Fluid dynamics instabilities are usually investigated in two types of situations, either confined in cells with fixed boundaries, or free to grow in open space. In this article we study the Faraday instability triggered in a floating liquid lens. This is an intermediate situation in which a hydrodynamical instability develops in a domain with flexible boundaries. The instability is observed to be initially disordered with fluctuations of both the wave field and the lens boundaries. However, a slow dynamics takes place, leading to a mutual adaptation so that a steady regime is reached with a stable wave field in a stable lens contour. The most recurrent equilibrium lens shape is elongated with the Faraday wave vector along the main axis. In this selforganized situation an equilibrium is reached between the radiation pressure exerted by Faraday waves on the borders and their capillary response. The elongated shape is obtained theoretically as the exact solution of a Riccati equation with a unique control parameter and compared with the experiment.

The surface of a fluid becomes unstable when submitted to a vertical oscillation
of sufficient amplitude. This is the Faraday instability

This dependence of the Faraday instability on the geometry is a general feature of
fluid mechanics instabilities when they grow in confined cells imposing rigid boundary
conditions. We can take as an example the widely investigated Rayleigh?BeŽnard
convection

Here we wish to investigate the Faraday instability of a classical fluid in a case intermediate between the confined and opened situations. We sought a situation where the instability would grow in a domain of finite area but with flexible boundaries. This aim was achieved by triggering the Faraday instability in liquid lenses of weak viscosity floating on an immiscible fluid of larger viscosity. In the absence of forcing, the equilibrium shapes of such floating liquid lenses are well known, having been analysed experimentally and theoretically by Langmuir (1933), then Noblin, Buguin & Brochard-Wyart (2002) and de Gennes, Brochard-Wyart & QueŽreŽ (2002). They have a circular contour in the horizontal plane. The shape of their vertical section depends on their size. For small floating drops it is completely fixed by the laws of capillarity for triple line (Young?Laplace laws). The larger lenses (that we will mostly use) have a shape that results from the equilibrium of capillarity and gravity forces. They have a pancake shape of practically constant thickness bordered by a meniscus of well-defined shape.

In the presence of surface acoustic waves the capillary wave motion on sessile drops
has been investigated in a recent work

Video camera Semi-transparent

mirror Accelerometer

Vessel

Filter (b)

LED lamp Vibration exciter

Video camera

More viscous fluid 2 Razor blade

Less viscous fluid 1 Vessel

Filter LED lamp acting on its border, which tends to remain circular as a consequence of the capillary response. Moreover, because of the dissipation at the lens border, the wave patterns with the largest longitudinal extension have the larger amplification rate. Long patterns are encouraged to set up and the lens transits to a stable elongated shape. This phenomenon of mutual adaptation between the wave pattern and the lens boundaries is explained in § 4. In contrast, in situations where the capillary forces are weak, the radiation pressure exceeds the boundaries response so that no steady regime is reached. The lens keeps being stretched and disrupted by the effect of the waves. Small fragments of the initial lens then keep propagating and interacting on the fluid interface, merging and dividing again. In the present article we present in detail the experimental and theoretical investigations of the first archetype only.

The basic experimental set-up is standard and shown in figure 1. A vibration generator forces a cell containing the two fluids with a vertical vibration having tunable frequency and amplitude. An accelerometer is placed vertically on the top of the cell wall and measures the effective sinusoidal acceleration of the cell .t/ D m cos 2 f0t. In all of our experiments the investigated frequency and amplitude ranges are 30 Hz 6 f0 6 250 Hz and 0 6 m=g 6 10, respectively.

We use several types of photograph and video recordings: (i) using top views we investigate the horizontal shape of the lenses and measure the

Faraday wavelengths; (ii) side views are recorded to amplitudes; measure contact angles, thickness and wave (iii) a high-speed videocamera (acquisition frequency 2000 Hz) is used for the investigation of transients.

For top views, the videocamera is placed vertically above the box and records the evolution of the surface topography (figure 1a). Horizontal zones of the surface reflect directly the vertical light beams, so that white regions on the photographs correspond either to the crests or the troughs of the waves. The Faraday wavelength is measured as the distance between two crests.

During side view imaging the videocamera is placed in front of the cell and collects the light coming from behind it. In order to observe the emerged and submerged part of the floating lens we have used transparent Plexiglas cells. The liquid substrate forms a meniscus on the cell walls which impedes direct observation of the lens. This problem is surmounted by glueing a razor blade on a cell top wall so that the liquid pins onto it (figure 1b). We also need to measure the amplitude of the Faraday waves that is of the order of 0.5 mm. This is done by strobing with a large magnification the oscillation close to the Faraday frequency.

We wish to trigger the Faraday instability in a liquid lens of fluid of weak viscosity floating on a more viscous liquid forming the substrate. Much attention has been paid to the choice of the two liquids as they must satisfy some criteria.

