Derivation of nonlinear equations for surface of fluid
adhering to a moving plate withdrawn from liquid pool
ІVAN V. KAZACHKOV1,2
1Dept of Information Technologies and Data Analysis,
Nizhyn Gogol State University,
UKRAINE
2Dept of Energy Technology, Royal Institute of Technology,
Stockholm, 10044, SWEDEN,
Abstract: - The processes of the magnetic tape producing, wire adhering, as well as many other important
technological processes, include preparing some special materials’ adhering to a product surface. For a surface
withdrawn from the molten metal or the other liquid material there is a problem to determine a profile of a film
surface. In this paper, the mathematical model developed for simulation of the adhering process of viscous liquid
film to a slowly moving plate, which is vertically withdrawn from the molten metal or the other fluid capacity.
The Navier-Stokes equations for a film flow on a surface of the withdrawn plate are considered with the
corresponding boundary conditions, and the polynomial approximation is used for the film flow profile. The
equations, after integration across the layer of a film flow, result in the system of partial differential equations for
the wavy surface ζ(t,x) of a film flow, of flow rate q(t,x) and of flow energy Q(t,x).The derived equations are used
for analysis of the nonlinear film flow that determines the quality of a fluid adhering on a surface of the withdrawn
plate.
Key-Words: - Liquid Pool, Withdrawn Plate, Interface, Non-Linear Waves, Fluid Adhering, Equations
Received: May 27, 2021. Revised: April 15, 2022. Accepted: May 13, 2022. Published: July 6, 2022.
1 Introduction
The problem of the surface coating by material with
a thin liquid film has been considered for a long time
in various practical problems, and then it has also
gained interest theoretically. This is due to the need
to apply uniform coatings of a given thickness, to
estimate the entrainment of a liquid from the pool
after removing the object from it, to determine an
amount of a liquid on the walls of the container after
pouring out the liquid from it, and many other
practically important tasks.
Also, many technological processes are doing
some special materials’ adhering to a product’s
surface. For example, this problem is important for
the magnetic tape’s producing, wire adhering, etc.
For the surface withdrawn from the molten metal or
from the other liquid material there is a problem to
determine the film surface profile as much as possible
precisely and to control it. This is a subject of the
present paper.
The theory of the free coating of a Newtonian
liquid on a plate was developed based on a scale
analysis of the flow [1] with an analysis isolated on
the flow in the apical part of the meniscus where the
film is captured, and the bulk liquid is transported
from the depths of the basin to the surface. The film
thickness and the characteristic curvature of the
meniscus were expressed in terms of the capillary
number Ca=μu0/σ (viscous forces to capillary forces)
and the dimensionless parameter Po=μ(g/ρσ3)1∕4. The
first is the dynamic criterion and the second is the
kinematic criterion (it is determined only by physical
properties: gravitational, viscous and capillary). Here
μ, ρ, σ are the dynamic viscosity coefficient, density,
surface tension coefficient of the liquid, respectively,
g is acceleration due to gravity; u0 is the characteristic
velocity of a fluid flow.
The authors [1] used two adjustable constants,
determined by least-squares fitting with the
experimental data [2], which were surprisingly
“universal” for free-coating on a plate. For a range of
Ca the film thickness was scaled to d0=(μu0/ρg)1/2
asymptotes independent of Po. But in the small-
Ca limit the classical Landau-Levich law [3] is duly
recovered.
The roles of capillary, inertial, and gravity forces
in the various regimes are playing depending on their
ratio changing the regimes. The theoretical and
experimental results are correlated well spanning
over three orders of magnitudes both of Ca and Po. It
is interesting to note that a non-monotonous
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behaviour of the characteristic meniscus curvature
scaled to the reciprocal film thickness, with a growth
followed by a drop as a function of Ca, is predicted,
in qualitative accordance with earlier experimental
observations and computational results.
2 Accomplishments and Challenges in
the Film Coatings
Derjaguin [4] proposed a "load" h, i.e. the thickness
of the film for the liquid adhering to the plate
h=(μu0/ρgsinα)1/2 (h=d0(sinα)1/2), assuming that the
effects of inertia and surface tension are weak. In [2],
an infinite plate (at an angle α to the horizontal at a
constant velocity u0) is considered from an infinite
pool of a viscous liquid, where the above formula
obtained from the Stokes equations within the
boundaries of small slopes of the plate (without this
assumption, the formula is invalid).
