# Damping of liquid sloshing by foams

A. Sauret<EMAIL_ADDRESS>Department of Mechanical and
Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA  Surface
du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 93303 Aubervilliers,
France.  F. Boulogne  A. Sauret and F. Boulogne contributed equally to this
work.  Department of Mechanical and Aerospace Engineering, Princeton
University, Princeton, NJ 08544, USA  J. Cappello  Department of Mechanical
and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Ecole Normale Supérieure de Cachan, 94235 Cachan, France  E. Dressaire
Department of Mechanical and Aerospace Engineering, Princeton University,
Princeton, NJ 08544, USA  Department of Mechanical and Aerospace Engineering,
New York University Polytechnic School of Engineering, Brooklyn, NY 11201, USA
H. A. Stone<EMAIL_ADDRESS>Department of Mechanical and Aerospace
Engineering, Princeton University, Princeton, NJ 08544, USA

###### Abstract

When a container is set in motion, the free surface of the liquid starts to
oscillate or slosh. Such effects can be observed when a glass of water is
handled carelessly and the fluid sloshes or even spills over the rims of the
container. However, beer does not slosh as readily as water, which suggests
that foam could be used to damp sloshing. In this work, we study
experimentally the effect on sloshing of a liquid foam placed on top of a
liquid bath. We generate a monodisperse two-dimensional liquid foam in a
rectangular container and track the motion of the foam. The influence of the
foam on the sloshing dynamics is experimentally characterized: only a few
layers of bubbles are sufficient to significantly damp the oscillations. We
rationalize our experimental findings with a model that describes the foam
contribution to the damping coefficient through viscous dissipation on the
walls of the container. Then we extend our study to confined three-dimensional
liquid foam and observe that the behavior of 2D and confined 3D systems are
very similar. Thus we conclude that only the bubbles close to the walls have a
significant impact on the dissipation of energy. The possibility to damp
liquid sloshing using foam is promising in numerous industrial applications
such as the transport of liquefied gas in tankers or for propellants in rocket
engines.

##  I Introduction

Sloshing is the motion of the free surface of a liquid induced by variations
of the velocity of the container. The amplitude of the generated waves depends
on the frequency and the amplitude of motion of the container.McIver (1989);
Ibrahim (2005); Molin (2001); Herczyński and Weidman (2012); Royon-Lebeaud,
Hopfinger, and Cartellier (2012); Faltinsen and Timokha (2012); Turner and
Bridges (2013); Bouscasse  _et al._ (2014); Reclari  _et al._ (2014) When the
frequency of the motion corresponds to the resonance frequency of the surface
wave, the amplitude increases significantly. If the amplitude of sloshing
becomes large enough, spilling, splashing, and/or drop formation are
possible.Mayer and Krechetnikov (2012) Such phenomenon can lead to challenging
technical constraints in various applications including the transportation of
oil and liquefied gas in tankers. Indeed, the sloshing dynamics leads to
considerable pressure forces on the walls of a container and can be a source
of destabilization or failure of a container. Therefore, the characterization
of the amplitude of the waves and the exploration of mechanisms capable of
damping sloshing is of critical importance. To damp sloshing, various methods
have been considered including the use of elastic membranes attached to the
top of a container, Bauer (1995); Bauer and Chiba (2007); Kidambi (2009) or
baffles placed in a container.Maleki and Ziyaeifar (2008); Liu and Lin (2009)
The use of containers with elastic walls was also considered.Bauer, Hsu, and
Wang (1968) However, these methods can be difficult to implement in large
containers or require the design of new containers.

In the present paper, we study the damping effects of a liquid foam placed on
top of the liquid. The present method is versatile as it can be used in any
container. Foam is easy to produce and cheap as it requires a small quantity
of foaming solution to generate a large volume of foam. In addition, foams are
already used in various industrial applications and processes such as oil
recovery, soil remediation, fire containment or blast wave
mitigation.Prud’homme (1995); Stevenson (2012)

A foam, as used in the present study, is an assembly of densely packed bubbles
produced upon aeration of a surfactant solution. The flow of foam in two-
dimensional channels has been studied extensively both experimentallyCantat,
Kern, and Delannay (2004); Dollet and Graner (2007); Dollet (2010); Ben Salem,
Cantat, and Dollet (2013) and numerically. Cox, Weaire, and Glazier (2004);
Cox (2005); Boulogne and Cox (2011) Common foams exhibit viscous, elastic
and/or plastic responses depending on the external forcing. Bronfort and Caps
(2015); Marmottant and Graner (2007); Benito  _et al._ (2008) For small
deformations, foams behave as visco-elastic solids.Cohen-Addad, Höhler, and
Pitois (2013) In this regime, the effective viscosity of a foam is much larger
than the viscosity of the foaming solution.Cantat  _et al._ (2010); Höhler and
Cohen-Addad (2005) Therefore, foam is a good candidate to damp sloshing and
thus to decrease the pressure forces exerted on the walls of a container.

Since the seminal work of Bretherton on the motion of a bubble in a
capillary,Bretherton (1961) the viscous dissipation of a flowing foam has been
considered.Denkov  _et al._ (2005); Saugey, Drenckhan, and Weaire (2006);
Raufaste, Foulon, and Dollet (2009); Costa  _et al._ (2013); Cantat (2013); Le
Merrer  _et al._ (2014) It has been shown that the friction coefficient
depends on the capillary number (ratio of viscous forces to surface tension
forces) but varies with the nature of the foam. For three-dimensional foams,
two sources of dissipation need to be taken into account. The first
contribution is the viscous friction induced by the displacement of the
Plateau borders associated with the displacement of the bubbles in contact
with the walls of a container and relative to one another. The second
contribution depends on the interfacial rheology of the foam and is induced by
the presence of surfactant molecules at air/liquid interfaces.Denkov  _et al._
(2005); Cantat (2013) To characterize the relative magnitude of the different
contributions, a distinction should be made between low and high surface
moduli. At high surface modulus, interfaces behave as incompressible
surfaces.Langevin (2014) In the present study, we rely on a foam with low
surface modulus where the interfaces are elastic.

