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European Journal of Applied Sciences – Vol. 12, No. 5
Publication Date: October 25, 2024
DOI:10.14738/aivp.125.17580.
Islam, M. R. & Nasrin, S. (2024). Hall and Ion-Slip Current Effects on Micropolar Fluid Flow over a Vertical Plate with an Inclined
Magnetic Field. European Journal of Applied Sciences, Vol - 12(5). 169-187.
Services for Science and Education – United Kingdom
Hall and Ion-Slip Current Effects on Micropolar Fluid Flow over a
Vertical Plate with an Inclined Magnetic Field
Md. Rafiqul Islam
Department of Mathematics,
Bangabandhu Sheikh Mujibur Rahman Science and
Technology University, Gopalganj-8100, Bangladesh
Sonia Nasrin
Department of Mathematics,
Jagannath University, Dhaka-1100, Bangladesh
ABSTRACT
In this study, we explore how Hall and Ion-slip currents influence the flow of
micropolar fluid over a vertical plate in the presence of an inclined magnetic field.
To simplify the analysis, we consider a small magnetic Reynolds number, allowing
us to exclude the magnetic induction equation. The equations governing the flow
are derived from principles of linear momentum equation, angular momentum
equation, and energy equation which are made dimensionless through similarity
analysis. Then these non-linear, dimensionless equations are solved by using the
explicit finite difference method for finding the primary velocity, secondary
velocity, microrotation, and temperature. This investigation extensively examines
the impact of key parameters on velocity, microrotation, and temperature
distributions. It also provided concise graphical explanations of skin friction and
heat transfer rate at the plate.
Keywords: Micropolar Fluid, Heat Transfer, Inclined Magnetic Field, Hall Current, Ion- slip.
INTRODUCTION
Micropolar fluids are characterized by their micro-structured composition, consisting of rigid
spherical particles endowed with spin inertia, dispersed within a viscous medium. These
fluids deviate from the behavior of traditional Newtonian fluids due to the presence of
suspended particles. Examples of micropolar fluids abound in various non-Newtonian fluid
systems such as blood flow, fluids traversing neural networks and the brain, polymer
solutions, and liquid crystals, all of which exhibit inherent polarities. Micropolar fluid
dynamics finds applications in modeling phenomena involving the presence of dust or
particulate matter in gases, especially in environments where traditional fluid dynamics fail to
capture the complexities. This framework enables the description of physical phenomena at
micro and nanoscales, providing an additional degree of freedom for rotational motion.
Understanding the influence of magnetic fields on micropolar fluid flows, particularly in the
context of Hall and Ion-slip currents, holds significant practical and theoretical importance
across diverse fields including magnetic material processing, astrophysics, nuclear
engineering, geophysics, and industrial processes.
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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 5, October-2024
Eringen [1,2] first proposed the general theory, which illustrates specific microscopic effects
arising from the microstructure and micro motions of flow are micropolar fluids that exhibit
the microrotational effects and micro-rotational inertia so that clench body couples and
couple stress. Whereas the Navier-Stokes theory cannot express the phenomena at micro and
nanoscales, but MFD can explain. After that, many authors investigated and developed the
MHD micropolar fluid flow with or without hall current and ion slip effect. The extensive
developments on micropolar fluids, their behavior, and applications in engineering and
technology are studied by Ariman et al. [3,4]. Rees and Pop [5] discussed the characteristics
and nature of a micropolar fluid flow that is set in a vertical plate with steady free convection.
Hsu and Wang [6] represented the laminar micropolar fluids flow with mixed convection in a
square dent with an induced stream influenced by temperature. Olajuwon et al. [7] have
analyzed the unsteady viscoelastic micropolar fluid flow upon an infinite moving plate in a
porous medium in a magnetic field with Hall effect and thermal radiation. Zakaria [8]
explained the nature of an electrically conducting micropolar fluid flows with the presence of
a transverse magnetic field over a porous medium in one-dimensional and used the Laplace
transformation with e –algorithm technique to find its key with the Laplace transformation
domain numerically. Nadeem et al. [9] are obtained the effect of heat and mass transfer with
thermal radiation on magneto-hydrodynamic micropolar fluids flow over an oscillating plate
set in a porous media. Seddeek and Abdelmeguid [10] have been explained the mixed
convection boundary layer flow of viscous, incompressible, and electrically conducting
micropolar fluids with Hall current and Ion-slip effect along with a horizontal plate. Gorla [11,
12] analyzed the mixed convection of a micropolar fluid from a semi-infinite vertical plate
with the heat transfer rate and showed the comparison of Newtonian and micropolar fluids.
Jha and Malgwi [13] studied the free convection viscous incompressible fluid flow within a
vertical microchannel with an induced magnetic field with hall current and ion-slip effect.
