# Generation of Giant Soap Films

Marina Pasquet1, Frédéric Restagno1, Isabelle Cantat2, Emmanuelle Rio1,
1 Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France.
2 Université de Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251,
35000 Rennes, France.

###### Abstract

Artists regularly make soap bubbles several meters long. In this article we
make soap films up to two meters high by pulling a horizontal fishing line
driven by belts out of a soapy solution at velocities ranging from 20 cm/s to
250 cm/s. We characterize the thickness profile of the central part of the
film that behaves like a rubber band under tension. We show that its thickness
profile is well described by a static model in which a homogeneous elastic
film is stretched by its own weight. This leads to an exponential thickness
profile with a characteristic length given by a competition between gravity
and surface elasticity. The prefactor is fixed by the shape and area of the
film, governed by the fishing line motion but also by a continuous extraction
of foam film from the lateral menisci, thicker than the central part, and that
progressively invades the film from its lateral boundaries. The model we
propose captures the subtle interplay between gravity, film elasticity and
film extraction and leads to predictions in good agreement with our
experimental data.

##  1 Introduction

Bubble artists are able to create giant soap bubbles and films, i.e. larger
than one meter, on a daily basis in their artistic performances. One example
is Graeme Denton’s world record for the largest vertical soap bubble exceeding
10 m in height, generated by pulling a ring out of a bath containing a soap
solution at a velocity of about three meters per second Denton (2020). In the
artistic field, these records are regularly broken and this one dates from
2020. From a fundamental point of view, the existence of these giant objects
raises many questions that are still to be elucidated, such as what limits
this maximum size, the role of the entrainment velocity to make them, or even
what sets the thickness profile of these giant films. Inspired by these
ephemeral giant objects, we studied the generation of planar soap films up to
two meters in size.

In the literature, soap films of comparable size are always fed from above
Ballet and Graner (2006); Kellay and Goldburg (2002); Kellay (2017); Rutgers
et al. (2001); Sane et al. (2018); Salkin et al. (2016); Kim and Mandre
(2017). Therefore we have developed a new experimental setup, presented in
detail in a previous article Mariot et al. (2021). The principle of the film
generation is identical to what has been developed to study small soap films
Lionti-Addad and Di Meglio (1992); Cohen-Addad and di Meglio (1994); Adelizzi
and Troian (2004); Berg et al. (2005); Saulnier et al. (2011, 2014); Seiwert
et al. (2017); Champougny et al. (2018) up to 10 cm: a frame made by a fishing
line is pulled out of a bath at a controlled velocity and drags a soap film.
At the small velocities (up to 10 cm/s) usually used in such experiments, the
thickness at the bottom of the films $h_{\text{Fr}}$ is given by a balance
between viscous stress and capillarity and follows Frankel’s law Mysels et al.
(1959):

$h_{\text{Fr}}\ =\ 1,89\ \ell_{c}\ \text{Ca}^{2/3}$ (1)

where $\ell_{c}=\sqrt{\gamma_{0}/(\rho g)}$ is the capillary length and
$\text{Ca}=\eta V/\gamma_{0}$ the capillary number. $\gamma_{0}$ is the
surface tension of the solution, $\rho$ its density, $\eta$ its viscosity, $g$
the gravitational acceleration constant and $V$ the entrainment velocity. The
good agreement between this law and experimental data has been studied
extensively as well as the deviations observed with complex fluids (Cohen-
Addad and di Meglio, 1994), at high velocities (Berg et al., 2005; Seiwert et
al., 2014a; Champougny et al., 2015) or at the top of the film (Saulnier et
al., 2011).

During the film generation, the fluid velocity is much smaller close to the
bath than close to the top fishing line. This leads to a stretching of the
film and to a dilution of surfactants at the interface. In the limit of poorly
soluble surfactants, the creation of the film surface area leads to a
diminution of the surfactant concentration $\Gamma$ and thus to a higher
surface tension $\gamma$. As a consequence, the film behaves as an elastic
membrane which pulls on its sides to relax towards its initial area and
surface tension. Surface tension and surfactant surface concentration are
linked by the Gibbs surface elasticity Gibbs (1928); Prins et al. (1967),

$E=-\frac{\partial\gamma}{\partial\ln\Gamma},$ (2)

which quantifies the surface tension variation subsequent to a change in
surface concentration. This results in a surface tension in the film higher
than in the meniscus, where the tension is assumed to remain close to its
equilibrium value $\gamma_{0}$. The interfacial stress denoted
$\gamma_{0}+\Delta\gamma$ leads to the extraction of a film from the menisci,
especially along the lateral ones Cantat (2013); Seiwert et al. (2014b). The
hydrodynamic laws in the vicinity of the meniscus impose a coupling between
$\Delta\gamma$, the extracted film thickness and the extraction velocity. In
the limit of high Gibbs modulus and in a steady state regime, this coupling is
derived from the Frankel’s law (Eq. 1) Mysels et al. (1959); Seiwert et al.
(2014a); Cantat (2013):

$\Delta\gamma\ =3.85\ \gamma_{0}\ \text{Ca}^{2/3}=2.03\ \gamma_{0}\
\frac{h_{\text{Fr}}}{\ell_{c}}$ (3)

A film element at a given altitude is stretched not only by the entrainment of
the fishing line but also by the weight of the film underneath (Lucassen-
Reynders, 1981; Couder, 1981; de Gennes, 2001). This leads to an additional
vertical gradient of surface tension in the film and therefore to a gradient
of extraction velocity along the vertical meniscus. The main goal of this
article is to measure the thickness profile in the central part of the film
during its generation, and to predict its evolution with a model that takes
into account these different effects.

In this article, we first present in the part 2 the setup used to generate
giant vertical films (up to 2 m) by pulling a fishing line out of a liquid
bath at high velocity (between 20 and 250 cm/s). Our experimental results
obtained by visualization and spectrometry will be described in part 3 and we
will propose an in-situ measurement of the surface tension gradient in a
static film. The hydrostatic model, which predicts the film thickness profile
and of the surface tension gradient is proposed in the part 4 together with a
numerical resolution of the proposed model. A direct comparison with
measurements will be presented at the end of the part 5.

##  2 Presentation of the Experimental Setup

Figure 1: Schematic representation of the experimental setup allowing the
generation of giant films.

