Technical Simulation for the Hydromagnetic Rotating Flow of
Carreau Fluid with Arrhenius Energy and Entropy Generation
Effects: Semi-Numerical Calculations
1 M. G. IBRAHIM, 2HANA ABDELHAMEED ASFOUR
1Basic and Applied Science Department, International Academy for Engineering and Media Science,
11311, Cairo, EGYPT
2Basic and Applied Science Department, Thebes Academy, 11311, Cairo, EGYPT
Abstract: - The present study aimed to investigate the influence of activation energy on the MHD Boundary
layer of Carreau nanofluid using a semi-numerical/analytical technique. The governing formulated system of
partial differential equations (PDEs) subject to appropriate boundary conditions is shortened to ordinary
differential equations (ODEs) by convenient transformations. Generalized Differential Transform (GDTM) is
used and compared with the Runge–Kutta Dahlberg method to find the results of the proposed system.
GDTM is chosen to cure and overcome the highly non-linear differentiation parts in the present system of
ODEs. Gradients of velocity, temperature, and concentration are computed graphically with different values
of physical parameters. The solutions are offered in two cases, the first in the case of non-Newtonian fluid
(=0.2) and the other in the case of base fluid (=0.2), which is concluded in the same figure. The
accuracy of GDTM is tested with many existing published types of research and found to be excellent. It is
worth-mentioned that the distribution of velocity growths at high values of power index law relation. This
fluid model can be applied in solar energy power generation, ethylene glycol, nuclear reactions, etc.
Key-Words: - Activation energy; Boundary layer; GDTM; Entropy Generation; Carreau nanofluid.
Received: August 18, 2021. Revised: October 14, 2022. Accepted: November 17, 2022. Published: December 31, 2022.
1 Introduction
Non-Newtonian fluids studies captured the
attention of scholars and researchers in the last
years, because of their engineering and industrial
applications in various fields. These applications
like cosmetics, molten polymers, food products,
oils, certain paints, drilling mud, fluid suspensions,
volcanic lava, etc.… Interests of
scholars/researchers increase with the stress tensor
in such fluids which accompany the deformation
rate tensor by relationships of highly nonlinear
differentiation. This non-linear mechanism of non-
Newtonian fluids gives rise to complicated non-
linear equations. Ghasemi and Hatami,
scrutinized the solar effects on the MHD boundary
layer flow of a nanofluid; they found that the
distribution of velocity is an increasing function in
heat generation. Khalil et al,. deliberated the
Eyring-Powell fluid with an inclined stretching
cylinder. Shafiq and Sindhu, suggest a new
radiative effect on the flow of a non-Newtonian
fluid. Mabood et al, deliberated the heat and
generation effects on the flow of Hybrid fluid.
Shafiq et al and Zari et al studied new
effects of Darcy and MHD effects on the Casson
flow with nanoparticles. Malik et al,
scrutinized the numerical investigation of
paramount non-Newtonian fluid on a sheet. There
are many papers introduces a non-Newtonian
fluids and its applications, [8-14]. In physics and
fluid mechanics, the boundary layer constitutes an
important concept and refers to the fluid layer near
an ocean surface where viscosity effects are
significant. Even in the flow where viscosity is
low and can be ignored, there is a thin film whose
viscosity cannot be ignored at a boundary such as
an object surface or wall surface, this layer is
called the boundary layer. Since the actual liquid
is viscous, the relative velocity is at the
boundary between the liquid and the wall. Ideal
with viscosity set to ideal fluids although the
theory is mathematically brief when dealt with, the
state of the adhesive cannot be satisfied, and the
sliding velocity parallel to the wall is generally
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shown. Also, in an ideal fluid, the resistance acting
on a body moving at a constant velocity in it
becomes zero. The boundary layer idea was
introduced by L. Prandtl in 1904, to mitigate
these contradictions and lay the foundation for
modern fluid dynamics. A timed method on
boundary layer flow problems was addressed by
Pasha et al, they found that radiation flux is
an increasing function in heat transfer. Khan et al,
studied the influences of solar radiation and
heat generation on non-Newtonian nanofluid over
a stretching sheet with variable thickness, they
found that the temperature distribution is enhanced
at high values of heat generation. In chemistry
and physics, activation energy is the energy that
must be supplied to a chemical or nuclear system
of latent reactants that lead to a chemical reaction,
a nuclear reaction, or various other physical
phenomena. It is denoted by the symbol, and the
unit kilojoule/mole is used to measure it. The term
was coined by the Swedish chemist Svante
Arrhenius in 1889. Shafique et al,
discretized the influences of activation energy on
boundary layer flow in a rotating frame; they
found that the activation energy is an increasing