In this paper, we study the damping effect of a foam placed on top of a liquid
bath in an oscillating rectangular container. In section II we begin by
presenting the experimental setup, the physical properties of the system and
the two methods of excitation considered. Then, in section III we characterize
the sloshing of a single Newtonian fluid. We recall analytical results
previously derived and validate the experimental approach. We also introduce
the damping coefficient $1/\tau$ and provide a scaling law to rationalize the
experimental results. Section IV is devoted to the sloshing in presence of a
layer of foam of varying thickness. We extend our observations to three-
dimensional foams and present a phenomenological model that predicts the
damping of the wave as a function of the foam thickness. We summarize our
conclusions and discuss avenues for future research in section V.

##  II Experimental methods

###  II.1 Experimental setup

To perform visualization of the different interfaces (liquid/air, liquid/foam,
foam/air) and track the motion of the foam, we use a rectangular cell in which
the displacement of the fluid remains mainly two-dimensional. The vertical
walls of the container are made of borosilicate glass (purchased from Ted
Pella and McMaster-Carr) and a rigid rubber sheet (McMaster-Carr) is used to
close the bottom of the cell. This configuration allows the introduction of a
needle at the center of the container to generate the monodisperse foam. The
cell has a length of $L=70\,\rm{mm}$, a height of $92\,\rm{mm}$ and a varying
width $w\in[3;\,19]\,\rm{mm}$. The experimental apparatus, represented in
figure 1(a), consists of a container, a LED panel and a high-speed camera
mounted on a moving stage. The moving stage is on rails and set in motion by a
mechanical vibrator (LDS 319024-3) controlled by a power supply (Stanford
research system Model DS345) and an amplifier (LDS PA100E). The lighting is
provided by the LED panel (Phlox, 10 cm $\times$ 10 cm) placed behind the
cell. The oscillations of the liquid surface are recorded using a high-speed
camera (Phantom V9.1) operating at typically 100 frames per second. Examples
of the visualizations are shown without and with foam, respectively, in
figures 1(b) and 1(c).

Figure 1: (a) Schematic of the experimental setup. Example of visualization
for (b) a Newtonian fluid and for (c) a Newtonian fluid with a $2$D liquid
foam on the top. The images are recorded at the maximum amplitude of the
oscillation. Scale bars are $1$ cm.

We first study the sloshing dynamics of a single Newtonian fluid. We use
different fluids to vary the viscosity, the density and the surface tension of
the liquid. We compare the behavior of water, mixtures of water and glycerol
(50% v/v, 60% v/v and 80% v/v of glycerol), dodecane, decanol and a foaming
solution (described in detail in the next paragraph). We are therefore able to
study how the physical parameters of the system influence the sloshing
dynamics of a single Newtonian fluid. The viscosity and the surface tension of
the Newtonian fluids are measured, respectively, with an Anton Paar rheometer
(MCR 301, CC50 geometry) and the pendant drop method at $20^{\rm o}{\rm C}$
and compared with tabulated values.Sheely (1932); Lide (2008); Soller and
Koehler (2008) The physical properties of the different fluids used in this
study are summarized in Table 1.

Fluid | kinematic viscosity $\nu$ | surface tension $\gamma$ | density $\rho$
---|---|---|---
| ($\rm{\times 10^{-6}\,{m}^{2}\,{s^{-1}}})$ | $\rm{(\times 10^{-3}\,\,N\,m^{-1})}$ | $\rm{(\times 10^{3}\,\,kg\,m^{-3})}$
water | $1.01$ | $72$ | $1$
water-glycerol (50 vol%) | $5$ | $67$ | 1.13
water-glycerol (60 vol%) | $9$ | $67$ | 1.15
water-glycerol (70 vol%) | $19$ | $66$ | 1.18
water-glycerol (80 vol%) | $46$ | $66$ | 1.21
foam solution | $1.4$ | $25$ | $1.02$
dodecane | $1.8$ | $25$ | $0.75$
decanol | $13.1$ | $28$ | $0.83$
Table 1: Physical properties of the fluids measured at $20^{\rm o}\,{\rm C}$.

To perform sloshing experiments in the presence of foam, we partially fill the
cell with a foaming solution well above the critical micelle concentration
(CMC). This solution is made of 90% v/v water, 5% v/v glycerol and 5% v/v of
commercial surfactant (Dawn dish-washing liquid, Procter & Gamble). The
surfactant molecules lead to a stable foam with low surface modulus.Denkov
_et al._ (2009) Glycerol is used to slow down the aging of the foam associated
with the drainage and the coarsening processes.Cantat  _et al._ (2010) In
addition, the experiments are performed on short time scales, typically less
than few minutes, to avoid such aging. The bubbles are generated using a
$1.3\,\rm{mm}$ inner diameter needle and a syringe pump operating at constant
flow rate, $Q=20\,\mu\rm{L.min^{-1}}$. This method leads to a monodisperse
foam with bubbles of diameter $D=3\,{\rm mm}$. In addition, to control the
number of bubble layers, the foam is generated sequentially, one layer at a
time. We also use a non-return valve to prevent the foam solution from
entering in the tubing between each step.

###  II.2 Oscillating forcing or impulse

We use two methods to investigate the sloshing dynamics. In the first method,
we impose a lateral harmonic excitation $X(t)=X_{0}\,\cos(2\pi ft)$ with a
mechanical vibrator. The frequency $f$ and the amplitude $X_{0}$ of
displacement of the moving stage are constant leading to the oscillation of
the interface at the frequency $f$ as reported in figure 2(a). This procedure
allows the determination of the maximum amplitude of sloshing for different
liquids, liquid heights $H$, frequencies $f$ and amplitudes of displacement of
the moving stage $X_{0}$. In this example, a small phase shift can be observed
between the harmonic lateral excitation of the container and the displacement
of the free surface. This shift is likely due to the presence of nonlinear
effects, which are stronger for a large container and large amplitude
excitation.

In the second method, we study the response of the fluid to a mechanical
impulse and record the oscillations of the interface as illustrated in figure
2(b). We thus estimate the damping coefficient $1/\tau$ of the system. In
addition, a temporal analysis of the oscillations provides the resonant
frequencies of the fluid corresponding to normal modes of the
container.Ibrahim (2005)

As the aim of the present study is to determine the influence of the foam
layer on the damping of the oscillations, the results presented in this
article are mainly obtained using the impulse method as shown in figure 2(b).
However, determining the velocity of the bubbles appeared to be much more
accurate with a forced lateral harmonic oscillation.

