Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 5.7.4 - High speed coating of a fiber - consequences of the inertia/surface tension duality
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Title: 5.7.4 - High speed coating of a fiber - consequences of the inertia/surface tension duality Interfacial Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Vermaut, D.
Lepetitcorps, Y.
Arquis, E.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
Subject: coating
surface tension
Abstract: A new way to produce reinforced titanium metal matrix is studied. Using high speed coating of a continuous ceramic fiber to manufacture such a material ensures a cost effective process. Some issues concerning this process and the stability of the coating thickness are discussed. Regimes for the thickness dependency with withdrawal rate for low viscosity fluids are identified and detailed in this paper. Studies point out a dependency of coating thickness destabilization with velocity and the role of surface tension in drop transportation out of dynamic meniscus is also discussed. An analysis of these results is proposed in this article.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

High Speed coating of a fiber consequences of the liquid internal flow

D. Vermaut*t, Y Lepetitcorpst and E. Arquis*

Laboratory TRanferts Ecoulements FLuides et Energ6tique (TREFLE), University of Bordeaux, Pessac, 33600, FRANCE
t Laboratoire des Composites Thermostructuraux (LCTS), University of Bordeaux, Pessac, 33600, FRANCE and
Keywords: Coating, inertia, surface tension


A new way to produce reinforced titanium metal matrix is studied. Using high speed coating of a continuous ceramic fiber to
manufacture such a material ensures a cost effective process. Some issues concerning this process and the stability of the
coating thickness are discussed. Regimes for the thickness dependency with withdrawal rate for low viscosity fluids are
identified and detailed in this paper. Studies point out a dependency of coating thickness destabilization with velocity and the
role of surface tension in drop transportation out of dynamic meniscus is also discussed. An analysis of these results is
proposed in this article.


High speed coating of a continuous ceramic fiber by liquid
titanium is an innovative way to produce reinforced titanium
materials. Achieving such a material by this process has not
only an economical advantage but also reduce material
weight increasing mechanical properties. Nevertheless, first
production tests showed discontinuities in titanium coating
thickness. The produced fiber shows anomalies such as
titanium pearls and uncoated fiber. Those discontinuities
induce fiber breakdown in process which sets materials
useless. A study of physical phenomena occurring in high
speed coating process was necessary to understand pearl
origins and to know if they can be avoided.
The coating process was studied for the first time by Landau
and Levich (1942) with the withdrawn of a plate out of a
liquid bath at low velocity. The followed studies were
focused on the improvement of their results and on the
expansion of their law to higher capillary numbers, Ca, or
higher velocities. In this paper, the theoretical conclusions
of those studies, for cylindrical conformation, are presented.
The issue of high speed regimes is also detailed. The study,
proposed in this paper, concerns the coating process for
fibers from 80 to 400 ltm diameters dragged out of low
viscosity fluid baths. Two aspects were studied: the coating
thickness of fluid along the fiber and the thickness

g Acceleration of gravity (m. s-2)
e Coated fluid thickness (m)
R Fiber radius (m)
V Withdrawal rate (m.s 1)
H Bath depth (m)
t time (s)
p pressure (Pa)
C phase function
g gravity field
F force
u local velocity field
n normal to the interface
t tangent to the interface
Dimensionless number
Bo Bond number
Ca Capillary number
Re Reynolds number
We Weber number
Greek letters
y Surface tension of the fluid (N.m 1)
T Viscosity of fluid (Pa.s)
p Density of the fluid (kg.m-3)
K1 capillary length (m)
K local curvature of the interface
6 indicating interface in CSF model
pl plate
f fiber
1 liquid
g gas
st surface tension
i mesh index
n at t time
n+1 at t+At time
BL boundary layers


Paper No

Theoretical background

The first law established to predict the coating thickness of
fluid on a moving plate is the LLD law named after theirs
authors Landau & Levich (1942) and Derjaguin (1943).
They studied the slow withdrawn of a plate out of liquid
bath and noticed that the curvature of the static meniscus,
inherent in solid/liquid/air contact, hardly changes. So, the
setting of a plate in motion forms a slightly bended dynamic
meniscus from which a thin film of fluid is dragged out
along the plate. If the fluid has a low viscosity and the
thickness of fluid is thin, the coating process is in
accordance with the lubrication approximation. The
Navier-Stokes equations can be simplified. In this paper, we
are considering the withdrawn of vertical plates (for the
LLD theory) and vertical fibers (Figure 1).
At low withdrawal rate, forces allowing the coating are the
capillary one, the viscous one, and the gravity one. As the
withdrawn is vertical, film drainage could occur because of
gravity. Moreover, the gravity force can be neglected if it
remains much smaller than the capillary suction. Therefore
we use the Bond number, Bo (1), which compares capillary
forces with gravity.

