Squeezing Flow of an Electrically Conducting Casson Fluid by Hermite
Wavelet Technique
PREETHAM M. P.1, KUMBINARASAIAH S.1, RAGHUNATHA K. R.2
1Department of Mathematics,
Bangalore University,
Bengaluru,
INDIA
2Department of Mathematics,
Davangere University,
Davangere,
INDIA
Abstract: - The squeezing flow of an electrically conducting Casson fluid has been occupied in the report. The
governing magneto-hydrodynamic equations transformed into highly nonlinear ordinary differential equations.
The Hermite wavelet technique (HWM) resolves the consequential equation numerically. The outcomes of the
Hermite wavelet and numerical approaches are remarkably identical. Through this, it is confirmed that we can
solve such problems with the help of the Hermite wavelet method. Flow properties involving material
parameters are additionally mentioned and defined in the element with the graphical resource. It is determined
that magnetic subject is used as a managed occurrence in several flows because it normalizes the drift property.
In addition, squeeze range theatre is a crucial responsibility in these sorts of issues, and an increase in squeeze
variety will increase the velocity outline.
Key-Words: - Normal differential equations, Squeezing flow, Casson fluid, Hermite wavelet method, Casson
fluid, Numerical method.
Received: December 19, 2022. Revised: October 26, 2023. Accepted: November 21, 2023. Published: December 22, 2023.
1 Introduction
During the end of the 20th century, the lavish and
philosophical theory of wavelets was formed due to
the efforts of mathematicians, physicists, and
engineers. The idea of wavelets is constantly
sophisticated to attempt various problems arising in
different branches of sciences and engineering.
Wavelet theory is one of the current up-and-coming
facts in applicable mathematics. It has applications
in subsequent fields, such as mathematical
modeling, image processing, signal analyses,
computer science, and applied sciences. The
primary goal of this research is to provide a forum
for multidisciplinary conversation among scientists
working on diverse projects related to wavelets,
fluid mechanics, and their applications. The wavelet
techniques to solve nonlinear equations in fluid
problems are among the recently created
methodologies for the numerical solution of an
equation that has received much attention, [1], [2],
[3], [4], [5].
Many mechanical system paintings are beneath the
principle of poignant pistons wherein plates show
off the squeezing motion that is normal to their
surfaces. Hydraulic lifters, engines, electric
vehicles, and also have this clutching glide in a
number of their components. Because sensible
consequence squeezing goes with the flow between
two horizontal parallel plates, its biological
packages are also of identical significance. Flow
interior nasogastric tubes and syringes are likewise
compressing flows, [6].
Initial work on squeezing flows can be named to
Stefan, who provided the fundamental method of
these flows underneath the lubrication hypothesis
[7]. Following him, many researchers have
acknowledged that they are more at ease with
squeezed flows and have achieved much technical
study to understand those flows. Several
contributions are noted in imminent strains [8], [9].
After that, exceptional scientists made numerous
attempts to apprehend squeezing flows with an
improved technique. Earlier research on squeezing
flows has been based on the Reynolds equation,
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whose lack of a few suitcases has been proven, [10].
Due to efforts of, [11], [12], greater supple and
helpful well-known similarity transforms are
availing a position. These transforms convert the
Navier–Stokes equation into a highly nonlinear 4th-
order normalized ordinary differential equation.
Undertaking non-Newtonian electrically fluid
flow is an especially significant occurrence. In most
realistic situations, we must cope with the glide of
electrically conducting fluid, revealing exclusive
behaviors that affect magnetic forces. In those
instances, the MHD characteristic of the glide
likewise had to be well thought-out. The Homotopy
solution for 2D MHD squeezing float between
horizontal parallel plates has been decided with the
aid of, [13]. Mass and heat transfer for squeezing
drift between parallel plates using the HAM is
investigated, [14]. Mainly of sensible fashions, the
fluids worried aren't effortless Newtonian. Highly
complex rheological homes of non-Newtonian
fluids cannot be studied through an available
version. Different arithmetical models have been
used to study diverse kinds of non-Newtonian
fluids. One of the important models is the Casson
fluid version. The main well-matched system to
reproduce blood-like fluid flow can be studied in,
[15]. It is obvious from the creative writing review
that the squeezing drift of a Casson fluid among the
plates shifting ordinary to their possess floor is but
to be investigated. Due to the intrinsic highly
nonlinearity of the governing equations, the fluid
glide actual results are extremely unusual. Still,
significant oversimplification assumptions had been
obligatory where they may be obtainable. Those
exaggeratedly obligatory suppositions may not be
second-hand for greater sensible flows.