The problem was shown to have infinitely many
stable solutions; all of them are stable but only one
corresponds to the above formula. This stable
solution can be distinguished only by comparing it
with a self-similar solution describing the non-steady
part of the film flow between the pool and the tip of
the film. Although the area of the near-pool region in
which the stable state is established expands with
time, the upper non-steady part of the film (its
thickness decreases to the tip) expands faster as it was
shown. It occupies most of the plate; therefore, an
average thickness of the film is 1.5 times smaller than
the load.
For the case of thick films, the formula [4] has
been given without strict derivation, showing that in
this case the thickness of the layer is independent of
a surface tension of a liquid. In [2]it was derived more
in detail considered the profile of a liquid layer which
remains on the wall of a vessel, inclined at an angle
to the horizon, at a time t after the level of the liquid
has begun to recede. It was supposed that the
condition dh/dx<<1 for a thickness h of a film at the
given point, is satisfied everywhere, except at the
place, where the film goes over into the free volume
of a liquid. Publication [4] was delayed due to the
discovery of divergences from experiment, the
explanation for which was found later. The
experimental data [2] fully confirm the theory
including the numerical coefficients.
The work of Landau and Levich [3] (1942)
initiated the fundamental theoretical, as well as
experimental investigation of a flow of the thin liquid
film entrained by a steady withdrawal of a flat plate
from a liquid bath. The existing theories are based on
a linearization of the problem and differ substantially.
They give relationships between the film
thickness h and the capillary number Ca. For
example, the paper [5] demonstrated theoretically
that different physical properties for the different
liquids result not in a single function but in a family
of the functions h(Ca). The complete set of previous
experimental work fitted the family of curves, while
the previous theories could satisfy just some of this
experimental data. The solution was found applying
the nonlinear theory [5]. The inertial terms and two‐
dimensional flow together with the parameter of
liquid physical properties were accounted. The direct
method of Galerkin was applied for solution of the
nonlinear problem; therefore, the new theory has got
an advantage of accurately determining the shape and
size of the upper meniscus profile. With the complete
set of the available experimental data achieved an
excellent agreement with the theoretical results. The
classical formula [4] was derived more in detail in
[6]. For the case of thick films, it has been given
without strict derivation, showing, in particular, that
in this case the thickness of a layer does not depend
on the surface tension of a liquid.
The classical coating problem of determining
the asymptotic film thickness on a flat plate, which is
being withdrawn vertically from an infinitely deep
liquid pool, has been examined through a numerical
solution of the stationary Navier-Stokes equations
[7]. For the creeping flow, the dimensionless load q
was determined as a function of the capillary
number Ca. For Ca<0.4, an agreement of the
Wilson’s extension [8] with the Levich’s well-known
expression was found. But for Ca→∞, q asymptotes
to 0.582, below the value of 2∕3 by Deryagin and Levi
[9]. For the finite Reynolds numbers Re≡mCa3∕2,
where m is a dimensionless number involving only
the gravitational acceleration g and the properties of
the fluid, q was found independent of the m at a
given Ca. Nevertheless, it was revealed correct only
up to a critical capillary number Ca*, dependent
on m, beyond which their numerical scheme failed.
Similarly, the corresponding nondimensional
flow rate qα≡q(cosα)1∕2 depends on both Ca and α
for the creeping flows in case of the inclined plate (at
an angle α to a vertical). Its maximum has been found
to increase monotonically with α up to 2∕3 when
α exceeds a critical angle αc∼π∕4, where the plate was
inclined midway to the horizontal with its
coating surface on the topside.
Nonlinear free coating onto a vertical surface was
studied theoretically in [10]. When a vessel of liquid
has been emptied and put aside, a thin film of liquid
clings to the inside and gradually drains down to the
bottom under the action of gravity [11]. The layer
being thin, the motion is very nearly laminar flow,
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and the curvature of the surface in a horizontal
direction may be ignored. Thus, the problem for a
cylindrical vessel is reducible to that of a wet plate
standing vertically.
Experimental study [12] enhanced the
fundamental understanding of the coating processes
in a wide range of the varying parameters. They
revealed the phenomena of the formation of an
asymptotic meniscus profile leading to a
development of a cusp at an interface. The
dimensionless description of such phenomena
allowed identification of the main parameters. And
flow visualization revealed the entire flow structure
with using a visible laser. The two phenomena of a
free coating have been shown depending on the
property number Po. By parameter Po over about
0.5, the dimensionless final film thickness h0is
constant up to the capillary number Ca of about 1.