Bo, =- << 1

pg(R +- e)2
Bof -- << 1

So the coating thickness is the result of the competition
between the antagonist effects of viscosity and capillarity.
Considering those hypotheses and the pressure matching at
the junction of the static meniscus and the dynamic one, the
Navier-Stokes equations can be solved and the thickness of
fluid coated on a plate predicted (2).

e-K-1Ca2/3 (2)

where K-1 is the capillary length, K-1 = ,y/pg, and Ca
is the capillary number, Ca=ilV/y.


Fiber -->I Coated thickness, e

Static meniscus I Dynamic meniscus

Fluid bath

Figure 1: Example of low velocity coating of a fiber by a
wetting fluid.

Because of those hypotheses, the LLD law is limited to low
capillary numbers and for low viscosity fluids. This
approximation is valid as long as Ca13 << 1 (for small
capillary numbers Ca<10-3).
Bretherton(1961) was the first one to consider the
cylindrical conformation in coating. He studied the
emptying of a capillary tube and showed that the curvature
pressure, imposed by cylindrical geometry, has to be taken
into account to predict the coating thickness (3).

e-RCa2/3 (3)

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

He applied the LLD law to an other geometry but its
limitation remains the same. This expression governs the
visco-capillary regime.
It appears that with increasing the withdrawal rate a
thickness divergence with equation (3) appears. White &
Tallmadge (1966) pointed out that the coating thickness is
not longer negligible in comparison whit fiber radius when
Ca tends to unity. They proposed a correction of the LLD
law (4) considering the previous remark.
e = (4)
1 1.34Ca2/3
This expression diverges for Ca=0.64 but before achieving
such a capillary number for low viscosity fluids, a larger
divergence is ascertained. Because of the withdrawal rate
increase, the inertia effects are no longer negligible.
De Ryck & Qu6rd (1996), (1998) studied the inertia effects
respectively in fiber and plate coating processes. They
showed that the divergence between the real coating
thickness and the theoretical one appears for capillary
numbers smaller than 0.64, for low viscosity fluids. In such
a range of withdrawal rate, the coating thickness is no
longer the result of the competition between only viscosity
and capillarity but also inertia. The influence of the inertia
forces can be quantified comparatively to viscosity thanks to
the Reynolds number, Re (5) and to capillarity thanks to the
Weber number, We (6).
pV(R + e)
Re = (5)

pV2 (R + e)
We =- (6)

Considering De Ryck & Qu6rd (1996) experiments, it's the
competition between inertia and capillarity which causes the
thickness divergence. That divergence appears for Weber
number in order of unity. By dimensional analysis, taking
into account inertia effects, they were able to write a new
expression for the coating thickness:
e ~-- (7)
1 We
The regime described by equation (7) is named the
visco-inertial regime. We notice that if inertia is negligible
in comparison with capillarity forces, the equation (7)
becomes the equation (3).
That regime has a quite small domain of existence, due to
the coating thickness drastic increase. At the end of
visco-inertial regime, a maximum of thickness is achieved
and two tendencies are observed by increasing withdrawal
* The coating thickness remains stable (Rebouillat & al.
2002, Vermaut & al. 2009). We will call this regime:
the asymptotic regime.
* The coating thickness becomes thinner because of the
viscous boundary layers less developed in the fluid
bath. The contact time between static bath and the fiber
in motion decreases with withdrawal rate increasing
(De Ryck & Qu6rd 1996, Vermaut & al. 2009). This
regime is called: the boundary layer regime.
In this article, we will show that those two regimes exist and
co-exist but are depending on process parameters.

Paper No

Experimental Facility

To study the influences of process parameters on coating, an
experimental apparatus (Figure 2) was specially designed to
run at various fiber velocities (from 0.25 m.s-1 to 10 m.s'1),
to use several fiber diameters (experimental diameters:
140 m and 240 ~ in), and to study different fluids
(demineralised water and tin-lead eutectic). This apparatus
allows us to study high speed coating in vertical
configuration. A large bath of fluid is used to avoid
squeezing effects in the development of the dynamic
meniscus and the depth of the fluid bath is kept constant
during one experiment. To avoid discrepancies in our results
due to fiber roughness, the ones used in experiments are
drawn copper fibers and silver coated copper fibers.

