Micropolar Fluid Flow Through a Porous Stretching/Shrinking Sheet
with Mass Transpiration: An Analytical Approach
RISHU GARG1,3, JITENDER SINGH1,3, U. S. MAHABALESHWAR2,3,
OKHUNJON SAYFIDINOV3, G. BOGNAR3*
1Department of Mathematics,
Guru Nanak Dev University,
Amritsar, Punjab, 143005,
INDIA
2Department of Studies in Mathematics,
Davangere University,
Shivagangothri, Davangere, 577 007,
INDIA
3Institute of Machine and Product Design,
Faculty of Mechanical Engineering and Informatics,
University of Miskolc,
Miskolc-Egyetemváros, H-3515 Miskolc,
HUNGARY
*Corresponding Author
Abstract: - In this paper, the flow of a micropolar fluid over a stretching or shrinking sheet is investigated under
magnetohydrodynamic (MHD) conditions. Such a flow is described by highly nonlinear PDEs. Using the
similarity transformation technique, the PDEs governing the flow are reduced to a system of nonlinear ODEs,
which further allows a closed-form analytical solution. The effect of the microrotation on the skin friction
coefficient, the dimensionless forms of the velocity, and the temperature flow fields in the neighborhood of the
stretching or shrinking sheet are discussed for various combinations of the dimensionless parameters. The
numerical results reveal that the micropolar flow may accelerate or deaccelerate depending upon the numerical
values of the mass transpiration and the permeability of the porous sheet. An increase in the tangential and the
angular flow velocities is found to occur with an increase in the microrotation. Further, it is observed that the
increase in the microrotation increases the skin friction coefficient.
Key-Words: - Micropolar fluid, porous medium, magnetohydrodynamics (MHD), dual solutions, stretching
sheet, shrinking sheet
Received: October 11, 2022. Revised: May 14, 2023. Accepted: June 15, 2022. Published: July 18, 2023.
1 Introduction
Dynamics of boundary layer flow due to stretching
or shrinking sheets has been the subject of active
research for decades since these boundary layer
flows have witnessed practical applications such as
in the drawing of plastic sheets, films, wires,
entropy generation, etc., [1]. The stretching sheet
problem was first discussed by, [2], [3]. In, [4], the
author extended recent work for Newtonian fluids
varying from the slit. Inspired by this research,
many investigations have been done concerning the
flow and heat transfer problems under various
physical conditions. In, [5], the authors have
analyzed the combined effect of the heat source or
heat sink parameter and the stress for both viscous
as well as inviscid fluids along with the MHD
conditions and chemical reaction parameters. In the
presence of a porous medium, analytical solutions
for the boundary layer flow for a variety of
boundary conditions have been obtained by, [6]. In,
[7], the authors discussed mass transpiration in
nonlinear MHD boundary layer flow due to a porous
stretching sheet. In, [8], the author investigated heat
transfer enhancing features with the non-Fourier
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Cattaneo Christov model. In, [9], the author
investigated the electro-osmotic flow of a third-
grade fluid in a micro-channel in the presence of
MHD. In, [10], the authors investigated the
Cattaneo-Christov double diffusion and radiative
heat flux in the bio-convective flow of Maxwell
liquid. In, [11], [12], the authors investigated the
stretching sheet problems for nanofluid boundary
layer flows. In, [13], the authors analyzed the
chemically reactive aspects of the flow of tangent
hyperbolic material. Later, in the study, [14], the
authors investigated the flow of a second-grade
nanofluid in view of optimizing entropy generation.
A micropolar fluid is a fluid with microstructure.
Such fluids belong to the class of fluids having
asymmetric stress tensors. In this paper, we shall
call them polar fluids. So, polar fluids include as a
special case, the well-known classical fluids. In
practice, a micropolar fluid is a suspension of rigid,
randomly oriented (or spherical) particles in a
viscous carrier, where the deformation of the
suspended particles is assumed to be negligible.
The present research focuses on the flow problems
related to viscous as well as inviscid fluids. Some
researchers have shown interest to analyze
stretching sheet problems in micropolar fluids. In,
[15], the author is credited to have initiated the
theoretical model of micro fluids. Eringen’s theory
of micropolar fluids played a central role in the
early analysis of the practical features of numerous
complex flows. Micropolar fluids are a type of
microfluid, and that has been studied by, [16].