Nevertheless, numerous analytical methods have
been urbanized to address this obstacle that have
typically been used in recent times. The variation of
parameters technique (HWM) is the currently
developed numerical strategy to remedy exclusive
problems. Several motivating fluid flow problems
are studied with the help of different wavelet
methods, [16], [17], [18], [19], [20].
As per the present literature review, the above
model is not considered by any mathematicians with
the wavelet method. This motivates us to explain
such equations via HWT. HWM is second-hand in
the current work for the solution of model highly
nonlinear equations. The calculated outputs are
compared with the results in the literature through
graphs and tables.
2 Problem Formulation
The squeezing flow of an electrically conducting
Casson fluid is explained and given in, [6]
03
1
12
1
FMFFFFFFSFiv
(1)
with suitable boundary conditions
(0) 0, (0) 0, (1) 0, (1) 1F F F F
(2)
The relevant parameters of equations (1) and (2)
Parameter
Notation
Velocity function
F
Casson fluid parameter
Squeeze number
S
Magnetic number
M1
2.1 Process of Hermite Wavelet Matrix
The Hermite wavelet is an incessant polynomial
basis wavelet, and its approximations are discussed
in, [21].
2.1.1 Preparation of Operational Matrix by
Integration
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Where,
9 1,0 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8
( ) [ ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )]T
x x x x x x x x x x
Integrate the above first nine basis about x limit
from 0 to x, then express as a linear combination of
Hermite wavelet basis as:
1,0 9
0
11
() 0000000 ()
24
xxx
1,1 9
0
11
( ) 0 0 0 0 0 0 0 ( )
48
xxx
1,2 9
0
11
( ) 0 0 0 0 0 0 0 ( )
3 12
xxx
1,3 9
0
51
( ) 0 0 0 0 0 0 0 ( )
4 16
xxx
1,4 9
0
21
( ) 0 0 0 0 0 0 0 ( )
5 20
xxx
1,5 9
0
23 1
( ) 0 0 0 0 0 0 0 ( )
3 24
xxx
1,6 9
0
116 1
( ) 0 0 0 0 0 0 0 ( )
7 28
xxx
1,7 9
0
103 1
() 0000000 ()
2 32
xxx
1,8 9 1,9
0
2680 1
( ) 0 0 0 0 0 0 0 0 ( ) ( )
9 36
xx x x
Hence,
9 9 9 9
0
( ) ( ) ( )
xx dx H x x
where,
11
24
11
48
11
3 12
51
4 16
21
9 9 9
5 20
23 1
3 24
116 1
7 28
103 1
2 32
2680
91,9
0
0 0 0 0 0 0 0
0
0 0 0 0 0 0 0
0
0 0 0 0 0 0 0
0
0 0 0 0 0 0 0
0
0 0 0 0 0 0 0 , ( )
0
0 0 0 0 0 0 0
0
0 0 0 0 0 0 0
0
0 0 0 0 0 0 0
1
0 0 0 0 0 0 0 0 ()
36
Hx
x
Next, the double integration of the above nine bases
is given below.