Figure 2: Sketch of experimental apparatus
In parallel, the use of different fluids will help us to
understand the influence of physical and chemical
parameters on deposited thickness and fluid stability.
Table 1 sums up these parameters. Because of the nature of
used fluid: water and metallic fluid, two methods are used to
analyse the results of experiments.

Fluid Fiber Wetting Influence

Water Copper Yes Physical

Tin-lead Copper No* Physical
Silver coated Physical and
Tin-lead Reactive
copper chemical
*Considering according to short contact time in high speed coating
Table 1: Sum up of the influence study considering the
fluid/fiber contact.

Materialsfor water study
For water experiments, the experimental tank is made in
PMMA and allows us to observe the liquid bath depth and
the dynamic meniscus formation through the tank. We use a
104 mm objective (Nikkon) combined with a high-speed
camera (Ultima APX-RS) fixed on a tripod. Using a tripod
allows us to observe the fiber at different levels and to study
the coated fluid (thickness and destabilisations). The
analysis of deposited thickness on fiber is processed with an

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

interface realized through the Image J software (free online
Materials for tin-lead eutectic study
For tin-lead experiments, we change the experimental tank
and use a stainless steel one wrapped by a heating ribbon
instead. The tank is heated at 200 OC (the Tin-Lead eutectic
melting point is: T=183 C) and the bath depth is estimated
through fluid weight. That weight is measured before and
after the experiment to verify if the bath depth remained
constant during the experiment or if it varied. Weight loss is
around 30 g for 1 kg of tin-lead eutectic during one
experiment, which can be neglected.
In this coating case, the deposited thickness stays around the
fibre and so, is easy to measure. We use a micrometer
calliper with an accuracy of 1 tm to measure the tin-lead
thickness. Those measures are confirmed by a microscopic
analysis of the coated fiber section.

Numerical Procedure

Our studies on high speed coating of fibers are completed
thanks to numerical simulations of the process. Numerical
simulations are only done in order to better understand
physical phenomena occurring in coating process.
The numerical approach is done with the code "Th6tis",
software developed at the TREFLE Laboratory
( Th6tis is a set of discretization tools
and solvers able to investigate numerous cases of continuum
fluid mechanics. A previous work, done in the department,
had used and proved the code validity to predict zinc
thickness deposited on plate (Lacanette & al., 2005). In our
case, we had to adapt the way of solving to cylindrical
conformation under axisymetric coating hypothesis. The
axisymetric hypothesis is important here because we are
simulating macro-structural experiment in order to describe
and study micro-structural phenomena. The gain of
simulation time (and so costs) using a 2D axisymetric
simulation instead of a 3D one is quite large.
The numerical modelling of two-phase flows can be
achieved by a 1-Fluid model (Vincent & Caltagirone, 2000,
Pianet & al., 2010). The 1-Fluid method consists in solving
the Navier-Stokes equations in both phases with interfacial
jump conditions. The Navier-Stokes equations are solved
once, like for a single phase resolution, but including
extra-terms. Those terms represent the local: viscosities,
densities and surface tension. So a phase function C,
C(z,r,O,t), is then necessary in order to follow the
multiphase description of the whole domain by solving an
advection equation. New forms for viscosity, q =
rl (1 C) + ,g C and density p = pi (1 C) + pg C are
taking into account the atmospheric phase defined as C = 0
and the continuous phase (coating fluid) as C = 1. The
interface between the phases is defined as the condition,
C = 0.5.
Finite volume discretizations on fixed Cartesian grids are
applied to solve the Navier-Stokes equations (8), an
advection equation (9) on the phase function and the
incompressibility constraint (10).
The surface tension force is expressed considering the
surface tension coefficient, y, the local curvature of the
interface, K, and the Dirac function, 61, indicating interface
according to the CSF model of Brackbill & al. (1992).

