Investigation of the Impact of a Chemical Reaction on the
Magnetohydrodynamic Boundary Layer Flow of a Radiative Maxwell
Fluid over a Stretching Sheet Containing Nanoparticles Employing the
Variational Iteration Method
AMINE EL HARFOUF1,*, ABDERRAHIM WAKIF2, SANAA HAYANI MOUNIR1
1Multidisciplinary Laboratory of Research and Innovation (LaMRI),
Energy, Materials, Atomic and Information Fusion (EMAFI) Team,
Polydisciplinary Faculty of Khouribga,
Sultan Moulay Slimane University,
MOROCCO
2Faculty of Sciences Aïn Chock, Laboratory of Mechanics,
Hassan II University,
Casablanca,
MOROCCO
*Corresponding Author
Abstract: - The heat and mass transmission properties of a 2-D electrically conducting incompressible Maxwell
fluid past a stretched sheet were studied under thermal radiation, heat generation/absorption, and chemical
reactions. This issue has a variety of real-world applications, most notably polymer extrusion and metal
thinning. The transport equations account for both Brownian motion and thermophoresis during chemical
reactions. Using similarity variables allows for non-dimensionalization of the stream's PDEs and associated
boundary conditions. The resulting modified ODEs are solved with the variational iteration approach. The
impact of embedded thermo-physical variables on velocity, temperature, and concentration was studied
quantitatively. When compared to the RK-Fehlberg approach, the findings are very similar. Raising the
chemical reaction parameter narrows the concentration distribution, whereas increasing the temperature
increases thermal radiation's impact. As the amount of increases, the thickness of the boundary layer
develops, causing the surface temperature to rise, resulting in a temperature increase.
Key-Words: - Chemical reaction, Nanoparticles, Maxwell fluid, Stretching sheet, Thermal radiation, the
variational iteration approach, the RK-Fehlberg approach.
Received: January 12, 2024. Revised: July 11, 2024. Accepted: August 8, 2024. Published: September 23, 2024.
1 Introduction
Nanotechnology has enabled the creation of
particles smaller than . Nanoparticles are
added to form a stable suspension and potentially
improve the thermal characteristics of the base
fluid. Adding small amounts of metal or metal
oxide nanoparticles to a fluid can improve its
thermal conductivity. Nanofluids, like current
working fluids, may absorb and transport heat.
Nanofluid flow has gained interest among
researchers in various fields, including technology,
science, biomechanics, chemistry, and nuclear
engineering. Nanofluid technology can address
technical issues such as solar energy collection,
heat exchangers, and engine cooling, [1]. [2],
studied the effect of Williamson nanofluid flow on
an exponential stretching surface. Their findings
show that the rate of heat transfer decreases with
increased Brownian motion and increases with
increased thermophoresis parameters. [3], studied
the heat and mass transport properties of copper-
aluminum oxide hybrid nanoparticles flowing over
a porous medium. Their findings indicate that a
porous media increase shear stresses for both pure
and hybrid nanofluids. [4], propose a heat and mass
transfer analysis of MHD nanofluid flow with
radiative heat effects in the presence of spherical
Au-metallic nanoparticles. Several scholars have
studied the impact of flow-governing parameters on
heat and mass transfer fluid flow problems
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involving nanoparticles in various models, [5], [6],
[7].
It is widely used in industrial operations such
as aerodynamics, metal spinning, plastic extrusion,
and condensation. The study of hydromagnetic
flow past a stretching sheet has received a great
deal of attention. [8], explored MHD 3-D heat and
mass transfer across a stretching sheet using a
water-based alumina nanofluid. They found that
increasing the nanoparticle volume fraction
parameter enhances heat transfer rate. [9],
simulated boundary-layer flow using a nonlinear
curved stretching sheet and convective mass
conditions. Recent research in this area has yielded
mixed results, [10], [11].