1,0 9
00
3 1 1
( ) 0 0 0 0 0 0 ( )
16 8 32
xx x dxdx x
1,1 9
00
1 1 1
( ) 0 0 0 0 0 0 ( )
6 16 96
xx x dxdx x
1,2 9
00
1 1 1
( ) 0 0 0 0 0 0 ( )
16 12 192
xx x dxdx x
1,3 9
00
3 5 1
( ) 0 0 0 0 0 0 ( )
5 16 320
xx x dxdx x
1,4 9
00
7 1 1
( ) 0 0 0 0 0 0 ( )
12 10 480
xx x dxdx x
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1,5 9
00
22 23 1
( ) 0 0 0 0 0 0 ( )
7 12 672
xx x dxdx x
1,6 9
00
81 29 1
( ) 0 0 0 0 0 0 ( )
8 7 896
xx x dxdx x
1,7 9 1,9
00
148 103 1
( ) 0 0 0 0 0 0 0 ( ) ( )
9 8 1152
xx x dxdx x x
1,8 9 1,10
00
773 670 1
( ) 0 0 0 0 0 0 0 ( ) ( )
5 9 1440
xx x dxdx x x
Hence,
9 9 9 9
00
( ) ( ) ( )
xx x dxdx H x x
where,
311
16 8 32
1 1 1
6 16 96
1 1 1
16 12 192
35 1
5 16 320
711
9 9 9
12 10 480
23
22 1
7 12 672
81 29 1
8 7 896
148 103
98
773 670
59
0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 , ( ) 0
0 0 0 0 0 0 0
0 0 0 0 0 0 1
0 0 0 0 0 0 0 11
0 0 0 0 0 0 0
Hx
1,9
1,10
()
52
1()
1440
x
x
The triple integration of the above nine bases is
given by,
1,0 9
000
5 3 1 1
( ) 0 0 0 0 0 ( )
96 64 64 384
xxx x dxdxdx x
1,1 9
000
7 1 1 1
( ) 0 0 0 0 0 ( )
128 24 128 1536
xxx x dxdxdx x
1,2 9
000
1 1 1 1
( ) 0 0 0 0 0 ( )
80 64 96 3840
xxx x dxdxdx x
1,3 9
000
19 3 5 1
( ) 0 0 0 0 0 ( )
96 20 128 7680
xxx x dxdxdx x
1,4 9
000
13 7 1 1
( ) 0 0 0 0 0 ( )
56 48 80 13440
xxx x dxdxdx x
1,5 9
000
65 11 23 1
( ) 0 0 0 0 0 ( )
64 14 96 21504
xxx x dxdxdx x
1,6 9 1,9
000
133 81 29 1
( ) 0 0 0 0 0 0 ( ) ( )
36 32 56 32256
xxx x dxdxdx x x
1,7 9 1,10
000
193 37 103 1
( ) 0 0 0 0 0 0 ( ) ( )
40 9 64 46080
xxx x dxdxdx x x
1,8 9 1,11
000
1211 773 335 1
( ) 0 0 0 0 0 0 ( ) ( )
22 20 36 46080
xxx x dxdxdx x x
Hence,
9 9 9 9
000
( ) ( ) ( )
xxx x dxdxdx H x x
where,
53
11
96 64 64 384
71 1 1
128 24 128 1536
1 1 1 1
80 64 96 3840
19 3 5 1
96 20 128 7680
13 7 11
99 56 48 80 13440
65 23
11 1
64 14 96 21504
133 81 29
36 32 56
193 37 103
40 9 64
773 335
1211
22 20 36
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0
H
9
1,9
1,10
1,11
0
0
0
0
0
0
, ( )
1()
32256
1()
46080
0 0 0 0 0 1()
63360
x
x
x
x
The fourth integration of the above nine basis is
given by,
1,0 9
0000
19 5 3 1 1
( ) 0 0 0 0 ( )
1536 384 512 768 6144
xxxx x dxdxdxdx x
1,1 9
0000
7 7 1 1 1
( ) 0 0 0 0 ( )
480 512 192 1536 30720
xxxx x dxdxdxdx x
1,2 9
0000
1 1 1 1 1
( ) 0 0 0 0 ( )
1152 320 512 1152 92160
xxxx x dxdxdxdx x
1,3 9
0000
17 19 3 5 1
( ) 0 0 0 0 ( )
336 384 160 1536 215040
xxxx x dxdxdxdx x
1,4 9
0000
55 13 7 1 1
( ) 0 0 0 0 ( )
768 224 384 960 430080
xxxx x dxdxdxdx x
1,5 9 1,9
0000
53 65 11 23 1
( ) 0 0 0 0 0 ( ) ( )
216 256 112 1152 774144
xxxx x dxdxdxdx x x
1,6 9 1,10
0000
497 133 81 29 1
( ) 0 0 0 0 0 ( ) ( )
480 144 256 672 1290240
xxxx x dxdxdxdx x x
1,7 9 1,11
0000
127 193 37 103 1
( ) 0 0 0 0 0 ( ) ( )