By Po less than 0.1, film thickness h0 depends
on Ca and the Reynolds number Re but it becomes
constant when the Weber number We=Ca Reis less
than about 0.2. In both cases h0 is constant when the
effect of a surface tension on the meniscus becomes
weak. A cusp formation is caused by the inertia
(Re). By large Re, the effect of applicator
dimensions on h0 was investigated for flows too.
The thin liquid sheet was transported by a vertical
flat plate, which was stationary moving upwards
under an action of gravity [13]. Some liquid flowed
down, a trend that can be increased by blowing up the
air jet on the side layer. The preformed analysis of
possible solutions of the stationary flow led to the
correlation of the thickness of final layer with the
strength of a jet. For the testing of stability, the
corresponding non-stationary flows have been
investigated, which have shown that the stripped flow
is resistant to the long-wave perturbations.
The size and shape of meniscus profiles, which
were enlarged by flow, have been measured
experimentally [14] and photographically for a range
of flow conditions. The free coating of flat sheets
withdrawn from a pool of wetting liquids was
studied. The withdrawal speeds were varied over
several capillary numbers Ca below 1 using and oil
with viscosity 0.194 Ns/m2 (194 cP). The deformed
profiles were modeled by three-parameter analytical
expression. The parameters may be used to study
influence of Ca, coating speed, surface tension,
viscosity, density on the profile size and shape.
Influence of Re on the profile was noted at Re above
2.
By withdrawn of a body from a liquid bath a liquid
film is kept on a surface of the body. In a review [15],
after recalling the predictions and results for pure
Newtonian liquids coated on the simple solids, an
analysis of the deviations to this ideal case was done
exploring successively three potential sources of
complexity: the liquid-air interface, the bulk
rheological properties of the liquid and the
mechanical or chemical properties of the solid. For
these different complex cases, the significant effects
on the film thickness were observed experimentally
and summarized the theoretical analysis from the
literature.
The propagation of hydrodynamic modes on the
surface of agarose gels in the frequency range 101–
103Hz has been studied [16] using the electrically
excited surface waves; and rheometry determined the
bulk rheological behavior in the frequency
range 10−2–102 Hz. Propagation of the two surface
modes, the capillary and the elastic one, was
observed at low frequencies, while the regular
capillary behavior was detected above a well-defined
crossover frequency. Both, theoretical analysis, as
well as the measured bulk viscoelastic properties
revealed an excellent agreement with the
experimental data.
From the recent study of the new soft-micro
technologies, the hydrodynamic theory of surface
waves propagating on viscoelastic bodies enforced
this field of technology with the interesting
predictions and the new available applications [17].
Presently many soft small objects, deformable meso-
and micro-structures, and macroscopically
viscoelastic bodies fabricated from colloids and
polymers are produced. Therefore, the new soft
products fabricated by functional dynamics based on
the mechanical interplay of the viscoelastic material
with the medium through their interfaces. In this
review, the author recapitulated the field from its
birth and theoretical foundation in the latest 1980s up
today, through its flourishing in the 90s from the
prediction of extraordinary Rayleigh modes in
coexistence with ordinary capillary waves on the
surface of viscoelastic fluids, a fact first confirmed in
experiments with soft gels [16]. With this
observational discovery at sight, it was not only
settled the theory previously formulated, but mainly
opened a new field of applications with soft materials
where the mechanical interplay between surface and
bulk motions matters.
Also, the new unpublished results from surface
wave experiments performed with soft colloids were
reported in this contribution, where the analytic
methods of wave surfing synthesized together with
the concept of coexisting capillary-shear modes were
claimed as an integrated tool to insightfully scrutinize
the bulk rheology of soft solids and viscoelastic
fluids.
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Considerable work has been done on coating
films, summarized in the review articles [18-20].
Work on coating flows includes [12, 21]. They found
that the final film thickness depends on the
physicochemical properties of the liquid and the
withdrawal rate. Pre-metered coating processes
attempt to overcome the limitations of free coating.
Within certain operating limits, the size of the final
film thickness becomes an independent parameter
[22-26]. Other experiments on coating flows are
available [14, 27, 28]. Based on those studies much
is known about the final film thickness and
the interfacial profile over a wide range of capillary
number.
The main interest of work [2] was dip coating at
high Reynolds number. Although some investigators
analyzed coating flows at high capillary and
Reynolds number by approximate methods, e.g. [29,
30], no systematic experimental studies have been
performed in the past under those conditions. By the
high Reynolds number, the flow in the film and in the
coating applicator becomes important. Schweizer
[31] has experimentally determined the two-
dimensional flow field for a slide-coating device at a
maximum capillary number of 0.25. No detailed
study of the flow field in dip coating is available.