Paper No

Forces due to capillarity are so defined by equation (11).

p + (u. V)u)
= pg Vp + V. (p(Vu + Vtu))
+ Fst


9C (9)
S+ u. VC = 0(9)

V. u = (10)

FST = yKni6i (11)
where u is the local velocity field, t the time, p the pressure,
g the gravity field, Fst is the surface tension force, and n, is
the normal to the interface.
The set of equations is solved by the augmented Lagrangian
method of Fortin and Glowinski (1982) to consider the
velocity-pressure coupling on a second order accurate
centred scheme of discretization. The discretization of the
convective term (u. V)u requires a linearization to be
solved, (un. V)un+l. The resulting algebraic system is
inverted by the direct solver called MUMPS (Amestoy & al.,
2000). The advection equation (9) is solved by the
implementation of a monotone Total Variation Diminishing
scheme, TVD (see Vincent & Caltagirone, 1999).
As said before, the simulated physical domain is a 2D
axisymetric one. That domain is discretized by an
exponential mesh. For the radius coordinate from Rfto Rtank,
the mesh is refined near Rf on 100 points with a 1 gm grid
spacing. Then, the grid spacing grows exponentially to Rtank
which is of 3 cm. For the z coordinate, the simulated size
and the number of points are function of the bath depth.
Taken as a whole, the mesh is refined at the fluid/air
interface with a minimum for grid spacing of 10gm and
along the fiber, the maxima for grid spacing is of 1 mm. On
this mesh, the discretization of the set of equations is solved
considering the following boundary conditions:
at Rf, fiber motion is here simulated
V.i= o
V. t = Vwthdrawa
at Rtank, the emptying of the bath is possible
V.i= o
V.-t = Vfee
at z=0, we impose a wall condition
V.i= 0
V.t= 0
at z=Zmax, we impose a Neumann condition
V.ii = Vee
V.=o 0
The time step used to solve the equations is of 1.10-6 s which
is the better compromise between the numerical calculated
divergence of velocity (-1.107) and simulated physical time
(500 ms, the thickness achieve its stable value after 20ms of
simulated coating).

Results and Discussion

Experimental facility and numerical procedure validation.
The first step of our study was to estimate the validity of

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

both numerical simulation and experimental apparatus. As
previous experimental works were essentially done with low
viscosity fluids as pure or demineralised water, we choose
to work with this fluid in a first place. As see previously in
the experimental facility section, we have a set of
continuous metallic fibers with diameter of 140 [tm and of
240 gm. Figure 3 sums up the results obtained
experimentally and numerically with both fibers and
compares them to the results obtained by De Ryck & Qu6re
(1996) with a 127 gm diameter fiber. As fiber radii are
different, we use dimensionless numbers for the comparison.
In the domain of low capillary numbers, the LLD tendency
is well observed with a slow increase of the e/R ratio with
capillary number (and so withdrawal rate). The tendency is
the same for all the compared cases. The direct comparison
between the "Qu&re [96] r63.5 nm" curve and the
"numerical data r70 gm" one allows us to validate the
numerical procedure described in the previous section. In
both cases, the visco-capillary regime and the visco-inertial
one are observed in the same range of capillary numbers.
The comparison between the "experimental data 120 gnm"
curve and the "numerical data 120 gm" curve leads us
towards to the same conclusion. The experimental apparatus
is then acknowledged as reliable.

0.8 -

0.2 H

numerical data r70pim
numerical data rl 20m /
Quere[96] r63.5pm
experimental data rl20 pm


Figure 3: Comparison between personal and published
experimental data and numerical ones by the plot of e/R in
function of Capillary number.
Figure 3 also allows us to point out great fiber radius
influence in high speed coating behaviours. In the case of
70 gm fiber radius, we can see the drastic increase of the
dragged thickness of fluid, feature of the visco-inertial
regime. The asymptotic regime is observed for higher
capillary numbers. That regime is then followed by a
smooth decrease of fluid thickness with withdrawal rate. It's
the beginning of the boundary layer regime. For the larger
fiber radius, the situation is different. The coating thickness
jump, observed in previous visco-inertial regime, starts for
smaller capillary number. The asymptotic regime is
observed as in previous case but the boundary layer regime
starts for lower capillary numbers and is more abrupt.
As numerical procedure had been validated for the study of
fluid coating, we will analyse according to the simulated
results the differences in the De Ryck & Qu6re (1996) and
the Rebouillat & al. (2002) conclusions. On one hand, De


