Theoretical Analysis of Magnetohydrodynamical Waves for Plasma
based Coating Process of Isothermal Viscous-Plastic Fluid
UMER REHMAN1,*, MUHAMMAD BILAL2, MUBBASHIR SHERAZ2
1Department of Physics, Air University E-09 Sector Islamabad 44000, PAKISTAN
2 Department of Mathematics, University of Wah Rawalpindi, PAKISTAN
Abstract: A mathematical formulation on coating of a thin film for a compressible isothermal
magnetohydrodynamic (MHD) viscous-plastic fluid flowing across a narrow gap between two rotating
rolls is described in this article. The lubrication approximation theory is used to create and simplify the
equations of motion required for the fluid injected for coating. The relation explaining MHD wave
dynamics and instability is obtained by analytical calculations. According to the current investigation, the
growth rate in the unstable MHD waves are numerically evaluated as the function of the concerned
parameters, It's worth noting that the Lundquist and Prandtl's numbers are growth rate control parameters
in unstable MHD modes. The results show that the viscous-plastic parameter and ratio of diffusion rate
have a significant impact on the fundamental MHD dynamics. It is also concluded that the MHD effects
have had a significant impact on the coating of Casson material.
Keyword: MHD waves, Viscous-Plastic Plasma Fluid, MHD theory, Coating
Received: June 13, 2021. Revised: January 4, 2022. Accepted: January 25, 2022. Published: February 21, 2022.
1. Introduction
Surface coating using a fluid film is
common in the magnetic tape, photographic,
wrapping industries, polymer sheet, hot
rolling, glass-fabric, wire drawing, as well
as for manufacturer of paint and paper, to
protect a large surface region with one/many
uniform layers [1-4]. Despite the fact that
the products utilized in various coating
sectors vary greatly, while the same basic
technologies are used to create the necessary
coatings. A fluid deposited film should be
thin, continuous, and homogeneous in
thickness in general. Of contrast,
instabilities in films are detected under
particular operating conditions, which can
only be investigated by delving into the
challenges of fluid dynamics associated to
the coating process, as demonstrated in the
approach [5, 6] for Newtonian flows.
The concept of MHD three-
dimensional Casson fluid across a porous
linear stretching sheet was presented by
Mahanta et al. [7]. Later, Mustafa et al. [8]
investigated boundary layer flow for a
Casson nano fluid using a nonlinear
stretching surface. The boundary layer flow
for a steady incompressible laminar free
convective MHD Casson fluid across an
exponential stretching surface grounded in a
thermal stratified medium was examined by
Animasaun et al. [9]. Further, Das et al. [10]
investigated the influence of mass and heat
transmission in a vertical plate for an
unstable Casson fluid. In physics, chemistry,
and engineering, the study of boundary layer
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flow across a continually extending sheet
has practical applications. Many
metallurgical processes, such as plastic film
drawing, annealing, copper wire thinning,
etc, play a vital role in managing momentum
and heat transfer boundary layer flow for a
stretching sheet. In the applied magnetic
field, Ganesh et al. [11] examined the
numerical solution for nano-fluid across a
linearly semi-infinite stretched sheet. Dessie
et al. [12] proposed the MHD flow theory
for incompressible viscous fluid and heat
transfer phenomena over a stretching sheet
embedded in porous medium with heat
source/sink and viscous dissipation. The
effect of partial slip on hydro-magnetic
boundary layer flow and heat transfer due to
stretched surface with thermal radiation was
investigated by Hakeem et al. [13]. For
better understand of MHD fluid flow, Hayat
et al. [14] studied a hydromagnetic third-
grade fluid across a continually stretched
cylinder and used a homotopic technique to
solve the problem. The approximate solution
for MHD flow of tangent hyperbolic fluid
model on a stretching cylinder was
investigated by Malik et al. [15]. The idea of
two-dimensional MHD radiative fluxes and
heat transmission of a dusty nano fluid over
an exponentially extending surface was
discussed by Sandeep et al. [16]. From
published literature it is observed that steady
MHD wave dynamics and instabilities of
Casson fluid over a stretching sheet has not
been discussed so far. In Maxwellian fluid
dynamics, nonetheless, MHD waves and
instabilities remain to be of key relevance.
