Heat and Mass Transfer Analysis of an unsteady Non-Newtonian
Magnetohydrodynamic Fluid Flow (Semi-Analytical Solution)
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: - In the presence of a time-dependent chemical reaction, this work investigates unsteady squeezing
Casson nanofluid flow and heat transfer between two parallel plates under the influence of a uniform magnetic
field. Considering the effects of viscosity dissipation, heat generation from friction resulting from flow shear,
Brownian motion, Joule heating, and thermodiffusion. The problem's nonlinear differential equations are solved
using the Runge–Kutta (RK-4) technique and the Homotopy Perturbation technique. The excellent accuracy of
the results is evident since they have been compared with other results from earlier research. In the form of
graphs and tables, flow behavior under the many physical factors that are modified is also covered and well
described. This work has shown that, by normalizing flow behavior, magnetic fields may be utilized to manage
a variety of flows. Additionally, it is demonstrated that in every situation, the effects of positive and negative
squeeze numbers on heat and mass transfer flow are opposite. Moreover, the thermophoresis parameter
decreases as the concentration field increases. However, when the Brownian motion parameter is increased, the
concentration profile gets better. Additionally, several other significant factors were examined. The results of
this study can facilitate quicker and easier research and assist engineers.
Keywords: - Casson nanofluid, squeezing flow, Homotopy perturbation method (HPM), Runge–Kutta (RK-4),
magnetic field, Brownian motion, Thermophoresis.
Received: April 22, 2023. Revised: September 25, 2023. Accepted: November 21, 2023. Published: December 31, 2023.
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DOI: 10.37394/232012.2023.18.15
Amine El Harfouf, Rachid Herbazi,
Sanaa Hayani Mounir, Hassane Mes-Adi,
Abderrahim Wakif
E-ISSN: 2224-3461
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Nomenclature
Specific heat at constant pressure
Magnetic field
Thermophoretic diffusion coefficient
Hartmann number
Distance of plate
Greek Symbols
Thermophoretic parameter
Dimensionless temperature
Pressure
Density (kgm-3)
Squeeze number
Dynamic viscosity of nanofluid
Fluid temperature
Electrical conductivity of nanofluid
Nanofluid concentration
Casson fluid parameter
Brownian diffusion coefficient
Dimensionless concentration
Eckert number
Thermal conductivity
Rate of squeezing
1 Introduction
The many mechanical and organic applications of
heat and mass transfer squeezing flow, such as
polymer processing, water cooling, hydro-
mechanical gear, and chemical processing
equipment, have drawn a lot of attention to the field
in recent years. In [1], the authors reported the first
study in this field utilizing lubrication
approximation. Numerous academics examined
these flows for various scenarios and geometries. In
their investigation, considered the effects of
injection or suction at the walls when analyzing the
analytical solution for the squeezing flow of
viscous fluid between parallel disks, [2]. In the
same context, the authors have addressed the
squeezing flow across a porous surface, [3]. Also,
in [4], looked at the combined impacts of heat and
mass transfer in the squeezing flow between
parallel plates while taking chemical reaction
effects and viscous dissipation into account.
Additionally, taking thermal radiation into
consideration. On the other hand, they investigated
the MHD nanofluid flow and heat transfer in a
stretching/shrinking convergent/divergent channel,
[5]. Additionally, in [6], investigated the heat
transmission and nanofluid flow between two
parallel plates in a spinning device.
Non-Newtonian fluids are more practical to
utilize in actual industrial applications than
Newtonian fluids, such as paints, tomato sauce,
condensed milk, and shampoos. Non-Newtonian
fluids exhibit unique characteristics that defy
explanation by Newtonian theory.
The shear stress and shear rate in a Newtonian
fluid have a linear relationship that passes through
the origin, with the coefficient of viscosity serving
as the constant of proportionality. The connection
between the shear stress and the shear rate is
different in a non-Newtonian fluid, though. The
fluid can display viscosity in relation to time. As
such, it is impossible to establish a constant
viscosity coefficient. Many non-Newtonian models
on various physical characteristics have been
proposed to close this gap. One example of a non-
Newtonian fluid used to mimic blood in arteries is
the Casson fluid; its formulation may be acquired
from, [7], and Casson fluid has been the subject of
multiple investigations. In other work, [8], [9] have
investigated the Casson fluid for both constant and
oscillatory blood flow as well as the flow between
two revolving cylinders. While in [10], examined
the unstable Boundary Layer Flow of a Casson
Fluid Due to an Impulsively Started Moving Flat
Plate, they examined the constant flow of a Casson
fluid in a tube, [11].