7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Ryck & Qu6r6 were working with a size limited bath,
Rta,=2 mm and large fiber radius, from R=63.5 gtm to
R=110 gum. They pointed out the boundary layer regime for
capillary numbers greater than Ca > 4.10-2 (corresponding to
a withdrawal rate around 2.9 m.s-1 for pure water) which is a
smaller capillary number as those we observed. On the other
hand, Rebouillat & al. were working with a large size bath
(several centimetres) and small fiber radius (R=27 ntm) to
study regimes induced by withdrawal rate. They came to the
conclusion that the high speed regime is a steady state one
and that the tendencies, observed in previous works, are due
to squeezing effects "which could interfere with the free
uptake of the fiber withdrawn from the bath".
As first analyse in regard to our results, we understand that
the bigger the fiber radius is, the quicker the asymptotic
regime exists in capillary number range. So those authors
could not observe the same regimes considering the fiber
radii they used to study high speed coating. The next section
proposes an analysis of the bath size influence on deposited
thickness to understand both points of view. In this section,
the dynamic meniscus development for high withdrawal rate
and the boundary layers developed in the liquid bath are
Dynamic meniscus and viscous boundary layer
To know if a 2 mm tank radius, is large enough to allow the
complete development of the dynamic meniscus, we use one
of the pictures made to validate experimental data. The
picture can be seen in Figure 4.a. On this picture, we clearly
see the fiber of 120 gm radius, the coating thickness, and the
dynamic meniscus size. Thanks to the treatment with the
"Image J" software, we calculate the ratio pixel/gm and thus
are able to estimate the dynamic meniscus size of 4.3 mm.
That meniscus is much larger than the static meniscus which
size is around the capillary length K-1, K-1 = 2.72 mm for
pure water. The bigger size indicates that the coating is not
in the visco-capillary regime anymore but at least in the
visco-inertial one. That regime is characterized by the
enlargement of the dynamic meniscus. We also notice that
the dynamic meniscus is larger than the development size
allowed by the De Ryck & Qu&rd's tank. As their studies
concerned fibers from 63.5 gm to 110 gm radius, we can
conclude that all phenomena were not observable because of
used materials.
The second point we need to study, is the development of
the boundary layers in fluid bath to know if the bath size has
another influence on the observation of the coating
thickness regimes. The study of the boundary layers
developed in the tank, is realized with the same numerical
procedure as described in the previous section but only
considering the bath. The simulated domain is an
axisymetric Cartesian mesh of 256 x 256. So we are now
solving the equation (8) without the surface tension forces
because of the one phase flow modelling and the equation
(10). The boundary conditions remain the same. Figure 4.b.
shows the boundary layers developed in a static bath of
water destabilised by a fiber of 120 gm in motion at
1.6 m.s-1. A larger study of those boundary layers allows us
to conclude that the boundary layer development has the
same behaviour as the viscous boundary layers described by
Blasius (12) for the interaction between a flow and a thin

0 0005 001
a (a)

0015 002

Figure 4: Analyse of squeezing effects (a.) Example of
water coating of 120 gm fiber at v=1.6 m.s-1 for a 10 mm
bath depth (b.) Isovelocity plot of the boundary layers
developed in R-axis in the case of water coating of 120 gm
fiber at v=1.6 m.s-1 in Z-axis for a 20 mm bath depth.

eBL (12)

The calculated thickness eBL is the thickness for which the
flow is back to 99 % of the starting incoming flow velocity
in Blasius case. In the coating process case, we are
considering that the boundary layer thickness is the
thickness at which the flow is back to 1 % of withdrawal

Fiber Depth of Grid
radius measurement sBL spacing Discrepancy
(gnm) (mm) (m) (gnm)

70 10 807 38.8
= 8 pm, smaller
120 10 815 38.6 than space foot

Table 2: Viscous boundary layers developed in bath with
a withdrawal rate of 1.6 m.s-' by R=70 [um and R=120 [um
The study of the boundary layers developed in bath, for
fiber coating cases, shows us the independency between the
boundary layer thickness and fiber radius (Table 2). That
independency was tested for the range of withdrawal rate
from 1 m.s-' to 4 m.s-' and a range of radius from 40 gm to