132 160 72 768 2027520
xxxx x dxdxdxdx x x
1,8 9 1,12
0000
4277 1211 773 335 1
( ) 0 0 0 0 0 ( ) ( )
288 88 160 432 3041280
xxxx x dxdxdxdx x x
Hence,
9 9 9 9
0000
( ) ( ) ( )
xxxx x dxdxdxdx H x x
where,
19 5 3 11
1536 384 512 768 6144
77
1 1 1
480 512 192 1536 30720
1 1 1 1 1
1152 320 512 1152 92160
17 19 3 5 1
336 384 160 1536 215040
55 13 7 11
99 768 224 384 960 430080
53 65 23
11
216 256 112 1152
497 133
480
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0 0
H
1,9
9
81 29 1,10
144 256 672
127 193 37 103
132 160 72 768
1,11
4277 773 335
1211
288 88 160 432
1,12
0
0
0
0
0
1()
, ( ) 774144
1()
0 0 0 0 0 1290240
0 0 0 0 0 1()
2027520
0 0 0 0 0
1()
3041280
x
x
x
x
x
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Analysis of the operational matrix method studied in
detail, [3], [21].
2.1.2 Method of Solution
Let us assume that
)()(
Tiv Af
(3)
integrate Eq. (3) concerning
from
0
to
, we get,
)()()0()(
PAff T
. (4)
Integrate (4) concerning from
0
to
)()()0()(
PAff T
(5)
Integrate (5) concerning
from
0
to
′ ′
‴ ″ ″
(6)
Integrate (6) concerning
from
0
to
′
‴ ‴ ‴
(7)
Put
1
in (6) and (7) we get
‴ ‴ ‴
″ ″
)1()1(1)1()1(
2
3
)1()1()0(
PAPAPAf TTT
(8)
Substitute these in (4) to (7)
)()()1()1((1)1()1(3)(
PAPAPAf TTT
(9)
)()()1()1(1)1()1(3)(
PAPAPAf TTT
(10)
)()()1()1(1)1()1(
2
3
)1()1(1)1()1(
2
3
)1()1()(
2
PAPAPA
PAPAPAf
TTT
TTT
(11)
)()()1()1(1)1()1(
2
)1()1(1)1()1(
2
3
)1()1()(
3
PAPAPA
PAPAPAf
TTT
TTT
(12)
Fit (3), (10), (11), (12), and (13) in (3) and collocate
the resultant equation by subsequent collocation
points
Mi
M
i
i...,,2,1
2
12
. Then, solve this
system with the Newton-Raphson method, which
yields unknown coefficients. Substitute these
coefficients in (4.10), which gives the Hermite
wavelet numerical solution.
3 Results and Discussions
The Hermite wavelet method is functional to solve
the nonlinear differential equations arising in non-
Newtonian heat transfer problems, and the
disadvantages and advantages of this method are
discussed, [16]. Acceptable comparison is made
with the earlier published work and validates the
correctness of the numerical results, as shown in
Table 1 (Appendix) and Table 2 (Appendix). The
effects of the Casson fluid parameter
, squeeze
number
S
, and the magnetic variety at radial (
()F
) and axial (
()F
) velocities are
characterized.
a) Plates moving apart
0S
In appendix section the Figure 1, Figure 2, Figure 3,
Figure 4, Figure 5, Figure 6, Figure 7, Figure 8,
describe how the squeeze number
S
behaves when
the plates move apart. Figure 1 (Appendix) shows
the properties of increasing values of
S
on the axial
speed
()F
. It is evident increasing
S
effects in a
decreased axial velocity. The property of increasing
S
on radial pace is shown in Figure 2 (Appendix).