Further, [17], [18], have analyzed several flow
problems considering micropolar fluids.
In the flow equation, micropolar fluids have both a
vector of classical velocity as well as a vector of
microrotation. The mass and momentum connection
illustrates the effect of the couple stress, the spin-
inertia, and the microrotation on some
characteristics of the fluid under consideration, [19],
[20]. Several researchers have investigated the flow
of micropolar fluids in the presence of MHD and
porous medium. In, [21], the authors investigated
the flow of a micropolar fluid due to a stretching
sheet while taking into account the impact of the
temperature-dependent viscosity and the variable
surface temperature. Micropolar fluids and heat
transmission caused by porous shrinking sheets have
been studied, [22]. In, [23], the author used an
analytical approach to study the effect of a
micropolar fluid over a linear stretching sheet.
The micropolar fluid model as introduced by
Eringen elegantly describes the dynamics of such
fluids. Eringen’s model is a generalization of the
existing Navier-Stokes model and has a much wider
range of applicability than the classical one, both in
theory as well as in practice. Further, the micropolar
fluid model is simple to apply, which makes it
interesting and suitable for use by researchers.
Finding closed-form solutions for boundary layer
flows regarding stretching or shrinking sheet
problems is another challenge. In view of this, in,
[24], [25], [26], the authors have obtained analytical
solutions for some of the related flow problems.
The most popular technique in dealing with closed-
form solutions of the boundary layer equations is to
apply the similarity transformations. The present
work is motivated by the earlier research, [27], [28],
and it focuses on investigating micropolar fluid flow
due to stretching, or shrinking sheet under mass
transpiration. Using the well-known method of
similarity transformations, the PDEs governing the
underlying boundary layer flow are transformed into
a system of ODEs, which along with the appropriate
boundary conditions lead to a two-point boundary
value problem in ODEs. The resulting boundary
value problem is solved analytically to obtain
closed-form solutions. The closed-form solution can
be unique, or exhibit dual behavior. The dual
behavior is expected for the case of the shrinking
sheet, which is also investigated numerically. The
present and stated fluid, micropolar fluid flow, has
several applications, particularly in the study of
rheological complex fluids, such as colloidal fluids,
polymeric suspension, liquid crystals, animal blood,
etc. Micropolar fluid flows have practical
applications in lubrication and flow through porous
media.
2 Nomenclature
Variable
Description
SI Units
a
Constant
C
Permeability of
porous medium
N
Microrotation
j
Microrotation per
unit mass
S
Suction/injection
parameter
K
Microrotation
parameter
Dimensionless
transverse velocity
Dimensionless
Tangential velocity
Flow velocity in
Cartesian
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coordinates
Stefan blowing
velocity
Stretching/shrinking
parameter
Solution domain
Porous medium
parameter
m
Vortex viscosity
Kinematic viscosity
Spin gradient
viscosity
(ms)
Fluid density
w
Wall condition
B.Cs
Boundary conditions
MHD
Magnetohydrodyna
mics
ODEs
Ordinary differential
equations
PDEs
Partial differential
equations
3 Problem Formulation
We consider a steady, incompressible two-
dimensional boundary layer flow of a micropolar
fluid through a porous medium. The Cartesian
coordinates and are taken along the surface and
are normal to it with and as respective velocity
components. The governing boundary layer
equations are
(1)
(2)
(3)
where is the fluid density, is the kinematic
viscosity, is the microrotation or angular velocity,
is the microinertia per unit mass,
and are the spin gradient viscosity
and the vortex viscosity, respectively.
3.1 Boundary Conditions
The boundary conditions for the proposed model are
the following:
at ;
as (4)
at
as (5)
where is the surface mass transfer velocity with
for the case of suction and > 0 for the
case of injection. Here, denotes the microrotation
or angular velocity. The boundary parameter n in
Eq. (5) satisfies . Here n = 0
corresponds to the situation when microelements at
the stretching sheet are unable to rotate and denotes
weak concentrations of the microelements at the
sheet. The case n = ½ corresponds to the vanishing
of the anti-symmetric part of the stress tensor and it
shows weak concentration of microelements.
Finally, the case n = 1 is for turbulent boundary
layer flows.