In the following we systematically denote by subscript 1 the physical characteristics of the floating liquid lens and by subscript 2 those of the liquid bath. We use , and to denote the density, surface tension and viscosity, respectively. We use pairs of liquids satisfying the following criteria.

(i) The two liquids must be immiscible. (ii) In order to obtain floating lenses, 2 > 1. (iii) In order to trigger Faraday waves in the lens only and avoid significant waves in the liquid 2 bath, 2 1. The threshold amplitude of Faraday instability increases with the fluid viscosity. The larger the difference between the viscosities, the larger the range of amplitude for which the waves are triggered in the lens only while the bath remains at rest.

We performed the first experiments using water lenses floating on a fluorinated oil bath. We also varied the surface tension by adding surfactants and noticed that it plays a crucial role in the system behaviour. We chose to switch from water to alcohols whose surface properties are stabler and less sensitive to the environment. Alcohols evaporate rapidly so we had to close the cell with a transparent cover for long-lasting experiments. We tested many pairs of liquids, usually an alcohol and an oil. We finally distinguished two global archetypes of evolution of the system.

In this paper we focus on the behaviour leading to the quasi-equilibrium of the floating lens shape, in which wave radiation pressure and capillary pressure jump are comparable. We have found that this behaviour is common to many liquid pairs, in particular alcohols (ethanol, butanol, pentanol, propanol) and water on fluorinated oils. The behaviour of all of the alcohol lenses deposited on fluorinated oil is similar once the Faraday instability is triggered. We choose the most viscous fluorinated oil in order to obtain a large viscosity contrast, and use isopropanol because the lens shapes we obtain look more stable in a wide range of forcing parameters. Throughout this paper we present results obtained for fluorinated oil with density 2 D 1850 kg m 1, surface tension 2 D 16:2 0:3 mN m 1 and viscosity 2 D 26 mPa s and isopropanol with density 1 D 785 kg m 1, surface tension 1 D 20:3 mN m 1 and viscosity 1 D 2:26 mPa s. The interfacial tension measured by the pendant drop method is 1 cm

D

E

HP

12 D 6:4 0:1 mN m 1. Throughout the paper we use the term ?surface? for the contact surface between liquid 1 or 2 and air and the term ?interface? for the contact surface between liquid 1 and liquid 2.

We deposit a lens of isopropanol on a fluorinated oil bath and increase the forcing amplitude. Above threshold waves are disordered and generate fluctuations of the lens boundary. In return the fluctuations of the boundary generate an unsteadiness of the wave pattern.

The phase diagram on figure 3 gives the various behaviours observed for a lens of volume 1 ml submitted to forcing of increasing amplitude m=g at various frequencies f0. Below the Faraday threshold the lens is circular (C) in the horizontal plane and presents a flat surface. The region of interest is located between region (C) of the phase diagram where no Faraday waves have formed anywhere and region (F) where they cover both the lens and the bath. It is the area located between the black and the red line, corresponding respectively to the lens and the bath Faraday thresholds (see figure 3). In this area we investigate the Faraday instability in the deformable domain constituted by the lens.

When the forcing amplitude is increased, Faraday waves appear in the lens above a first onset m > mD (transition (C) to (D) in figure 3). A circular wave appears as a transient followed by complex unsteady patterns. They deform the lens (figure 2, D), its average shape remaining circular. Sometimes we observe that the lens shape becomes stable being slightly deformed from the initial circle, while definite patterns establish on it. These states are described in § 3.1. Increasing the amplitude, lens boundary deformations become larger and wave paths try to elongate the lens in some directions.

Above a second threshold mE (transition (D) to (E) in figure 3) the lens elongates into a stable shape (figure 2, E). The waves strengthen and start organizing, resulting into the elongation of the lens in the direction parallel to the wave vector. Correlatively this elongation favours a further organization of the waves. By a slow (t 1=f0) evolution the lens reaches a stable elongated shape with a standing wave pattern. The wave vector is parallel to the long side of the lens, as is usual for Faraday waves in elongated rigid cells. By increasing the forcing amplitude the lens undergoes a further elongation. Above a third threshold mHP (transition (E) to (HP) in figure 3) the waves become disordered and the envelope of the domain undergoes very large fluctuations. The shape is no longer elongated and returns to being circular on time average (figure 2, HP). Finally at a fourth threshold mF Faraday waves form on the viscous substrate.

In the following we focus on the deformed and elongated regimes, the latter being described via the equilibrium between the wave radiation pressure and the boundary capillary response.

Above the Faraday threshold the lens is deformed by waves from its initial circular
shape. It is possible to observe stable states when an eigenmode of the Faraday wave
?fits? in the size of the circular lens. Most of them are circular cavity modes already
observed for the Faraday instability

We also observe non-axisymmetric modes, their stability being due to the stabilization of curved Faraday paths (figure 4d,e). They are stable and become axisymmetric if the forcing amplitude is increased.

