Numerical Analysis on Stagnation Point Flow of Micropolar Nanofluid
with Thermal Radiations over an Exponentially Stretching Surface
FERAS M. AL FAQIH1, KHURAM RAFIQUE2, SEHAR ASLAM2, MOHAMMED Z. SWALMEH3
1Department of Mathematics,
Al-Hussein Bin Talal University,
Ma’an 71111,
JORDAN
2Department of Mathematics,
University of Sialkot,
Sialkot 51310,
PAKISTAN
3Faculty of Arts and Sciences,
Aqaba University of Technology,
Aqaba 77110,
JORDAN
Abstract: - Several industrial developments such as polymer extrusion in metal spinning and continuous metal
casting include energy transmission and flow over a stretchy surface. In this paper, the stagnation point flow of
micropolar nanofluid over a slanted surface is presenting also considering the influence of thermal radiations.
Buongiorno’s nanoliquid model is deployed to recover the thermophoretic effects. By using similarity
transformations, the governing boundary layer equations are transformed into ordinary differential equations.
The Keller-box approach is used to solve transformed equations numerically. The numerical outcomes are
presented in tabular and graphical form. A comparison of the outcomes attained with previously published results
is done after providing the entire formulation of the Keller-Box approach for the flow problem under
consideration. It has been found that the reduced sherwood number grows for increasing values of radiation
parameter while, reduced Nusselt number and skin friction coefficient decreases. Furthermore, the skin-friction
coefficient increases as the inclination factor increases, but Nusselt and Sherwood's numbers decline.
Keywords: - Micropolar nanofluid, Stretching sheet, Stagnation point flow, Thermal Radiations, Keller-Box
Technique.
Received: January 26, 2023. Revised: November 15, 2023. Accepted: December 14, 2023. Published: February 1, 2024.
1 Introduction
The concept of nanofluids is not new, [1], initially
proposed it when they were looking for new coolants
and cooling technologies. It quickly gained popularity
due to its many uses in nuclear reactor systems, heat
exchangers, electronic cooling, boilers, and energy
storage devices. A fluid called a nanofluid contains
microscopic quantities of nanoparticles or nanofibers,
which are particles with a diameter of less than 100
nm. Emerging heat transfer base fluids including
water, ethylene glycol, toluene, and motor oil are
mixed with nanoparticles or nanofibers to create
nanofluids. [2], investigated the base fluid's thermal
conductivity may be increased by the inclusion of
nanoparticles, which is expected to increase the free
convection heat transfer of the nanofluid relative to the
base fluid. The capability to conduct heat is increased
by up to when added nanoparticles to a base
liquid, with a volume proportion of 5%, according to a
study by [3]. Additionally, they claimed that adding a
1% volume fraction of copper nanoparticles to the
regular fluid boosted heat conductivity by up to 40%.
In his research, [4], identified seven pathways that are
crucial for improving the base fluid’s thermal
conductivity. The two most important of these
processes are thermophoresis and Brownian motion.
He concluded that due to the impact of the temperature
differential and thermophoresis, the boundary layer's
nanofluid characteristics may differ dramatically.
These effects can significantly reduce the viscosity
inside the boundary layer for a heated fluid, improving
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Feras M. Al Faqih, Khuram Rafique,
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heat transmission. [5], examined the key elements that
might improve thermal conductivity. It has been
shown that several variables, including particle size
and shape, base fluids, fluid PH, temperature,
surfactant type, hydrogen bonding, solvent type, and
others, directly affect how well nanofluids transmit
heat. On the other hand, viscosity is one of the other
elements that affects heat conductivity in an "indirect"
manner.
A stretched sheet's boundary layer flow has
various technical applications, including skin friction,
grain storage, paper manufacturing, and drag
reduction. The initial study was done by [6], on the
boundary layer flow with unchanged velocity over a
continuous solid surface. Further, [7], studied and
published closed-form solutions for the boundary layer
flow of a viscous fluid through a stretched surface. [8],
investigated the impact of nonlinear radiation and heat
generation or absorption on the flow of nanofluid at its
stagnation point along a moving surface. Coupled
nonlinear PDEs were converted into nonlinear coupled
ODEs using similarity transformations, and a finite
difference method was then used to derive the
unknown functions for velocity, temperature, and
nanoparticle concentration. [9], talks about the
microorganism-containing nanofluid's stagnation
point flow. By using suitable transformation, the
system of pde's was transformed into a system of ode's,
and the resulting equations were then solved
numerically using the bvp4c MATLAB tool. [10],
researched micropolar nanofluid stagnation point flow
for slanted surfaces.