Figure 2: Temporal spectrum $P(f)$ of the free-surface displacement measured
$1$ cm away from the edge of the cell for (a) a continuous sinusoidal forcing
at frequency $f$=3.3 Hz and (b) an impulse excitation. The vertical dashed
line shows the first resonant frequency $f_{1}\simeq 3.2\,{\rm Hz}$. Insets:
the corresponding time evolution of the free surface $\eta$ is represented by
a blue continuous line (scale: left axis). The red dotted line shows the
motion of the container $X(t)$ (scale: right axis). The black dash-dotted line
in the inset of (b) corresponds to an exponential damping,
$\eta_{0}\,\text{exp}(-t/\tau_{\ell}),$ where $\eta_{0}$ and
${1}/{\tau_{\ell}}$ are fitting parameters. Experiments are performed with a
depth $H=4\,{\rm cm}$ of foaming solution in a container of width
$w=1.59\,{\rm cm}$ and length $L=7\,{\rm cm}$.

##  III Sloshing of a Newtonian fluid

The sloshing dynamics of a single Newtonian fluid resulting from a forced
lateral excitation of a container has been the subject of extensive studies in
various geometries such as rectangular, cylindrical or conical containers (for
a review, see e.g. IbrahimIbrahim (2005)). In addition to the calculation of
the normal modes of the liquid in the container, the hydrodynamic pressure and
the forces and moments acting on the container walls have been estimated. We
should note that other forced excitations such as pitching or rolling are also
possible. In the present study, we focus on the lateral excitation of a
rectangular container. Before considering the influence of the foam on the
damping of the oscillations, we characterize the sloshing dynamics of a single
Newtonian fluid.

###  III.1 Resonant frequencies

Assuming that the flow is incompressible, inviscid and irrotational, we use
potential flow theory. The detailed derivation of the dispersion relation and
velocity potential for a Newtonian fluid in a rectangular container of length
$L$ and width $w\ll L$ is presented in Appendix A. The resonant frequency of
order $n$, $\omega_{n}$, is

${\omega_{n}}^{\\!2}=\left(g\,k_{n}+\frac{\gamma}{\rho}\,{k_{n}}^{\\!3}\right)\tanh(k_{n}\,H)$
(1)

where $k_{n}={n\pi}/{L}$ is the wavenumber, $g$ is the gravitational
acceleration, $\gamma$ the surface tension and $\rho$ the density of the
fluid. For a lateral time periodic forcing and a rectangular container, only
the odd modes with $x$ are present in the Fourier development. Therefore, we
only observe the resonant frequencies corresponding to $\omega_{2n+1}$.

The two experimental methods exhibit the same resonant frequency as
illustrated in figure 3(a). For an impulsive motion, we record the time
evolution of the free-surface elevation, $\eta(x,t)$, at a given position $x$.
The resulting signal, presented in figure 2(a), is a damped harmonic
oscillation. The time series of this signal exhibit several peaks, each
corresponding to a resonant frequency ($f_{1}$, $f_{3}$, $f_{5}$, $f_{7}$) as
illustrated in figure 3(a). We also observe that for the forced lateral
excitation at resonant frequencies, the maximum free-surface elevation reaches
local maxima. Indeed, for a harmonic forcing at a frequency $f$, the free
surface oscillates at the same frequency $f$ as the excitation. Then, the
amplitude of sloshing becomes large when the excitation frequency $f$ is equal
to a resonant frequency of the cell ($f_{1}$, $f_{3}$, $f_{5}$, …). Therefore,
the results obtained with the two methods of excitation are complementary and
consistent with each other.

According to the dispersion relation (1), the resonant frequencies depend on
the height of liquid $H$ but are independent of the width of the cell $w$. The
analytical expression of the resonant frequencies is consistent with the
experimental results (see Appendix A). In addition, we find that for
sufficient height of fluid ($k_{n}\,H\gg 1$), i.e. when the deep-water
assumption is valid, $\omega_{n}$ does not depend on $H$ (see Appendix A).

Figure 3: (a) Temporal spectrum $P(f)$ of the surface displacement resulting
from an impulse excitation (blue continuous line, left axis), and with a
harmonic excitation at a given frequency (red squares, right axis). The
vertical dashed lines are the resonant frequencies from equation (1). The
experiments are performed with a depth $H=4\,{\rm cm}$ of water in a container
of width $w=1.59\,{\rm cm}$ and length $L=7\,{\rm cm}$. (b) Evolution of the
damping coefficient ${1}/{\tau_{\ell}}$ for $w=1.6$ cm and
$\nu=5.26\,\rm{mm}^{2}\,\rm{s}^{-1}$ (blue filled circle), $w=1.6$ cm and
$\nu=9.17\,\rm{mm}^{2}.\rm{s}^{-1}$ (blue hollow circle), $w=1.6$ cm and
$\nu=18.6\,\rm{mm}^{2}.\rm{s}^{-1}$ (red filled square), $w=1.6$ cm and
$\nu=46.3\,\rm{mm}^{2}.\rm{s}^{-1}$ (red hollow square), $w=1.6$ cm and
$\nu=1.01\,\rm{mm}^{2}.\rm{s}^{-1}$ (green filled diamond), $w=0.48$ cm and
$\nu=1.01\,\rm{mm}^{2}.\rm{s}^{-1}$ (purple hollow diamond), $w=1.1$ cm and
$\nu=1.01\,\rm{mm}^{2}.\rm{s}^{-1}$ (green filled triangle), $w=1.9$ and
$\nu=1.01\,\rm{mm}^{2}.\rm{s}^{-1}$cm (orange filled star). Inset: first
resonance frequency as a function of the kinematic viscosity $\nu$.

Note that we also consider the influence of the viscosity on the value of the
first resonant frequency measured experimentally from the impulse excitation.
For the range of parameters considered, the viscosity does not influence
significantly the value of $f_{1}$.