The experimental setup for the generation of vertical soap films has been
described previously Mariot et al. (2021) and we will only outline the main
features here. The films are formed by pulling a fishing line (in fluorocarbon
with a diameter of 0.74 mm) out of a glass container, as we can see in the
Fig. 1. This wire is driven out of the soap solution by two transmissions
toothed belts. These belts are themselves driven by two coupled stepping
motors (SM56 3 18 J4.6 from Rosier Mecatronique) associated with pulleys. The
motors were chosen in order to be able to control the velocity of the
generation of the films between 20 and 250 cm/s. The maximum size of the films
we can generate with this device is 2 m high and 70 cm wide.

The setup allows automatic detection of the presence of the films, by
measuring the reflection of a white light by three photoresistances. In
addition, thickness measurements are also possible using a UV-VIS spectrometer
(Ocean Optics Nanocalc 2000) associated with an optical fiber of 400 $\mu$m
diameter. This spectrometer, placed at the center of the films in the $y$
direction (as defined in Fig. 1), allows a measurement of the local thickness
and can be moved on the whole height of the films. To control the humidity in
the environment where the films are generated, we used two commercial
humidifiers (one located at the bottom of the device, the other at the top of
the chamber). The bottom humidifier (Okoia AH450) allows us to select the
target humidity and the top humidifier (Bionaire BU1300W-I), whose flow rate
is adjustable, allows us to have a more homogeneous humidity in the chamber,
especially when we want to work in a very humid environment. In the following,
the humidity is fixed at $85\pm 4$ %, so we will neglect evaporation. The
temperature is measured throughout the experiments with a typical uncertainty
of 0.1 ∘C and corresponds to the ambient temperature, which is about 22 ∘C.

To make direct visualizations of the soap films, we illuminate them thanks to
a large white photo studio background paper (BD 129 Super White 2.72 × 11 m),
which serves as a light reflector for four halogen lamps. We made a hole in it
to fit the wide-angle lens (Nikon AF-P DX 10-20 mm) of a camera (Nikon D7200).

For each experiment, we use 5 liters of soap solution made of 4 % dishwashing
liquid (Fairy from Procter & Gamble, containing between 15 % and 30 % of
anionic surfactants) with ultrapure water, with resistivity greater than 18.2
M$\Omega\cdot$cm.

##  3 Experimental characterization of giant soap films

In this section, we present experimental observations and thickness
measurement data obtained using the experimental setup presented in part 2.

###  3.1 Experimental observations

Figure 2: Photographs showing the evolution over time of a soap film generated
at a fixed velocity of 200 cm/s. Thicker films extracted from the lateral
menisci are observed: they are called Frankel films and are noted (F). The
central film is noted (C).

The qualitative behavior of the giant film during its generation is
illustrated in Fig. 2, the interferential colors providing us a direct
indication of the film thickness. We can see here that there is a central part
(C) in which the film color is uniform at a given height, indicating a
thickness stratification by the gravity. On the image on the left, on both
sides of this central film we observe two lateral parts (F), which are thicker
and not stratified. They correspond to the Frankel’s films, extracted from the
lateral menisci. The drawing of the boundaries in Fig. 2 was done by hand,
under ImageJ. In the following, we will focus on the measure and prediction of
the central film thickness profile. We will show that this necessitates a
description of the lateral films extraction dynamics.

###  3.2 Film thickness

Figure 3: Evolution of the thickness $h$ of soap films during the generation
as a function of time $t$. These measurements were performed by spectrometry
at the center of the films, at 13 cm above the surface of the bath. (a) Time
evolution of the thickness for $V=60$ cm/s. (b) Time evolution of the
thickness for all the velocities probed between 20 cm/s (purple) and 250 cm/s
(red).

Fig. 3 (a) shows the evolution of the thickness of a film entrained at a
velocity of 60 cm/s as a function of time, for a spectrometer position fixed
at 13 cm above the surface of the solution, at the center of the film (in the
$y$ direction). The time $t=0$ corresponds to the moment when the fishing line
leaves the bath. We observe that up to $t\simeq 1$ s there is a thinning of
the central part of the film. After reaching a minimum value around 1.2
$\mu$m, the thickness $h$ increases linearly with time. This non-monotonic
behavior is a robust phenomenon, as we can see in Fig. 3 (b), for all the
measured velocities. We will show that this can be recovered by the model
developed.

Figure 4: Evolution of the thickness $h$ of the soap films during the
generation as a function of the height $H$ reached by the films, for all the
velocities probed between 20 cm/s and 250 cm/s. These measurements are made by
spectrometry (in the center of the films in the $y$ direction) at 13 cm above
the solution surface (a) and at 30 cm above the surface (b). In the figure
(a), this is the same data as shown in Fig. 3 (b). With this type of
representation, all the data are superimposed on the same master curve.

.

In Fig. 4, we have plotted these thicknesses as a function of the height
$H(t)$ reached by the films during the generation, i.e. as a function of the
product $V\times t$, for two different spectrometer positions. This leads to a
reasonable collapse of all the data on a master curve and a similar non-
monotonic behavior is observed. The thickness evolution thus does not depend
on the imposed velocity, which rules out inertial and air friction effects in
the dynamics. The model will show that the central film can indeed be
described by a hydrostatic equilibrium, which can be disconcerting at first
sight since we generate the films at high velocities.

###  3.3 Film thickness profiles

Figure 5: (a) Thickness $h$ of the soap films measured during the generation
as a function of $z$ (oriented downwards, with the origin at the top of the
film), in semi-logarithmic scale, for a velocity fixed at 100 cm/s. For a
given instant, corresponding to a color, the value of $\ell$ extracted from
the exponential fit and the height of the films $H$ are written in the legend.
The average value is $\ell\simeq 0.82$ m. The points at the bottom of the
films (3 cm from the bath) not described by the exponential are shown in the
grayed area. (b) Measurement of the characteristic length $\ell$ on the
thickness profiles for different velocities (ranging from 20 cm/s to 140
cm/s). The values of $\ell$ are plotted as a function of the height reached by
the soap $H$ films during this generation. The average value of $\ell$
measured for all the probed velocities is indicated by a blue line. The
experimentally determined standard deviation is represented by the colored
rectangle. All our measurements lead to $\ell=0.80\pm 0.09$ m.

