Water-based Ferrofluid Flow over a Rotatable Plate
ANUPAM BHANDARI*, AKMAL HUSAIN
Department of Mathematics,
School of Advanced Engineering,
University of Petroleum and Energy Studies (UPES), Dehradun,
Energy Acres Building, Bidholi,
Dehradun- 248007, Uttarakhand,
INDIA
Abstract: - In the current work, the influence of rotational viscosity as a result of an external magnetic field on
water-based ferrofluid flow over a rotating plate is investigated. The governing equations of the physical
model are transformed into a set of ordinary differential equations. The numerical solution of the differential
equations is obtained by using the finite element method. The findings of the radial, tangential, and axial
velocity distributions are descriptively presented for the different range of rotational viscosity The outcomes of
this research demonstrate that the magnetic field has an important role in controlling the velocity profiles in the
flow. A comparative study of velocity distributions is presented for , , and
nanoparticles.
Key-Words: - Ferrofluid; rotating disk; velocity; magnetic field; rotational viscosity; differential equations.
Received: January 7, 2023. Revised: November 13, 2023. Accepted: December 11, 2023. Published: December 31, 2023.
1 Introduction
Ferrofluids are smart materials that can vary their
rheological and physical characteristics for the sake
of the external magnetic field. Ferrofluids are
colloidal suspensions of single-domain magnetic
nanoparticles (3-15nm approx.), [1], [2], [3]. It is
composed of iron oxide ,
ferrites, , and in carrier liquid after adding
some surfactant to avoid the agglomeration of
nanoparticles, [4], [5], [6]. Because of the nanosize
of the particles, ferrofluids contribute significantly
to nanoscience and nanotechnology. Ferrofluids can
achieve a broad range of viscosity subject to the
availability of a magnetic field. Ferrofluids are
generally needed in the sealing of hard disks,
electronic packaging, aerospace engineering,
loudspeaker coils, and biomedical engineering, [7].
Ferrofluids manifest rotational viscosity and the
origin of the additional viscosity becomes due to the
difference between the speed of the rotation of the
nanoparticles and fluid vorticity The medical
implementation such as drug targeting and
hyperthermia are gaining considerable attention, [8],
[9], [10]. [11], used the Keller-box approach to
solve the similarity differential equations while
studying the magnetohydrodynamic flow and heat
transport over a shrinking surface. [12],
investigated how thermal radiation affected
magnetohydrodynamic flow over a sheet that
was stretching exponentially and came up with
a numerical solution using the shooting method.
[13], measured the effects of activation energy
and sheet thickness variation by examining the
rotational hydromagnetic flow of Carreau fluid.
The flow behavior of magnetic fluid has attracted
many researchers due to its diversified applications.
Attia has studied MHD flow near a rotating disk
with various parameters, [14], [15], [16], [17]. The
researchers have examined the rheological
characteristics of ferrofluid, [18], [19]. In a
magnetic fluid, [1], introduced magneto-viscous
phenomena. An important aspect of rotational
viscosity due to the magnetic field named known as
negative viscosity has been explained in detail, [20].
Using a few new mathematical expressions, [21],
provides a theoretical explanation of negative
viscosity. [22], presented a novel magnetization
equation. [23], investigated the flow of nanofluids in
a vertical plate. The influence of magnetization
force and viscosity in ferrofluid flow in the presence
of a disk has been investigated, [24], [25], [26],
[27]. Under the influence of a strongly oscillating
magnetic field, the ferrofluid generated by a stretchy
rotating plate was examined, [28]. [29], investigated
the effects of rotational viscosity in water-based
ferrofluid through experimental work. [30],
analyzed the pipe flow of magnetic fluid in an
oscillating magnetic field with variable viscosity.
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[31], investigated microstructure and inertial
characteristics in ferrofluid flow over a stretching
sheet. Different classes of nanofluid flow and heat
transfer properties have been studied by researchers,
[32], [33], [34], [35], [36].