Finally we have also obtained stable regimes in which the Faraday pattern is quite distorted and the lens begins rotating (figure 4f ), a behaviour that can be ascribed to the wave radiation pressure exerted on the lens boundaries by the surrounding liquid.

Above a given threshold mE the lens reaches an elongated stable shape. By looking at the phase diagram (figure 3) we note that the area in which the elongated regime exists increases in width as the frequency exceeds f0 ' 110 Hz. The elongated state is more stable for f0 > 110 Hz, so we focus on high-frequency observations. For f0 < 110 Hz the parallel wave pattern is perturbed by a supplementary perturbation along the whole lens. Its origin can be ascribed to global modes more easily excited because of the low ratio between wavelength and lens size.

Above f0 ' 110 Hz in the zone indicated by ?E? in the phase diagram the lens achieves a stable elongated regime with a pattern of parallel waves in it. Observations by high-speed videocamera (figure 5a) show that waves are standing on the lens surface. Moreover, waves are also observed in the surrounding bath, mainly near the lens tips, and they are standing too. As the forcing amplitude is increased the elongation increases (figure 5b). We observe that the tip remains fixed while the body becomes thinner and longer (figure 5c). The amplitude of waves in the surrounding liquid increases too.

We have also observed particle motions inside and around the lens (figure 5d ). Inner
particles are strongly transported to the tip if they are on the surface, and return to
the middle of the lens length moving on the interface. The transport can be ascribed
to Stokes drift, already observed in Faraday instability even for small-amplitude waves
and largely ordered wave pattern

We have measured the wavelength in the elongated lens for different forcing
frequencies. Taking the wave frequency f as half the forcing frequency

Finally we have measured the variation of the lens surface in the deformed regime and for different elongations. At high frequency the surface variation does not exceed 5 %. This is also the relative error to surface measurements, so we suppose that the lens average thickness remains constant during the transition and the further elongation.

The equilibrium shape results from the competition between the radiation pressure exerted by waves on the lens border and the border response due to surface tension. We define a dimensionless parameter as the ratio between these two effects 2 2 1! 0 a0 D 4. 1=R0/ (3.1) where ! is the wave angular frequency, 0 the wave amplitude and R0 the lens radius at rest. For frequency and wavelength typical of our experiment and taking R0 ' 1 cm and 0 D 0:5 mm, we find a0 ' 1. The radiation pressure is then of the order of magnitude of the surface tension response. This explains the observed equilibrium shape. 1 cm

E 1 cm

P

Sq 1 cm 8 6 4 2

F

E

D

C 50 100 150 200 250

An increase of the wave amplitude 0 results in a larger value of a0. Experimentally this corresponds to a further elongation of the lens and a new equilibrium situation. This is because, as a result of the elongation, the horizontal curvature at the tips increases (R0 decreases), so the restoring force due to surface tension increases too.

Large floating lenses behave similarly to small ones with the addition of new regimes. We report the results of experiments performed with an isopropanol lens of volume 10 ml (figures 6?7).

At low frequencies the lens starts elongating at the Faraday onset. The response of
the border is very low because of the small horizontal curvature (large lens radius).
The elongation process is very slow (several minutes) compared with small lenses.
Moreover, we recognize transverse perturbations that continuously evolve but do not
perturb the shape (figure 6, E). They can be related to the competition among scarred
patterns observed in the case of stadium geometry and due to their different dissipation
rates

At higher frequencies the lens starts elongating only above a certain threshold and by increasing rapidly the amplitude we observe at least two different shapes: a pear shape (figure 6, P) and a square (figure 6, Sq). Such shapes cannot be obtained by slowly increasing the amplitude, because it will only lead to a further elongation of the lens (this is why we have E=P and E=Sq domains in the phase diagram).

The pear shape is obtained by setting the amplitude above threshold. A square
pattern takes place on one lens side but not on the other side, so that the waves
pushing action continues (figure 6, P). The transition from stripes to squares was
observed in the case of very large cavity by

If waves are triggered directly at very large forcing amplitude the shape obtained
is a square (figure 6, Sq). A square pattern takes place first on the lens while the
border remains quite disordered. Afterwards, the border tries to adapt to the pattern,
usually with a tilted square configuration in which wavevector forms an angle of 45
with the sides. This is what is expected by the superposition of two perpendicular
wave fields. Nevertheless, this configuration is unstable and borders slide in order
to be parallel to one wavevector and perpendicular to the other, that is the case of
figure 6(Sq). However, this configuration is only transiently stable. Global oscillations
of the lens destabilize it, then they disappear, the square forms again and so on. While
in small lenses the mutual adaptation between border and wave pattern is simultaneous,
here the square pattern takes place first, then borders adapt. This is because the
wavelength R0, so the lens behaves as a large cell and presents the same pattern
observed in a rigid square cavity

We have mentioned that the competition between the wave radiation pressure and the capillary response is essential to obtain equilibrium shapes. We note that the most recurrent equilibrium shapes are elongated with a unidirectional wave pattern. We explain their origin in this section.