We measured the thickness as a function of time for several positions of the
spectrometer (between 3 and 180 cm above the liquid bath, in the center film
in the $y$ direction) in order to determine the whole film profile, in the
central part of the film. These profiles are given for five successive times,
and therefore for five different film heights, in Fig. 5 (a). Two zones can be
identified, despite a limited spatial resolution due to the difficulty of the
thickness measurement in films moving at high velocity. At small $z$, close to
the top of the film (white background in Fig. 5 (a)), the profile can be
fitted with an exponential model, whereas at large $z$, close to the liquid
bath (gray background), the thickness is systematically larger than the
extrapolation of this exponential profile. In the following, the exponential
part will be referred to as the initial film. Indeed, at short time, a small
quantity of liquid is entrained by the fishing line and forms an initial film.
As we will see later, this initial film is a closed system, which is stretched
by the fishing line movement and by the film weight. The physical process
governing this initial film extraction seems to differ from the visco-
capillary law (Eq. (1)) relevant at larger times.

The measurements made for seven entrained velocities, ranging from 20 cm/s to
140 cm/s, lead to the same results. The top part of the profiles have been
fitted by the law

$h(z)=K\,\text{e}^{(z-H)/\ell}$ (4)

with $\ell$ a characteristic length and $K$ a prefactor. These lengths are
plotted in Fig. 5 (b) as a function of the film height (during the film
generation). The values of $\ell$ obtained suggest that the characteristic
length remains constant during the film’s generation and does not depend on
the entrained velocity. Taking into account all the experiments, we measure a
characteristic length $\ell=0.80\pm 0.09$ m.

In the following (part 4) we will develop a hydrostatic theoretical
description of the films during their generation to rationalize these
observations.

###  3.4 Area of lateral Frankel films

We determined the area $A_{\text{Fr}}$ of the lateral Frankel’s films by
direct visualization, by dividing the films as described in the section 3.1.
The ratio between $A_{\text{Fr}}$ ($A_{Fr}$ takes into account both sides of
the film) and the total area $A$ of the film is given as a function of the
film height $H$ for three different velocities in Fig. 6. This ratio, like the
central film thickness, is governed by the film height and is independent of
the entrained velocity.

Figure 6:  Evolution of the proportion of the area of the lateral Frankel
films $A_{\text{Fr}}$ to the total area of the films $A$ as a function of the
height $H$ reached by the soap films during their generation for three
different velocities: 60 cm/s, 100 cm/s and 140 cm/s.

###  3.5 Surface tension gradients in the films

As explained in the introduction, a vertical surface tension gradient due to
gravity is expected along the giant soap film, which we propose to measure in
this section. We perform this measurement for a static (i.e., after the
generation) soap film of 120 cm high.

The surface tension is deduced from the deformation of a small elastic object,
put at different heights in the film. This method was initially proposed by
Adami et al. (Adami and Caps, 2015), and used in 15 cm high fed films.

The elastic object used for the measure is shown in Fig. 7 (a). The sensor is
made with Silicone SKIN FX 10 (from Rougier & Plé), using a PTFE mold (see
Fig. 7 (b)). This matter is usually used to make prosthesis or special effects
in movies, and makes it possible to manufacture elastic objects in less than
an hour.

The obtained sensor is held at the chosen height in the plane of the film by a
thin wire attached to the horizontal fishing line entraining the films, and
the piece of film trapped inside is pierced to perform the measure. The
unbalanced tension forces outside the object deform the arms, as its shape
tends to get closer to a circle. This deformation is quantified by the
distance $\delta$ defined in Fig. 7 (a). The deformation $\delta$ varies
linearly with the tension around the object (Adami and Caps, 2015):

$\gamma\ =\ \frac{16e^{4}E}{l^{4}}\ \delta$ (5)

with $e$ the thickness of the arms of the elastic object ($e=0.8$ mm), $E$ the
Young’s modulus of the object ($E\simeq 0.4$ MPa) and $l$ the height of the
arms ($l=24$ mm). As the Young modulus of the object is not precisely known,
the prefactor of the linear relation (5) is obtained from a calibration with
horizontal soap films of known surface tension: a solution containing TTAB at
a concentration of 10 cmc ($\gamma=39$ mN/m), a solution containing Fairy
diluted to 4 % ($\gamma=25.2$ mN/m) and a solution containing Tween 20 at 1 %
($\gamma=33.7$ mN/m).

From an experimental point of view, this measurement is a real challenge
because the films last only a few seconds.

Figure 7: (a) Example of an image recorded during an experiment. The arms of
the elastic object are deformed due to the surface tension forces. (b)
Photograph of the PTFE mold used to fabricate the elastic object. (c) Surface
tension $\gamma$ measured at the center (in the $y$ direction) of a film of
height H=120 cm as a function of the coordinate $z$, just after the motors
stop. The black line corresponds to the surface tension $\gamma^{\text{th}}$,
obtained by integrating Eq. (6) using the film thickness profile given in
inset, with the equilibrium tension at the film bottom. Inset: film thickness
obtained by spectrometry at two different high. The whole profile (black line)
is the prediction of Eq. (4) with $\ell=0.8$ m taken from Fig. 5 and $K=2.4$
$\mu$m as adjustable parameter.

The results obtained for a 120 cm high static film, entrained at a velocity of
100 cm/s, are shown in Fig. 7 (c). The measurements begin just after the
motors stop, and last 10 seconds. The thicknesses given in the inset are
average during this time period. We measure a surface tension difference,
between the top and bottom of the film, of about 5 mN/m, which we will compare
to a theoretical prediction in the next paragraph.

##  4 Theoretical description of the soap films during their generation

###  4.1 Equilibrium film thickness profile under gravity

A static model build by Lucassen (Lucassen-Reynders, 1981) and used by de
Gennes (de Gennes, 2001) relies on the fact that the weight of the soap films
is balanced by the surface tension gradient so that

$\rho gh+2\ \frac{\partial\gamma}{\partial z}=0\,,$ (6)

with $h$ the thickness of the soap film and with the vertical axis $z$
oriented downwards.

This equation is simply a force balance in the vertical direction, per unit
area, on a film element of volume $\mathrm{d}V=h\,\mathrm{d}A$ spanning from
one interface to the other.

The validity of Eq. (6) can be tested using the tension measurements made in
120 cm high static film, described in the section 3.5. To obtain the
theoretical profile shown in Fig. 7 (c), inset, we assumed an exponential
variation of the thickness with $z$. The measurements are compatible with the
characteristic length $\ell=0.8$m, as obtained for all films during generation
(see Fig. 5). The theoretical tension profile is then obtained by integration
of Eq. (6), using the theoretical profile and $\gamma_{0}=25.2$ mN/m,
$\rho=997$ kg/m3 and $g=9,81$ m/s2. The measure, shown in Fig. 7 (c), is
slightly below the prediction proposed, but with an agreement close to the
experimental uncertainty.