Micropolar fluids are fluids made up of randomly
oriented and rigid particles suspended in a viscous
medium with micro-structure components. To
investigate the impact of micro rotations on fluid
motion, [11], [12], established the theory of micropolar
fluids. [13], used a fluid model to undertake numerical
analysis on incompressible, time-dependent
electrically-conducting squeezing flow/micropolar
fluid. Slip parameters were found to down the value of
the Nusselt and Sherwood numbers on both discs. [14],
used a computational model to investigate the mass
and energy transport behavior of micro-rotational flow
through a Riga-plate, taking into account suction or
injection as well as mixed convection. [15]
,investigated hydromagnetic micropolar nanofluid
flow via a nonlinear stretchy sheet and generated
entropy usingNavier slips. The results demonstrated
that when the Brownian motion was increased, the
momentum boundary layer improved while, the
concentration distribution decreased. Results of
several recent researches on micropolar nanofluid flow
are presented in the studies, [16], [17], [18], [19], [20],
[21], [22], [23], [24], [25].
Nanotechnology has gained a prominent place in
the current era research area due to which nanoliquid
becomes a very important liquid that trigger the
thermal efficiency of base liquids. The higher thermal
efficiency ability is very helpful in energy
transportation. The literature previously mentioned
inspired us to explore the stagnation point flow of the
micro-polar nanofluid towards a sloping surface with
Brownian motion and the Thermophoretic impacts
However, no researcher has to date taken into account
the slanted surface for the stagnation point flow of
micro-rotational nanofluid by incorporating the
considered effects, using the Keller box method. We
subsequently carried out this analysis to fill this gap in
the literature. There are different methods that can
utilize to find the numerical and graphical results of the
current research but the Keller box technique is easier
for simulation and to prepare the Matlab program.
Further this method is very friendly to use and gives
more accurate results. Moreover, Figure 1 presents the
physical Model and Coordinate System of the utilized
study.
Fig. 1: Physical Model and Coordinate System
2 Problem Formulation
In this study micro-rotation of the incompressible
nanoliquid flow is considered. The rotational effects of
micropolar liquid along with nanoparticles are
investigated. We examine a micropolar nanofluid's
two-dimensional stagnation-point flow towards a
slanted exponentially stretchable sheet. Additionally,
(magnetic field) along with inclination
effects are under examination. In this problem, the
stretching0and free stream velocitie’s are taken as
and
where,
and both are constants and is the axis measured
along0the stretching surface. Along with
nanoparticles, the base fluid also contains rotating
micropolar finite-sized particles. At the wall, and
remain constant and are denoted by the letters and
, where stands for temperature and for the
nanoparticle fraction.
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The equations that govern boundary layers given the
reference, [26], [27], [28], are,
, (1)
, (2)
, (3)
, (4)
. (5)
The Rosseland approximation is used to simplify the
equation (4), which reduces the radiative heat flux to:
, (6)
Where, respectively, σ*aand k* represents the
Stefan-Boltzmannmconstant and the mean absorption
coefficient. Ignoring higher-order terms, extending
about T∞ , T4 in a Taylor series results in:
,
Thus, simplified form of equation (4) is:
,
(7)
Where u and. v both are velocity components in x
and0y directions resp.,, depicts the electrical
conductivity, represents basenfluid’s density,
denotes viscosity,, shows spin gradient viscosity,
depicts vertex viscosity,, denotes micro-inertia
per unit mass, the thermal diffusivity parameter is
where, ,is the heat capacity of base fluid
and is. known as thermal conductivity, represents
radiation parameter,
is the relation between
heat capacity of nanoparticles and of liquids,
furthermore, stands for thermophoresis diffusion
coefficient and stands for Brownian motion.
Boundary conditions that are imposed in view of
[27], [28], are listed below.
,
. (8)
When, , it is implied that 0at the wall,
which stands for concentrated outline flow and deny
the rotation of micro-elements along the surface of
wall. In order to convert the nonlinear PDE’s into
nonlinear ODE’s, similarity transformations are
defined. The stream function σ is defined
as follows for this use:
. (9)
The exponentially stretching sheet velocity is used to
define the similarity transformations given, [27], [28],
as follows:
,
. (10)
Where,
.