For rising
S
, an increase
()F
is pragmatic
0.5 1
; nevertheless, there is a decrease in
()F
is for
0 0.5
. Figure 3 (Appendix)
depicts the behavior of
on
()F
. An increase in
slows down the axial flow. The effects of
developing the Casson fluid parameter on radial
velocity are proven in Figure 4 (Appendix).
Increasing the Casson fluid parameter decreases
()F
for
0 0.5
, and an upward thrust in
()F
is determined for
0.5 1
.
In Figure 5, Figure 6, Figure 7, Figure 8
(Appendix), the effects of
1
M
on
()F
and
()F
are explained. It may be determined from Figure 5
(Appendix) and Figure 6 (Appendix) that for
increasing magnetic number
1
M
, and there may be
a decrease in
()F
for somewhat lower values of
squeeze quantity
S
; while for
()F
, the growth in
1
M
offers a velocity sketch comparable to the case
of increasing
S
. Figure 7 and Figure 8 (Appendix)
are pinched to investigate the results of magnetic
area for barely better values of squeeze wide variety
S
. The conduct of radial and axial velocities
remainder nearly much lower
S
.
b) Collapsing movement of the plates
0S
In appendix section the Figure 9, Figure 10, Figure
11, Figure 12, Figure 13, Figure 14, Figure 15,
Figure 16 are for the case when collapsing
movement of the plates. In Figure 9 (Appendix),
tremendous axial acceleration is found for declining
S
. Figure 10 (Appendix), represents the results of
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decreasing
S
on radial velocity. It is understandable
that
()F
increases with the squeeze charge for
0 0.4
. A surprising exchange in
()F
is
found while
0.4 1
. Figure 11 and Figure 12
(Appendix) show the impacts of
on radial and
axial velocities, respectively. The same conduct is
determined for
and
S
when plates are
approaching together.
Figure 13, Figure 14, Figure 15, Figure 16
(Appendix) gift properties' of the float while plates
are approaching collectively
0S
and the
1
M
is
changing. In Figure 13 (Appendix), the results of
increasing
1
M
on
()F
are shown, and a lower in
()F
is discovered for larger values to some extent
S
. Figure 14 (Appendix) gives us a diagrammatical
exhibition of
()F
for increasing
1
M
. It represents
()F
decreases for
0 0.4
however for
0.4 1
it behaves in any other case, i.e., for
increasing values of magnetic quantity, there's a
speedy growth in radial speed of the liquid. A
comparable behavior is determined for growing
magnetic wide variety while
10S
has more
well-known consequences. Likewise, in Figure 16
(Appendix), a pretty speedy modification can be
located for increasing values of the magnetic
quantity. Also, the backflow can come out with a
lower squeeze variety, and a physically powerful
magnetic field is needed to decorate the stream, as
proven in Figure 16 (Appendix).
4 Conclusion
An electrically conducting non-Newtonian fluid
flow between two parallel plates is studied using the
Hermite wavelet method. The basic equations are
condensed using a similarity model to a single
regular, highly nonlinear ordinary differential
equation. Considering two cases, i.e., One while
plates are transferring apart and the other when
plates are approaching nearer. HWM is applied to
resolve the basic equation that goes with the flow.
The properties of up-and-coming known parameters
on glide are verified graphically, and a complete
dialogue is provided. A numerical answer is also
acquired using the RK-four method, VPM, to
evaluate the effects received by HWM, and some of
the answers determine remarkable conformity. It
can be seen from the figures that a robust magnetic
field may be second-hand to decorate the float while
plates are approaching jointly, and squeeze variety
increases the velocity sketch for both cases, i.e.,
while plates are approaching nearer and while plates
are leaving aside. Further, Squeeze flow is studied
by considering different types of non-Newtionain
fluids.