Non-Newtonian fluids' boundary layer flows
have received significant interest due to their
increasing use in engineering. The only way to
completely understand non-Newtonian fluids and
their applications is to study their flow. Engineers,
physicists, and mathematicians face particular
challenges when dealing with non-Newtonian
fluids. Nonlinearity can manifest in diverse
industries, including food, drilling, and
biotechnology. Multiple models have been
developed to characterize non-Newtonian fluids
due to their diverse nature, making it hard to use a
uniform stress-strain rate relationship. The
Maxwell model describes the simplest subclass of
rate-type fluids. Fluid rheology offers an alternative
to Newtonian fluids. The Maxwell fluid model
simplifies the physics of diluted polymeric liquids.
The papers [12], [13], [14] provide instances of
several approaches to addressing Maxwell fluid
flows.
Thermal radiation plays a significant role in
developing novel energy conversion technologies
for extreme temperatures. Thermal radiation can
significantly impact heat transfer in the polymer
processing industry, where the quality of the end
product is heavily influenced by heat management
variables. Thermal radiation has a greater impact
when the surface temperature differs significantly
from the surrounding environment. [15], studied
the thermal energy and mass transport of thinning
fluids with varying shear rate viscosity. [16],
investigated the effects of radiation, velocity, and
thermal slippage on MHD boundary layer flow in
Williamson nanofluid with porous media. Our
selection of research on heat radiation's impact on
MHD boundary layer movement, [17], [18], [19]
has been thoroughly reviewed by scholars and
analysts.
Differential equations are commonly used to
explain many mechanical issues, thus the method
used to solve them has a significant impact in
different circumstances. Analytical solutions to
low-order and simple differential equations are
easy to find; however, analytical solutions to high-
order or complex differential equations are difficult
to obtain or may not even exist. Numerical
techniques are so commonly used in practice to
obtain useful findings, particularly in engineering.
Ordinary differential equations (ODEs) are
equations with only one independent variable that
can be classified into two forms based on their
boundary conditions: initial value problems (IVPs)
and boundary value problems (BVPs). BVPs are
more difficult to solve than other types of problems
because their boundary conditions are defined at
multiple points. The gunshot technique, Galerkin
method, finite difference approach, finite element
method, and others are popular numerical solutions
for BVPs. These approaches are all capable of
solving ODEs, but each has significant
disadvantages. The shooting approach entails
repeatedly solving the BVP's estimated IVP, which
can be time-consuming for complex situations. The
Galerkin technique necessitates determining a
sequence of trial functions that are compatible with
the boundary conditions, which are not always
straightforward to calculate. In the finite difference
and finite element methods, the correctness of the
solution is strongly dependent on the mesh density
or quality. Furthermore, all of these approaches are
approximate, even those for linear ODEs.
Variational iteration method is a relatively
recent approach for solving differential equations
that are theoretically accurate, uses basic concepts,
and converges quickly. It is based on the generic
Lagrange multiplier approach for solving nonlinear
equations in quantum physics, which [20], [21],
[22] and [23] adapted into the variational iteration
method (VIM). The VIM is essentially a
generalized version of the Newton-Raphson
iteration method. Furthermore, [24] recently
demonstrated that when the largest derivative
component is treated as the linear part in the VIM
while solving nonlinear differential equations, the
VIM is comparable to the well-known classical
sequential approximation approach. The VIM is an
excellent tool for solving differential equations,
massive linear systems, and so on. It has been used
to solve numerous differential equations in diverse
disciplines, such as Boltzmann equation [25] in
statistical mechanics.
The heat and mass transport features of an
incompressible steady Maxwell fluid were
investigated as it flowed across a stretched sheet in
the presence of thermal radiation and chemical
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interaction with nanoparticles. The study of
hydromagnetic flow and heat transfer over a
stretching sheet has received a lot of attention
because of its practical applications and major
impact on a variety of manufacturing processes,
including aerodynamics, plastic sheet extrusion,
condensation processes, and metal spinning.
The variational iteration method is used to find
solutions that are similar to those that already exist
and to solve reduced ODEs. When compared to the
RK-Fehlberg approach, the findings are very
similar.