(1
0)
Equations,(2, 3, 5 and 7) are converted to the
following nonlinear ODE’s when Eq. (10) is
substituted:
, (11)
, (12)
,
(13)
, (14)
Where,
, (15)
Here, prime denotes dervatives w.r.t ,
represents magnetic .parameter,.the Prandtl number.is
depicted by , K depicts dimensionless vortex
viscosity, velocity ratio parameter, is radiation
parameter, is kinematic viscosity of the fluid, the
Lewis number is. , =
where, .is
thermophoresis parameter and is .Brownian
motion parameter, .is buoyancy parameter and .is
local Grashof number, is solutal buoyancy
parameter and is local Grashof number. The
imposed boundary conditions (8) are transformed to
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,
(16)
The Sherwood number , the coefficient of
skin friction and the Nusselt number are
defined as:
. (17)
Where,
The relations of the coefficient of skin friction
the reduced0Sherwood.
number and the reduced0Nusselt number.
are defined as:
.
(18)
Here, .the local Reynolds number is
.
3 Results and Discussion
The Keller-box approach is used to solve the
transformed nonlinear ODE’s (11–14) that are
subjected to BC’s (16). The results for the relevant
physical parameters, such as , ,
, and are presented in tabular form by
using tables 1 and 2. When , , Nb, and
are equal to zero and . Table 1 compares the
current findings for the reduced Nusselt number
to the findings from, [26] and [27]. Here, a
good consensus can be seen. In this work Keller box
technique utilized. Since last few years, this
numerical technique is very effective and
accomplished of making correct numerical results
of the flow problems by being categorically stable
up to second-order convergence. This technique is
very friendly in use and coding, easy to program,
and give unconditional convergence in reasonable
time at second order.
Table 1. Comparison of when
Bidin and Nazar [26]
Ishak [27]
Present results
1.0
0
0
0.9548
0.9548
0.9548
2.0
0
0
1.4714
1.4714
1.4714
3.0
0
0
1.8691
1.8691
1.8691
1.0
0
1.0
0.5312
0.5312
0.5312
1.0
1.0
0
--
0.8611
0.8611
1.0
1.0
1.0
--
0.4505
0.4505
Table 2. Values of , and .
0.1
0.1
6.5
5.0
0.1
1.0
1.0
0.1
0.1
0.5
0.8984
2.5492
1.1925
0.3
0.1
6.5
5.0
0.1
1.0
1.0
0.1
0.1
0.5
0.6010
2.6985
1.1930
0.1
0.3
6.5
5.0
0.1
1.0
1.0
0.1
0.1
0.5
0.7167
2.6100
1.1859
0.1
0.1
9.0
5.0
0.1
1.0
1.0
0.1
0.1
0.5
0.9902
2.5505
1.1949
0.1
0.1
6.5
9.0
0.1
1.0
1.0
0.1
0.1
0.5
0.8701
3.8017
1.1979
0.1
0.1
6.5
5.0
0.5
1.0
1.0
0.1
0.1
0.5
0.8943
2.5407
1.2716
0.1
0.1
6.5
5.0
0.1
3.0
1.0
0.1
0.1
0.5
0.9188
2.5914
1.6193
0.1
0.1
6.5
5.0
0.1
1.0
3.0
0.1
0.1
0.5
0.6908
2.5851
1.1879
0.1
0.1
6.5
5.0
0.1
1.0
1.0
0.5
0.1
0.5
0.9030
2.5591
1.0894
0.1
0.1
6.5
5.0
0.1
1.0
1.0
0.1
1.0
0.5
0.9051
2.5638
1.0230
0.1
0.1
6.5
5.0
0.1
1.0
1.0
0.1
0.1
1.5
1.0615
2.8750
-1.7549
0.1
0.1
6.5
5.0
0.1
1.0
1.0
0.1
0.1
0.5
0.8978
2.5480
1.2057
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To illustrate how , , and vary
for different values of , , , , , , , , ,
and , Table 2 is constructed. It has been found that
when, , , , and are increased,
lowers, whereas , and are increased,
grows. The table, however, clearly demonstrates that
is decreasing while rising , and M. While,
rising with higher values of , , , , ,
and . Additionally, it has been discovered that
decreases as , , and increases when
, , , and increases values rise. The
negative values of signify a drag force being
applied to the motions of the micropolar nanofluid by
the stretching sheets. This is not unexpected
considering that stretching is the only factor
responsible for the boundary layer's development. It
can be seen from this table that the increasing value of
gives higher values of . Logically the
inclination factor improves the skin friction. Further it
is seen clearly from Table 2 the increment in the
magnetif effect the energy and mass transmission rates
diminishes. Physically the magnetic causes reduction
in the speed of the liquid. Moreover, the effect of the
magnetic field on the velocity outline for
and is depicted in Figure 2. It demonstrates that
as the strength of the magnetic field is increased,
decreases for and increases for . This
figure matched with already published research work
of Alkasasbeh [29] which validatesur current utilized
numerical method. Additionally, as seen in Figure 3,
gets better with the growth of for both
and . This occurs because a boundary layer
forms in the flow when , or the free stream
velocity, exceeds the stretchable velocity. Physically,
the fluid motion increases as it approaches the point of
stagnation, which increases the external stream's
acceleration. In turn, as rises, the boundary layer
thickness decreases. On the other hand, a reversed
boundary layer develops when the velocity of the
stream is lower than that of the stretching, or .