In the present work, the influence of rotational
viscosity on the steady and incompressible flow of
as a result of a rotating plate has been
investigated. The magnetization equation derived by
Shliomis from irreversible thermodynamics has
been employed to calculate an increase of the
ferrofluid viscosity in the presence of an oscillating
magnetic field. In response to the magnetic field,
fluid, and magnetic nano-particles rotate with
different angular velocities, which creates an
additional resistance on the velocity distribution.
Similarity transformation has been used to transform
a set of differential equations in dimensionless form.
The shooting method has been used to solve a
system of nonlinear-coupled differential equations.
This resistance to the velocity profiles for different
values of field-dependent viscosity parameters has
been investigated.
Fig. 1: Flow configuration of nanofluid flow over a
rotatable plate
2 Problem Formulation
Figure 1 shows the flow configuration for the
nanofluid. The plate rotates with a uniform angular
velocity about axis. The components of the radial,
tangential, and axial velocity distributions are ,
and , respectively. The disk is kept in a
uniform magnetic field which is balanced by the
pressure gradient. The governing equations for the
ferrohydrodynamic nanofluid flow are as follows:
, (1)
, (2)
, (3)
, (4)
where exhibits the density, indicates the fluid
velocity, represents the pressure, exhibits the
reference viscosity, exhibits the magnetic
permeability of free space, exhibits the
magnetization, represents the magnetic field
intensity, indicates the angular velocity of the
particle, indicates the vorticity of the flow,
indicate the time, indicates the Neel relaxation
time and indicates the sum of the particle moment
of inertia.
Instantaneous equilibrium Magnetization
is defined in terms of Langevin function as:
,
, , (5)
where indicates the Brownian relaxation time,
indicates the magnetic moment of the particle,
indicates the Langevin parameter, indicates the
Boltzmann constant and indicates the temperature.
Considering
, Eq. (3) can be expressed
as:
(6)
Eq. (6) reduces Eq.(1)-(2) as:
(7)
(8)
Where
is called Brownian relaxation
time. Using Eq. (6), the equilibrium of magnetic and
viscous torque can be presented as:
(9)
Where denotes the volume concentration. The
non-equilibrium actual magnetization is considered
to be the equilibrium as:
(10)
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Where indicates the effective magnetic field
parameter and denotes the Langevin function.
The effective magnetic parameter is a ratio of
effective magnetic force and thermal force i.e.
. Owing to the slow and oscillating magnetic
field, Eq. (8) can be written as, [17], [18]:
(11)
The effective field parameter by the equation of zero
approximation from Eq. (11) as, [21]:
, (12)
is the amplitude of the magnetic field. The linear
approximations in
, Eq. (11) gives
where and is a
unit vector, [21]. Then the function is
determined by Eq. (11) as:
, (13)
where denotes the effective magnetization
function and denotes the angular frequency of
the magnetic field.
The mean magnetic torque is as:
,
(14)
=
(15)
Thus
the magnetic field-dependent
viscosity and the angular velocity of the nano-
particle may be expressed [21].
Eq. (1) can be written with the help of Eq. (15) as:
(16)
Magnetization force and pressure force can be
merged as:
(17)
The momentum equation can be expressed in the
reduced pressure form as:
(18)
For the incompressible flow as a result of the
rotating plate, Eq. (18) and Eq.(4) have been
converted into three-dimensional cylindrical form
as:
(19)
(20)
(21)
Subject to the boundary conditions:
;
(22)
Density and viscosity of nanofluid are
,
(23)
The following transformation is used to convert
the given system into the dimensionless form:
, ,
, ;
(24)
From Eqs. (19)-(23) and by applying Eq. (24),
the transformed nonlinear differential equations
are as follows:
(25)
(26)
(27)
,
, (28)
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3 Problem Solution
In the framework of this study, the nonlinear-
coupled differential equations have been obtained
for the flow of water-based nanofluid owing
to a rotating plate subject to the availability of an
oscillating magnetic field. The set of nonlinear
differential equations has been converted into solved
with the help of COMSOL Multiphysics Software.