2 Problem Description and
Mathematical Formulation
In this study, we investigate a non-Newtonian
Maxwell nanofluid flowing over a stretched surface
that coincides with the plane , with a stable
and laminar boundary layer. The circulation is
limited to the region , as shown in Figure 1,
where y is the coordinate measured perpendicular
to the stretched surface. Furthermore, convection
from a hot fluid with a temperature of is thought
to heat the stretching plate's bottom surface,
producing a heat transfer coefficient of . First
order homogeneous chemical reaction processes
involving species have been considered throughout
this study. As the plate is stretched along the
, the linear velocity where is a
positive constant applied to it. An applied magnetic
field with a constant strength is placed parallel
to the flow direction along the . The
magnetic Reynolds number is thought to be low.
As a result, the induced magnetic field is not as
strong as the magnetic field that is provided from
outside.
The studied transport phenomena are
quantified by Wakif's-Buongiorno formulation as
follows, [26], [27], [28], [29], [30]:
Fig. 1: Physical configuration of the model
(1)
(2)
(3)
(4)
According to the boundary conditions:
, ,
, , at
(5)
,
,
at
(6)
is the thermal diffusivity,
is the
ratio of nanoparticle heat capacity to base fluid heat
capacity, is the dimensional heat generation and
absorption coefficient, is Wakif’s coefficient. The
concentration of nanoparticles at the surface is
higher than that of the surrounding fluid
.
Using the Rosseland estimator, we may
approximate the radiative heat flux:
(7)
is the Stefan-Boltzmann constant, while is the
coefficient of mean absorption. Assuming little
flow temperature variance, the equation for is
linear function. Expanding in a Taylor series
about
and eliminating the higher-order terms
beyond the first degree in
yields:
(8)
Consider the following nondimensional variables:
,
,
(9)
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Where
and
, the continuity equation
(1) is satisfied by the stream function .
Equations (2), (4), and (8) are translated into the
ODEs shown in Eq. (9).
(10)
(11)
(12)
Here are the appropriate boundary conditions:
, ,
,
,
,
(13)
The prime represents differentiation with relation to
. The similarity function is denoted by, while the
Prandtl number is
.
Represent the
Lewis number,
is the Biot number,
Is the elastic parameter,
is the
heat generation/absorption parameter,
is the magnetic parameter,
is
the Brownian motion parameter,
is the thermophoresis
parameter,
the radiation parameter,
The skin friction coefficient, local Nusselt
number, and local Sherwood number are
evaluated as follows:
(14)
In this equation,
represents shear stress,
whereas and represent surface heat and
mass flux.
(15)
The similarity transformation converts the non-
dimensional versions of skin friction, nusselt
number, and sherwood number as follows:
(16)
Represents the local Reynolds number.
3 The Runge-Kutta-Fehlberg Fourth-
Order Method
The region has been replaced by the limited
region , where is an acceptable real
number. The solution must confirm the domain
standard to solve the ODEs from (10) to (12), as
well as the initial and boundary conditions (13). To
solve (10) - (12), we have seven first-order
problems, each with seven variables.
(17)
To create the most successful numerical
strategy, we combine the shooting methodology
with the Runge-Kutta-Fehlberg fourth-order
method. Maple, a symbolic software, was used to
obtain the numerical results. The model requires
seven initial conditions, but only four are available:
, , , and . The remaining three,
, , and , were unavailable. To
determine the necessary end boundary conditions,
we use the shooting strategy, estimating the three
beginning circumstances first. To obtain a result
from the mathematical simulation, set the step
length to and the convergence
condition to . Figure 2 depicts the subsequent
process.
Fig. 2: chart of the computational process
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4 The Variational Iteration Method
Consider the differential equation.
(18)
and are linear and nonlinear operators,
respectively, with representing the source
inhomogeneous term. The variational iteration
method allows for the employment of a correction
functional for equation (17) in the form.
