Non-Fourier Heat Flux Model for the Magnetohydrodynamic
Casson Nanofluid Flow Past a Porous Stretching Sheet using the
Akbari-Gangi Method
AMINE EL HARFOUF1,*, RACHID HERBAZI2,3,4, SANAA HAYANI MOUNIR1,
HASSANE MES-ADI5, ABDERRAHIM WAKIF6
1Multidisciplinary Laboratory of Research and Innovation (LaMRI),
Energy, Materials, Atomic and Information Fusion (EMAFI) Team,
Polydisciplinary Faculty of Khouribga,
Sultan Moulay Slimane University,
MOROCCO
2Intelligent Systems and Applications Laboratory (LSIA), EMSI,
Tangier,
MOROCCO
3ENSAT, Abdelmalek Essaâdi University,
Tangier,
MOROCCO
4ERCMN, FSTT, Abdelmalek Essaâdi University,
Tangier,
MOROCCO
5Laboratory of Process Engineering, Computer Science and Mathematics,
National School of Applied Sciences of the Khouribga University of Sultan Moulay Slimane,
MOROCCO
6Faculty of Sciences Aïn Chock, Laboratory of Mechanics,
Hassan II University,
Casablanca,
MOROCCO
*Corresponding Author
Abstract: - The Casson fluid flow with porous material in magnetohydrodynamics is examined in this work.
Additional semi-analytical results are investigated using the Silver-Water nanofluid. The Akbari-Ganji Method
(AGM) is used to solve the semi-analytical Cattaneo-Christov heat flux model after taking thermal radiation
into account. With the use of appropriate parameters, such as the relaxation time parameter, Prandtl number,
radiation parameter, magnetic parameter, and so on, the normalized shear stress at the wall, temperature profile,
and rate of heat flux may be examined. This issue has numerous industrial applications and technical
procedures, such as the extrusion of rubber sheets and the manufacture of glass fiber. The main physical
application is the discovery that a rise in the thermal relaxation parameter and Prandtl number maintains a
constant fluid temperature.
Key-Words: - Magnetohydrodynamics; Porous medium; Couple stress; Nanofluid; The Akbari-Ganji Method
(AGM); porous stretching sheet.
Received: March 13, 2023. Revised: December 27, 2023. Accepted: February 26, 2024. Published: April 2, 2024.
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DOI: 10.37394/232013.2024.19.16
Amine El Harfouf, Rachid Herbazi,
Sanaa Hayani Mounir, Hassane Mes-Αdi, Abderrahim Wakif
E-ISSN: 2224-347X
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1 Introduction
Non-Newtonian fluids have drawn a lot of attention
because of their wide range of applications, which
include cooling engines and extracting crude oil
from petroleum products, [1]. Like this, stretching
sheet issues are important in the engineering
domains. The authors [2], were the first to pioneer
the study of laminar flow problems, and researchers
later expanded on this work, [3]. In this work, fluid
flow occurred because of stretching the sheet. As a
result of their work, numerous scholars investigated
the problems associated with stretching sheets.
Stretching sheet difficulties were studied by [4] and
[5]. In addition, in other work [6], using various
media. Fluid flow happened in the presence of a
porous media. There are numerous industrial uses
for this phenomenon. Subsequently, nanofluids are
used to solve stretching sheet problems in
conjunction with other fluids and boundary
conditions. The thermal properties of nanofluids
were studied in [7], [8], [9] and some of the
magnetorheological properties were also reviewed.
In addition to the computational time, the authors
[10] and [11] investigated nanofluids with suction
and laminar natural convection. Additional
instances about nanofluids are enumerated in [12],
[13] and [14]. Only the momentum and energy
equations with the classical Fourier law are covered
by the books. As a result, research is done on the
temporal relaxation parameter, [15], [16]. The
derivatives of the usual type are transformed into
Oldroyd's upper convicted derivative, which is
known as the Cattaneo-Christov heat flux and was
enhanced by [17].
The current work, which discusses the heat
transfer properties of Casson fluid flow through a
porous material with radiation, was inspired by
previous research. Non-Newtonian fluid behavior is
typically described by Casson fluid models. The
current work is unusual in that it uses analytical
tools to characterize the flow behavior of the
Casson fluid and adds nanoparticles to the fluid's
surface to increase thermal efficiency. Additionally,
the primary methodology describes how to solve
the stretching sheet problem analytically using the
Appell hypergeometric technique and a time
relaxation parameter. The Cattaneo-Christov
equation is used in this case to transform ordinary-
type derivatives into Oldroyd's upper convicted
derivative. then straight integration is used to solve
the temperature equation. Industrial, biomedical,
and engineering processes are the primary physical
components of the present. Refer to the Casson
fluid flow research conducted by [18] and [19].