###  III.2 Damping of the interface motion

An important feature of the sloshing dynamics is the ability of the system to
damp the waves generated by a sudden impulse excitation. The damped harmonic
oscillations resulting from an impulse allow us to define the damping
coefficient ${1}/{\tau_{\ell}}$ that depends on the physical parameters of the
system: the cell width $w$, the kinematic viscosity $\nu$ and the frequency
$f$. The damping coefficient is defined as Lam and Caps (2011); Bronfort and
Caps (2012)

$\frac{1}{\tau_{\ell}}=\frac{\left\langle|\dot{E_{\ell}}|\right\rangle}{2E}$
(2)

where $|\dot{E_{\ell}}|$ is the rate of energy dissipation in the fluid and
$E$ is the input energy. The notation $\langle\,\,\,\rangle$ indicates that
the quantity is averaged over one period of oscillation. Experimentally,
${1}/{\tau_{\ell}}$ is determined using an exponential fit of the time
evolution of the-free surface elevation at the position $x$:
$\eta(x,t)=\eta_{0}(x)\,\text{exp}(-t/\tau_{\ell})\,\sin(\omega t)$. An
example of an experimental fit to determine the damping coefficient
${1}/{\tau_{\ell}}$ is shown in figure 2(b). The dependence of
${1}/{\tau_{\ell}}$ on the parameters of the system is reported on figure
3(b). For different fluid viscosities and cell widths, we use the rescaled
parameter $\sqrt{2\pi f\nu}\,(1/L+1/w)$ where $\nu$ is the kinematic viscosity
of the fluid.

We assume in the modeling of the damping rate that only the first frequency of
resonance contributes significantly to the damping of the sloshing dynamics
for an impulse. Indeed, we did not observe a pattern of the liquid-bubble
interface smaller than the cell length, which indicates that only the first
mode is dominant contrary to Faraday’s instability. Bronfort and Caps (2012)
To estimate analytically the damping coefficient, two sources of dissipation
need to be taken into account.Landau and Lifshitz (1959) The first one
corresponds to viscous forces in the bulk of the fluid:

$\left\langle|\dot{E_{\ell}}|\right\rangle_{1}=\frac{1}{2}\rho\,\nu\int_{\rm
bulk}{\left(\bm{\nabla}\bm{v}+\bm{\nabla}\bm{v}^{T}\right)^{2}\rm{d}V}\sim\rho\,\nu\left(vk\right)^{2}\,\frac{L\,{w}}{k}$
(3)

The term proportional to $1/k$ arises from the sum of the velocities along the
$z$-axis in the deep water regime. The second source of damping comes from the
viscous effects in the boundary layers, namely the oscillating Stokes layers
of thickness $\delta=\sqrt{{\nu}/{\omega}}$ near the two walls:Bronfort and
Caps (2012)

$\left\langle|\dot{E_{\ell}}|\right\rangle_{2}=\frac{1}{2}\rho\,\nu\int_{\rm
BL}{\left(\bm{\nabla}\bm{v}+\bm{\nabla}\bm{v}^{T}\right)^{2}\rm{d}V}\sim\rho\,\nu\,\left(\frac{v}{\delta}\right)^{2}\left(L+w\right)\frac{\delta}{k}$
(4)

Considering $k=k_{1}=\pi/L$ and $w\ll L$, the ratio of equations (3) and (4)
is typically of the order of $\pi\delta w/L\sim 10^{-2}$ in our system.
Therefore, the damping in the bulk is negligible compared to the contribution
in the wall boundary layers in our experiments. Consequently, the dissipated
power can be estimated as:

$\left\langle\dot{|E_{\ell}|}\right\rangle\sim\rho\,\nu\,\left(\frac{v}{\delta}\right)^{2}\left(L+w\right)\frac{\delta}{k}$
(5)

The initial energy injected in the system is equal to the initial kinetic
energy and is defined as Landau and Lifshitz (1959)

$E=\frac{1}{2}\rho\int{\bm{v}^{2}\rm{d}V}\sim\frac{1}{2}\,\rho\,v^{2}\frac{L\,{w}}{2\,k}$
(6)

for which the integral is taken over the volume $w\,L/(2\,k)$. Substituting
equation (5) and (6) in the definition (2) of the damping coefficient
${1}/{\tau_{\ell}}$ leads to

$\frac{1}{\tau_{\ell}}\sim\,\sqrt{2\pi\,f\,\nu}\left(\frac{1}{L}+\frac{1}{{w}}\right).$
(7)

We observe on the figure 3(b) that this scaling law is in good agreement with
the experimental measurements for various fluids and cell thicknesses. These
results confirm that the dissipation in the wall boundary layer is the main
source of damping of the sloshing in our system.

##  IV Sloshing damped by a liquid foam

We now consider the influence of a liquid foam placed on top of a vibrating
liquid bath.Cappello  _et al._ (2014) The experiments presented in this
section were performed with a liquid height $H=4$ cm of foaming solution in a
container of length $L=7\,{\rm cm}$. The liquid foam is made of monodisperse
bubbles of diameter $D=3$ mm and the height of the foam layer is $\xi D$ where
$\xi$ represents the number of layers.

###  IV.1 Experimental observations in a Hele-Shaw cell

To allow visualization of the motion of the bubbles, we first perform
experiments in a confined cell of width $w=3$ mm equal to the bubble diameter
$D$. The foam is generated by injecting air at a constant flow rate through a
needle located at the bottom of the cell. In all experiments, no topological
rearrangements (T1) are observed.Weaire and Fu (1988) As a result, the foam
behaves like a visco-elastic fluid without plastic deformation.

Experimentally, we consider the influence of the foam on the resonant
frequency of the dominant mode of sloshing, i.e. the mode corresponding to the
frequency $f_{1}$. Figure 5(a) shows the temporal spectrum extracted from the
oscillations of the liquid-foam interface following an impulse motion. In the
presence of a foam, the spectrum exhibits a peak located around $f_{1}\simeq
3.1$ Hz. This value is similar to the frequency of the first mode in the
absence of foam, which is predicted to be $3.3$ Hz from equation (1). This
result suggests that the presence of foam does not significantly modify the
characteristic frequency of the system. To characterize the foam thickness, we
introduce $\xi$, the number of layers of bubbles, as the ratio of the foam
height and the bubble diameter. The resonant frequency $f_{1}$ is measured
experimentally for different foam thicknesses, i.e. values of $\xi$ (see the
inset of figure 5(a)). As no significant change is observed, the frequency of
the first mode is assumed constant. This result indicates that the elasticity
at the interface between the air and the foam can be neglected at first order
thus leading to no supplementary contribution in the dispersion relation
(1).Planchette, Lorenceau, and Biance (2012)

Figure 4: Time evolution of the amplitude of sloshing for a harmonic lateral
forcing at $f_{1}=3.125$ Hz (first resonant frequency) and an amplitude of
excitation $X_{0}=0.1$ cm. The number of layers of bubbles $\xi$ increases
from left to right. The arrows indicate the velocity field. The photographs
are taken at different moments during half a period of the container
oscillation. Scale bars are $1$ cm. (Multimedia View). Figure 5: (a) Influence
of the foam on the temporal spectrum of the fluid response to an impulse
excitation motion $P(f)$ for the foam solution alone (black continuous line)
or with one (red dashed line), two (blue dashed-dotted line) and three (light
green continuous line) layers of bubbles. Inset: evolution of the first
resonant frequency $f_{1}$ as a function of the number of layers of bubbles
$\xi$, the horizontal dashed line shows the theoretical frequency. (b)
Evolution of the damping coefficient $1/\tau$ as a function of the number of
layers $\xi$. The black continuous line shows the fit by equation (8). Both
figures are obtained with experiments performed with a depth $H=4\,{\rm cm}$
of foaming solution in a container of width $w=0.30\,{\rm cm}$ and length
$L=7\,{\rm cm}$.