This issue has been addressed in current
investigation.
Although, the roll coating procedure
is simulated in this theoretical work to
categorize the characteristic time and length
scale associated with MHD wave dynamics,
nevertheless the formulation is applicable to
Casson charged particle fluid in any plasma-
based process in the presence of an
externally applied magnetic field. Roll
coating is a diverse procedure that requires
technical and operational expertise to
maintain consistency. The use of a coated
steel strip in the manufacturing industry is
currently expanding to include more novel
features such as acoustical protection
coatings and photovoltaic coatings that
absorb solar energy [17]. In recent decades,
experimental studies, theoretical approaches,
and numerical analysis have all been used to
investigate the flow challenges of roll
coating processes. The necessity in coating
manufacturing to coat stably, quickly, and
uniformly thin layers by improving coater
operating parameters and coating
microstructure spurred the work's
innovation. When both the roll and the sheet
are moving at the same speed, Zafar et al.
[18] discussed the roll-coating analysis of an
incompressible nanofluid. Using lubrication
approximation theory, the simplified form of
the equation of motion was developed.
Recently, Zheng et al. [19] explored reverse
roll coating of non-isothermal MHD
viscoplastic fluid theoretically. To the best
of our understanding and belief, there is no
reference in the theoretical formulation on
MHD wave’s dynamics and associated
instabilities for roll coating with the Casson
fluid model. The aim of this research is to
develop a mathematical model for MHD
flow instabilities in viscous-plastic fluids so
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that key MHD waves can be studied
theoretically in the presence of an external
magnetic field.
The rest of manuscript is arranged in
the following manner. Section 2 is
apportioned for model and governing
equations, in the section normal mode
analysis has been carried out. In Section 3
results have been interpreted and discussion
on relevant physics has been presented.
Section 4 contains brief summary of the
work.
2. The Model and Governing
Equations
Assume a laminar, and a compressible
isothermal flow of viscous-plastic MHD
fluid in externally applied magnetic field
between two rolls revolving in opposite
directions with velocities
, respectively, where
r is the radius of each roll and the subscripts
fr and rv denote forward and reverse rotating
rolls. Applied magnetic field is
along x-direction in rectangular coordinate
system. The nip area is the shortest distance
between two revolving rollers, and the range
at the nip region is denoted by.
Moreover, the x-direction is taken parallel to
the flow movement, whereas the y-direction
is taken perpendicular to the direction of
flow, as shown in Figure 1.
Figure 1. Geometry and schematic
representation of the process under
consideration.
The mathematical model for
geometry described above is presented, and
it is based on a set of equations as shown
below:
, (1)
(2)
.
(3)
Equation (1) is the momentum
conservation, Equation (2) is the mass
conservation and Equation (3) is the Ohm’s
law for the Maxwellian fluid. Where v is the
velocity of the fluid; ρ is the mass density;
is the pressure (T is constant for
isothermal fluid); j is the current density; E
is the electric field; the magnetic field B =
B0 + b composed of externally applied and
induced component, respectively; is the
fluid’s electric resistivity; and the stress
tensor is represented by τ for the fluid model
[20, 21]. The equation of the stress tensor
composed of two parts gyro-viscous part
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and isotropic part
,
where represents the viscosity of plastic
properties for non-Newtonian
fluids. Deformation rate based on the
model of non-Newtonian fluid, so is the
critical value of off diagonal product terms.
In two-dimensional flow for
rectangular coordinate system, we use the
formula as follows:
(4)
The Lorentz force becomes
from Equation (3). The Lorentz
force becomes: in this circumstance, due to
the magnetic field:
(5)
Where triple vector product identity
is used along with drift velocity relation
. Equations (1) can be expressed
as follows using Equations (2), (4), and (5):
(6)
(7)
These are the equations for dynamical
Casson fluid flow retains all effect of
electric and magnetic force in the dynamics
of fluid, in the case of incompressible steady
state flow with zero electric field and very
small induced magnetic field compared to
the applied magnetic field, so that the
magnetic Reynolds number is negligible, set
of equation given in Ref. [22] are retrieved.