Kudrolli et al. (2001) already noticed that patterns selection at threshold was
also determined by dissipation at the walls of the container. An explanation for the
selection of modes has been provided by

In a complementary experiment we have investigated patterns at threshold in square cavities (figure 8). In order to avoid meniscus effects at the walls, the cavity has been immersed in the liquid bath as shown in figure 8(a). The small layer of thickness h above the walls inhibits the formation of waves on them.

We have used cavities of different sizes and observed that the threshold pattern always develops along the diagonals, showing that the pattern length is the dominant factor for the pattern selection (figure 8b). We have also observed the pattern formation in rectangular cavities and cavities of arbitrary shape. At the onset, patterns maximize their length and usually they are not straight. They curve in order to achieve 0.02 0.04 0.06 0.08

H
the longest possible path. We have measured the threshold in square cavities and we
have noticed that it varies linearly with the inverse of the cavity size (figure 8c), then
as the inverse of the pattern length, in agreement with

In the case of a floating lens, as soon as a deformation has affected the circular shape, the creation of a long array of waves will be favoured. In the perpendicular direction the short axis tends to shrink so that the role of the dissipation at borders increases. Waves having a wavevector in that direction become more damped and vanish. This is the origin of the observed transition from a pattern of waves in all directions (figure 2, D) to a unidirectional one in the elongated lens (figure 2, E). We have checked this idea by performing complementary experiments and measuring the Faraday threshold in lenses of different size. The measured thresholds are linear function of the inverse lens radius, indicating that pattern length and dissipation at borders play an important role even in this system (figure 8d ).

Equilibrium and non-equilibrium shapes observed when the Faraday instability is triggered in the lenses are the result of the competition between the action of waves on the lens border and the restoring force due to the curvature dependent capillary pressure at the menisci.

The lens shape is given by the solution of a three-dimensional free boundary problem. In fact, the surface wave deforms the lens shape from the circle to some ovoid shape, but also the lens shape modifies the flow itself. As all of the threedimensional free boundary problems, the solution is extremely difficult to find even numerically, especially for finite size domains. For this reason we present here an oversimplified model, in which many simplifications are suggested by experimental observations.

We assume as an ansatz that the surface waves organize into standing waves, a
surprising result if we consider that the initial circular geometry is itself a solution
of Faraday waves. Moreover, as shown by the experiments there exists a range of
parameters for which the border of the lens is static on average. For this reason we
average the pressure field and obtain a wave radiation pressure that only depends on
the horizontal length scale of the lens. Finally we take advantage of the fact that
the lens has thickness at the centre H D 2:18 0:05 mm which is small compared
with the horizontal scale of the order of the initial radius R0 D 12:5 0:5 mm, in
order to perform a lubrication approximation in the spirit of successful results obtained
in viscous fingering

We consider the static surface of the upper part of the lens. The pressure jump between the interior and the exterior can be written

P 1gz C Pr D 1H (5.1) where P is the pressure at the horizontal mid-plane and Pr is the pressure due to waves that we will call radiation pressure, inspired by the work of Longuet-Higgins & Stewart (1964) on the radiation stress of water waves. The right-hand side of (5.1) represents the capillary pressure, with H the local average curvature of the lens.

5.1. Waves on a nite liquid layer overlying a nite liquid bath In order to obtain the radiation pressure on the lens border we have to take into account the structure of the waves in a floating liquid layer of finite thickness. We have thus solved the hydrodynamical problem of standing gravity-capillary waves produced at the surface of a liquid layer resting on a second liquid layer. As a consequence of the lens finite thickness the waves could perturb the bottom interface between the lens and the bath.

The vertical structure of surface waves in the thin layer of the lenses was clarified by specific additional experiments performed on an extended thin layer of alcohol deposited on a bath of fluorinated oil. The cell was closed so as to minimize the evapouration. We triggered the Faraday instability in the system and we measured the wavelength as a function of the excitation frequency. The experimental data of the dispersion relation are reported in figure 9(a). Moreover, we performed observations from the side of surface and interface displacement (figure 9b). We observed a barotropic mode in which the interface separating fluid 1 from fluid 2 oscillates in phase with the surface of fluid 1.

With these observations we solve the complete hydrodynamical problem of gravitycapillary waves produced at the surface of a liquid 1 layer overlying a bath of a denser and more viscous liquid 2 in the case of small surface displacement.