Finally, the results have been obtained for the radial,
azimuthal, and axial velocity profiles. The densities
of the carrier liquid and nano-particles
are 1024 and 5170 , respectively.
Figure 2 demonstrates the convergence plot of the
numerical solution. The results in the numerical
solution are corrected up to six decimal places. If we
consider the parameters and , the
present problem reduces to Karman Swirling flow
problems, [37], [38], [39]. In this case, Table 1
represents the validation of the present numerical
solution with the previous theoretical model of
Karman flow.
Fig. 2: Convergence plot of the numerical solution
Table 1. Validation of the numerical solution
Kelson and
Desseaux, [37]
0.510233
-0.65922
Pop et al., [38]
0.5102
-0.6159
Turkyilmazoglu
[39]
0.5102326
2
-
0.61592201
Present
Result
0.5102134
911
-
0.61590974
66
The authors are required to look over and verify
whether the in-text citations exist in the reference
list and whether all the references mentioned in the
reference list exist in the in-text citations.
Also, they need to look over if the in-text
citations of the Tables, Equations and Figures are
properly connected with the Tables, Equations and
Figures.
4 Results and Discussion
Figure 3, Figure 4, and Figure 5 reveal the profiles
of and for various values of . In the
absence of an effective magnetic parameter there
is no magnetic torque acting on the fluid. Increasing
the values of the parameter , the magnetic torque
favors the swirling flow of ferrofluid. The
higher range of increases the velocity
distributions. The Minus range of the axial velocity
points out that the flow of nanofluid is oriented
toward the plate. In Karman swirling flow, the
velocities of ferrofluid are higher due to magnetic
torque. Figure 6, Figure 7 and Figure 8 shows the
distributions of and for different sizes of
volume concentrations of . Increasing the volume
concentration of decreases the velocity
distribution in the flow increases. As we increase
the concentration, the fluid becomes more
magnetized. Therefore, the magnetic torque
dominates over the flow which causes a significant
enhancement in the velocity profiles. The results for
indicate that only the flow of carrier liquid.
With increasing values the magnetic fluid
becomes magnetized and changes its behavior under
the influence of the field. Figure 9, Figure 10 and
Figure 11 exhibit the profiles of and for
different types of varieties of ferrofluids. The
comparison has been made among ,
, and nano-particles in a carrier
liquid. The densities considered for these particles
are 3594 , 5170 , 6670 and
8860 , respectively. The velocity of the
ferrofluid depends on the types of nanoparticles and
the carrier liquid. The velocity of the water-based
CO ferrofluid is less than other nanofluids. The flow
characteristics of ferrofluid can be controlled by
changing the strength of the magnetic field as well
as changing the ferromagnetic material.
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Fig. 3: Profile of for different values of and
fixed value of
Fig. 4: Profile of for different values of and
fixed value of
Fig. 5: Profile of for different values of and
fixed value of
Fig. 6: Profile of for different values of and
fixed value of .
Fig. 7: Profile of for different values of and
fixed value of .
Fig. 8: Profile of for different values of and
fixed values of
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Fig. 9: Profile of for different ferromagnetic
nanoparticles
Fig. 10: Profile of for different ferromagnetic
nanoparticles
Fig. 11: Profile of for different ferromagnetic
nanoparticles
5 Conclusion
In this investigation, the outcome of ferromagnetic
nanofluid flow over a rotating plate under the
influence of the oscillating magnetic field has been
studied numerically with the help of the finite
element method. The main results of the present
investigations are as follows:
Growing volume concentration and
effective magnetic parameter increases
the velocity distributions in the flow.
Water based Cobalt ferrite
ferrofluid can be used as low viscous
nanofluid and Cobalt ferrofluid can be
used as a highly viscous nanofluid.
These results could be useful for studying
the rheological behavior of ferrofluids under
the influence of the oscillating magnetic
field.
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Contribution of Individual Authors to the
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The authors equally contributed to the present
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problem to the final findings and solution.
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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.
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