function in the temperature of the fluid. Gowda et
al, studied the heat and mass transfer effects
on the boundary layer of non-Newtonian fluid,
they found that the growing values of the magnetic
parameter develop the velocity gradient and decay
the heat transfer. In nearly times, applications of
activation energy appeared more and more in
different fields like nuclear reactions in
engineering, various physical phenomena in
physics, and many applications can be
found in . The current scientific
eagerness to find new numerical, semi-numerical,
and analytical methods for finding the solutions to
fluid problems that have a high degree of
nonlinearity is in the interest of mathematics
sciences. One of the methods that appeared to
solve the problem of nonlinearity in partial or
ordinary differential equations is called the
generalized differential transform method, which
is shortened to GDTM as offered in this
manuscript. The differential transform method was
first introduced at the end of the last century by
Zhou, which is defined as a semi-analytical
method for solving nonlinear partial differential
equations. In early times, Odibat et al studied
the non- non-chaotic or chaotic systems by using a
new modified technique called the multi-step
differential transform algorithm. Also, a large
number of investigations have proven the
effectiveness of the DTM and its modification
techniques, . On another side, the main
idea for GDTM is to choose/ divide a suitable
number of intervals from solution intervals, as
well as make a generalization of the resulting
differential transform series solution. To illustrate
the GDTM in detail, section 3 is made to compute
the graph solutions of the MHD boundary layer
flow of Carreau nanofluid.
The main novelty of this paper is to introduce
a new simulation of the Arrhenius activation
energy of the hydromagnetic rotating flow of
Carreau fluid entropy generation effects using
GDTM. Combined solutions of the proposed
model in the cases of Newtonian and non-
Newtonian fluid are discussed to clarify the fluid
behaviors. Comparisons have been made with
previously existing published results by Khan et
al, [24], Wang, [25], Gorla and Sidawi, [26], and
Mabood et al, [27], and found to be in good
agreement. In addition, comparisons of
temperature and concentration distributions have
been made with results given by Khan et al, [24]
and found to be excellent as stated/indicated. In
the next subsection, the formulation of the
problem is introduced; in section 3, the
generalized method is presented; discussions of
the graphed results are offered in section 4;
eventually, the conclusion and main points are
concluded in the last section of the current paper.
2 Mathematical Formulations
The boundary layer flow of Carreau nanofluid
with variable thickness is considered as a
mathematical system of differential equations
is to be assumed the thickness of the
sheet, where is constant and is the index of
the power law. The magnetic field is
supposed to vary with the strength and vertical to
plate as shown in.
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The Carreau nanofluid combined with the
boundary layer approximation mode l is
written as:
, (1)
,
(2)
,
(3)
,
(4)
The appropriate boundary conditions for Carreau
nanofluid are chosen as follows:
(5)
.
(6)
Here and are the specific heat, the
temperature of the fluid, the fluid density, the heat
generation, the concentration of fluid, the radiative
heat flux and the electric conductivity,
respectively. are the components along
direction, are the thermophoretic
diffusion coefficient and the kinematic viscosity.
and are the near and distant away
concentrations, is the applied magnetic field. A
similarity transformation is introduced as the
following.
By using self-similarity transformations
, (7)
By using transformations in Eq. (7), the system of
Eqs. (1-4) with boundary conditions (5-6) become
, (8)
, (9)
(10)
The transformed boundary conditions are
,
(11)
Here,
and,
The Hartmann number
, (12)
The Weissenberg number:
, (13)
The Prandtl number:
, (14)
The heat generations parameter:
, (15)
The Brownian motion parameter:
, (16)
The thermophoresis parameter:
, (17)
The chemical reaction species:
, (18)
Thermal radiation (heat flux) parameter:
, (19)
The Lewis number:
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, (20)
The temperature difference parameter:
, (21)
The Activation energy:
, (22)
3 GDTM Applications
The solution of a system of differential equations
(8-10) with boundary conditions (11) using the
GDTM is presented in this section in detail. The
solution of the proposed system is illustrated step
by step to show the quality of the technique with
aid of.
The first step mentioned the system of
differential equations written at the start of
Mathematica algorithms. In the same step, we
choose the appropriate limit to the solution at the
point at which solutions are proven and do not
change, no matter how much this value increases.
In the second step, we transform the system of
equations (8-10) with boundary conditions (11)
using the differential transform technique
as follows:
,
(23)
,
(24)
.