To solve exactly this problem we refer to

The critical wavelength for surface waves in the experiment is c D 8 12= 1 1 '
10 3 mm. As we have c the dispersion relation is not affected by viscosity

We consider an unidirectional standing surface wave described by the surface displacement function .x; t/ D 0 cos kx cos !t (5.2) where 0 is the surface wave amplitude, k is the wavenumber and ! the wave angular frequency. We assume that 0 is small, so that 0= 1 and 0=H 1. Moreover, we assume that the interface is perturbed by the surface displacement so that it oscillates at the same frequency. The interface displacement is described by

.x; t/ D 0 cos kx cos !t with 0 the interface wave amplitude and we assume again We assume the following potential functions in liquid 1 0= 1 and 0=H0 1. and liquid 2

H: For incompressible fluids both potentials must satisfy the Laplace equation r2 1 D 0, r2 2 D 0 and the following boundary conditions

C g at z D

H0 at z D 0 2g at z D

H: (5.10) Equations (5.6)?(5.8) concern the continuity of the normal component of the velocity, while the two last equations are the Bernoulli relations to apply respectively at the (5.3) (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) ekzi cos kx sin !t for ekH .ek.zC2H0/ C e kz/ cos kx sin !t for z 6

H 0 D 0.e kH C F sinh kH/ F D 1 gk C Experimentally we obtain 0= 0 ' 20 % at f0 D 59 Hz, that is comparable to the value 0= 0 D 15 % obtained from (5.13). The magnitude of the interface perturbation depends on the product kH where k D 2 = is the wavenumber and h the lens thickness. When kH > 1, that is the thickness is larger than the wavelength, the waves are sufficiently damped so that the interface can be considered at rest. This is the case of floating lenses in the first experimental archetype. When kH < 1 the interface perturbation becomes important.

Here F is a factor which takes into account finite layer effects on the dispersion relation. It is the dimensionless deviation from the dispersion relation we will have in the case of an infinitely deep fluid !2 D gk C 1 3

k : 1 The superposition with experimental data in figure 9 shows that for H D 1:92 mm waves disperse as if the bath was infinitely deep.

5.2. Radiation pressure of surface gravity-capillary waves Faraday waves deform the lens by exerting a radiation pressure on its border and the shape at equilibrium is elongated. Waves are unidirectional and standing (figure 11a), while at the extremities a spatially damped wave is excited into the liquid substrate 2 (figure 11b).

We calculate the radiation pressure exerted by a standing unidirectional wave (equation (5.2)) acting on the border at the horizontal mid-plane (figure 12). The (5.11) (5.12) (5.13) (5.14) (5.15) x x (c) 0.08 thickness of the lens H is observed to be constant during the process so this will be our assumption in the calculation which follows.

Lens observations from the side do not allow us to measure any interface displacement (figure 11a). Moreover, from (5.13) this displacement is supposed to be 2 % the surface displacement in the range of frequencies the steady elongated state is observed (figure 11c), so we assume 0 D 0.

Condition (5.8) then simplifies and gives We find

H:

ekz cos kx sin !t where we have considered e kH 1. Looking at the lens tip we observe waves in liquid 2 (figure 11b), so the velocity potential around the lens is not constant. We calculate it by imposing the continuity of the horizontal velocity at the interface where h1 is the emerging height of the lens (figure 12a). Integrating over x u1 D u2 at

H 6 z 6

h1 we find the amplitude of the surface displacement of liquid 2

h1 where we have considered the wave amplitude at the tip 0=2, as suggested by observations, and e 2kH 1. We estimate 0=. 0=2/ ' 20 % at f0 D 130 Hz, in agreement with the experimental observation of figure 11(b). The radiation pressure on the lens border is then given by the difference between the pressure of the waves from the interior and the exterior of the lens, but the latter is only a few per cent of the first one so it can be neglected.

Since the lens shape is strongly coupled to the wave fields in both fluids, we will
consider only the upper cap of the lens. Obviously, the horizontal lens contour we are
interested in pertains simultaneously to both caps, the upper one in contact with air
and the lower one in contact with fluid 2. Local equilibrium of the curved horizontal
triple line gives the so-called wetting angles in the case of partially wetting fluids

We define n D .nx; ny; nz/ as the unit vector laying perpendicularly to the cap border (figure 12b). The momentum flux density tensor is (Landau & Lifshitz 1987) We calculate it in the local frame defined by the normal n and a tangent vector and consider only the 101 component, acting perpendicularly to the border. We have ik D p ik C

u1iu1k: 101 D p C 1u12nx2 where 0ik is the local momentum flux density tensor. We average over one period and one wavelength. The average on space assumes that the wavelength is small compared with horizontal shape variations, a hypothesis that breaks down for sharp tips. The average on time assumes that the dynamic of the lens horizontal shape is slow compared with the period of the Faraday waves, in agreement with experimental observations. In our system p is varying with time because of vibrations, but the time-dependent term will disappear by the time averaging and does not enter the tensor components. Finally we obtain the radiation pressure

1 2 2 Pr D 4 ! 0 1 C y 0.x/2 with y.x/ the profile function of the lens contour in the horizontal mid-plane.