This paper addresses the squeezing flow of a
two-dimensional incompressible Casson nanofluid
between two parallel plates under the influence of a
uniform magnetic field. It considers various factors
such as viscous dissipation effect, heat generation
due to friction caused by shear in the flow, Joule
heating, Brownian motion, and thermophoresis
effect with time-dependent chemical reaction. The
research is motivated by the above investigations.
Since the most significant engineering problems
heat transfer equations are nonlinear, some of them
may be solved numerically, while others can be
solved analytically using various techniques. To get
the optimum outcome, both approaches were
applied in this study. Researchers in this field have
used analytical techniques such as the differential
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transformation method (DTM) [12], [13]
Homotopy perturbation technique HPM [14],
adomian decomposition method ADM [15],
homotopy analysis method HAM [16], and
variation of parameter method VPM [17]. The
HPM, which He invented and refined, is an
analytical simulation technique that does not need
tiny parameters [18]. According to this approach,
the answer is found by adding up an infinite series,
which often converges quickly to the precise
answer. Thus, the fourth order Runge–Kutta
technique (RK-4) has been used as a numerical
approach and HPM as an analytical method to
solve the reduced ordinary differential equations in
this paper.
2 Mathematical Formulation of the
Problem
This work examines the heat and mass transfer in
the unstable two-dimensional Casson nanofluid
squeezing flow between two infinite parallel plates,
as seen in Figure 1. The distance between the two
plates
, where is the
initial location (when time ) for
the two
plates are squeezed until they touch
and
for
There is a separation between the two
plates. It is assumed that a constant magnetic field
is applied in the direction of . Assuming a low
magnetic Reynolds number, the induced magnetic
field may be insignificant when compared to the
applied magnetic field. As a result, the following
define the electromagnetic force and the electric
current:
=
(×
), =×
then =
(×
) ×
.
The rheological equation of state for an isotropic
flow of a Casson fluid can be expressed as, [19]:
(1)
Where is the component of stress tensor,
represent the product of deformation
components, indicates the deformation rate on
the component, indicates the critical
value. The Casson fluid's dynamic plastic viscosity
is represented by and represents the yield
stress.
The deformation rate is equal to:
(2)
For the sake of this issue, the equations controlling
the Casson nanofluid flow for mass, momentum,
thermal energy, and nanoparticle concentration are
expressed as:
(3)
(4)
(5)
(6)
(7)
The boundary conditions are:
(8)
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(9)
Where and are the velocities in the and
directions respectively, is the temperature, is
the pressure, is the electric conductivity, is the
effect thermal conductivity, is the concentration,
is the dynamic viscosity, and are the
Brownian motion coefficient and the
thermophoretic diffusion coefficient ,
is the
mean fluid temperature
is the Casson
fluid parameter and
is the time-dependent reaction rate.
The following dimensionless groups are
introduced:
(10)
Equations 3 through 7 are obtained by substituting
the parameters, and the results yield:
(11)
(12)
(13)
The boundary condition in the new similar
variables becomes:
(14)
(15)
Where is the squeeze number, is the
Hartmann number, is the Brownian motion
parameter, is the thermophoretic parameter, is
the Prandtl number,is the Schmidt number and
the chemical reaction parameter which are defined
as:
,
,=
,
,=
,
=
.=
,=
,
It is important to note that the movement of the
plates is described by squeezing number S (
is known as a "squeezing flow" and denotes that the
plates are moving apart, whereas denotes
that the plates are moving together, [20], [21].
Moreover, it should be noted that and
, corresponds to the situation in which there
is no magnetic force effect or viscous dissipation.
While represents the destructive chemical
reaction and characterizes the generative
chemical reaction, the Prandtl number is used to
assess the velocity at which momentum and energy
propagate through the nanofluid and the Schmidt
number is used to characterize fluid flows in which
momentum and mass transfer processes are
occurring simultaneously, [22].