(19)
In this equation, represents a generic
Lagrange's multiplier that can be ideally discovered
using variational theory.
represents a restricted
variation, with
.The Lagrange multiplier
plays a significant role in the procedure and can be
either a constant or function. After determining ,
we apply an iteration method to calculate
consecutive approximations of the answer :
; . The zeroth approximation can
be any selective function. However, the initial
values , , and are preferred for
the selected zeroth approximation , as will be
demonstrated later. Thus, the solution is given by:
(20)
We officially derived the various forms of the
Lagrange multipliers in [XX], therefore we will
skip the details. We just provide an overview of the
acquired results.
For first-order ODEs of the type:
(21)
The correction functional yields the iteration
formula when is equal to .
(22)
For the second-order ODE.
(23)
, and the correction function yields the
iteration formula.
(24)
Furthermore, in the third-order ODE
(25)
We discovered that
, and the
iteration formula takes the form
(26)
Typically, for the nth-order ODE
,…,
(27)
We discovered that
, and the
iteration formula takes the form
(28)
The zeroth approximation can be any
selected function, although it is recommended to
select it in the following form:
(29)
Where is the order of the ODE.
The three Figure 3, Figure 4 and Figure 5 represent
the comparison of the results given by RK and
VIM, therefore the results given by VIM are
validated.
Fig. 3: Comparison between the results given by
(RK) and (VIM) for
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Fig. 4: Comparison between the results given by
(RK) and (VIM) for
Fig. 5: Comparison between the results given by
(RK) and (VIM) for and
5 Results and Discussions
Figure 6, Figure 7, Figure 8, Figure 9, Figure 10,
Figure 11, Figure 12, Figure 13, Figure 14, Figure
15, Figure 16, Figure 17, Figure 18, Figure 19,
Figure 20, Figure 21 show a model of a
Maxwell fluid moving across a stretched sheet,
including heat radiation and nanoparticles. Skin
friction, Nusselt number, and the Sherwood number
were calculated numerically, together with velocity,
temperature, and concentration. This was
accomplished through the use of numerous discrete
nondimensional flow parameters. Figure 6 shows
the temperature circulation of the thermal boundary
layer with different thermophoresis parameter
values. When there is a temperature differential, a
transport force called thermophoresis occurs.
Increasing the amount of increases the thickness
of the boundary layer, which raises the surface
temperature.
Fig. 6: Impact of on temperature
Figure 7 shows the temperature dispersion of
the thermal boundary layer with various Brownian
motion values. Brownian motion depicts how
particles in a fluid move randomly and erratically
due to repeated collisions with other molecules.
This erratic mobility accelerates collisions between
nanoparticles and molecules in the fluid. The
molecules' kinetic energy converts to thermal
energy, leading to an increase in temperature.
Fig. 7: Behavior of on temperature
Figure 8 illustrates the temperature distribution
of the thermal boundary layer with different Biot
number approximations . The Biot number refers
to the ratio of the body's internal to external heat
resistance. The ratio determines how much an
external thermal gradient affects a body's internal
temperature fluctuations over time. Problems with
Biot numbers less than 1 are easily solved
thermally due to homogenous temperature fields
within the body. Higher Biot numbers indicate
more complex challenges due to varying
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temperature fields within the object. As the Biot
number increases, the temperature near the
boundary rises significantly. Convective heat
transmission at the plate's surface leads to thicker
thermal boundary layer.
Fig. 8: Behavior of on temperature
Figure 9 and Figure 10 depict temperature
distributions for positive and negative scenarios
based on altering chemical reaction parameter . In
the boundary layer, damaging chemical reactions
generate heat or thermal energy. This
improves the thermal boundary layer. A generative
chemical reaction causes the fluid's
temperature to decrease.
Fig. 9: Behavior of
on temperature
Fig. 10: Behavior of
on temperature
Figure 11 and Figure 12 demonstrate how Le
affects temperature and concentration. The Lewis
number represents the relationship between heat
and mass diffusivity. Fluid flow is defined by the
simultaneous transmission of heat and mass. The
Lewis number compares the thicknesses of the
thermal and concentration boundary layers.
Increasing the Lewis number leads to increased
thermal diffusivity and decreased Brownian
diffusion. This means that the concentration
decreases as the temperature increases.