Paper No

_ I

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

120 gm for baths of depth between 10 mm to 40 mm. So the
developed boundary layers in coating are function of the
withdrawal rate and the bath depth for the same fluid. The
viscous boundary layers representing the fluid in motion in
the bath, the black line in figure 4.b., is around 800 pm for a
120 tpm fiber coating at 1.6 m.s-' withdrawal rate for a
10 mm bath depth. That dimension is much smaller than
tank dimensions. So in all tank size cases, mechanical
energy which bumps into the free surface to form the
dynamic meniscus is the same (for equivalent withdrawal
rate and bath depth conditions).
The kinetic energy transported by its layers is thus divided
into several parts. The most intense part, the one near by the
fiber, imposes the dynamic meniscus thickness and is
directly dragged along the fiber. It represents the coated
fluid. For the coating cases detailed previously, this part
represents over 9% of the viscous boundary layers. This
percentage depends on kinetic energy incoming and coating
conditions. The rest of the viscous layers bumps into the
free surface and predominantly flows back into the bath,
generating the dynamic meniscus length. So the dynamic
meniscus cannot develop itself as effectively as it could in a
size limited bath. The back flow is developed side by side
with the viscous boundary layers and can affect their
development. The deposited thickness is then more affected
by the decrease of the incoming energy with the withdrawal
rate increasing. That is why the transition between the
visco-inertial regime and the boundary layer regime could
occur at a smaller capillary number in size limited bath.
As the boundary layer regime in coating process occurs for
greater capillary numbers in small fiber coating and for
smaller capillary number in size limited bath, the direct
comparison between Rebouillat & al. and De Ryck & Qu6rd
cannot be done. Now, the comparison between coatings in
the same bath size and depth conditions for various radii is
necessary to precise the previous arguments.
Influence offiber radius on c .,I,, i thickness
The previous authors had shown that the fiber radius has a
great influence in the transition between the visco-capillary
regime and the visco-inertial one. The visco-inertial regime
starts when the inertia forces are influent enough to counter
a part of the Laplacian pressure imposed by the surface
tension forces and the pressure imposed by the fiber
curvature. That appends when the Weber number (6), We, is
greater or equal to unity. As the Weber number is function of
fiber radius, the withdrawal rate imposed to reach the
visco-inertial regime is smaller for larger fibers (figure 5).
In the previous section, we had shown the independency of
the viscous boundary layers with fiber radius. But we also
see that the deposited thickness along the fiber varies in the
same time. The coating potential is the same for different
fiber radii; the incoming flow of fluid is the same. But the
pressure imposed by the fiber curvature is less important in
large radius case. So the dynamic meniscus is easier to
enlarge. The coating thickness is bigger. This last argument
also explains the earlier transition between the
visco-capillary regime and the visco-inertial one. The fluid
in motion will need less energy to destabilize the dynamic
meniscus and to enlarge it. The transition appears for
smaller withdrawal rates.

0.9 A R-40pm ,
S :- ------ R-70pm
S0.8 -.---... R=120pnm
0.7 -


cR 0.3


Figure 5: Plot of dimensionless coating thickness as
function of capillary number for R=40 to 200 pum.
Figure 5 show us the direct comparison of the dimensionless
coating thickness in function of capillary number for
different radius. For a large fiber radius, all regimes are
observed in a smaller range of capillary numbers. The
asymptotic one has a small domain of existence. It also
allows us to see that the asymptotic regime is conserved for
a larger range of capillary numbers for a small fiber. In this
case, the pressure imposed by the fiber curvature is strong
enough to resist to the fluid incoming intensity. The balance
between the growing fluid energy resulting in the dynamic
meniscus enlargement and the viscous layer thickness
decrease is kept. The coating thickness remains steady. We
also observe that to reach the asymptotic regime at low
withdrawal rate, it will be better to work with large radius
fiber. But we will lose in coating efficiency. Even if the
coating thickness is larger with a bigger fiber radius, the
thickness gain is not as large as radius enlargement. The
ratio e/R is smaller. The comparison between figure 3 and
figure 5 leads to a better understanding of the coating
efficiency. We can see that, for a same capillary number, the
dimensionless coating thickness is greater for the R= 120 pm
radius fiber than R=70pm one in figure 5 but the ratio e/R is
smaller in figure 3. The coating process loses in efficiency
by increasing fiber radius.
The boundary layer regime is observed for coating with
fiber of R=70 um, R=120 rpm and R=200 pm radii in the
simulated range of withdrawal rates (from v=0,5 m.s-1 to
v=6,0 m.s-1). We noticed that the coating thickness tends to
the same value with withdrawal rate increasing. If we had
simulated a larger radius of fiber, we certainly would have
found the same behaviour as plate coating with an abrupt
transition between the visco-inertial regime and the
boundary layer one. To conclude with the radius influence
analysis, we want to point out that using small fiber radius
in high speed coating process allows to soft regime
transition process and a better coating efficiency, e/R, with a
same withdrawal rate. Indeed the coating efficiency is better
for small fiber coating and the asymptotic regime is kept
longer which is a good argument for an industrial point of
view. The coating thickness depends on the influences of the
fluid moving forces such as viscosity and inertia and the
retaining fluid forces imposed by fluid surface tension and
fiber curvature. Influences, we found out to be responsible

Paper No

Paper No

of numerous coating thickness destabilisations.