The most significant dynamic phenomenon
occurs in the roll coating at the nip,
according to the problem's geometry. 2H0
between rolls characterizes the smallest gap
at the nip region. As a result, it may be
advantageous to suppose that the flows are
virtually parallel, with the common fluid
motion largely in the x-direction and the
fluid's minimal velocity in the y-direction.
Additionally, the y-direction velocity
increase takes precedence over the x-
direction flow velocity. To make things
easier, we can use an order of magnitude
analysis to determine the scale of pressure
and velocity parameters, identifying U, x,
and y as, , and ,
respectively. In view of these relations, we
arrive at where , hence
Equation (6) can be written as;
(8)
In Equation (8) gyro-viscous cancellation
has been incorporated as given in Ref. [20],
where is the Casson
parameter for the viscous-plastic fluid. To
describe the MHD wave dynamics, we can
apply Fourier analysis to Equation (2) and
(8). For that purpose, applying the
linearization to the Eq. (2) and (8) with
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respect to first order perturbation
proportional to ,
where is the angular frequency, is the
perpendicular component of the wave
vector, and is the component of the wave
vector along the magnetic field, the
equations become,
(9)
(10)
Where and are equilibrium mass and
number densities, respectively. Putting the
value of from Equation (10)
to (9), we get;
(11)
The following relation can be found when
rearranging above Equation (11).
(12)
Here,
is the acoustic speed in
the Casson fluid. The dimensionless form of
above Equation (13) can be derived by
defining dimensionless variables such as
, ,
(here,
is the Alfven
speed), while
, and are resistive diffusion
(associated with magnetic field diffusion)
and Alfven (associated with MHD wave)
time scales respectively. Using above
definitions, we get dimensionless form of
above relation describing unstable MHD
wave dynamics in Casson fluid flowing in
the reverse roll for coating purpose;
(13)
For the first time, we have presented MHD
wave dynamics for the non-Newtonian fluid.
The relation might be called as viscous-
resistive-acoustic (VRA) mode, which is a
polynomial having two roots.
3. Results and Discussion
The two roots of Equation (13) defining the
dynamics of two MHD waves, one acoustic
wave and the other Alfven wave, are
coupled because of the presence of an
externally applied magnetic field. The
complex frequency consists of two parts,
real and imaginary. The real part describes
only the propagation of acoustic wave in the
fluid, while imaginary part narrates the
growth in unstable coupled acoustic-Alfven
wave. Acoustic waves are hydrodynamical
propagation of disturbance through adiabatic
compression and decompression that
propagate energy across a fluid. While, the
Alfvén wave is a MHD wave in which
charged particles oscillate in response to a
restoring force given by effective tension on
magnetic field lines. The presence of a free
energy source may result in oscillations that
are unstable. In the current investigation, we
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found coupled unstable MHD wave
dynamics are caused by magnetic energy.
To estimate the growth rate of unstable
modes we have plotted the imaginary part of
the Equation (13) as a function of different
controlling parameters of fluid dynamics in
Figures 2 to 8. Parallel and perpendicular
dynamics are characterized in terms of
propagation of parallel and perpendicular
wave number with respect to magnetic field,
which is extremely useful in fluid dynamics.
We have solved the Eqn. (13) numerically
for fluid dynamics controlling
parameters, , , and .
Figure 2 displays the variation of the growth
rate of the unstable mode against the
parameter, this parameter is actually
describing the ratio of kinetic pressure of the
charge fluid to magnetic pressure associated
with the externally applied magnetic field.
By keeping the perpendicular wave number
fixed at, increasing the wave
number associated with parallel propagation
causes a decrease in the growth of the
unstable MHD wave. The dashed and dotted
lines represent lower values of parallel wave
number.
Figure 2. Variation of the growth rate of unstable MHD mode against fluid beta.
0.0
0.2
0.4
0.6
0.8
1.0
0
10
20
30
40
50
___k 10
....k9
k8
.. k7
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Figure 3. Variation of the growth rate against parallel wave number.
Figure 4. Variation of the growth rate of MHD mode against perpendicular wave number.
0
2
4
6
8
10
0
10
20
30
40
50
k
___ 0.2
.... 0.4
0.6
.. 0.8
0
2
4
6
8
10
0
500
1000
1500
k
___k 10
....k9
k8
.. k7
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Figure 5. Variation of the growth rate of MHD mode against parallel wave number.