Fig. 11: Impact of on temperature
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Fig. 12: Impact of on concentration
Figure 13 illustrates the temperature profiles
for various Prandtl values. As the Prandtl number
increases, the thermal boundary layer thins. Fluids
with lower Prandtl numbers have better thermal
conductivity and larger thermal boundary layer
structures, allowing heat to escape the sheet faster
than those with higher Prandtl numbers and thinner
boundary layers. Changing the Prandtl number
adjusts the cooling rate.
Fig. 13: Impact of on temperature
Figure 14 displays the temperature curve for
different radiation parameter values. boosting
the radiation parameter value increases temperature
distribution by boosting surface heat transfer,
resulting in hotter fluids.
Figure 15 illustrates how the concentration
profiles change with the chemical reaction
parameter . As the pace of a chemical reaction
increases, the distribution of concentrations
decreases. Rising predictions minimize the
effects of buoyancy on concentration, resulting in a
decline.
Fig. 14: Effect of on temperature
Fig. 15: Behavior of
on concentration
Figure 16 shows that increasing the amount of
at the surface leads to a drop in the fluid
concentration. The Brownian motion warms the
boundary layer, causing particles to move away
from the fluid regime. However, further away from
the surface, the opposite trend occurs.
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Figure 17 illustrates the distribution of
concentration in the thermal boundary layer for
different thermophoresis parameter values .
Stronger thermophoresis results in a broader
concentration boundary layer, leading to improved
concentration.
Figure 18 and Figure 19 demonstrates that
increasing the elastic parameter decreases the
velocity distribution and thickness of the boundary
layer. In terms of physics, relaxation time
influences the elastic parameter. Longer relaxation
times cause the fluid to travel slower.
Fig. 16: Effect of on concentration
Fig. 17: Behavior of on concentration
Fig. 18: Impact of on velocity
Fig. 19: Impact of on velocity
This reduces the velocity field and thickness of
the boundary layer. As seen in Figure 15 and Figure
16, as the magnetic parameter increases, the
fluid velocity decreases. Persistent magnetic fields
generate an opposing force known as drag force,
which acts in the direction of fluid flow. Because of
this obstruction, the momentum barrier layer may
be thinner.
The velocity profile decreases when the
magnetic field parameter increases as shown in
both Figure 20 and Figure 21.
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Fig. 20: Effect of on velocity
Fig. 21: Effect of on velocity
6 Conclusions
We explored the heat and mass transport
parameters of an incompressible steady
Maxwell fluid as it flowed across a stretched sheet
under thermal radiation and chemical interaction
with Nanoparticles. The effect of embedded
thermo-physical parameters on velocity,
temperature, and concentration has been quantified.
The variational iteration method aims to solve
reduced ODEs by identifying equivalent solutions
to existing ones. Compared to the RK-Fehlberg
approach, the results are highly consistent.
The conclusions are as follows:
As the level of increases, velocities
decrease.
As rises, temperature distributions
flatten.
As approximations increase, Nusselt
number approximations decrease.
Sherwood predicts a rise in line with
estimations.
Concentration profiles decrease as Le rises.
Raising the estimations of improves the
temperature distribution.
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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. Conceptualization, Formal analysis,
Investigation, Methodology, Project administration,
Resources, Validation, Writing – review & editing.
Conceptualization, Investigation, Writing – review
& editing. Conceptualization, Formal analysis,
Investigation, Methodology, Project administration,
Resources, Validation, Writing – review & editing.
A. Wakif: Conceptualization, Investigation, Project
administration, Supervision, Writing – review &
editing. Conceptualization, Formal analysis,
Investigation, Methodology, Project administration,
Resources, Validation, Writing – review & editing.
Conceptualization, Investigation, Project
administration, Writing – review & editing. S.
Hayani Mounir: 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
n_US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.25
Amine El Harfouf, Abderrahim Wakif, Sanaa Hayani Mounir
E-ISSN: 2224-347X
281
Volume 19, 2024