For a continuous harmonic forcing at the resonant frequency $f_{1}$ and
lateral displacement amplitude $X_{0}=1\,{\rm mm}$, the maximum amplitude of
sloshing decreases significantly for increasing foam thicknesses as shown in
figure 4 (Multimedia View). Indeed, without foam, the free-surface has a
maximum amplitude of the order of $1$ cm; the addition of only $\xi=5$ layers
of bubbles leads to a maximum amplitude of the order $1$ mm, which is a factor
of ten reduction. To evaluate the damping coefficient for various foam
thicknesses, we perform experiments with an impulse motion and measure the
time evolution of the oscillations of the liquid-foam interface (see figure
2(b)). The experimental results reported in figure 5(b) show that the damping
coefficient $1/\tau$ increases exponentially toward a plateau over a typical
rescaled length corresponding to $5$ layers of bubbles. This result suggests
that the addition of a few layers of bubbles is sufficient to efficiently damp
the sloshing. These qualitative observations confirm that foam has a strong
damping effect on the sloshing dynamics, decreasing the amplitude of the
oscillations and increasing the damping coefficient of the system $1/\tau$.

To understand the effect of the foam, the damping coefficient is decomposed
into two contributions. The first contribution ${1}/{\tau_{\ell}}$ is due to
the liquid alone, and depends only on the frequency $f$, the kinematic
viscosity $\nu$ and the size of the cell. The second contribution to damping
$1/\tau_{f}$, is induced by the presence of the foam. Thus, we assume
$1/\tau={1}/{\tau_{\ell}}+{1}/{\tau_{f}}$. It is noteworthy that $\tau_{\ell}$
is independent of the position or velocity of the interface. This makes the
decomposition valid even in presence of the foam that reduces the amplitude of
the liquid-foam interface. Then, the experimental values of the damping
coefficient of the system can be fit by:

$\frac{1}{\tau}=\frac{1}{\tau_{\ell}}+\frac{1}{\tau_{f}^{\infty}}\,\left[1-\text{exp}\left(-\frac{\xi}{\xi_{c}}\right)\right]$
(8)

as illustrated in figure 5(b) where $1/\tau_{f}^{\infty}=9\pm
3\,\,\rm{s^{-1}}$, a plateau ${1}/{\tau_{\ell}}+1/\tau_{f}^{\infty}=11\pm
3\,\rm{s^{-1}}$ and $\xi_{c}=5\pm 2$ for the considered experimental
parameters. The damping coefficient of the foam solution alone is estimated to
be ${1}/{\tau_{\ell}}\simeq 2\,\rm{s^{-1}}$, which is in agreement with the
results presented in figure 3(b) for a height of foaming solution $H=4$ cm in
a cell of width $w=3$ mm and length $L=7$ cm. These experimental results show
that the foam-air interface is substantially stabilized with a sufficient
amount of foam.

We have also recorded the dynamics for a harmonic forcing for various numbers
of foam layers $\xi$. At sufficiently large foam thicknesses, $\xi>5$, the
damping coefficient $1/\tau$ and the maximum free-surface elevation $\eta$
reach a plateau value. This saturation is consistent with the observations
reported above: when the thickness of the foam is sufficiently large,
typically $\xi>5$, the foam-air interface remains nearly steady.

Also, we have measured the velocity of individual bubbles with bubble
labelingvan der Walt  _et al._ (2014) and particle tracking techniquesAllan,
Caswell, and Keim (2014) as shown in figure 4 (Multimedia View). We observe
that adding a large number of layers of bubbles allows to document the spatial
dependence of the velocity of the bubbles. For fewer than two layers of
bubbles the velocity is mostly vertical. The velocity becomes more horizontal
as the number of layers increases, except for bubbles near the vertical walls,
which have a non-negligible vertical velocity due to the boundary conditions.
As represented in figure 6(a), we can observe that the mean bubble velocity
decreases exponentially
$\langle|v_{b}|\rangle\propto\text{exp}(-z/(\xi_{c}D))$ with a characteristic
length scale of $\xi_{c}D=16.8\pm 1.8\,{\rm mm}$ corresponding to
$\xi_{c}\simeq 5$ layers of bubbles. We also noted that the mean velocity of
the interface is independent of the $x$ position.

###  IV.2 Phenomenological modeling

A full analytical description of sloshing dynamics in presence of the foam is
beyond the scope of this study as the physical description of the motion of
the foam remains mainly empirical. Instead, in this section we rationalize the
evolution of the damping coefficient $1/\tau$ presented in figure 5. To do so,
we describe the contribution of the addition of foam to the damping
coefficient using a phenomenological model. Here, our approach remains the
same as the calculation of the damping coefficient for a single Newtonian
fluid $1/\tau_{\ell}$ presented in section III.2. We consider a $2$D foam
confined in a cell of width $w$.