3.2 Similarity Transformations
In order to transform the governing PDEs into a
system of non-linear ODEs, we introduce the
following dimensionless and similarity variables for
Eqs. (2) and (3):
(6)
where is constant. Using Eq. (6) in Eqs. (1)-(3),
we get the following ODEs:
(7)
(8)
where the primes denote differentiation with respect
to , is the microrotation parameter, and is the
porous medium parameter. These parameters and
dimensionless numbers are defined as follows:
,
. (9)
The transformed boundary conditions (4)-(5)
become
(10)
(11)
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where is the suction/injection parameter, and is
the stretching/shrinking parameter.
The local skin friction coefficient is given by
=
/ (ρ
. (12)
In the dimensionless form, the local skin friction
coefficient can be expressed as
R = [1 + (1-n)K] , (13)
where R = a/ ν denotes the local Reynolds
number.
4 Analytical Solution
For weak concentration, that is, for
Eqs.
(7)-(8) along with boundary conditions (10)-(11)
have the exact solution of the form
(14)
Substituting Eq. (14) in Eq. (7), we get a quadratic
equation for :
. (15)
Solving Eq. (15) we get
(16)
Thus, the closed-form solutions of Eq. (7)-(8)
subject to the boundary conditions (10) -(11) are
given by
,
(17)
. (18)
The velocity profile is determined after
differentiating Eq. (17) once, and we have
. (19)
The skin friction coefficient in closed form is given
by
= (1 + K/2). (20)
5 Results and Discussion
Figure 1a shows the solution domain in the (S,)-
plane for a fixed parametric value of =0.2. The
solid and dotted lines in the Figure 1a correspond to
K=0, and K=1, respectively. The black and red
portions of the curves correspond to the upper and
lower branches of the dual solution, wherein
different curves have been drawn for seven different
values of , . We
observe that the upper branch of the solution domain
shifts upwards in the (S,)-plane by increasing the
values of . We note that for , the solution
exists only for , or . For the other
considered values of , the solution exists for all
values of S. The effect of shifting the curves
reverses in the case of the lower branch of the
solution domain. Further, it is observed that the
solution domain shifts downwards in the (S,)-
plane on varying microrotation parameter from 0
to 1.
The behavior of the solution domain in the (K,)-
plane is shown in Figure 1b for Λ =0.5. The solid
and dotted lines in the Figure 1a correspond to S =5,
and S S=-5, respectively. The black and red portions
of the curves correspond to the upper and lower
branches of the dual solution. The different curves
have been drawn for four different values of =-5, -
4, -3, -2. Here, the solution domain in the (K,)-
plane increases by increasing the values of
stretching/shrinking parameter for the upper
branch solution but the effect reverses in the case of
the lower branch solution. Also, it is observed that
the solution domain occurs for a larger value of in
the suction case as compared to the injection case.
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(a)
(b)
Fig. 1: The solution domain for as a function of
(a) mass transpiration S and (b) microrotation
parameter .
Figure 2a portrays the effect of various values of
stretching/shrinking parameter = 0, 0.1, 0.5, 1,
=0.2 on the tangential velocity
component. From these graphs, we observe
that the tangential velocity component
increases with an increase in the value of . Apart
from the usual behavior of a decrease in with
η, we observe that the rate of decrease of
with
increases sharply as the parameter is
varied from 0 to 1.
Similar variations of the profile with
are
found to occur on varying K, which have been
depicted in Figure 2b for K= 0, 0.1, 0.5, 1, = S =1,
Λ=0.2.
(a)
(b)
Fig. 2: The velocity profiles for various values
of (a) stretching/shrinking parameter and (b)
microrotation parameter .
Figure 3a shows the effect of various values of =
1, 2, 3, 4, , K=0.2 on the profile of .
Clearly the profile of shifts upwards on
incrementing from 1 to 4.
Similar variations in the profile of can be
observed from Figure 3b when S is incremented
from -1 to 1 for , K=0.2.
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(a)
(b)
Fig. 3: The tangential velocity profiles for
various values of (a) porous medium parameter
and (b) mass transpiration S.
Figure 4a shows the effect of various values of
, 0.1, 0.5, 1, on the profile of the angular
velocity for fixed parametric values of
, and . We find that is an
increasing function of .