(25)
With transformed boundary conditions using
theories of DTM as:
(26)
In the third step, we make a table of differential
operations for each variable according to the
degree of differentiation for each variable as an
algorithm in Mathematica (suppose the numberof
required solution equal 10, that chosen as we
need)
(27)
Where the value of represents the number of
solutions to be found, is defined as values of
non-dimensional parameters, and refer to
the system of equations, respectively. In the fourth
step, we introduce the initial values to the
solutions and make a table of contents for the
required solutions as follows:
(28)
,
(29)
In the fifth step, the direct substitution step is used
to find the complement of missing real
solutions. In addition, we get the following shape
of solutions.
(30)
And then results take the shape
, (31)
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, (32)
. (33)
When testing the accuracy of the solutions
that we obtained from this step, we found that the
solutions, in the beginning, are identical to the
exact solution. Then the solutions deviate from the
correct path. So it had to generalize of given
results to get more accurate solutions in the sixth
step
,
(34)
Solutions that are obtained after applying this
algorithm are found to be accurate with the exact
solution. In the next section, solutions that are
compared with the nearest/
existing published results are given by Khan et
al. In addition, the solution of the present
system of differential equations are offered in
different two cases, the first in the case of base
fluid at and other in non-Newtonian fluid
(Carreau fluid).
4 Results
This section is divided into four subsections
formed and designed as follows: In the first
subsection, the accuracy of presented results with
recently published results by Khan et alis
proven/ verified through figures and table
contents. The second subsection offers the
distribution of velocity against penitent physical
parameters to show the variance and physical
meaning of increased fluid velocity. In the third
subsection, the variance of physical parameters on
the temperature and concentration profiles is
studied.
4.1 Accuracy of Presented Results
It is common knowledge that to prove the
effectiveness of a new method, we compare the
graphs or tables obtained with recently published
results by researchers recently. In the present
article, results of velocity and temperature
distributions are compared with existing published
results by Khan et al. It found that the
solutions computed by GDTM are in good
agreement with the results given by the Rung-
Kutta Fehlberg method. It is worth mentioning in
Fig. 2 that the velocity profile is an increasing
function at high values of the Hartmann number.
Also, Fig. 3 shows that as usual large numbers of
Prandtl numbers cause a rising in the temperature
profile.
Fig. 2: Comparison of velocity behavior against
Hartmann number [Present results versus Khan et
al [24] results]
Fig. 3: Comparison of temperature profile against
Prandtl number [Present results versus Khan et al
[24] results]
Table 1 is representing a numerical comparison of
Nusselt Number values at different values
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of the Prandtl number. This comparison has been
made with previously existing published results by
Wang [25], Gorla and Sidawi, [26], Mabood et al
[27], and Khan et al, [24], and found to be in good
agreement. In addition, comparisons of
temperature and concentration distributions have
been made with results given by Khan et al, [24],
and found to be excellent as stated/indicated in
Table 2. It’s worth mentioning that the standard
values of parameters for all obtained figures
are
.
Table 1: Comparison of at
and
Table 1. numerical comparison of Nusselt Number
′ values at different values of the Prandtl
number.
Khan et
al [24]
Wang
[25]
Gorla and
Sidawi
[26]
Mabod
et al
[27]
Present
results
0.07
0.0645
0.0656
0.0656
0.0665
0.0655
0.20
0.1663
0.1691
0.1691
0.1691
0.1691
0.70
0.4554
0.4539
0.4539
0.4539
0.4534
2.00
0.9100
0.9114
0.9114
0.9114
0.9113
7.00
1.8929
1.8954
1.8905
1.8954
1.8903
20.00
3.3505
3.3539
3.3539
3.3539
3.3538
70.00
6.4598
6.4622
6.4622
6.4622
6.4627
Table 2. Comparison of (obtained
resent results compared with given by Khan et al
[24])
Khan et
al [24]
Present
results
Khan
et al [24]
Present
results
0.0
1
1
1
1
0.7
0.787089028
0.787089028
0.87298093
0.872980938
1.4
0.582520232
0.582520232
0.746262989
0.746262989
2.1
0.407245275
0.407245275
0.612085865
0.612085865
2.8
0.265684286
0.265684286
0.469935625
0.322200746
3.5
0.155088686
0.155088686
0.322200746
0.322200746
4.2
0.070658319
0.070658319
0.171806533
0.171806533
4.2 Velocity Distribution Study
As stated above, the governing equations of the
activation energy effects on nanofluid flow over a
stretching sheet are transformed and
solved analytically using GDTM. The velocity
gradient is graphically displayed by changing the
values of the power index law relation and
Hartmann number. It's depicted in Fig. 3 that
the high values of the power index law cause an
increase in stretching velocity, which creates
supplementary deformation in the fluid. It can
realize from Fig. 4 that the gradient of velocity
diminishes by growing values of, which causes
the momentum boundary layer thickness to
improve as upturns, whilst the dissimilarity in
Lorentz force decreases the distribution of
velocity.