The foaming solution used in the experiments leads to a foam with a low
surface modulus.Denkov  _et al._ (2009) Thus, the friction force of the
Plateau borders on two solid walls can be estimated as
$f_{b}=K\,\gamma_{f}\,{\rm Ca}^{2/3}$, where ${\rm
Ca}=\mu_{f}\,v_{b}/\gamma_{f}$ is the Capillary number and $K$ is a constant
for given geometrical parameters describing the Plateau borders.Raufaste,
Foulon, and Dollet (2009); Cantat (2013) The typical Capillary number
considered in our experiments is of order ${\rm Ca}\sim 3\times 10^{-3}$,
where $\mu_{f}$ and $\gamma_{f}$ are the viscosity and surface tension of the
foaming solution, respectively. Note that the present scaling for the friction
force is no longer valid for ${\rm Ca}>10^{-2}$ due to inertial effects, but
our experiments remain below this limit.Quéré and De Ryck (1998)

We denote $v_{b}(x,z,t)$, the velocity of the bubble located at $(x,z)$. The
rate of energy dissipation by the foam, averaged over a period, can be
written:

$\left\langle|\dot{E_{f}}|\right\rangle=\left\langle\sum
f_{b}\,v_{b}\right\rangle,$ (9)

where the sum is taken over all bubbles constituting the foam. The shape of
the Plateau borders is assumed to be independent of $x$ (translational
symmetry) and $z$, i.e., we neglect the variation of the liquid fraction along
the vertical direction, such that $K$ is independent of $z$. Upon
substitution, we obtain

$\left\langle|\dot{E_{f}}|\right\rangle=K^{\prime}\,\sum{\left\langle{v_{b}}^{5/3}\right\rangle}$
(10)

with $K^{\prime}=K{\gamma_{f}}^{1/3}\,{\mu_{f}}^{2/3}$.

Experimentally, we observe that the time-averaged velocity $\langle
v_{b}\rangle$ does not depend significantly on the horizontal position of the
bubble $x$. In addition, as noted previously, the velocity of the bubbles can
be fit by an exponential curve
$v_{b}(z,t)=v_{0}^{max}\,\text{exp}(-z/(\xi_{c}D))\,\cos(\omega\,t)$ where
$\xi_{c}D=16.8\pm 1.8\,{\rm mm}$ as illustrated in figure 6(a). We note that
the order of magnitude of the lengthscale over which the velocity decreases
has the same order of magnitude as the length scale associated with the first
mode of resonance $1/k_{1}=L/\pi\simeq 22\,{\rm mm}$. The maximum velocity
$v_{0}^{max}$ is obtained at $z=0$, i.e. for the first bubble layer on top of
the liquid bath (see inset of figure 6(a)).

Since we have assumed that the velocity of the bubbles is independent of the
horizontal position, the integral over the Plateau border in the $x$-direction
leads to a prefactor $L/D$, which corresponds to the number of bubbles. Thus,
equation (10) becomes

$\left\langle|\dot{E_{f}}|\right\rangle={K^{\prime}}\frac{L}{D}\int\limits_{0}^{\xi
D}\,\left\langle(v_{0}^{max})^{5/3}\,|\text{cos}(\omega
t)|^{5/3}\right\rangle\,\text{exp}\left(-\frac{5\,z}{3\,\xi_{c}D}\right)\,{\rm
d}z$ (11)

where $\xi D$ is the height of foam. Therefore, the dissipated power can be
approximated as

$\langle|\dot{E_{f}}|\rangle=\frac{3}{5}\xi_{c}\,L\,{K^{\prime}}\,(v_{0}^{max})^{5/3}\left\langle|\text{cos}(\omega
t)|^{5/3}\right\rangle\left[1-\text{exp}\left(-\frac{5\,\xi}{3\,\xi_{c}}\right)\right].$
(12)

The mechanical energy injected initially in the system can be approximated as
the maximum of the kinetic energy of the fluid, which is reached when the
liquid-foam interface is horizontal. Therefore, the injected energy is:

$E=\frac{1}{2}\rho\int{(v_{0}^{max})^{2}\rm{d}V}=\frac{\rho\,{w}}{4}\,\frac{L^{2}}{\pi}(v_{0}^{max})^{2}.$
(13)

Finally, combining equations (12) and (13), we obtain the damping coefficient
of the foam:

$\frac{1}{\tau_{f}}=\frac{12\pi}{5}\frac{\xi_{c}\,{K^{\prime}}\left\langle|\cos(\omega
t)|^{5/3}\right\rangle\,}{\rho\,w\,{L}(v_{0}^{max})^{1/3}}\left[1-\text{exp}\left(-\frac{5\,\xi}{3\,\xi_{c}}\right)\right].$
(14)

The value of the maximum velocity $v_{0}^{max}$ is extracted from experimental
measurements (see figure 6(a)). Substituting $(v_{0}^{max})^{1/3}$ with its
experimental fit, we can estimate the evolution of $1/\tau_{f}$. The
phenomenological law is reported in figure 6(b) and shows a fairly good
agreement with the experimental measurements for different container sizes and
different foam thicknesses.

The prefactor is determined by fitting the experimental data to find

$\frac{12\pi}{5}\frac{\xi_{c}\,{K^{\prime}}\left\langle|\cos(\omega
t)|^{5/3}\right\rangle\,}{\rho\,{L}(v_{0}^{max})^{1/3}}=2.76\,\textrm{cm/s}$
(15)

which leads to $K\approx 1.9$ m/s with $v_{0}^{max}\approx 0.1$ m/s. The value
of $K$ can be compared with the phenomenological lawCantat (2013)

$K=5.13\left(\frac{R_{p}}{\sqrt{A}}\right)^{-0.48}$ (16)

where $R_{p}$ is the radius of curvature of the Plateau border and $A$ is the
bubble area.

In the present study, we assumed that $K$ does not depend on the liquid
fraction as its measurement during the oscillations is difficult. In addition,
the strongest variation of the liquid fraction happens between the first and
the second layer of bubbles because the first layer is directly in contact
with the liquid bath (see figure 4). The change in liquid fraction is less
significant between the 2nd and the 5th layers, where most of the dissipation
occurs according to the velocity profiles. Thus, we estimate the ratio
${R_{p}}/{\sqrt{A}}$ to be of the order of $0.3$. Note that the dependency on
this ratio is only weak because of the power law in equation (16).
Consequently, with this order of magnitude estimate, we obtain a prediction of
$K=9.1$ m/s. These two values are compatible, especially as it is not possible
to estimate the bubble shape accurately.