Dynamic meniscus drop transportation

A phenomenon of sporadic transportation appears during the
coating process with the withdrawal rate increasing. It starts
with the Weber number of order of unity such as the
visco-inertial regime and is the result of the competition
between fluid inertia and pressure imposed to the dynamic
meniscus. We earlier explain, in the dynamic meniscus and
viscous boundary layer developments section, the
importance of fluid bumping into the free surface in the
dynamic meniscus formation.

t=0 ms

t=8 ms

t=32 ms

t=48 ms

Figure 6: Images of sporadic transportation out of the
dynamic meniscus for a R=120 tm fiber coating at
visco-inertial regime beginning.

The viscous boundary layers can be divided into three parts.
The part of fluid by the fiber, where kinetic energy exchange
has the best yield, is the fluid fraction which enlarges the
dynamic meniscus and is transported out forming the
coating thickness. The smooth part is the fluid fraction with
less energy, far from moving fiber. That part flows back into
the liquid bath. The last part is the entrapped one inbetween
the both named before. In this last part, the velocity gradient
is intense enough to enter in the dynamic meniscus but is
stopped in it by the surface tension and the curvature
pressure. It results in an enlargement of the dynamic
meniscus basis (figure 6.b.). This enlargement extends itself
with the incoming fluid, as we are considering continuous
process (figure 6.c.). But with such an extension, velocities
gradients appear because of a less effective viscous
transmission of the fiber energy though the thickened
meniscus. The external layers are slowed and the surface
tension forces can act to minimise the interfacial energy. A
drop is transported out of the dynamic meniscus (figure 6.d.).
This phenomenon is a periodic one and we characterized it
in function of coating regimes to define its domain of
existence and its consequences.
As transportation are the direct consequence of the
influences of the inertia forces and the pressures imposed to
the dynamic meniscus, we now work with Weber
numbers (6). Working with such a dimensionless number
will allow us to consider same intensity in influence
between the two forces for the different coating cases.
Transportation start with the visco-inertial regime but the
first withdrawal rates of that regime are not periodicals.

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

Some drops are expulsed but without rhythm and similitude
in expulsion size. So expulsions are sporadic. We also
remark that even if thickness destabilisations are observed
in the boundary layer regime, they are chaotic and of small
size in regard to the deposited thickness. So phenomenon is
principally occurring in the visco-inertial and asymptotic

Figure 7: Evolution of dimensionless coating thickness as
function of time for Weber number around 10 for coating of
20 mm depth bath with various radii.

The fiber radius has, as in regime identification, a great
influence on drop transportation. The pressures imposed on
the dynamic meniscus are less important in the case of large
fiber. So the coating thickness represents a larger fraction of
the developed viscous boundary layers. As a consequence,
the thickness of entrapped film, the part able to counter
pressures, is thinner; drop expulsions are of smaller
amplitudes but also less frequents. The entrapped thickness
has to enlarge itself to affect the meniscus curvature.
Figure 7 is a representation of those arguments: for a Weber
number around 10, expulsions are more frequent and thicker
for a R=40 gtm coating than the others.

0.7 -
S -- R-40pm
A0.6 ..------- R-70pm
S 0.6 ---.-'-.-.-.. R-l2Oprm
S'- ------- R- 20pm
0.5 -
o I

0 0.3 \


10 10' 102
0.1 i t


Figure 8: Evolution of the transportation size as function
of the Weber number for coating of 20 mm depth bath with
various radii.

Frequency of transportation and expulsion sizes decrease

Simulated time (s)

Paper No

with fiber radius. Moreover, we have to keep in mind that an
equivalent Weber number comparison imposes bigger
withdrawal rate for a small radius fiber. The coming flow
is more intense and the regeneration of entrapped fluid is
faster. The same argument is used to understand the
transportation frequency increase considering a given radius
but with various withdrawal rates. This frequency increases
until the drop size is small enough to consider the end of the
phenomena which corresponds to boundary layer regime
Now we consider the drop transportation in term of drop
sizes. Figure 8 shows the evolution of drop sizes,
instabilities of the coating thickness, in regard to the coating
thickness as function of the Weber number. We clearly
understand here that working in the beginning of the
visco-inertial regime is harmful in term of coating stability.
A better compromise is to work at high withdrawal rate near
to the end of the asymptotic regime: the destabilisations are
frequents and transportation sizes are around 5% of the
coating thickness.