Figure 6. Variation of unstable MHD mode vs perpendicular wave number at different Pr.
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
k
___k 10
....k9
k8
.. k7
0
2
4
6
8
10
0
21011
41011
61011
81011
11012
k
___Pr 104
....Pr 5 x103
Pr 103
.. Pr 5 x102
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Figure 7. Contour plot of growth rate variation.
Figure 8. Contour plot of growth rate.
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In Figure 3, the variation of the
growth rate of the unstable mode against the
parallel wave number is plotted. The
dependence of mode frequency follows
acoustic wave
at small
predominantly electrostatic in character. At
large frequency grows exponentially,
indicating the electromagnetic nature of the
mode, which is confirming the coupling of
MHD waves. The controlling parameters are
same as described in figure 2. Decreasing
the value of the fluid beta is enhancing the
growth of the unstable MHD wave due to
increasing free energy in magnetic field.
Figure 4 is the variation of the growth rate
of the unstable mode against the
perpendicular wave number . Cut offs and
resonances are found to the case of
perpendicular propagation of MHD wave in
the charged fluid flowing in the externally
applied magnetic field. There are no
propagation regions between 0 to 1 and
resonance at is found. The effect of
increasing the parallel wave number on the
MHD mode's growth rate is investigated.
The growth rate increases as the parallel
wave number increase. Further, in figure 5,
the growth rate of the MHD instability is
plotted from the Eqn. (13), with the same
fluid parameters (defined in in figure 1).
Growth rate vs normalized parallel wave
number are plotted for different values of
perpendicular wave number. The
dominant unstable MHD mode found at
lower values of perpendicular wave number.
The most unstable MHD modes are found
for the perpendicular wave number.
Increasing the Prandtl number significantly
enhance the growth of the unstable mode as
shown in figure 6.
The analytical relation (13) is now
treated to parametric analysis in figure 7 and
8. The imaginary part of the unstable mode
is numerically solved. The imaginary part of
the relation represents the growth rate
spectrum
of the unstable mode The
growth rate shows non-monotonic behavior.
Figure 7 and 8 are the contour plots of
relation () and illustrates the variation of the
growth rate with the fluid beta against
normalized parallel wave number and
normalized perpendicular wave number
respectively. Plot shows growth rate
dependence of the MHD mode on parallel
and perpendicular wave-number at fixed
Lundquist number , Prandlt number
and visous-plastic fluid parameter
as the fluid beta increases the
growth rate increases.
These images show that in the
presence of a magnetic field, MHD wave
dynamics are extremely relevant in charged
particle based coatings. We have
demonstrated MHD wave dynamics for the
first time, including the viscosity effect of a
non-Newtonian fluid. It's worth noting that
the unstable MHD wave dynamics have a
substantial impact on coating thickness
stability via control parameters. Although
fluid characteristics such as fluid velocity,
coating thickness, pressure gradient
distribution, flow rate, the graphical
relationship between the coating thickness
and the roll velocities ratio, and temperature
distribution have previously been examined
[23-24], none of the existing articles address
MDH wave dynamics and instability
conditions for such plasma-based processes.
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4. Conclusion
A mathematical model has been constructed
and addressed to explore the fluid film flow
of viscous-plastic fluids in the presence of a
magnetic field in relation to the many uses
of coating in engineering and industrial
processes. The Casson fluid is taken into
consideration in the flow model. Numerical
techniques were used to solve the obtained
equation.
The following are the work's main
concluding remarks:
1. Given the simplicity of the
lubrication model and the flow
pattern in the region of the separation
area, it may sound surprising that
there is a fluid flow instability that
affects coating thickness.
2. Numerically generated plots are used
to investigate the effects of magnetic
diffusion, viscous-plasticity, and
MHD instabilities on the flow of
Casson material.
3. The MHD instability has also been
found to have a significant impact on
the coating of Casson material.
4. The most unstable MHD mode is
found at higher value of Prandtl
number with respect to parallel
propagation of the MHD wave.
Scope: Engineers and scientists from related
industries throughout the world are invited
to validate our findings in a practical
situation. The theoretical MHD analysis of
viscous-plastic materials was the focus of
our research.
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