Figure 6: (a) Velocity of the bubbles $v_{b}$ averaged over one period and
normalized by $v_{0}^{max}=v_{b}(z/\xi=0)$ as a function of the vertical
position $z/\xi$ in a foam made of $9$ layers of bubbles. The curves
correspond to different locations $x$ and the black dotted line is the
exponential fit $\exp[-z/(5.2\,\xi)]$. Inset: Evolution of the rescaled
maximum value of the mean velocity of the bubbles $v_{0}^{max}/(X_{0}\,f_{1})$
as a function of $\xi$. The dotted line corresponds to an exponential fit of
the experimental measurements. These measurements are obtained in a
$w=0.3\,{\rm cm}$ wide cell of length $L=7\,{\rm cm}$ and a height of foaming
solution $H=4\,{\rm cm}$. (b) Rescaled damping coefficient $w/\tau_{f}$ as a
function of $\xi$ for various widths of the cell: $w=0.3$ cm (blue circles),
$w=0.6$ cm (green triangles) and $w=1.6$ cm (red squares) in $L=7\,{\rm cm}$
cells. The black dashed line shows the best fit adjusting the prefactor in the
equation (14).

###  IV.3 From 2D to confined 3D foams

We performed experiments with an impulse excitation for three values of the
cell thickness $w$: $3$, $6$ and $15.9$ mm which corresponds to $1$, $2$ and
$5$ bubbles diameters in the $z$ direction, respectively. We follow the
experimental procedure developed for the 2D foam to determine the damping
coefficient. We describe these foams as confined 3D foams since we consider
that the sloshing of the interface can be described by the two-dimensional
modeling.

The phenomenological model developed in the previous section relies on an
hypothesis: the dissipation that enhances damping is only induced by the
viscous friction between the bubbles and the walls of the container. To
consider a confined $3$D foam, we consider that only the Plateau borders in
contact with the walls contribute to the energy dissipation and that the
bubbles in the bulk are simply advected by the bubbles in contact with the
walls. Thus, neglecting the contribution to the damping arising from the bulk,
the damping coefficient, $1/\tau_{f}$, should scale as the inverse of the
width of the cell, $1/w$.

The evolution of the damping coefficient for a $3$D confined foam is reported
in figure 6(b) as a function of the number of layers $\xi$ of bubbles. The
results for the different width collapse well on a master curve. In addition,
the experimental results can be fit with the exponential behavior obtained in
the equation (14). The exponential trend leads to a value of the plateau
${w}/\tau_{f}^{\infty}=1.6\pm 0.3\,\,\rm{cm}\,\rm{s}^{-1}$ whereas the best
fit suggests that the plateau has a value of ${w}/\tau_{f}^{\infty}\simeq
3.1\,\,\rm{cm}\,\rm{s}^{-1}$. The phenomenological model gives a good order of
magnitude of the contribution of the foam to the damping coefficient. The
relative error in prefactor can likely be explained by the shear between the
bubbles which can induce some dissipation. We also observe in figure 6(b) that
the saturation value of the rescaled damping coefficient $w/\tau_{f}$ seems to
slightly increase with the width of the cell. This observation indicates that
the dissipation in the bulk could add a small contribution to the damping for
a $3$D foam.

These results and the phenomenological model confirm that the foam is
responsible for the damping of the free-surface oscillations. A model based on
the viscous dissipation of the foam against the walls of the container
captures the evolution and the order of magnitude of the damping coefficient
in 2D and confined 3D situations. It is important to stress that the cells
used in this study are narrow, which means that the number of bubbles in the
transverse direction does not exceed 6 bubbles. Thus, further investigation of
3D foams is necessary to fully characterize the role of the third dimension.
In addition, to refine the present model, the influence of the foam elasticity
as well as the dissipation in the bulk should be taken into account.

##  V Conclusion

In this paper, we have studied how the presence of foam on top of a sloshing
liquid increases the damping coefficient and reduces the amplitude of the
free-surface oscillations. First, we characterize the sloshing dynamics in the
linear regime for a Newtonian fluid in Hele-Shaw cells. We show that the
measured resonant frequencies are well described by a wave model based on the
inviscid and irrotational flow hypothesis. Moreover, the damping coefficient
of the interface motion can be attributed to the fluid viscosity through the
oscillating Stokes boundary layers on the two walls.

Then, the influence of the foam thickness is experimentally quantified for 2D
and confined 3D foams. We report that the resonant frequencies are not
modified significantly by the presence of foam. We show that the damping
effects of the foam is significant even with only $5$ layers of bubbles. For a
larger number of layers, the damping coefficient and the amplitude of the
oscillations become independent of the foam thickness. For both 2D and
confined 3D foams, we rationalize our experimental results using a
phenomenological model that accounts for the viscous dissipation induced by
the motion of Plateau borders against the walls.

This study demonstrates that a relatively thin layer of foam effectively damps
sloshing. Our findings suggest that foam could be used in various industrial
processes in which sloshing needs to be minimized. Future projects should
consider additional effects and parameters relevant to those applications. For
instance, when the width of the container becomes comparable to the height of
foam, the interface motion becomes truly three dimensional and the
contribution of the bulk to the dissipation need to be taken into account.
Other parameters can be considered to enhance the damping effect such as the
interfacial rigidity that depends on the physicochemical properties of the
surfactantsDressaire  _et al._ (2008) and the roughness of the walls that
could increase the friction.

###### Acknowledgements.

The authors thank Isabelle Cantat, Marie-Caroline Jullien and Jonathan Katz
for helpful discussions and comments. This research was made possible in part
by the CMEDS grant from BP / The Gulf of Mexico Research Initiative.

## Appendix A: Sloshing of a single Newtonian fluid

###  .1 Analytical potential model

In the following we consider the small lateral excitation
$X(t)=X_{0}\,\sin(\omega\,t)$ of a rectangular rigid container of length $L$
and width $w\ll L$ such that the motion of the fluid remains two-dimensional
(figure 7). The height of the liquid at rest is $H$. The theoretical analysis
of the response of the fluid relies on the inviscid, incompressible and
irrotational flow assumptions. In this framework, the velocity field of the
fluid derives from a potential $\Phi$ such that $\bm{v}=-\nabla\Phi$. The
potential $\Phi$ satisfies the Laplacian equation:

$\nabla^{2}\Phi=0$ (17)

Figure 7: Two-dimensional schematic of the sloshing dynamics in a rectangular
container.