The skin friction coefficient, Nusselt number,
and Sherwood number are physical parameters of
engineering relevance that are defined as:
(16)
(17)
(18)
Using variables (9), after the simplification we
get the skin friction coefficient and Nusselt,
Sherwood numbers:
′′
′
(19)
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=′
3 Homotopy Perturbation Technique
To illustrate the basic ideas of this method, we
consider the following equation:
(20)
With the boundary condition of:
(21)
Where is a general differential operator, a
boundary operator, a known analytical
function and
is the boundary of the domain .
can be divided into two parts which are and ,
where is linear and is nonlinear.
Eqn. (16) can, therefore, be rewritten as follows:
(22)
Homotopy perturbation structure is shown as
follows:
(23)
Were,
(24)
In Eq. (19), p belongs to [0, 1] is an embedding
parameter and is the first approximation that
satisfies the boundary condition. We can assume
that the solution to Eq. (21) can be written as a
power series in , as follows:
(25)
The best approximation for the solution is:
(26)
Applying the HPM to the problem, the equations
are rewritten as follows:
(27)
(28)
(29)
We consider f and as follows:
(30)
(31)
(32)
With substituting and from equations
(26-28) into equations (10-12) and some
simplification and rearranging based on powers of
, we have:
(33)
(34)
(35)
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Fig. 1: Geometry of the problem
(36)
The boundary conditions are:
(37)
(38)
Solving the above equations with their
corresponding boundary conditions using maple we
would obtain:
(39)
The terms , and when is too
large that is mentioned graphically. The solution of
equations is obtained when , will be as
follows:
(40)
(41)
(42)
Validation of code
To answer the problem, HPM and RK-4 are used in
the current paper. As can be seen in Table 1, the
current codes are validated by contrasting the
acquired findings with the previously published
results in the literature. This comparison shows that
the codes provide a very precise answer to this
issue. The findings show excellent agreement.
Table 1.
Ref [4]
Ref [5]
Present results
HAM
DRA
HPM
NM
-1.0
2.170090
3.319899
2.170091
3.319888
2.170092
3.319860
2.170090
3.319899
-0.5
2.614038
3.129491
2.617403
3.129491
2.617403
3.129491
2.617403
3.129491
0.01
3.007134
3.047092
3.007133
3.047091
3.007133
3.047091
3.007133
3.047091
0.5
3.336449
3.026324
3.336449
3.026323
3.336449
3.026323
3.336449
3.026323
2.0
4.167389
3.118551
4.167041
3.113386
4.168065
3.127819
4.167389
3.118550
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4 Discussion of Results
The results are interpreted in graphs and tables for
various values of relevant factors to aid in better
clarity and comprehension of the issue. Figure 2,
Figure 3, Figure 4 and Figure 5 show how
squeezing number affects velocity,
temperature, and concentration profiles. It's
important to note that all figures are depicted for
the expanding flow scenario and the
squeezing flow case . Figure 2, Figure 3,
Figure 4 and Figure 5 demonstrate that the impact
of positive and negative squeezing numbers differs:
According to Figure 2, the normal velocity profile
becomes higher for and lowers for .
This is because as plates separate, fluid is drawn
into the channel, increasing the velocity field. In a
different scenario, liquid within the channel is
released when plates move near one another. This
results in a liquid drop inside the channel and a
decrease in fluid velocity. Squeezing number,
however, depends on the flow region's velocity
field. Furthermore, as seen in Figure 3, the axial
flow velocity profile increased in the remaining
half for rising values of and dropped in the
area . Conversely, with rising
values of S < 0, it drops in the remaining section
and increases in the region . For
, the concentration profile rises, and the
temperature field continues to decrease in Figure 4
and Figure 5. In contrast, the concentration profile
decays, and the temperature field grows when
. The greater distance between the plates is the
cause of the temperature field's drop, which can be
related to a decrease in kinematic viscosity or an
increase in the speed at which the plates travel.
Similarly, Figure 6, Figure 7, Figure 8 and
Figure 9 show how the Hartman number affects
temperature, velocity, and concentration curves.
The pictures illustrate how the normal velocity
profile in Figure 6 reduces as Hartman number
grows because of the existence of Lorentz forces
acting against the flow, which causes resistance to
flow to occur and a corresponding decrease in the
velocity field. Additionally, Figure 7 shows that the
axial velocity field rises in the remaining portion
and decreases in the area .
However, Figure 8 shows that the temperature field
increases. Additionally, it can be shown in Figure 9
that the concentration increases.
Similarly, Figure 10, Figure 11, Figure 12 and
Figure 13 show how the Casson fluid parameter
affects temperature, concentration, and velocity
profiles. As the Casson fluid parameter grows,
the normal velocity drops, as seen in Figure 10.