5.3. Determination of the horizontal contour Our objective being the shape of the horizontal border, we take into account that the lens thickness is small compared with the horizontal size ( 1=10). To eliminate (5.20) (5.21) (5.22) (5.23) (5.24) the third dimension, we perform a lubrication approximation of the three-dimensional Laplace law of capillarity with the Faraday forcing, inspired by Park & Homsy (1984), Reinelt (1987) and Burgess & Foster (1990). The aim of these works was to analyse the correct boundary conditions for viscous drops or fingers confined by the two horizontal plates of a Hele-Shaw cell. In our case there is no real confinement but a small thickness due to the physical parameters.

We focus on the meniscus region that we assume to be scaled by the thickness h1, i.e. the maximum value of the upper cap height. We define a local coordinate system (rQ; sQ) at a point M of the border in the mid-plane with coordinates .x; y; 0/ (figure 12b). Being in the close vicinity of the border, rQ is defined on the outer normal of the contour at M and is scaled by h1. Here sQ is the arclength and it is scaled by R0. The small parameter of expansion in our treatment is the ratio of the upper cap height to the lens radius D h1=R0. We will show that h1 is indeed the height determined by Langmuir (1933) for a cylindrical lens by considering the interplay between gravity and capillarity h1 D 2 r 1 sin s 1g 2 (5.25) where s is the local wetting angle on the border in the horizontal mid-plane. For a non-circular border, s depends on the position of the point M on the contour. We do not expect a sharp variation to its value except for pointed ends. Experimentally, we have measured h1 D 1:04 0:06 mm, in agreement with the value h1 D 0:96 0:07 mm predicted by (5.25).

The height of the upper cap hQ being scaled by h1, P will be scaled by 1=h1 and we can define the dimensionless quantities

P D h1 p.s/

1 rQ D h1r sQ D R0s hQ D h1h.r; s/:

D 41h11 !2 02: With these definitions, equation (5.1) becomes pi D h C

y02 hr2 1 C y02 1 C hr2 C p.s/ D d hr dr .1 C hr2/1=2 C c.s/

hr .1 C hr2/1=2 where pi is the pressure below the surface with height h.r; s/, D 4sin2 s=2 and c.s/ is the curvature in the .x; y/ plane. Here is the radiation pressure factor In our system both terms and h.r/ as and

are of order 1=10. We expand the quantities p.s/ p.s/ D p.0/ C c.s/p.1/ C O. 2/ h.r/ D h.0/.r/ C c.s/h.1/.r/ C O. 2/ (5.26a) (5.26b) (5.26c) (5.26d ) (5.27) (5.28) (5.29a) (5.29b) in order to solve the master equation (5.27). For our purpose we need p.1/, then we have to compute p.s/ and h.r/ up to first order. The details of the calculation are reported in Appendix.

We can now write the equation for the lens basis in the horizontal mid-plane ssin2 s C p.0/ D c.s/p.1/: (5.30) In our experiments s D 34:3 2:0 is the value measured on side view images of the lens at rest, i.e. without waves. Being a slowly varying function of the forcing, p.1/ remains close to 1=4 (equation (A 18)).

Equation (5.30) can be expressed in terms of the curve describing the border At rest a D 0 D 0 and we obtain which has two solutions b C a y02.x/ d

y 0.x/ 1 C y02.x/ D dx p1 C y02.x/ where b is the unknown pressure, so a Lagrange parameter which will be fixed by volume conservation, and a measures the strength of the waves with respect to the surface tension effects where b0 D 1 and we recover the circular lens.

Now we turn to the complete shape (5.31). Taking we obtain a Riccati equation Taking y 0.0/ D q.0/ D 0 for x D 0 we obtain

y 0.x/ q.x/ D p1 C y02.x/ D r b a

p 1 tan. abx/ D p tan.b p

Ax/ A where A D a=b. We solve it with the condition of cancellation and vertical tangent on the x-axis at the lens extremities y. L=2/ D 0 and q. L=2/ D 1: We find the exact solution of (5.31): y.x/ D

1 r

A y l

L x (c) (d)

The problem reduces to one free parameter by applying the volume conservation. The height of the lens remaining approximatively constant, the area can be considered constant during the elongation. Fixing its value to we find the relation between b and A b D s log.1 C A/

Ap1 C A : The horizontal shape is given as a function of the sole parameter A which is the control parameter, in principle a tunable dimensionless number.

We compare the theoretical solution for the lens shape reported in (5.39) to experimental observations. We plot the analytic solutions for different values of A and superpose them on experimental images of the elongated lens obtained by increasing the forcing amplitude at fixed frequency (figure 13b?d ). Theoretical shapes superpose well to the experimental ones for many elongations. The model is no longer valid in the case of very strong elongation, such as that reported in figure 5(c). In this case, experimentally we observe that the tip shape remains fixed while the lens keeps elongating, with sides remaining linear and parallel. This situation of limiting value for the tip is also found in anisotropic viscous fingering where the tip radius cannot decrease below a value of order the thickness of the Hele-Shaw cell. The tip radius being of the order of the wavelength the initial hypothesis of small thickness and wavelength compared with the radius breaks down. Moreover, the hypothesis of standing waves is obviously inadequate at the tip, whose size is comparable to the viscous boundary layer. This explains why our model cannot account for strongly elongated shapes. 1 cm (5.40)