The surface tension involved in Eq. (6) depends on the interface surfactant
concentration $\Gamma$. For small coverage variations, an interface elasticity
$E$ is defined by Eq. (2) and a linearized expression of the surface tension
can be expressed as $\gamma=\gamma_{0}-E(\Gamma-\Gamma_{0})/\Gamma_{0}$.

Here, we will focus on films stabilized by a dishwashing liquid and we will
assume that the film elasticity is mainly due to poorly soluble surfactant
molecules, of negligible bulk concentration. In that case, a simple
relationship between $E$ and the film mechanical dilatationnal elasticity,
i.e. between the film tension and dilatation or compression can be obtained,
once the reference state of each film element is defined.

In this aim, we consider a film element of volume $\mathrm{d}V$ and define its
reference state, indicated by the subscript 0, as the state in which the
interface concentration is in equilibrium with the bulk liquid used to produce
the film. It is characterized by its area $\mathrm{d}A_{0}$, its thickness
$h_{0}$ and the interface concentration $\Gamma_{0}$. The relative motion of
the bulk phase with respect to the interfaces is a downwards Poiseuille flow,
with a velocity scaling as $\rho gh^{2}/\eta\sim 10\ \mu$m/s, which is
entirely negligible in our experiment. A film element is thus a closed
material system. Consequently, when the film element is stretched to reach an
area $\mathrm{d}A$, the volume conservation imposes
$h\mathrm{d}A=h_{0}\mathrm{d}A_{0}$. Moreover, the surfactant conservation
leads to $h\Gamma=h_{0}\Gamma_{0}$ and finally

$\gamma=\gamma_{0}\ +\ E\frac{\varepsilon}{1+\varepsilon}$ (7)

with the extension $\varepsilon$ defined by :

$\varepsilon=\frac{\mathrm{d}A}{\mathrm{d}A_{0}}\ -\ 1=\frac{h_{0}}{h}\ -\
1\,.$ (8)

As there are two interfaces, the film tension is simply $2\gamma$.

From these relations, we get $\partial_{z}\gamma=E\partial_{z}(h/h_{0})$, and
Eq. 6 becomes

$\dfrac{1}{h}\dfrac{\partial}{\partial z}\left(\frac{h}{h_{0}}\right)\
=\dfrac{\rho g}{2E}\,.$ (9)

In the case of soluble surface-active molecules, the film elasticity also
depends on the film thickness Prins et al. (1967); Van den Tempel et al.
(1965); Lucassen-Reynders (1981). In the frame of this model, the thickness
profile of the soap film is a 2D equivalent of a 3D compressible gas in a
gravity field.

The Eq. (9) governs the thickness profile of a film in a gravity field. It
predicts an exponential profile only if the reference thickness $h_{0}$ does
not vary with the position $z$. Indeed, in this case it can be integrated to
get

$h(z)=h(H)\,\text{e}^{(z-H)/\ell}\quad\text{with}\quad\ell=\frac{2E}{\rho
gh_{0}}$ (10)

the characteristic length of the problem. The exponential profile observed in
Fig. 5 is thus the signature of a homogeneous reference thickness in the
initial film.

This reference thickness is governed by the amount of surfactant trapped is
the film during its extraction. As it is constant in the upper part of the
film (defined as the initial film), its value does not depend on time during
the first moment of the film extraction. Even more surprisingly, it does not
depend on the motors’ velocity. This is an important experimental result that
we have not been able to elucidate yet, as discuss in the conclusion.

Indeed, as verified in Fig. 5 (b), the characteristic length $\ell$ is
constant with time, which is consistent with the fact that a film element is a
closed material system, keeping the same reference thickness at all times.

In order to determine the thickness profile $h(z)$ completely, it still
remains to determine the value of the thickness $h(H)$. We will show that it
only depends on the area and on the shape of the initial film.

###  4.2 Validity of the equilibrium force balance

The Eq. (6) is the projection in the vertical direction of the vectorial
equilibrium condition of the film $\rho gh\vec{e}_{z}+2\vec{\nabla}\gamma=0$.
Taking the curl of this equation, we obtain $\partial_{y}\gamma=0$, with $y$
the horizontal direction in the film plane. Now using $\rho
gh=-2\partial_{z}\gamma$ we obtain, after a derivation, the condition
$\partial_{y}h=0$. This condition is verified in the central part of the film,
but not in the lateral part, which is thus necessary in an out of equilibrium
situation (see Fig. 2).

The air friction on a moving film can be estimated using the Prandtl theory
Rutgers et al. (1996). It scales as $\eta_{a}V/\delta_{\text{BL}}$ with the
boundary layer thickness
$\delta_{\text{BL}}\sim[\eta_{a}H/(\rho_{a}V)]^{1/2}$, the air viscosity
$\eta_{a}=1.8$ kg/m/s, the air density $\rho_{a}=1.3$ kg/m3, and the film
height $H\sim 1$ m. In the initial film, the velocity varies from the motors’
velocity at the top to a much smaller velocity at the bottom (as will be shown
later). For the smallest motors’ velocity, $V=0.2$ m/s, the air friction is
thus at most of the order of 10-3 kg m-1 s-2 and should be negligible in
comparison with the gravity of the order $\rho gh\sim 10^{-2}$ kg m-1 s-2. The
highest motor velocities are close to the terminal velocities observed for
films of comparable thickness (see Fig. 4 in Rutgers et al. (1996)) and air
friction should begin to play a significant role. However, the rescaling
obtained in Fig. 4 shows that the influence of the motors’ velocity is small,
and the air friction will be neglected in the model. The inertia of the
central film scales as $\rho hV^{2}/H\sim 10^{-3}$ kg m-1 s-2 and will be
neglected too.

Finally, the surface shear viscosity leads to an in-plane friction force of
the order of $\eta_{s}\delta V/L^{2}$, with $\delta V$ a velocity difference
between two points of the film separated from a distance $L$ and $\eta_{s}\sim
10^{-7}-10^{-6}$ kg/s Stevenson (2005) the shear interface viscosity. The
thickness stratification in the central part of the film indicates a uniform
velocity in the $y$ direction, and this viscous friction does not contribute
to the force balance. The extracted films on the two film sides are thicker
than the central part and fall down under gravity. If the downward velocity
$V^{\text{Fr}}$ reaches 1 m/s, with a characteristic width
$L=w^{\text{Fr}}\sim 10^{-1}$ m we get an upward viscous term of the order of
$10^{-4}$ kg m-1 s-2 in the Frankel film, which is negligible. The higher
gravitational force in this thick part of the film is thus probably
compensated by the film inertia.

