing fluid velocity as shown in
Figure 2E, while a reverse trend is observed in
Figure 2F.
Fig. 3A:
versus Temperature Profile
Fig. 3B: Ra versus Temperature Profile
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Fig. 3C: M versus Temperature Profile
Fig. 3D: Re versus Temperature Profile
Figure 3 is sketched for various values of inclination
angle parameter (
), Radiation parameter (Ra),
magnetic field parameter (M) and Reynolds number
(Re). Generally, fluid temperature is enhanced by
the rising values of each of the parameters. In Figure
3A, the influence of the angle of inclination
parameter is displayed against fluid temperature. As
the parameter varies, fluid temperature appears to
have risen slightly. This is due to the fact that higher
fluid velocity, as shown in Figure 2A, tends to
increase the forces acting on fluid flow, resulting in
the enhancement of fluid temperature. Figure 3B
depicts an increasing trend in fluid temperature with
increasing values of radiation parameter.This is
attributed to the fact that increasing the radiation
parameter
Ra
decreases the fluid Rosseland
absorptivity parameter
c
k
, resulting in an increase
in fluid temperature.In Figure 3C, the effect of the
magnetic field parameter on fluid temperature is
presented. It is observed that as the value of M
increases, the temperature rises. Lorentz force is
produced when a magnetic field is applied to an
electrically conducting fluid and interacts with it.
Because the Lorentz force slows fluid motion,
kinetic energy is converted to heat energy (Joule
heating), and the fluid temperature rises as a result
of Joule heating. The influence of Reynolds number
on fluid temperature is shown in Figure 3D, it
depicts a rising trend as Reynolds number increases.
This can be attributed to increased frictional force,
the Re values improve the thermal distribution of
the fluid, hence a rise in fluid temperature.
Fig. 4A:
versus Entropy Generation
Fig. 4B: Weissenberg number versus Entropy
Generation
Fig. 4C: M versus Entropy Generation
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Fig. 4D: Re versus Entropy Generation
Fig. 4E: Pr versus Entropy Generation
Figure 4 depicts the effects of various governing
parameters on the entropy generation within the
channel. As the angle of inclination parameter
increases, Figure 4A shows a reduction in fluid
entropy generation at the lower wall and an increase
at the upper wall. The increase noticed at the upper
wall is because higher fluid velocity increases the
forces acting on the fluid flow, resulting in more
entropy formation. The effect of Weissenberg
number (
We
) on fluid entropy generation is
presented in Figure 4B. As
We
increases, entropy
generation enhances towards the channel's lower
wall, with a marginal increase at the upper wall.This
is due to the shear-thinning effect, which causes
higher flow motion at the lower wall, as discussed in
Figure 2B. Figures 4C and 4D present a falling trend
in entropy generation at the lower wall with the
reverse trend at the upper wall as magnetic field
parameter and Reynolds number take higher values.
With a high Reynolds number, disordered motion
develops because the fluid moves more randomly as
Re increases, and so the contribution of fluid
friction and heat transfer to entropy generation tends
to increase. Figure 4E shows that entropy generation
is significantly lowered at the lower channel but
yields to a sharp rise at the upper channel as Prandtl
number increases.
Fig. 5A:
versus Bejan number
Fig. 5B: Weversus Bejan number
Fig. 5C: M versus Bejan number
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Fig. 5D: Re versus Bejan number
Fig. 5E: Ra versus Bejan number
Fig. 5F: Pr versus Bejan number
Figure 5 shows the reaction of the Bejan number to
variations in the angle of inclination parameter,
Weissenberg number, magnetic field parameter,
Reynolds number, radiation parameter, and Prandtl
number. In Figure 5A, it is obvious that fluid
friction dominates entropy production because the
Bejan number decreases at the lower wall as the
angle of inclination parameter increases in value.
For Weissenberg number, fluid friction is effective
in contributing to entropy formation at the lower
wall, while heat transfer dominates at the middle
and upper walls, as seen in Figure 5B. Heat transfer
is the dominant contributor to entropy generation in
Figure 5C because values of Bejan number grow
with increasing magnetic field parameter throughout
the entire channel. Figure 5D, on the other hand,
shows a reversal of the pattern, since entropy
generation decreases as Reynolds number rises.
Finally, Figure 5E depicts that heat transfer
dominates entropy generation as the radiation
parameter increases.However, Figure 5F shows a
reversal of the trend with a rise in the Prandtl
number.
7 Conclusion
The heat irreversibility analysis of thermal radiation,
Ohmic heating, and angle of inclination on
Williamson fluid is investigated in this paper. The
flow model equations obtained are non-
dimensionalised and solved via homotopy
perturbation technique. The entropy generation and
Bejan number are analyzed using the results
obtained for velocity and temperature profiles, and
plots are presented to illustrate the flow
characteristics for the velocity, temperature, entropy
generation, and Bejan number. The main points are
summarized as follows:
The angle of inclination parameter and the
radiation parameter increase fluid velocity,
whereas the magnetic field parameter, Reynolds
number, and Prandtl number reduce flow motion,
The angle of inclination, magnetic parameter,
Reynolds number, and radiation parameter all
enhance fluid temperature,
All of the factors, except the radiation parameter,
promote entropy formation at the upper wall,
For increasing values of the inclination
parameter, Weissenberg number, Reynolds number,
and radiation parameter, fluid friction dominates
entropy generation, whereas the magnetic field
parameter and Prandtl number reveal the dominance
of heat irreversibility.
Future investigations can be directed to nonlinear
thermal and internal heat generation as well as mass
transfer rate.
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Acknowledgments:
The authors express their gratitude to Covenant
University for funding the article processing charge
(APC).
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Entropy, Vol. 15, 2013, pp. 220-233.
http://dx.doi.org/10.3390/e15010220
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Conception and design of the model by Opanuga
and Sobamowo
non-dimensionalised the model was carried out by
Ogunniyi,
Solution and analysis of the model was performed
by Opanuga,
Typesetting and proofreading by Okagbueand
Interpretation of results by Sobamowo.
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.en
_US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.20
Abiodun Opanuga, Gbeminiyi Sobamowo,
Hilary Okagbue, Peter Ogunniyi
E-ISSN: 2224-347X
227
Volume 17, 2022
Appendix
Nomenclature
,uv
Velocity components along x and y directions
respectively,
Kinematic viscosity,
k
Thermal conductivity,
p
c
Specific heat of the fluid at constant pressure,
Fluid density,
g
Acceleration due to gravity,
Dynamic viscosity,
T
Temperature of the fluid,
Dimensionless temperature,
2
0
B
Magnetic field parameter,
Angle of inclination,
We
Weissenberg number,
Re
Reynolds number,
M
Magnetic parameter,
Pr
Prandtl number,
Ra
Thermal radiation parameter,
Ec
Eckert number,
Be
Bejan number,
A
Constant pressure gradient,
Gr
Grashof number,
Material fluid parameter,
h
Channel width,
G
E
Local volumetric entropy,
Ns
Dimensionless entropy generation parameter,
r
q
Radiative heat flux,
c
k
Rosseland mean absorption coefficient,
c
Stefan-Boltzmann constant,
Electrical conductivity of the fluid,
Coefficient of thermal expansion,
L
Characteristic temperature ratio.
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.20
Abiodun Opanuga, Gbeminiyi Sobamowo,
Hilary Okagbue, Peter Ogunniyi
E-ISSN: 2224-347X
228
Volume 17, 2022