In order to have an accurate comparison with experimental results we define the lens aspect ratio as the ratio of the lens width over its length. The lens length derived from (5.37) is (5.41) (5.42) (5.43) (5.44) 2 L D bpA arctan pA while the lens width l given by y.0/ is

2 l D bpA.1 C A/

p log. 1 C A C p

A/: The lens aspect ratio R D l=L is then

logTpA C R D p 1 C A arcta1nCpAAU :

p a2p1 C A D A log.1 C A/ The measurable parameter is a given by (5.32). From (5.40) we have the relation between a and A which allows a numerical calculation of A for a given a. Here a is a function of the fluids properties, the lens profile and the wave amplitude. While the other characteristics have been reported in the previous sections, the wave amplitude must be measured as a function of the forcing amplitude and frequency.

Side views of the lens in the elongated regime show that the amplitude varies spatially. It is larger at the centre of the lens and lower at the tips and sides. We have measured the amplitude by pointing the videocamera to the lens centre and we have taken into account the effect of decreasing the amplitude on the lens side. The results we report are the arithmetic average of the amplitude measured at the centre and side of the lens. The effect of lowering amplitude while approaching the lens side disappears when the amplitude becomes important. We have measured the wave amplitude for lenses in the elongated regime at different forcing frequencies and varying the forcing amplitude. Measurements are directly performed on images acquired by side view and results are reported in figure 14(c). We have tried to measure the wave amplitude at f0 D 80 Hz, but it is irregular because of the aforementioned perturbation at low frequencies (figure 14a), its value being around 0.6?0.7 mm. At f0 D 120?140 Hz waves are regular and the amplitude can be properly measured with uncertainty evaluated to 1 0 D 0:03 mm. In this range of frequencies the wave amplitude lays in the interval 0 D 0:3?0.6 mm. We have found that the square wave amplitude is proportional to the forcing one as observed for usual Faraday waves and reported by Douady (1990). We extrapolate the amplitude for any forcing from linear fits (figure 14).

In order to fit the experimental data of the lens aspect ratio we calculate values of the parameters a and A at f0 D 120, 130, 140 Hz from (5.32) and (5.44). The wave amplitude 0 is taken from the linear fits in figure 14(c). The computed aspect ratio R is a decreasing function of the forcing amplitude as expected. With such values of a, R results underestimated with respect to the experimental ones. We fit the experimental points obtained at f0 D 140 Hz because we have the shorter wavelength, in agreement with the approximation of our model. We find that the best fit is obtained using a0 D 0:9a as shown in figure 15. The need for this corrective factor could be due to x x z z 1 mm 1 mm (c) the uncertainties in the measurements of the wave amplitude 0 and the wetting angle s entering the expression of the parameter a (5.32). It is more probably due to the simplified assumptions used in the model.

We have investigated experimentally the mutual adaptation of Faraday instability patterns and the flexible boundaries of a floating liquid lens. In our system the radiation pressure of Faraday waves can be balanced by the capillary response of the lens border, so that steady lens shapes are observed. They depend on the wave amplitude and the ratio of the wavelength over the lens horizontal size. Several equilibrium shapes were obtained. For small lenses and wave amplitude we have observed either Bessel or asymmetric modes as a result of slight adaptation of the lens shape to the waves. For larger lenses the wave field is initially disordered and its pressure generates random fluctuations on the boundaries. However, a slow dynamics is observed to take place, in which the wave field becomes more ordered as the lens elongates. Finally an equilibrium is reached with a stable field in a well determined contour. As it was shown this shape is given by the equilibrium between the wave radiation pressure and the capillary response of the lens border. It is an experimental evidence of the effect of the radiation pressure of gravity-capillary waves. Elongated shapes are predicted theoretically as the exact solution of a Riccati equation with one control parameter. The lens aspect ratio has also an analytical expression which fits the experimental results up to a factor of order one.

The observed phenomena have a similarity to the self-tuning by a slow dynamics
observed previously in several types of resonant systems. In all of them the existence
of an additional degree of freedom was essential to its ability to self adapt. The
simpler of these systems introduced by Boudaoud, Couder & Ben Amar (1999b) is
formed of a vibrating wire loaded by a small sliding mass. When set into vibration
at an imposed arbitrary frequency the small mass is observed to slide slowly until
it reaches the position where the whole system is at resonance at the imposed
frequency. Similarly in a vibrating soap film the mass distribution was shown to
adapt to set the film at resonance

We are grateful to A. Eddi, E. Fort and J. Moukhtar for fruitful discussions and M. Receveur and L. ReŽa for technical assistance. UniversiteŽ Franco-Italienne (UFI), ANR 06-BLAN-0297-03 and ANR FREEFLOW supported this work.