2 Mathematical Formulations
In the current investigation, a non-Newtonian fluid
flow with porous media and MHD is analyzed.
Silver-water nanofluid is another fluid that is added
to the flow. The schematic diagram utilized in this
investigation is described in Figure 1 and the
amounts of nanofluid are shown in Table 1.
The Maxwell’s equation
(1)
(2)
(3)
(4)
Here
stands for the induced
electric field, while other terms are defined
according to the nomenclature. The Maxwell
equations are usually integrated into a single
equation called the magnetic induction equation in
magneto-convection.
However, the magnetic Rayleigh number
can be obtained by applying
the constitutive Eq. (1). (here is
Characteristic velocity), and the Lorentz force
for weak conducting fluid can be
written as:
.
This is known as the Hartmann formulation of
the magnetohydrodynamic issue.
Table 1. Thermo-physical properties of water and nano particles.
Cp K
Pure water (H2O)
Silver (Ag)
997.1
10,500
4179
235
0.613
429
0.05
5.97*107
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Fig. 1: Illustration of the Casson fluid flow
schematic
The current problem's modified Navier-Stokes
equation is given as:
(5)
(6)
(7)
here,
indicates the Casson fluid parameter,
indicates permeability,
is the heat
flux.
The boundary conditions are:
(8)
Here is the linear velocity, is a positive
constant, and
are the wall and for field
temperature.
The appropriate transformation of similarity:
(9)
The variables used in the equations are defined
by the terminology.
Momentum equation:
(10)
The appropriate threshold condition decreases to:
(11)
Energy equation:
Equation obtained from the Cattaneo-Christov
model is as follows:
(12)
It is possible to calculate the term K by
applying Rosseland's approximation.
(13)
Where
is the Stefan–Boltzmann constant and
is the mean absorption coefficient of the
nanofluid. Moreover, we suppose that the
temperature difference within the flow is such that
may be expanded in a Taylor series. Hence,
expanding about
and ignoring higher order
terms we get:
(14)
The solution of the equations takes on the
following form of an ordinary differential equation
upon substitution of the similarity variables defined
in Eq. (9) into Eq. (15).
(15)
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(16)
Where
is the Darcy number,
is the magnetic parameter,
is the Prandtl number, is the
thermal relaxation parameter,
is the
radiation parameter.
The energy equation's boundary conditions
were reduced to:
(17)
The amounts of nanofluid utilized in the
findings might be described as:
Here is the effective density of the nanofluid,
is the effective dynamic viscosity of the
nanofluid, is the heat capacite of the
nanofluid and is the thermal conductivity of the
nanofluid are given as:
(18)
Here , , , and are dimensionless
constants given by:
(19)
3 Akbari–Ganji’s Method Basic Idea
of (AGM)
To comprehend the given method in this paper, the
entire process has been declared clearly.
In accordance with the boundary conditions,
the general manner of a differential equation is as
follows:
The nonlinear differential equation of p, which
is a function of u, the parameter u which is a
function of x, and their derivatives are considered
as follows: Boundary conditions:
(21)
To solve the first differential equation
concerning the boundary conditions in in
Eq. (21), the series of letters in the nth order with
constant coefficients which we assume as the
solution of the first differential equation is
considered as follows:
(22)
The more choice of series sentences from Eq.
(22) causes a more precise solution for Eq. (20).
For obtaining solution of differential Eq. (20)
regarding the series from degree (n), there are (n +
1) unknown coefficients that need (n + 1) equations
to be specified. The boundary conditions of Eq.
(21) are used to solve a set of equations that
consists of (n + 1) ones.
Applying the boundary conditions
The application of the boundary conditions for the
answer of differential Eq. (21) is in the form of:
When :
(23)
And when :
(24)
After substituting Eq. (23) into Eq. (20), the
application of the boundary conditions on
differential Eq. (20) is done according to the
following procedure:
(25)
Regarding the choice of n, () sentences
from Eq. (21) and to make a set of equations which
is consisted of (n+ 1) equations and (n +1);
unknowns, we confront with several additional
unknowns which are indeed the same coefficients
of Eq. (21). Therefore, to remove this problem, we
should derive m times from Eq. (20) according to
(20)
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the additional unknowns in the afore-mentioned
sets of differential equations and then apply the
boundary conditions on them.