Ram [14] considered a free convective heat-generating on a rarefied gas in a rotating frame
with a strong magnetic field attributed perpendicular to the plate with the effects of Hall and
ion-slip current that has various industrial applications, tangential filtration, crystallization,
turbo machinery, petroleum industry, bio-reaction, and liquid-liquid extraction. Uddin and
Kumar [15] investigated the micropolar fluids flow over a non-conducting wedge with hall
and ion-slip effects with a strong magnetic field. Singh [16] expressed the joint effect of Joule
heating and thermal diffusion on MHD free convection viscous fluid flow with the Hall current
of an electrically conducting fluid. The influence of the Hall current effect in a viscous and
conducting fluid with the generalized Couette flow under an inclined magnetic field is
analyzed by Prasada et al. [17]. Seth et al. [18] investigated a steady, incompressible, and
rotational Couette flow along an inclined magnetic field. It has significant use in geophysics,
agriculture, astrophysics, and chemical, mechanical and fluid engineering. Krishna et al. [19]
investigated the angle of inclination of an unsteady magneto-hydrodynamic free convective
flow with hall and ion slip current effects in a rotational system through an accelerated
inclined plate entrenched in a porous medium. Opanuga et al. [20] considered Hall current
and ion-slip effects on the rate of entropy generation of couple stress fluid with velocity slip
and temperature wall. Hanvey et al. [21] examined the flow of a viscoelastic, electrically
conducting fluid between two parallel plates filled with a porous medium, positioned in an
inclined magnetic field. Initially, the flow is driven by a pressure gradient parallel to the
bounding fluid. However, once the system reaches a steady state, the pressure gradient
diminishes, and a magnetic field inclined simultaneously is applied to ascertain the velocity
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Islam, M. R. & Nasrin, S. (2024). Hall and Ion-Slip Current Effects on Micropolar Fluid Flow over a Vertical Plate with an Inclined Magnetic Field.
European Journal of Applied Sciences, Vol - 12(5). 169-187.
URL: http://dx.doi.org/10.14738/aivp.125.17580
profile of the fluid. Javeri [22] investigated the associated impact of Hall and ion slip effect on
MHD laminar viscous fluid flow with heat transfer. Beg et al. [23] developed a numerical
solution of an incompressible magneto-hydrodynamic fluid flow past in a channel between
two infinite parallel plates with rotation, inclined magnetic field, and hall currents. Singh et al.
[24] examined the inclined magnetic field effect with variable temperature on unsteady
viscous, incompressible fluid flow past through a vertical movable plate. Sharma et al. [25]
expressed the incompressible unsteady viscous mixed convective fluid flow along with an
infinite porous plate with the Hall Effect and heat source (or sink).
Inspired by the studies outlined earlier, numerical solutions were pursued for the unsteady
micropolar fluid flow over a vertical plate influenced by an inclined magnetic field, with
consideration given to Hall and Ion-slip currents. The findings have undergone
comprehensive discussion, elucidating different facets of the flow dynamics. Furthermore,
detailed explanations and graphical representations are provided for crucial parameters like
shear stress, Nusselt number, and Sherwood number, facilitating a deeper understanding of
the obtained results.
MATHEMATICAL FORMULATION
Consideration is given to the unsteady two-dimensional flow of a viscous, electrically
conducting micropolar fluid in the direction of the -
+
x
axis, oriented vertically upward, while
the -
+
y
axis is perpendicular to the -
+ +
x z
plane. An electrically non-conducting plate is placed
parallel to the -
+
x
axis. Initially, at time
+
0 t
, it is assumed that the fluid velocity,
microrotation, and temperature of the fluid are respectively.
= 0,
+
u = 0,
+
w 0, 1 =
+
0, 3 =
+ +
+
T = T
At the time
0
+
t
, the plate moves in its plane with a velocity
u0
along x-axis, at the plate the
microrotation is defined as
+
+
+
=
y
u
1
and
+
+
+
=
y
w
3
Additionally, the temperatures of the plate increase from
+
T
w
to
+
T
with time t
+. Here, the
constant η signifies the boundary parameter on microrotation lying in the range 0≤η≤1. When
η=0, it corresponds to the no-spin condition, indicating that the fluid particles exhibit strong
concentration and the microelements near the plate are unable to rotate. For η=0.5, it denotes
weak concentration, signifying the vanishing of the anti-symmetric part of the stress tensor.
Another scenario, η=1, leads to the modeling of turbulent boundary layer flows. The scope of
present analysis is limited to 0≤η≤0.5.
The assumption is made that the magnetic field Bo is applied in a direction forming an angle ψ
with the normal to the plate. Consequently, the induced magnetic field is defined as