Here we show how to compute the parameter p.1/ which enters the shape equation of the liquid lens under forcing (5.30). Owing to their complexity equations (5.29a) and (5.29b) cannot be solved exactly, so we will distinguish the close vicinity of the border (what we call the inner region) from the far field (outer region), always remaining in the meniscus region.

For the zero-order we obtain p.0/

D df .r/ dr

C

A.1. The zero-order results h.0/ s D where with solution df .r/ dr s p.0/

D s where p1 D pp.0/= s and p2 D p sp.0/l.0/ is defined by f .0/ D sin s, that is

sin s D p1 tan.p2l.0//: We can integrate (A 4) obtaining a rather complicated expression, that is not necessary for our purpose. This solution has to be matched to the outer solution deduced from (A 1) with hr 1. We obtain for the outer region which can be integrated giving h.0/ r D s s q p.0/

D h.0/ rr C h.0/

s.h.r0//2 .1 h.0// e2 s.1 h.0// with

D 1 1 2 s p.0/ : Then is a profile parameter which depends on the position on the lens contour. (A 2) (A 3) (A 4) (A 5) (A 6) (A 7) (A 8) (A 9) !)

: In order to recover the behaviour predicted by Langmuir (1933) for r ! 1 and h ! 1, we fix the value of to 1=.2 s/. We obtain p.0/ D , that is the result for gravity-capillary lens.

The asymptotics of the outer solution is h.0/.r/ 1 e p r for r ! 1 where is an integration constant (with respect to r) to be determined by matching this outer solution with the inner one. A small value of the parameter corresponds to a large size of the meniscus. We match the slopes using the inner solution given by (A 5) and we obtain p s D tanTpp.0/ s.l.0/ l1/U where l1 is the matching point.

In order to match the height we integrate (A 4). We do not report the solution of this equation, but we give the result for as a function of p1 D pp.0/= s D 2 sin. s=2/=p s .1

1 / s D p1 C p1 2

2 1 C p1 log s

2 4 p1 C p1 p12 C sin2 s C pp21 C sin2 s p1 cos s Here s (so p1) varies along the contour and we consider the limits s ! 0 and s ! 1. Experimentally, the lens shape in the horizontal plane is slightly curved except at both tips. Hence, the limit s ! 0 is valid except for very strong forcing, while the limit s ! 1 corresponds to the tips. If s is small we recover the gravitycapillary limit

1 D .1 cos s/=p.0/ D 21 (A 11) a result that we can compare to the work of Burgess & Foster (1990). For s large we find (A 12) (A 14) (A 15) 1

1

D sin s then the lens profile at the tips depends on the wetting angle.

Now we consider the first order of the expansion (5.29a). This will give us p.1/, which is the parameter we really need to fix the horizontal shape of the lens, being the prefactor of the border curvature c.s/. The first-order term of (5.27) gives the following linear ordinary differential equation d h.r1/ hr.0/ dr .1 C .h.r0//2/3=2 C 2 s .1 C .h.r0//2/1=2 h.r1/ C

D Looking for a solution of the homogeneous equation for h.1/, we notice that a possible solution is h.0/ since the left-hand side is simply a linearization of the fully nonlinear r equation. In order to check that the asymptotics is not destroyed by the first order, we consider the outer solution first. We have h.1/ rr C 2 sh.r0/h.1/ r C h.1/ Since y1.r/ D h.r0/ behaves as e p r at infinity, y2.r/ behaves as ep r. This proves that the asymptotics is controlled by the interplay between gravity and capillarity. The general solution of (A 14) is then

Z r

Z 1 h.1/ C p.1/=

D y1.r/ h.r0/y2.x/=W.x/ dx C y2.r/

h.r0/y1.x/=W.x/ dx C c1y1.r/ (A 16) We integrate twice to derive the profile perturbation h.1/. Requiring that h.1/ vanishes for r D 0 and r D l.0/ we obtain the correction to the pressure p.1/ D p1 C p1 2 cot 1..p1=p1 C p12/ cot s/

.p1p1 C p12=.p21 C sin s2// sin s cos s 1 C sp12 p21 cos s=.p12 C sin2 s/

: (A 18) Here p.1/ can be calculated numerically once s is fixed. In the capillary limit and taking s D =2 we recover the result of Burgess & Foster (1990).

R E F E R E N C E S

LANDAU, L. D. & LIFSHITZ, E. M. 1987 Fluid Mechanics. Butterworth-Heinemann/Elsevier. LANGMUIR, I. 1933 Oil lenses on water and the nature of monomolecular expanded films. J. Chem.

Phys. 1, 756?776.

LONGUET-HIGGINS, M. S. & STEWART, R. W. 1964 Radiation stress in water waves; a physical discussion, with applications. Deep-Sea Res. 11, 529?562.

MARQUARDT, F., HARRIS, J. G. E. & GIRVIN, S. M. 2006 Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities. Phys. Rev. Lett. 96, 103901.