(26)
Application of the boundary conditions on the
derivatives of the differential equation Pk in Eq.
(75) is done in the form of:
(27)
(28)
(n+ 1) equations can be made from Eq. (22) to
Eq. (27) so that (n + 1) unknown coefficients of Eq.
(21) such as , , . . ... Be computer. The
solution of the nonlinear differential Eq. (20) will
be gained by determining coefficients of Eq. (21).
To comprehend the procedures of applying the
following explanation we have presented the
relevant process step by step in the following part.
Application of Akbari–Ganji’s Method (AGM)
According to the mentioned coupled system of
nonlinear differential equations and by considering
the basic idea of the method, we rewrite Eqs. (10) –
(16) in the following order:
Due to the basic idea of AGM, we have utilized
a proper trial function as solution of the considered
differential equation which is a finite series of
polynomials with constant coefficients, as follows:
(31)
(32)
4 Validation of Numerical Results
and Discussion of Results
In this work, the steady MHD nanofluid flow and
heat transfer past a porous stretching sheetin the
attendance of thermal radiation impacts and
considering the Christov Cattaneo heat flux model
of heat conduction are studied Semian-alytically by
using the Akbari Ganji’s Method (AGM). To verify
the present analytical solution, we compared our
results with results given by using Runge-Kutta.
They are in excellent agreement as they have been
demonstrated in Figure 2.
Fig. 2: Comparison between results given by AGM
and RK for
and
(29)
(30)
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The current analysis looks at an incompressible
Casson fluid flow with MHD and a porous
medium. We look into the Silver-Water nanofluid
in more detail. The Akbari Gangi Method is then
used to solve the ensuing equations that are
acquired. The results can be verified using the solid
volume fraction, magnetic parameter, radiation
parameter, and so on. The current challenge uses a
range of physical parameters: , ,
with all other parameters modified
with suitable values to obtain appropriate
parameters. The Cattaneo-Christovs idea, which
considered the significance of downtime, serves as
the foundation for the current investigation. These
changes in thermal conductivity are characteristics
that are temperature dependent.
Figure 3 shows the relationship between
and
for various values. In this, when values
rise,
values decrease. Figure 4 shows a similar
effect of
when we change the
values, i.e.,
the
lowers as the
values increase. Figure 5
illustrates how
affects ETA when
values are taken in ascending order. It can be
observed that the
increases in value with
small values of and declines with increasing
values.
The influence of
vs
for various
values and was shown in Figure 6 and Figure 7,
respectively.
in Figure 6 has an inverse
relationship with 's values. Figure 7 illustrates
this same effect, where
decays as values
rise. Figure 8 illustrates the impact of
vs
for the values of
in ascending order. This
indicates that
is inversely proportional to
,
meaning that
is greater for lower
values.
The impact of
versus
for changing the
values of and , respectively, is shown in Figure
9 and Figure 10. We deduce from Figure 9 that
when values rise,
also rises. Like this,
rises as increases, as shown in Figure 10.
Figure 11 shows the comparison between
and
for different values. In this case, the
rises as the values do, the same effect is observed
in Figure 12 for the the impact of
on
.
Fig. 3: The impact of on
Fig. 4: The impact of on
Fig. 5: The impact of on
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Fig. 6: The impact of on
Fig. 7: The impact of on
Fig. 8: The impact of on
Fig. 9: The impact of on
Fig. 10: The impact of on
Fig. 11: The impact of on
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Fig. 12: The impact of on
4 Conclusion
The current study investigates the flow of an
incompressible, non-Newtonian fluid in the
presence of heat radiation, a porous material with
an inverse Darcy number, and MHD. Using AGM,
the resultant equations are solved analytically. The
result is also analyzed using a silver water
nanofluid. Using this solution, we get the following
conclusions.
When M is added, the tangential and
transverse velocities drop.
As
values increase,
also rises.
and
have an inverse relationship.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
A. EL Harfouf: Conceptualization, Formal
analysis, Investigation, Methodology, Project
administration, Resources, Validation, Writing –
original draft, Data curation, Software,
Visualization. R. Herbazi: Conceptualization,
Formal analysis, Investigation, Methodology,
Project administration, Resources, Validation,
Writing – review & editing. S. Hayani Mounir:
Conceptualization, Investigation, Project
administration, Writing – review & editing. H.
Mes-adi: Conceptualization, Formal analysis,
Investigation, Methodology, Project administration,
Resources, Validation, Writing – review & editing.
A. Wakif: Conceptualization, Investigation, Project
administration, Supervision, Writing – review &
editing.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
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