On the basis of the previous discussion and of the observed rescaling in Fig.
4, the dynamics of the initial film will be rationalized using the equilibrium
model of section 4.1. The modelization of the Frankel film motion is beyond
the scope of this paper, and its downwards flow will be taken into account
phenomenologically.

###  4.3 Modelization of the time evolution

The goal of this section is to predict the time evolution of the film
thickness profile, on the basis of the model leading to Eq. (9). The thickness
profile evolves because of the extraction of film from the menisci, which
allows the relaxation of the initial film towards its reference thickness.
Actually, this means that the area and shape of the initial film are the only
ingredients to take into account.

We thus propose to describe completely the generation of soap films by taking
into account the extraction of film from the menisci, at the bottom and on the
lateral sides.

####  4.3.1 Film parametrization and equilibrium equation

The central film is defined as the material system extracted from the bottom
meniscus stretched by the fishing line entrainment. We assume that it remains
invariant by translation in the horizontal direction, so the relevant
elementary systems are horizontal film strips of homogeneous properties. These
film elements are identified by their distance $x$ from the top meniscus,
measured in the reference state of the film, using a Lagrangian approach. $x$
is the position the film element would have in the absence of gravity and with
a film tension $2\gamma_{0}$ imposed at its boundary (see Fig. 8). With this
definition the film element of height $\mathrm{d}x$ is a closed material
system, that is followed during its motion and deformation. This Lagrangian
approach leads us to introduce the spatial variable $\xi$ defined such that
the position $z$ of the tracked element in the film is related to the variable
$x$ via the relation :

$z(x)=x+\xi(x)$ (11)

The shape of the film element is a rectangle
$\mathrm{d}x(1+\partial_{x}\xi)w(x)$, with $w(x,t)$ the width of the central
film, from one boundary with the lateral extracted film to the other. The
reference thickness $h_{0}(x)$ of the element of reference position $x$ is
determined by its extraction from the bath, which occurs at the time
$t^{\text{ex}}(x)$. On the basis of our experimental results, it will be
assumed uniform in the top part of the film, and then governed by the visco-
capillary film extraction (Eq. (1)). Note that the Lagrangian variable $x$
corresponds to a material system which keeps a reference thickness constant
with time, so $h_{0}$ may depend on $x$ but not on time. The width of the
element at $t^{\text{ex}}(x)$ is $w_{0}$, as the film extraction occurs along
the whole bottom meniscus length. This length is chosen as the reference width
for the film element.

The volume $\mathrm{d}V$ of a film element is conserved during a deformation,
which allows us to write

$w_{0}\ h_{0}(x)=(1+\partial_{x}\xi)\ w(x)\ h(x).$ (12)

Figure 8: Scheme of the film profile and notations used in the text. (Left)
Reference state of the film with a thickness $h_{0}(x)$ and a height $H_{0}$.
An elementary material system is defined between $x$ and $x+dx$. (Right)
Actual state of the film, in which the same elementary system is located
between $z(x)$ and $z(x)+dz(x)$.

Its area $\mathrm{d}A=w(x)(1+\partial_{x}\xi)\mathrm{d}x$ may in contrast vary
and the film extension $\epsilon(x)$ (Eq. (8)) is:

$\epsilon(x)=\dfrac{w(x)\ (\ 1+\partial_{x}\xi)}{w_{0}}\ -\ 1$ (13)

Within the central part of the film, the force balance is given by Eq. (6) and
(7) which becomes:

$\rho gh(x)\ (1+\partial_{x}\xi)\ =\ -2\ E\
\partial_{x}\left(\frac{\epsilon(x)}{1+\epsilon(x)}\right)\,.$ (14)

Using Eq. (12) we can reformulate this equation as a function of $h_{0}$, $w$
and $\varepsilon$ only

$\partial_{x}\left(\frac{\epsilon(x)}{1+\epsilon(x)}\right)=-\frac{1}{\ell}\frac{h_{0}(x)}{h_{00}}\frac{w_{0}}{w(x,t)}\,,$
(15)

with $h_{00}$ an arbitrary reference and $\ell=2E/(\rho gh_{00})$ the
characteristic length already defined in Eq. (10), (where $h_{00}$ is taken as
the uniform reference thickness $h_{0}$ of Eq. (10)).

This determines the extension $\varepsilon$ and thus the whole film profile if
the function $w(x,t)$ and $h_{0}(x)$ are known, as well as a boundary
condition at the film bottom. These quantities are determined from the film
extraction at the bottom and lateral menisci, and from the variation of the
film height $H$ imposed by the motors.

####  4.3.2 Film extraction

In the limit of high film elastic modulus, and in a steady state regime, a
film extraction at a velocity $U_{Fr}$ is associated with a film tension
$2\gamma_{0}+2\Delta\gamma$ (Eq. (3)).

The extraction rate can thus be related to the local tension in the film, and
using Eq. (7), to its deformation:

$U_{\text{Fr}}(x)=\beta\left(\frac{\epsilon(x)}{1\
+\epsilon(x)}\right)^{3/2}\quad\text{with}\quad\beta\ =\ k\
\frac{E^{3/2}}{\eta\gamma_{0}^{1/2}}$ (16)

and $k$ a numerical prefactor which theoretical value, deduced from Eq. (3),
is $k=0.13$.

The above equation assumes that the surface tension in the menisci is
everywhere equal to the reference surface tension $\gamma_{0}$. This is a
strong assumption, especially for the lateral menisci. If these menisci are
not sufficiently effective surfactant reservoirs and become depleted, there
can be a large error in the extraction velocity of the side films. To overcome
this difficulty, we have chosen to use $k$ as a fitting parameter. For the
physical situation studied in this article, the theoretical prediction of
$\beta$ with the Eq. (16) is: $\beta=2.63$ m/s, with $E=22$ mN/m Kim and
Mandre (2017), whereas the fitted value will be $\beta=0.25\pm 0.05$ m/s. Note
that this expression is only valid for a positive extraction velocity, so for
a stretched film, at a tension higher than its surrounding meniscus.