While numerous authors have studied falling thin
films, studies by [30] show that rising films are
uniquely different.
The submersible coating is to immerse the
substrate in a reservoir containing a film forming
fluid, and then withdraw from the bath to produce a
film. The purpose of [32, 33] was a development of a
mathematical model for the hydrodynamic process of
immersion, given that the film-forming fluid behaves
as a generalized Newtonian fluid. An analytical and
simple mathematical model that binds the main
parameters of a liquid with the use of the generalized
Herschel-Bulkley model was proposed. This model
was obtained based on strict balance of mass and
momentum applied to the homogeneous on-
evaporative system, where the main forces are
viscous and gravitational. The parameters that can be
evaluated are the velocity profile, flow rate, local
thickness, and average thickness of the coating film.
Finally, sufficient conditions for the model were
obtained. Experimental testing and sensitivity
analysis have been presented in the supplementary
article as part 2.
3 Statement of Problem by Nonlinear
Wave Flow on Withdrawn Surfaces
3.1 Description of Physical Situation
It is well known that the thin liquid sheet on the
withdrawn surface decreases to the constant
thickness h0 determined by the surface moisten
quality, its moving velocity u0 and physical properties
of fluid: viscosity μ, density ρ, surface tension ϭ, etc.
[34].But earlier investigations did not take into
account that the film flow effected by gravitational
forces are marked by nonlinearity and has many
different regimes including solitary waves [35-39]
that strongly influences on the surface covering
properties and their quality. The coordinate system
x,y shown in Fig. 1 is used.
Fig. 1. Model of fluid adhering to a moving
withdrawn surface
We consider the problem accounting influence of
the nonlinearity phenomena. It is supposed that the
fluid is Newtonian, and the process is isothermal.
Gravitational force acts against the surface moving
direction x. It could be also organized using the
electromagnetic systems for the process control if the
fluid is electroconductive [37-39], e.g. by application
of the crossed E, H fields.
Despite the above-mentioned works, investigation
of the film flow is still interesting being unknown in
many aspects because an interplay of the different
forces creates a lot of combinations of the diverse
regimes. For example, in a considered in this paper
flow there are two specific peculiarities: the flow is
going against the surface moving direction (vortex
flow) and the static meniscus determines the flow
beginning part, except the fact that by different
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Reynolds numbers the situation is changing
dramatically.
3.2 Mathematical Model of the Problem
The substitutive equation array describing the
process was derived, assuming that fluid flow is two-
dimensional and considering the film flow in a
boundary layer approach. The system of the
boundary layer equations with the corresponding
boundary conditions is
uv
xy
,
0
p
y
, (1)
22
22
1u u u p u u
u v g
t x y x x y
, (2)
0y
, u = u0, v=0; (3)
( , )y x t
,
uv
yx
,
vu
tx
, (4)
2
12 2
2v
pp yx
, (5)
where
,uv
are components of the fluid velocity in
coordinate system x,y; p is the total hydrodynamic
pressure, g- gravitational acceleration,
( , )y x t
-
the free surface equation for the film flow, p1- inside
pressure value on the free surface, p2- outside
pressure value on the free surface, t- time,
/
.
Thus, from (1) follows p=p(x,t) – inside pressure
of the film flow, p2=const – outside pressure of the
film flow (atmospheric).
4 Derivation of Integral Correlations
and Differential Equations
4.1 Integral Correlations for Film Flow
The differential equation array (1), (2) with the
boundary conditions (3) - (5) was integrated across
the boundary layer using the Leibniz's rule for
differentiation under the integral sign
00
u
udy dy u
x x x
.
For the first equation (1), with account of (3) and
the last boundary condition (4) we get
0, 0,
qq
uu
t x x x t x
where q is determined as
0
q udy
. Indexes ζ and 0
indicate here that correspondent values are taken by
y=ζ and y=0.
Then integrating the equation (2), with account of
the boundary conditions (3) - (5) results in
22
0
qu u dy u vu g
t t x x
0
1up
u
y x t x x
,
vu u u
tx
,
22
22
22
aa
u
v
p p p
y x x x
,
where pa is the atmospheric pressure. We used the
correlations:
2
1
2
uv uv
u v u
vu
y y y y x
,
0
vu
x x t x
,
uv
yx
,
2
2
,
u v u v v
x y x x y y x
.