Fig. 4: Velocity behavior versus values
Fig. 5: Velocity behavior versus values
4.3 Temperature and Concentration Study
Distributions of temperature and nanoparticle
concentration are fully illustrated through this
subsection versus different values of
thermophoresis parameter thermal radiation
parameter , chemical reaction species,
power index law parameter, Hartmann
number and activation parameter in
Figs. 6-17. It's visualized from Figs. 6 and 7 that
the temperature gradients are growing at high
values of and. Noticeably, thermophoresis
and thermal radiation parameters acquire the fluid
particle more energy making the temperature have
high values, [24] and [27]. As expected, the
temperature distribution is a decreasing function
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on and by looking at their definitions.
So, Figs. 8-10 are depicted to approve that the
high values of and reduce the fluid
temperature. In addition, chemical reaction and
activation energy parameters do not directly affect
on temperature of fluid accordingly, the effect is
not clear as found in the case of the power index
law parameter and Hartmann number [11], [12].
All results show that the influences of physical
parameters on the distribution of temperature
become more sight in non-Newtonian fluid than
found in the base fluid.
Nanoparticle concentration distribution have
usually a contradictory behavior to temperature
distribution with the same parameters, as the
inverse relationship between them. However, it
should be noted from Figs. 12-13 that the
concentration of fluid diminutions at high values
of and actually as in [25] and [26]. Whilst,
the concentration and temperature distributions are
considered to have an increasing function in
thermophoresis parameter [27], as seen in Figs.
6 and 14. In addition, Figs. 15-17 show that the
fact of nanoparticle concentration distribution rises
at high values of and.
Fig. 6: Temperature behavior versus values of .
Fig. 7: Temperature behavior versus values of .
Fig. 8: Temperature behavior versus values of .
Fig. 9: Temperature behavior versus values of .
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Fig. 10: Temperature behavior versus values of .
Fig. 11: Temperature behavior versus values of .
Fig. 12: Concentration behavior versus values of
.
Fig. 13: Concentration behavior versus values of
.
Fig. 14: Concentration behavior versus values of
.
Fig. 15: Concentration behavior versus values of
.
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Fig. 16: Concentration behavior versus values of
.
Fig. 17: Concentration behavior versus values of
.
5 Conclusion
In this article, a semi-analytical solution to the
MHD Boundary layer of Carreau nanofluid is
computed using the generalized differential
transform method. Entropy generation, activation
energy, and variable thickness sheet are taken into
consideration. Combined solutions of the proposed
model in the cases of Newtonian and non-
Newtonian fluid are discussed to clarify the fluid
behaviors. The obtained solution is compared with
those of Wang, Gorla and Sidawi,
Mabood et al, and Khan et al, and a
good agreement was found. Activation energy and
thermal radiation effects are considered. The main
outcomes of the present study can be conceived as
follow:
Analytical methods like GDTM are the best
way to get more accurate solutions of fluid
models without linearization or perturbation
assumptions.
The distribution of velocity growths at high
values of power index law relation.
This fluid model can be applied in solar energy
power generation, ethylene glycol, nuclear
reactions, etc.
High values of activation energy increase the
concentration distribution.
GDTM is an effective method for solving a
highly non-linear system of differential
equations.
Thermal radiation has opposite effects on the
distributions of temperature and concentration.
Power index law relation and Hartmann
number have to contradict influences on the
velocity distribution.
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WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2022.17.21
M. G. Ibrahim, Hana Abdelhameed Asfour
E-ISSN: 2224-347X
239
Volume 17, 2022
[38] W. Hasona, Nawal H. Almalki, Abdelhafeez
A. El-Shekhipy and M. G. Ibrahim,
Combined Effects of Variable Thermal
Conductivity and Electrical Conductivity on
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Conflict of Interest
The authors declared that there is no conflict of interest
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
M.G. Ibrahim: Conceptualization, Methodology,
Software, Data duration, Writing – original draft, Hanaa
Abdel Hameed Asfour: Visualization, Investigation,
Validation, Writing – review & editing.
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: 10.37394/232013.2022.17.21
M. G. Ibrahim, Hana Abdelhameed Asfour
E-ISSN: 2224-347X
240
Volume 17, 2022