As previously discussed, the first moments of the film extraction is not
described by this process and an initial film of uniform reference thickness
is used as the initial condition. During the dynamics, some new film elements
are added in the system, thus increasing the range value for the Lagrangian
coordinate $x$ which varies between $x=0$ at the top and $x=H_{0}(t)$ at the
bottom. The film extraction at the velocity $U_{Fr}(H_{0})$ is associated to
the increase of $H_{0}$ at the rate

$\frac{\mathrm{d}H_{0}}{\mathrm{d}t}=\frac{U_{\text{Fr}}(H_{0})}{1+\epsilon(H_{0}(t))}$
(17)

The thickness $h_{\text{Fr}}$ of the extracted film elements is governed by
Eq. (3) and thus depends on the local deformation $\epsilon(H_{0})$ through
Eq. (16). The reference thickness $h_{0}$ can then be computed at $x=H_{0}(t)$
using Eq. (8)

$h_{0}(H_{0})=h_{\text{Fr}}(U_{Fr}(H_{0}))\ \times\ (1+\epsilon(H_{0}))$ (18)

The mechanism at the origin of film extraction from the lateral meniscus is
the same as the one at the origin of the extraction from the bottom meniscus.

The width of the lateral extracted film $w_{Fr}$ thus increases at the rate
$\partial_{t}w_{Fr}(x)=U_{\text{Fr}}(x)$ and its area varies as

$\frac{\mathrm{d}A_{Fr}}{\mathrm{d}t}=\int_{0}^{H_{0}}U_{\text{Fr}}(x)(1+\partial_{x}\xi)dx$
(19)

With this model the lateral film width $w_{Fr}(x)$ should be an increasing
function of $x$, first because the extraction begins sooner at smaller $x$
(because the film element is created earlier), and second because the tension,
and thus the extraction velocity is higher at the top of the film.

We observe experimentally that these lateral films are actually wider in the
central part (see Fig. 2). This is due to the fact that they are thicker than
the central part of the film, which leads to their fast downward motion. As
this dynamical effect is beyond our theoretical description, we choose to take
it into account through a phenomenological correction of the model. The area
evolution of the lateral extracted film is still the one predicted by Eq.
(19), but its shape is assumed to be a rectangle of uniform width
$\bar{w}_{Fr}$ verifying $H\bar{w}_{Fr}=A_{Fr}$. This artificial
redistribution of the lateral film elements mimics the downward lateral motion
in the simplest possible way, and keeps a realistic shape for the central part
of the film, which is the system of interest. The central film width is thus
assumed to be uniform, with $w(t)=w_{0}-2\bar{w}_{Fr}$.

###  4.4 Numerical resolution

####  4.4.1 Equations set

The film profile is determined by its reference thickness $h_{0}(x)$ and by
its deformation $\epsilon(x,t)$. As the model is based on an equilibrium
equation, it only evolves because of the boundary conditions evolution: the
variation of its width $w(t)$ and of its reference height $H_{0}(t)$. At each
time, $h_{0}$ is obtained at the bottom of the film from Eq. (18) and $w(t)$
is deduced from Eq. (19). The Eq. (15) can thus be numerically integrated if
the tension is known at a given point of the film.

The missing information is obtained from the total height of the film, imposed
by the motors motion:

$H(t)=\int_{0}^{H_{0}}(1+\partial_{x}\xi(x))\mathrm{d}x$ (20)

By using Eq. (13), we can express $\partial_{x}\xi(x)$ as a function of
$\epsilon(x)$, leading to

$H(t)=\int_{0}^{H_{0}}\frac{w_{0}}{w(x)}(1+\epsilon(x))\mathrm{d}x$ (21)

Introducing the auxiliary function

$f(x)=\frac{\epsilon(H_{0})}{1+\epsilon(H_{0})}-\frac{\epsilon(x)}{1+\epsilon(x)}$
(22)

we get, from Eq. (15),

$f(x)=-\frac{1}{\ell}\int_{x}^{H_{0}}\frac{h_{0}(x)}{h_{00}}\frac{w_{0}}{w(t)}\mathrm{d}x$
(23)

The function $f$ is thus expressed as a function of the known quantities
$h_{0}$ and $w$. The Eq. (21) leads to

$H(t)=\int_{0}^{H_{0}}\frac{w_{0}}{w(t)}\frac{1}{1-f_{0}+f(x)}\ \mathrm{d}x,$
(24)

wich provides an implicit equation for the parameter
$f_{0}=\epsilon(H_{0})/(1+\epsilon(H_{0}))$. Once $f_{0}$ has been determined
from Eq. (24), the film extension can be deduced from Eq. (22), which closes
the model.

The problem can in principle be solved from an initial reference film height
$H_{0}(0)=0$, thus addressing the first moments of the film extraction from
the bottom meniscus. However, using our experimental parameters, this model
leads to unrealistic prediction for the first 10 cm, and we assume that
inertial effects are non-negligible at this stage. We thus initialize the code
with a small film of height $H_{0}$, and of uniform thickness $h_{0}$ and
width $w_{0}$. This initial film corresponds to the liquid entrained by
inertia and viscosity at very short time.

####  4.4.2 Numerical scheme

The numerical resolution of the model was carried out under Python. The film
is discretized in $n$ strips. Their reference height $\mathrm{d}x$ is uniform,
except for the bottom strip which has a variable height $\mathrm{d}x^{\ast}$.
At each time step, the height $H$ is increased by $V\mathrm{d}t$, with $V$ the
velocity of the motors. We compute $f(x)$ in each film element via Eq. (23).
We determine the tension at the bottom meniscus $\epsilon(H_{0})$ (Eq. (22))
by solving Eq. (24) to find the value of $f_{0}$. The tension $\epsilon(x)$ in
each element is deduced from $f(x)$ using Eq. (22). The thickness $h(x)$ is
computed using Eq. (8). If the tension at the bottom of the film is larger
than $\gamma_{0}$, the film length $\mathrm{d}z_{\text{Fr}}=U_{\text{Fr}}\
\mathrm{d}t$ is extracted from the bottom meniscus. The height of the last
element $\mathrm{d}x^{\ast}$ is thus increased by

$\mathrm{d}x_{\text{Fr}}=\frac{U_{\text{Fr}}\
\mathrm{d}t}{1+\epsilon(H_{0})}\,.$ (25)

and the reference thickness $h_{0}(H_{0})$ of the last segment is recalculated
so that $h(H_{0})=h_{\text{Fr}}$ (Eq. (1)) and
$h(H_{0})=h_{0}(H_{0})/(1+\epsilon(H_{0}))$. As soon as
$\mathrm{d}x^{\ast}\geq 2\mathrm{d}x$, the strip $n$ is divided in two: the
strip $n$ with a reference height $\mathrm{d}x$ and the strip $n+1$ with the
reference height $\mathrm{d}x^{\ast}\ -\mathrm{d}x$. The height of the film at
rest is recalculated as $H_{0}=n\mathrm{d}x$. The increase of the lateral
extracted film is computed from Eq. (19). The central film width is then
obtained from

$w=w_{0}-2\ \frac{A_{\text{Fr}}}{H}.$ (26)

Figure 9: Evolution of the proportion of the area of the lateral Frankel films
$A_{\text{Fr}}$ to the total area of the films $A$ as a function of the height
$H$ reached by the soap films during their generation for three different
velocities: 60 cm/s, 100 cm/s and 140 cm/s (data identical to those shown in
Fig. 6). The black dashed line is the numerical resolution of the model for
$V=100$ cm/s and $\beta=0.26$ m/s.