0
/0vx
because
0v
on a surface of the
plate. The above-considered yields the following
equation array for the film flow:
0
q
tx
,
2
2
2
a
u
pp xx
, (6)
2
0
12
qp
u dy g u
t x x x t x
.
Substituting the pressure into the last equation, we
can get the following equation array of two
equations:
0
q
tx
,
qQ g
tx
(7)
23
23
22
uu
x x x t x
.
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The function
2
0
Q u dy
is introduced (kinetic energy).
4.2 General Case of the Uneven Surface of the
Withdrawn Plate
Instead of the boundary condition (3) the more
general condition may be considered for the uneven
plate:
yx
, u=u0, v=0; (8)
here
yx
is equation of the surface of the
withdrawn plate (e.g. wavy).
Then similar to the above:
uu
udy dy u u dy
x x x x x
0
uu
xx
,
0,
q
u u u
t x x x x
00
qu
t x x
,
0
t
,
0
u
y
,
0
uu
,
2 2 2
0
qu u dy u u vu
t t x x x
uu
y x t x
1pg
x
.
In this general case of uneven surface of the plate,
the equation array (7) with the boundary conditions
(8) is transformed as follows
00
qu
t x x
, (9)
23
2
023
2u
qQ
u
t x x x x
2gu
x t x
.
Here q and Q are determined as
q udy
,
2
Qu dy
.
4.3 Unique Peculiarities of the Derived
Equation Array
First, as shown above, the similar nonlinear terms in
the differential equations obtained after integration
across the film layer have mutually reduced. Thus,
only the second equations contain the nonlinear terms
in their right hands in the equation arrays (7) and (9).
The other unique feature has concern to the first
equations of the system, (7) and (9), which are
presented in the following form:
0
q
tx
,
00
qu
tx
,
where from follows
/1 1q f x t
, (10)
01/1q u F x t
,
so that the hyperbolic-type equations of the mass
conservation in a film flow have general solution f
and F – any arbitrary functions of the argument
1xt
, which means that a speed of the waves
in both cases is the same and is equal 1.
In the solutions (10), the 1 is introduced as a unity
of velocity (e.g. 1 m/s) to correlate the dimensional
values. In a dimensionless form, of course, it is not
needed. Thus,
qf
,
0
q u F
,
xt
.
5 Dimensionless Equation Arrays for
film flow on a surface of the plate
5.1 The Equation Array for Film Flow with
Approximate Flow Profile
The first simple equation (7) shows that with
decrease of the flow rate at the current point of x the
film thickness has tendency to grow, and inversely
with increase of the flow rate – like in the Bernoulli
equation: with increase of the width of a flow (tube)
the velocity is going down.
Here the speed of a plate is constant, therefore
with increase of thickness of a film the flow rate may
decrease only due to changes in velocity profile. The
second equation, except the functions q and ζ
contains the values: velocity of a film flow at the free
surface
u
, its second derivative by x and to the left
– integral from square of u.
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With account of the above mentioned, for further
analysis of the wave processes in a film flow, we need
to approximate the film flow profile with a
polynomial function: u=u0[1+(a2+a3y)y2]. We
obtained this profile using the boundary condition (3)
and condition y=0,
/0uy
as a requirement of
the smooth velocity profile since some finite layer of
a liquid is kept on a surface of the plate, so that close
to a plate in some sublayer velocity is nearly constant.
Due to action of gravity against direction of the
plate’s movement, there is vortex flow in a film.
Liquid on a plate is going up, while the free surface
is prone to an action of gravity down, up to the point,
where gravity becomes small comparing to adhesive,
capillary and viscous forces. Thus, profile of the film
flow velocity becomes nearly uniform by its thin
cross section.
We have done approximation up to a third order
by y because it looks reasonable due to vortex flow
and change of sign in a layer of a film. For this
reason, the parabolic profile seems to be too rough.
The peculiarities of the above equations (6), (7) is
that that we do not know the initial conditions, neither
boundary at the pool, we only can request stationary
parameters far away from the surface of pool:
x=∞, ζ=h0, u=u0, q=q0=u0h0. (11)
As the initial condition, we can state capillary
meniscus on the plate at the initial moment of time.
Equations (6) or (7) with boundary conditions (11)
can be used for analysis of the nonlinear film flow
that determine the quality of a fluid adhering to the
withdrawn surface.