On the other hand, when the parameter K is
incremented from 0 to 3, the variations in the profile
of are dramatic, which can be observed from
Figure 4b, for the fixed parametric values of
, . Here, increases on
incrementing K at any fixed location near the slit,
while decreases with K at a location
sufficiently away from the slit.
(a)
(b)
Fig. 4: The angular velocity profiles for
various values of (a) stretching/shrinking parameter
and (b) microrotation parameter .
Figure 5a shows the effect of various values of
, on the profile of the
microrotation for the fixed parametric values of
, K=0.2. Clearly increases with an
increase in the values of for small values of and
decreases with for all sufficiently large . So,
microrotation is favored by the porosity of the
medium near the slit while the microrotation is
hindered by the porosity of the medium away from
the slit.
A similar variation of the profile of by
varying S which has been shown in Figure 5b for the
fixed parametric values of , .
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(a)
(b)
Fig. 5: The angular velocity profiles on
similarity variable for various values of (a) porous
medium parameter and (b) mass transpiration S.
Figure 6 shows the variation of the skin friction
with for various values of
, and the fixed parametric values
of = 0.01, K = 1, . For a fixed value of S,
the black and red parts of the corresponding graph
represent the respective upper and lower branches of
the dual solution. For the upper branch of the dual
solution, increases with an increase in
for and decreases with
for . On the other hand, in the case of the
lower branch of the solution curve, is a
decreasing function of for , and
increases with for . These observations
show that the skin friction may decrease or increase
depending upon .
Fig. 6: The effect on of for various
values S.
Figure 7a and Figure 7b show the variations of the
skin friction parameter with the
microrotation parameter for both stretching and
shrinking cases, respectively for the fixed
parametric values of , n=0.5, and various
values of . For a fixed value of ,
the black and the red parts of the graph correspond
to and , respectively. Clearly, in the
case of stretching, the skin friction is
greater for suction and smaller for injection but the
effect reverses in the case of the shrinking sheet,
that is, increases with for both
stretchings as well as shrinking cases.
(a)
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(b)
Fig. 7: The effect on of for various
values of suction/injection parameter for (a)
shrinking and (b) stretching cases.
6 Conclusion
A micropolar fluid flow with a porous stretching or
shrinking sheet in the presence of mass transpiration
is investigated analytically and numerically. The
highly nonlinear PDEs governing the flow field are
transformed into a system of highly nonlinear ODEs
by using similarity transformations. The parametric
domain for the existence of the unique as well as the
dual solutions are investigated. The unique solution
is observed at the stretching sheet and dual behavior
is observed at the shrinking sheet. Based on the
results, the following conclusions can be drawn:
The solution domain for increases by
increasing the values of the stretching/shrinking
parameter for the upper branch solution but
the effect is reversed in the case of the lower
branch, that is, the solution domain for
decreases by increasing the values of the
stretching/shrinking parameter for lower
branch solution as a function of both mass
transpiration S and microrotation parameter .
The solution domain for is wider for fewer
values of microrotation parameter .
Each of the tangential and the angular velocity
components is an increasing function of the
stretching/shrinking parameter and the
microrotation parameter .
Each of the tangential and angular velocity
components is found to decrease with an
increase in the value of the porous medium
parameter and the mass transpiration S.
The skin friction coefficient increases with an
increase in microrotation parameter K.
Acknowledgment:
The authors have been supported under project grant
no. K_18-129257 provided by the National
Research, Development, and Innovation Fund of
Hungary, financed under the K18 funding scheme.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Rishu Garg carried out the formal analysis and
simulation.
-Jitender Singh has written the original draft.
-U.S. Mahabaleshwar and Okjunjon Sayfidinov
have been participated in the writing and review of
the paper.
-Gabriella Bognár has participated in the writing
and the review of the paper and in the organization
of funding.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The authors have been supported under project grant
no. K_18-129257 provided by the National
Research, Development, and Innovation Fund of
Hungary, financed under the K18 funding scheme.
Conflict of Interest
The authors have no conflict 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
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_US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.3
Rishu Garg, Jitender Singh,
U. S. Mahabaleshwar, Okhunjon Sayfidinov, G. Bognar
E-ISSN: 2224-347X
33
Volume 18, 2023