This is because the applied stresses cause the slight
increase in to increase the viscosity of the
nanofluid, which in turn causes the normal velocity
to decrease. Furthermore, Figure 11 shows that the
axial velocity profile increases in the remaining
section while decreasing in the portion
. Additionally, the temperature field drops in
Figure 12 and the concentration rises in Figure 13.
Fig. 2:
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Fig. 3:
Fig. 4:
Figure 14 and Figure 15 show the effects of the
Brownian motion parameter and the
thermophoresis parameter on the concentration
profile. It is evident from Figure 14 that when
increases, the concentration profile diminishes.
Stronger thermophoresis forces that have emerged
in the flow zone are the cause of this condition. In
contrast, Figure 15 shows that it rises with .
Figure 16 and Figure 17 show how the time-
dependent chemical reaction parameters and
Schmidt number Sc affect the concentration profile.
Figure 16 shows that the concentration profile
is affected differently by both positive and negative
time-dependent chemical reaction parameters. The
concentration profile decreases for due to
the chemical reaction's breakdown. Conversely, it
rises when because of the chemical
reaction. In contrast, the concentration distribution
decreases as rises in Figure 17.
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Fig. 5:
Fig. 6:
Fig. 7:
The Skin Friction coefficient , Nusselt
number , and Sherwood number were found
to vary with the Squeeze number S, Casson
parameter β, and Hartman number in Table 2. It
is evident from Table 2 that as and values rise,
the skin friction coefficient drops, and when
values rise, it rises. Additionally, Tab. 3 shows that
the Nusselt number rises for but falls for
growing values of S and . Additionally, Table 2
demonstrates that the Sherwood number rises as
and values rise while falling as values rise.
Fig. 8:
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Fig. 9:
Fig. 10:
Fig. 11:
Fig. 12:
Fig. 13:
Fig. 14:
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Fig. 15:
Fig. 16:
Fig. 17:
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-1
1
2
-6.0858218460
1.0113136395
-1.0622791120
-0.5
-6.4318509673
0.8088745369
-0.8655341258
0.0
-6.7579586889
0.6703420350
-0.7247119376
0.5
-7.0665588433
0.5707471689
-0.6398027050
1.0
-7.3592614969
0.4851209735
-0.6107965858
0.5
0.5
-10.094364079
0.7422135661
-0.8106079572
1.0
-7.0665588433
0.5707471689
-0.6398027050
1.5
-6.0507360057
0.5130723151
-0.5845460590
1
0.0
-6.3462955265
0.3484822973
-0.3954251196
0.5
-6.3934787417
0.3626192293
-0.4099299356
1.0
-6.5332004275
0.4048828001
-0.4540328386
5 Conclusion
In the presence of a uniform magnetic field, this
study investigates the impact of thermophoresis and
Brownian motion on the heat and mass transfer of a
two-phase, unstable squeezing flow of MHD
Casson nanofluid with time-dependent chemical
reaction. The governing equations are solved using
HPM, and the correctness of our solutions is
confirmed using a shooting method and the fourth
order Runge-Kutta approach. Therefore, the
previously published study and the analytical and
numerical results agreed in a fascinating way, as
expected. Furthermore, the impacts of additional
significant variables have been examined and
displayed in tables and graphs, accordingly. The
current study has led to the following significant
findings:
When , the velocity field is
suppressed; for , it is augmented.
When , the temperature profile
falls, and when , it rises. On the
other hand, the concentration field shows
the opposite tendency.
As β increases, the velocity field decreases.
Additionally, when β grew, the temperature
profile decreased, and the concentration
profile rose.
Concentration and velocity fields get
smaller when the values of. In contrast,
the temperature profile rises with the
values of.
As the thermophoresis parameter increases,
the concentration field shrinks.
As the Brownian motion parameter
increases, the concentration profile rises.
Concentration field ultimately becomes
strengthened for and and
eventually muted for .
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Abderrahim Wakif
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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
n_US
WSEAS TRANSACTIONS on HEAT and MASS TRANSFER
DOI: 10.37394/232012.2023.18.15
Amine El Harfouf, Rachid Herbazi,
Sanaa Hayani Mounir, Hassane Mes-Adi,
Abderrahim Wakif
E-ISSN: 2224-3461
194
Volume 18, 2023