Using the obtained film flow profile, we get the
following equations:
0
q
tx
, (12)
2
0 2 3
1()u a a
x t x
0 2 3
23()u a a
, (13)
23
2
0 2 3
23
2()
qQ u a a
t x x x
0 2 3
2 2 3()u a a g
. (14)
The equation (13) was obtained from the first
boundary condition (4). With the introduced
polynomial profile, the equation array could be
presented totally through the function ζ but we use it
only for the functions
22
/ux
and
u
. After
getting the solution, we can substitute into the
approximations for calculation of the constants
23
,aa
:
2
0 2 3
1()u u a a y y
,
2
0 2 3
1()u u a a
,
2
0 2 3
()
uu a a
xx
,
222
0 2 3
22
()
uu a a
xx
,
23
3
2
0
0
134
a
a
udy u
,
2
2 2 2 3 4
32
02
0
2
13 2 5
(aa
u dy u a
2
56
2 3 3
37
)
a a a
.
The boundary and initial conditions for the
equations (12) - (14) are following:
x=0,
00
h
,
00
qq
,
00
QQ
; (15)
x=∞,
*
h
,
*
qq
,
*
QQ
,
0
qQ
x x x
;
t=0,
0
hx
,
0
q q x
,
0
QQx
, (16)
where
0()x
is the well-known static meniscus
equation, and the other parameters are stated
according to this, and the speed of the plate
withdrawn from a pool. The stars assign the
stationary parameters far away from the pool.
The equation array (12) - (14) is transformed to
the following dimensionless form:
0
q
tx
, (17)
223
1()aa
x t x
23
23()aa
, (18)
2223
2
2
Re ()
qQ aa
t x x
3
23
32
12
23
Re ()aa
We x Fr
. (19)
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Here Re and We are the Reynolds and Weber
numbers, Re=u0l0/
,
2
00 0 /We h u
,
00
/Fr u gl
- the Froude number, We=Ca Re. The scale values for
non-dimensional equations (17) - (19) are the
following: ζ , x- l0, u- u0, t- l0/u0. We introduce h00 as
characteristic thickness of the film at the beginning
of the withdrawn process; l0 is the characteristic
distance by plate where the film flow is established.
5.2 Dimensionless Equation Array for Film
Flow with Approximate Flow Profile
Analysis of the linear equation (17) of the mass
conservation shows that η=
xt
is the complex
variable, so that any function f(η) satisfies this
equation. Obviously, η=
xt
is equation of the
simple wave moving with constant speed 1 (the same
as a plate is moving).
Thus, from (17) - (19) yields
q f x t f
, (20)
2
22 3 2 3
22 3 2 3
2 3 2 3 0
( ) ( )
a a a a
dd
d a a d a a
, (21)
32
23
32
43
Re ()
dQ d d
aa
d We d d
23
2
223
Re ()
daa
d Fr
. (22)
5.3 Solution of Dimensionless Equation Array
The solution procedure is a follows. First the non-
linear second-order equation (21) is solved, then a
solution obtained is substituted into the equation (22),
/dQ d
is computed through the velocity profile and
then the equation obtained is used for calculation of
the constants
2
a
,
3
a
. It is interesting that a parabolic
profile (
3
a
=0) yields from (21)
2
2
2
22
0
dd
dd
, (23)
so that function
does not depend on the flow
velocity profile, which influence is revealed only in
the equation (22), where from the profile is
determined.
Film flow to a moving withdrawn surface is not
as simple as considered in [34] because the
nonlinearity of the process may be strong; therefore,
it cannot be neglected. The mathematical simulation
of the process of fluid adhering to a moving
withdrawn surface in linear approach is rough
enough that explains poor correspondence between
the linear theory results and the experimental data.
This process also can be controlled by
electromagnetic fields [37-39] in case of
electroconductive fluid.
6 Conclusion
We have identical profiles (20) for the functions of
the film flow surface and the film flow rate. The
nonlinear second-order equation (21) was solved for
a range of available values of the constants. Then
from (22), after substitution of the solution obtained,
the constants of the polynomial approximation of the
profile, which satisfy the (22) can be computed. The
(21) shows that parabolic film flow profile leads to a
universal solution, which does not depend on the film
flow profile. But the constants of integration and
constant a2 in a film flow profile can be computed
afterwards from the equation (22). Further analysis of
the equations derived for a smooth withdrawn plate,
as well as the equations (9) for a wavy plate, and their
solution is a subject for further research. As shown
above and is known from the cited literature, there
are many very different regimes and the solutions
might differ correspondingly.
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