The potential $\Phi$ is the sum of the potential $\Phi_{0}$ associated to the
oscillating container, $\Phi_{0}=-X_{0}\,\omega\,x\,\cos(\omega\,t)$ and the
potential $\tilde{\Phi}$ that describes the relative motion of the fluid in
the reference frame of the container such that $\Phi=\Phi_{0}+\tilde{\Phi}$.
In the frame of reference attached to the container, the boundary conditions
on the bottom $z=-H$ and the sidewalls $x=\pm{L}/{2}$ are

$\frac{\partial\tilde{\Phi}}{\partial z}=0\quad\text{at}\quad
z=-H\qquad\text{and}\qquad\frac{\partial\tilde{\Phi}}{\partial
x}=0\quad\text{at}\quad x=\pm\frac{L}{2}.$ (18)

The linearized kinematic and dynamic conditions at the free interface
$\eta(x,t)$ lead respectively to Ibrahim (2005); Rocca  _et al._ (2005)

$\displaystyle-\frac{\partial\tilde{\Phi}}{\partial
z}=\frac{\partial\eta}{\partial t}\qquad\text{at}\qquad z=\eta(x,t),$ (19)
$\displaystyle{\rm and}\qquad\qquad g\eta-\frac{\partial\tilde{\Phi}}{\partial
t}+\frac{\gamma}{\rho}\kappa+\ddot{X}(t)\,x=0\qquad\text{ at}\qquad
z=\eta(x,t),$ (20)

where dots indicate time derivatives, $\eta(x,t)$ denotes the position of the
free interface measured from its equilibrium position, $g$ is the
gravitational acceleration, $\gamma$ is the surface tension and $\rho$ is the
density of the fluid. In the linear assumption, the curvature of the free
interface is $\kappa=-\partial^{2}\eta/\partial x^{2}$.

The solution of the equation (17) with the boundary conditions (18)-(20) is
given by:

$\Phi(x,z,t)=-X_{0}\,\omega\,\cos(\omega
t)\left[x+\frac{4}{L}\,\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}\sin(k_{2n+1}x)\cosh(k_{2n+1}z)}{{k_{2n+1}}^{2}\cosh(k_{2n+1}h)}\frac{\omega^{2}}{({\omega_{2n+1}}^{2}-\omega^{2})}\right)\right]$
(21)

where $\omega=2\,\pi\,f$ and $X_{0}$ are the angular frequency and the
amplitude of oscillation of the moving stage, respectively. In addition,
$k_{n}={n\pi}/{L}$ is the wavenumber and $\omega_{n}$ is the fluid-free-
surface natural frequency. Using the relation (20) we obtain

$\eta(x,t)=\frac{X_{0}}{g}\,\omega^{2}\,\sin(\omega
t)\left[x+\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}\sin(k_{2n+1}x)}{L\,{k_{2n+1}}^{2}}\frac{\omega^{2}}{({\omega_{2n+1}}^{2}-\omega^{2})}\right)\right]$
(22)

The resonant frequency of order $n$ of the system, $\omega_{n}$, satisfies

${\omega_{n}}^{\\!2}=\left(g\,k_{n}+\frac{\gamma}{\rho}\,{k_{n}}^{\\!3}\right)\tanh(k_{n}\,H)$
(23)

We notice that only the odd modes with respect to $x$ are present in the
Fourier developments for $\Phi$ (see equation 21) and $\eta$ (see equation
22); the $x$-component of the velocity field vanishes at $x=\pm L/2$, i.e.
$\partial\tilde{\Phi}/\partial x=0$ at $x=\pm L/2$.

###  .2 Experimental characterization

We first characterize the effect of the width of the cell on the amplitude of
sloshing and resonant frequencies using a Newtonian fluid, water. We set a
harmonic forcing $X(t)=X_{0}\,\cos(\omega\,t)$ and measure the maximum
amplitude of sloshing along the $x$-axis for various forcing frequencies and
cell widths. The results are reported in figure 8(a).

Figure 8: (a) Rescaled maximum amplitude $\bar{\eta}/{X_{0}}$ averaged over
$x$ as a function of the width of the cell $w$ for a forcing frequency $f=4$
Hz (blue diamonds), $f=5$ Hz (red circles) and $f=7$ Hz (green squares).
Inset: evolution of the first resonant frequency $f_{1}$ as a function of $w$.
(b) Maximum amplitude of sloshing $\bar{\eta}$ averaged over $x$ as a function
of the amplitude of the moving stage $X_{0}$ for a frequency $f=3.1$ Hz and a
width $w=0.3$ cm (green squares), a frequency $f=3.5$ Hz and a width $w=1.6$
cm (blue diamonds) and a frequency $f=4.8$ Hz and a width $w=1.6$ cm (red
circles).

Experiments with a Newtonian fluids are performed with cells of width
$w\in[3;15.9]$ mm, which are sufficiently thin to avoid three-dimensional
effects and sufficiently large to separate the boundary layer effects from the
flow in the bulk. The dispersion relation (23) shows that the resonant
frequencies do not depend on the width $w$ of the cell. This result is
confirmed experimentally in the inset of figure 8(a). Studies with two-
dimensional foam are performed with a cell of width $w=3$ mm to remain in the
regime where the potential theory can be used while allowing the visualization
of the bubbles.

The analysis presented in the previous section assumes that the response of
the fluid remains within the linearized dynamics. In this study we defined a
range of experimental parameters such that the fluid motions are in the linear
regime as shown in the left part of figure 8(b). In the following, we ensure
that the harmonic excitation or the impulse forcing are sufficiently low to
avoid wave breaking which would add complexity to the analysis.

Finally, we note that the height of the liquid $H$ is a parameter in the
dispersion relation (23). The influence of the height of the liquid on the
first resonant frequency $f_{1}$ is reported in figure 9. The resonant
frequency is obtained from harmonic forcing experiments by estimating the
frequency at which the amplitude of sloshing is maximal (inset of figure 9).
We observe a good agreement between our measurements and the analytical
expression. The dispersion relation reaches a plateau for $H\geq 4\,{\rm cm}$.
In this situation, the deep water model can be used and the dispersion
relation simplifies to
${\omega_{n}}^{\\!2}=\left(g\,k_{n}+{\gamma}\,{k_{n}}^{\\!3}/{\rho}\right)$.
Our measurements of the damping coefficient are focused on the deep water
regime.

Figure 9: Evolution of the resonant frequency $f_{1}$ as a function of the
water depth $H$ for a container of length $L=7\,{\rm cm}$ and two widths:
$w=1.59$ cm (in blue squares) and $w=0.30$ cm (red circles). The dashed line
shows the analytical expression of the first resonant frequency $f_{1}$.
Inset: evolution of the maximum amplitude of sloshing $\eta_{max}$ for forcing
close to the resonant frequency.

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