However, both velocities are identical and there is
no boundary layer when . Figure 4 depicts the
behaviour of the temperature profile in relation to the
radiation parameter (N). As the radiation parameter
increases, the temperature profile rises, which causes
the flow field to produce heat and raise the temperature
of the thermal boundary layer. Figure 5 represents the
effects of Brownian motion parameter on term. Profile
for and . The temperature profile rises in
response to rising values of Nb. Figure 6 compares and
shows the effects of thermophoresis on the
temperature profile against and . The
thermophoresis effect demonstrates a direct interaction
with the temperature field. Logically the increase in
Brownian motion factor enhances the movement of
particles which cacauseshe growth of temperature.
Figure 7 shows how a change in prandtl number results
in a drop in temperature and a corresponding reduction
in boundary layer thickness. Figure 8 describes the
thermophoretic effect on for and .
From the sketch, it is clear that the concentration is
reduced for changed values of A decrease.in
boundary layer thickness caused by an increase.in
against and results in a decrease.in the
concentration profile (Figure 9). Figure 10
demonstrates how the concentration.outline drops off
as increases. It is due to the decrease in boundary
stream viscosity against increment in .
Fig. 2: Variation of ) via Magnetic parameter
Fig. 3: Variation of ) via Velocity ratio parameter
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Fig.. 4: Variation of via Radiation parameter
Fig. 5: Variation of via Brownian motion
parameter
Fig. 6: Variation of via Thermophoresis
parameter
Fig. 7: Variation of via Prandtl number
Fig. 8: Variation of via Thermophoresis
parameter
Fig. 9: Variation of via Brownian motion
parameter
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Fig. 10: Variation of via Lewis number
4 Conclusions
The investigation of the stagnation point<flow of a
micropolar nanofluid>toward an inclined exponential
stretching surface has<been examined in this article.
The impacts of Brownian motion and thermophoresis
are incorporated in the flow field. Moreover, the
radiation impact has been considered in this
investigation. The micro-rotational effects on the
nanoliquid discussed. Additionally, the stagnation
flow by incorporating magnetic factor has been
discussed numerically with Keller box scheme. The
following are the key findings of this study.
when and are increased,
lowers, whereas , and are increased,
grows.
Radiation parameter diminishes the energy
transportation with growing values.
is decreasing while rising and .
decreases as, and increases,
increases for increasing values of while
and decreases.
increases as the inclination factor ( )
increases.
and enhances by the growth in
inclination. In the future, this work can be extended for
different geometries, for instance, disk, cylinder,
sphere, etc.
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Feras M. Al Faqih, Khuram Rafique,
Sehar Aslam, Mohammed Z. Swalmeh
E-ISSN: 2224-347X
47
Volume 19, 2024
Contribution of Individual Authors to the Creation
of a Scientific Article (Ghostwriting Policy)
- Feras Al Faqih, and Mohammed Swalmeh carried
out the simulation and the optimization and
implemented the Algorithm in MATLAB.
- Khuram Rafique, and Sehar Aslam have organized
and executed the experiments and they prepared the
final proof of the manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The corresponding author would like to acknowledge
Al Hussein Bin Talal University for the financial
support.
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.en_
US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2024.19.4
Feras M. Al Faqih, Khuram Rafique,
Sehar Aslam, Mohammed Z. Swalmeh
E-ISSN: 2224-347X
48
Volume 19, 2024