Water Tin-lead eutectic and Titanium c. ,,,,

The global influences of coating parameters are now
understood thanks to the previous sections. The idea is now
to understand which fluid parameters are influencing the
coating thickness and so going back to industrial issues.
Water, Tin-Lead eutectic (named Tin-Lead) and Titanium
are all low viscosity fluids but their other characteristics are
very different (see table 3).

Fluid Tfuson 11 P Y
(C) (mPa.s) (kg.m3) (N.m1)

Water 0 1 1000 0.07275

Tin-Lead 183 2 8843 0.52*

Titanium -1700 2 4506 1.5

*Howell & al. (21. '4)

Table 3: The fluid characteristics.


-- Titanium
------- Water



/I / I

0' 10D 10'

Figure 9: Coating thickness in function of Weber number
for the three fluids.

Considering the calculated Bond numbers (1) for coating
with those fluids, we can conclude that the fluid density

7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30-June 4, 2010

does not influence the coating. But considering all previous
conclusions, we know that the surface tension has a great
influence. Figure 9 sums up coating results obtained
experimentally for Tin-Lead and numerically for Titanium
and Water. As expected, the first role of surface tension is to
maintain the fluid into the bath. The deposited thickness is
larger for Water than for Tin-Lead eutectic for an equivalent
Weber number and is much larger than Titanium. So with
surface tension increasing, the coating efficiency decrease
for an equivalent Weber number.
The transition between regimes appears for equivalent
Weber numbers too; but we have to keep in mind that the
withdrawal rate is much higher for a titanium coating
process. So the surface tension forces do not interfere in
regime existence but the withdrawal rate to achieve to coat
in the asymptotic regime is much larger. In addition, at such
withdrawal rates, chemical problems are appearing because
of reduces hot time contact between the filament and liquid
By simulating titanium coating, we also wanted to know if
the surface tension forces affect the drop transportation
frequency and the transportation sizes. Concerning
transportation drop sizes in regard to the coating thickness,
the comportment is quite the same and is principally due to
the high withdrawal rate use to achieve the visco-inertial
regime and the steady state one. The withdrawal rate being
much higher, the entrapped fluid enters in the dynamic
meniscus speeder and destabilisations are occurring faster.
The transported drops remain small even if with such a
surface tension, we were expecting bigger drop considering
force needed to destabilize the dynamic meniscus. Indeed
the frequency of drop transportation is much higher than
for water for an equivalent Weber number.

Titanium drop frequency
500 ------. Water drop frequency
Titamium dimensionless drop size
450 ------- Water dimentionless drop size O.9
E400 0.8 .
5350 0.7
L300 0.6 8
0250 0.5
0200 0.4

100 0.2
S150 1.0 1


Figure 10: Frequency of transportation and relative
transportation size for titanium and water in same coating


Study of the fiber radius influence on coating thickness and
its destabilizations was studied. Four regimes are identified
in both cases. The visco-capillary regime is well identified
and the competition between viscous and capillary forces
leads to a small increase of the coating thickness with the
withdrawal rate. The coating thickness in this regime is

Paper No

quite steady at the exit of the dynamic meniscus but a
gradient of velocity could appear far from the meniscus.
That gradient could allow the destabilisation of the coating
thickness by surface tension forces into a Rayleigh-Plateau
modus. Increasing the withdrawal rate implies the
emergence of inertia forces. The fluid put in motion bumps
more effectively into the free surface and the dynamic
meniscus is thickened. A divergence in coating thickness is
observed and the entrapped fluid phenomenon begins. It
results in slow and large drop expulsions. As drops are large,
they can be affected by gravity and merge together to form
unsteady bigger drops. That regime is the worst one to coat
a fiber. It is followed by the asymptotic one which presents
the best compromise for fiber coating. That regime can have
a large domain of existence in choosing a good radius for
the fiber to coat. It also presents small and frequent
transported drops. In that regime, if the insides flow of the
coating thickness is strong enough, small drops will remain
steady and so avoid to growth. The last regime observed is
the boundary layer one. It concerns the higher withdrawal
rates and it characterized by a decrease of the coating
thickness proportionally to the viscous boundary layers.


This work was made possible with the support of the arming
general delegation for France Defense (DGA) and of


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