###  4.5 Results of the numerical resolution

####  4.5.1 Choice of the numerical parameters

There are only two numerical parameters in the equations set: the
characteristic length $\ell$ and the prefactor $\beta$ involved in the
Frankel’s law. Their values, as well as the film profile at short time, have
been determined from the experimental data, as discussed below.

The experimental profiles are exponential in the initial film, with a
characteristic length (Fig. 5) $\ell=0.80\pm 0.09$ m identical for all the
velocities. This implies that the reference thickness

$h_{00}=\frac{2E}{\rho g\ell}$ (27)

is uniform and independent of the motors’ velocity.

Using the experimental value of $\ell$ and $E=22$ mN/m (Kim and Mandre, 2017),
we obtain that $h_{00}\simeq 5.6\ \mu$m. We initiate the numerical simulation
with a film of reference height $H_{0}=10$ cm, and a reference thickness
$h_{0}=h_{00}=5.6$ $\mu$m. The motors’ velocity implemented in the simulations
is $V=100$ cm/s.

Figure 10: Evolution of the proportion of the area of the lateral Frankel
films $A_{\text{Fr}}$ to the total area of the films $A$ as a function of the
height $H$ reached by the soap films during their generation for three
different velocities: 60 cm/s, 100 cm/s and 140 cm/s (data identical to those
shown in Fig. 6). The black dashed line is the numerical resolution of the
model for $V=100$ cm/s and $\beta=0.26$ m/s.

As discussed before, we used the parameter $\beta$ in Eq. (16) as an
adjustable parameter. Its value is determined from the experimental data shown
in Fig. 10: the proportion of the Frankel film area as a function of time
obtained numerically fits the data for $\beta=0.26$ m/s. This value is ten
times lower than the one expected if the surface tension in the menisci was
equal to that of the bath, which indicates that the lateral menisci are at a
tension much higher than the equilibrium tension and that they are depleted in
surfactants.

####  4.5.2 Numerical results

The numerical results allow us to quantify the various processes involved in
the dynamics. We will first discuss the properties of the initial film, which
can be easily followed at different times, using our Lagrangian formalism. A
salient feature is the competition between the film extension, due to the
motors’ motion, and the film extraction, which mitigates this extension. This
competition is clearly illustrated by the initial film area, plotted in Fig.
11 as a function of time: it increases during the first second, remaining
close to the total film area imposed by the motors’ motion, and decreases at
longer time, as the Frankel films are invading the films. The largest part of
the extraction occurs in the lateral menisci, and the total area of the
central film remains close to the area of the initial film.

Figure 11: (Left) Evolution of the areas of the initial film $A_{\text{i}}$
(solid black line) and of the total film area $A$ (dashed black line) and
(Right) evolution of the width $w$ (dotted green line) as a function of time,
obtained numerically.  Figure 12: Numerical resolution results for the
generation of a film of a width $w_{0}=70$ cm and an initial height for the
reference state $H_{0}=10$ cm, at a velocity $V=100$ cm/s. Each color
corresponds to a different time $t$, indicated by the vertical color bar. The
difference between two represented instants is $\Delta t=0.19$ s. (see next
page)  Figure 13: Evolution of (a) the extension $\epsilon$ as a function of
the $z$ coordinate, (b) the tension $\Delta\gamma=\gamma-\gamma_{0}$ as a
function of the $z$ coordinate, (c) the extraction velocity of the lateral
Frankel films $U_{\text{Fr}}$ as a function of the $z$ coordinate, (d) the
dimensionless thickness $\widehat{h}=h/h_{00}$ for the initial film as a
function of the $x$ coordinate, (e) the dimensionless thickness $\widehat{h}$
for the initial film as a function of the $z$ coordinate, (f) the
dimensionless thickness $\widehat{h}$ as a function of the $z$ coordinate for
the whole film: the initial stretched film (whose data are represented in the
figure (e)) and the Frankels films extracted from the meniscus with the bath.
The circles correspond to the points measured in each element. These
resolutions are conducted by dividing the initial film into $n=$ 500 elements
and setting the time step $\mathrm{d}t=10^{-4}$ s.

This non-monotonic evolution is also visible in Fig. 13 (a), where the film
extension is plotted as a function of $z$ for different times. The slope
discontinuity in the curves corresponds to the boundary between the initial
film and the film extracted from the bottom meniscus.

In the initial film, the average extension is related to the total area of the
initial film, whereas the extension gradient is due to gravity. The film
tension shown in Fig. 13 (b) is directly deduced from the film extension,
using Eq. (7), and governs the film extraction. Its value at the different
heights in the film is a complex interplay between (i) the non-monotonic
evolution of the mean tension, due to the motors motion and the film
extraction, and (ii) the increase of the tension difference between the top
and the bottom of the initial film due to the gravity and to the increase of
the initial film height. The resulting tension profile leads to the extraction
rate from the lateral menisci plotted in Fig. 13 (c): the higher we are in the
film, the higher the tension and thus the higher the extraction velocity. It
can be noted that, as we will explain later, when the extension values are
negative, we have chosen to set a zero extraction velocity at the bottom, and
thus prevent the film from entering the bath meniscus. This explains the
plateaus observed at long times in Fig. 13 (b,c).

The thickness profile of the initial film is the most important prediction, as
it can be compared to the experimental data. Its dimensionless value
$\widehat{h}=h/h_{00}$ is plotted using two representations in Fig. 13 (d,e).
In Fig. 13 (d), we use the Lagrangian parameter $x$ so that a given piece of
film can be followed during its motion and deformation. At each place in the
initial film the non-monotonic behavior is observed, with a thinning followed
by a thickening, but the transition occurs at a longer time at the top of the
film. The profile is plotted as a function of the actual position in the film
$z$, in Fig. 13 (e), and the exponential profile is recovered. This profile is
a direct consequence of the initial condition we used: as a uniform reference
thickness is imposed in the initial film, the profile remains exponential at
all times. The prefactor is, however, an important prediction of the model,
and shows again, a non-monotonic evolution.