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APPENDIX
Fig. 1: Variation of
()F
for different values of
S
.
Fig. 2: Variation of
()F
for different values of
S
.
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Fig. 3: Variation of
()F
for different values of
.
Fig. 4: Variation of
()F
for different values of
.
Fig. 5: Variation of
()F
for different values of
1
.
Fig. 6:. Variation of
()F
for different values of
1
.
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Fig. 7: Variation of
()F
for different values of
1
.
Fig. 8: Variation of
()F
for different values of
1
.
Fig. 9: Variation of
()F
for different negative
values of
S
.
Fig. 10: Variation of
()F
for different negative
values of
S
.
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Fig. 11: Variation of
()F
for different values of
.
Fig. 12: Variation of
()F
for different values of
.
Fig. 13:. Variation of
()F
for different values of
1
.
Fig. 14: Variation of
()F
for different values of
1
.
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Fig. 15: Variation of
()F
for different values of
1
.
Fig. 16: Variation of
()F
for different values
of
1
.
Table 1. Comparison between the VPM, HWM, and numerical results for
4.0
and
.1
1M
5S
5S
)(
F
)(
F
)(
F
)(
F
VPM
HWM
Numerical
VPM
HWM
Numerical
VPM
HWM
Numerical
VPM
HWM
Numerical
0
0
0
0
1.359393
1.359393
1.359393
0
0
0
1.677216
1.677216
1.677216
0.1
0.139081
0.139081
0.139081
1.348452
1.348452
1.348452
0.166839
0.166839
0.166839
1.650804
1.650804
1.650804
0.2
0.276358
0.276358
0.276358
1.357517
1.357517
1.357517
0.328444
0.328444
0.328444
1.572994
1.572994
1.572994
0.3
0.409918
0.409918
0.409918
1.310148
1.310148
1.310148
0.479861
0.479861
0.479861
1.447971
1.447971
1.447971
0.4
0.537628
0.537628
0.537628
1.239953
1.239953
1.239953
0.616685
0.616685
0.616685
1.282424
1.282424
1.282424
0.5
0.657014
0.657014
0.657014
1.142869
1.142869
1.142869
0.735286
0.735286
0.735286
1.085120
1.085120
1.085120
0.6
0.765125
0.765125
0.765125
1.013414
1.013414
1.013414
0.832992
0.832992
0.832992
0.866366
0.866366
0.866366
0.7
0.858383
0.858383
0.858383
0.844480
0.844480
0.844480
0.908218
0.908218
0.908218
0.637365
0.637365
0.637365
0.8
0.932408
0.932408
0.932408
0.627096
0.627096
0.627096
0.960506
0.960506
0.960506
0.409532
0.409532
0.409532
0.9
0.981819
0.981819
0.981819
0.350136
0.350136
0.350136
0.990529
0.990529
0.990529
0.193804
0.193804
0.193804
1.0
1
1
1
0
0
0
1
1
1
0
0
0
Table 2. Numerical values and HWM for skin friction coefficient.
S
M
)1(
1
1F
[6]
)1(
1
1F
HWM
-5.0
-6.298708
-6.298700
-3.0
-8.320727
-8.320731
-1.0
-9.970376
-9.970303
1.0
-11.376240
-11.376224
3.0
-12.610669
-12.610675
5.0
-13.718095
-13.718073
-3.0
0.1
-30.991005
-30.991088
0.3
-10.873387
-10.873323
0.5
-6.771549
-6.771515
3.0
0.1
-35.260196
-35.260155
0.3
-15.149577
-15.149564
0.5
-11.078736
-11.078727
-3.0
0.4
2
-13.101572
-13.101587
4
-14.908219
-14.908243
6
-17.501183
-17.501128
3.0
0.4
2
-9.038196
-9.0381932
4
-11.531983
-11.531954
6
-14.819321
-14.819334
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
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.
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
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