All the results discussed above help describing the physics of the initial
film evolution during the motors motion. In Fig. 13 (f), we have represented
the thickness profile of the whole central film (the initial film and the film
extracted at the bottom at longer times). The transition leads to a non-
physical discontinuous film thickness, due to the fact that the initial film
thickness has been chosen at the measured value, and the extracted film
thickness is imposed by the dynamics. As the tension gradient is governed by
the film thickness, the thickness discontinuity leads to the slope
discontinuity observed in Fig. 13 (a) (extension). The model leads to
compression at the bottom of the film, and to a tension lower than the
equilibrium one. This corresponds to a tendency of the system to extract film
at the top at high tension, and to absorb some film at the bottom meniscus,
where the film tension is lower. Such film absorption is difficult to evidence
experimentally, and could be associated with the marginal regeneration
observed at the bottom of vertical soap films (Gros et al., 2021). However,
the extraction velocity has been set to zero in that case in order to keep the
model simple.

##  5 Comparison with experimental data

We propose in this part to compare the results of the numerical resolution
presented in the previous section with the experimental results presented in
the section 3.3.

Figure 14: Evolution of the thickness $h$ as a function of $z$, in semi-
logarithmic scale, for a entrained velocity fixed at 100 cm/s and two
different heights: $H=0.8$ m (pink) and $H=1.3$ m (purple). The solid line
corresponds to the numerical resolution while the points (triangles or stars)
are experimental data identical to those in Fig. 5.

In Fig. 14, we have plotted two thickness profiles obtained numerically and
experimentally for a film generated at a velocity fixed at 100 cm/s. A good
agreement is obtained for the initial film profile. Note that the
characteristic length has been fitted on the experimental data, but the
prefactor of the exponential profile is quantitatively predicted, which is an
important validation of the model.

In the film extracted at the bottom, the thickness obtained numerically is
more than 10 times larger than the experimental one, which is a failure of the
model. This may be due to the fact that we did not take into account the
Poiseuille flow induced by gravity. Indeed, the downwards Poiseuille flow
scales as $\rho gh^{2}/\eta$. Thus, for thicknesses between 10 and 100 $\mu$m
gravity become non-negligible in the extracted film ($\rho gh^{2}/\eta\sim
0.1-10\,\ $cm/s). This explains that the measured thicknesses are smaller than
the prediction. However, the width of this film is reasonably captured.

Importantly, the dynamics of the initial film, on which this paper focuses, is
not affected by this non-physical film thickness at the bottom. The tension
profile in the initial film is governed by the gravity and by the constraint
that the initial film must reach a total height, determined by the top wire
position and by the location of the boundary with the extracted film at the
bottom. Only the size of the bottom extracted film matters for the initial
film dynamics, not its thickness. As the bottom extracted film has a
negligible size, both experimentally and numerically, the prediction made for
the initial film are independent of this extraction at the bottom. The only
consequence of the overestimated film thickness is the strong decrease of the
tension, which reaches non-physical small values at the bottom meniscus.

Figure 15: (a) Comparison between the prefactor of the exponential profiles
$K$ of the films measured experimentally for different velocities (ranging
from 20 cm/s to 140 cm/s) and the one determined using the numerical
resolution (black dashed line). The values of $K$ are plotted as a function of
the height reached by the soap films $H$ during the generation. (b) Evolution
of the thickness $h$ of the soap films during generation as a function of the
height $H$ reached by the films for all velocities probed between 20 cm/s and
250 cm/s. The colored points correspond to measurements taken at 30 cm above
the liquid bath. These data are identical to those shown in the Fig. 4 (b).
The black dotted line corresponds to the evolution of $h$ given by the
numerical resolution.

In the following, we restrict the analysis to the initial film. The
experimental and numerical film profiles are fitted by the exponential shape
$Ke^{(z-H)/\ell}$, and the obtained values of the prefactor $K$ are compared.
They are plotted as a function of the film height in Fig. 15 (a). Over the
range of heights we can access in our experiment, between 0.5 and 1.9 m, the
values of $K$ predicted by the numerical resolution are in agreement with our
data. However, a discrepancy can be noticed when the height of the films
becomes close to 2 m. This may be related to our method of redistributing the
films in rectangular form, which becomes too far from the redistribution
actually observed experimentally for the highest films.

This observation is confirmed if we compare the measurements directly made by
spectrometry, for example at 30 cm above the bath, with the numerical
resolution. Indeed, in Fig. 15 (b), we observe a good qualitative agreement
between the model and the experiment. First a decrease of the thickness is
noted, then a minimum value is obtained at a similar height of film, and
finally a thickening is observed. However, this thickening is much faster than
the increase in thickness measured experimentally.

##  6 Conclusion

The aim of this article is to describe the thickness profile of giant foam
films during their generation. Our main result is the successful description
of the thickness profile in the central part of the film. This necessitated a
phenomenological description of the extraction of thick film from the lateral
and bottom menisci. Our current understanding of these foam films generation
is summarized below.

At short time, the liquid is entrained by viscosity around the wire, as it
would be the case with a much more viscous liquid, like silicon oil
(Champougny et al., 2017). A reasonable hypothesis is that the reference state
is prescribed by the moment at which this viscous flow is counterbalanced by
the apparition of a strong enough Marangoni stress allowing entering the
regime described in this article.

This initial film is then stretched and we have shown that its shape is an
exponential profile predicted the hydrostatic model. This is true though the
pulling velocities are large, up to meters per seconds. Additionally, the
characteristic length of the exponential depends neither on the pulling
velocity nor on the time. This is a strong indication that an equilibrium
shape of the film exists with a reference thickness $h_{0}$ and a reference
area $A_{0}$. What fixes this reference remains unknown.

We show that the relaxation of the central films starts during the generation
and is governed by the extraction of thicker film from the lateral menisci.
The extraction law we use in the model is nevertheless not quantitative. As
explained in the text, this can be due to the fact that we do not know the
value of the surface tension in the menisci.

Finally, when the motors stop, we expect a competition between the relaxation
of the central film due to further extraction of thick films from the lateral
meniscus and a thinning due to both drainage and evaporation. A quantitative
description of these mechanisms would be necessary to predict the lifetime of
the soap film.

## Acknowledgments

We are grateful to François Boulogne for the design of the Oospectro library,
and Vincent Klein and Sandrine Mariot for the design of the experimental
setup. Funding from ESA (MAP Soft Matter Dynamics) and CNES (through the GDR
MFA) is acknowledged. IC has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (grant agreement No 725094).

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