M Maza-Cuello

C Frétigny

L Talini

Probing interfacial effects with thermocapillary flows

We report on the thinning of supported liquid films driven by thermocapillarity. The liquids are oil films, of initial thicknesses of a few tens of microns. A local and moderate heating of the glass substrate on which they are spread on induces a thermocapillary flow, which allows the formation of ultra-thin films. We show that, within a given time range, the thinning dynamics of submicron thick films is governed by the thermocapillary stress. It results in a simple dependency of the thickness profiles with time, which is evidenced by the collapse of the data onto a master curve.

The master curve only depends on the liquid properties and on the thermal gradient, and allows a measurement of the latter. As the films further thin down to thicknesses within the range of molecular interactions, a deviation from the master curve appears. Although all investigated oils are supposedly fully wetting glass, the nature of the deviation differs between alkanes and silicone oils. We attribute the observed behaviors to differing signs of disjoining pressures, and this picture is confirmed by numerical resolution of the thin-film equation. We suggest thermocapillary flows can be used to finely probe molecular interactions.

I. INTRODUCTION

The surface tension of a pure liquid decreases with increasing temperature and a temperature gradient at its free surface results in a thermocapillary stress, inducing a flow away from the hottest region. The migration of liquids induced by a temperature gradient has been known and exploited for a long time [START_REF] Hershey | Ridges in a liquid surface due to the temperature dependence of surface tension[END_REF]. It is used for instance for the patterning of surfaces, in the context of micromachining with molten metals [START_REF] Anthony | Surface rippling induced by surface-tension gradients during laser surface melting and alloying[END_REF][START_REF] Willis | Transport phenomena and droplet formation during pulsed laser interaction with thin films[END_REF]. In this process, a strong laser beam is shined on the surface of a body of metal, producing the melting of an upper layer which is driven mainly by the surface tension gradient. In recent years, a similar principle has been applied in the micropatterning of polymer melts for lithography techniques [START_REF] Singer | Thermocapillary approaches to the deliberate patterning of polymers[END_REF].

In contrast, in other applications thermocapillary flows may be undesired. For instance, they are known to cause failure of lubrication in tribo-systems in which thermal gradients spontaneously appear; therefore, the design of devices must be thought in consequence [START_REF] Dai | Directional interfacial motion of liquids: Fundamentals, evaluations, and manipulation strategies[END_REF].

From a more fundamental point of view, the behavior of a supported liquid film submitted to thermal gradients has been studied in different configurations. Theoretical works have focused on the dimpling of the free surface of the film that results from a thermocapillary stress, and to which both capillarity and gravity oppose [START_REF] Hershey | Ridges in a liquid surface due to the temperature dependence of surface tension[END_REF][START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF][START_REF] Pimputkar | Transient thermocapillary flow in thin liquid layers[END_REF][START_REF] Yeo | Marangoni instability of a thin liquid film resting on a locally heated horizontal wall[END_REF]. Experimentally, controlled thermal gradients at the free surface can be created by locally heating the liquid by radiative transfer [START_REF] Hitt | Radiation-driven thermocapillary flows in optically thick liquid films[END_REF], but more usually it is the substrate they lie on that is heated. Focusing a laser at an absorbing substrate provides an easy way to locally increase its temperature and create a temperature gradient at a liquid interface by heat transfer [START_REF] Chraïbi | Thermocapillary flows and interface deformations produced by localized laser heating in confined environment[END_REF][START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF][START_REF] Pottier | Boundary condition in liquid thin films revealed through the thermal fluctuations of their free surfaces[END_REF][START_REF] Klyuev | Thermocapillary deformation induced by laser heating of thin liquid layers: Physical and numerical experiments[END_REF]. Whatever the effect inducing the thermocapillary stress, the latter allows the formation and control of thin liquid films, in a region whose size is given by the extent of the thermal gradient [START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF]. Within this region, in which the films can reach submicron thicknesses even with a moderate temperature rise, intermolecular forces have non negligible effects, which in particular condition the rupture of the film. Thermocapillary-induced dewetting of liquids on solid substrates has thus been investigated by monitoring the thickness at the central hottest point [START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF][START_REF] Klyuev | Thermocapillary deformation induced by laser heating of thin liquid layers: Physical and numerical experiments[END_REF].

However, up to date, poor attention has been paid to the dynamics of the whole thin film region, in which intermolecular forces are large. Here, we show that interfacial effects at very small scales can be probed through accurate measurements of thickness variations in this region.

The paper is organized as follows. We describe the experimental setup and methods in section II. A review of the hypotheses that are made leading to a well-known thin film equation describing the evolution of a thin liquid film under the action of thermocapillarity are discussed in section III. We further present the experimental results in section IV. Finally, section V provides a conclusion and outlook for future work.

II. EXPERIMENTAL SET-UP AND METHODS

A. Experimental set-up

The liquid films are spread by spin-coating on substrates that are rectangular glass slides of thickness 100 µm. Prior to deposition, the top side of the substrate is washed by spincoating isopropanol and acetone over it several times, letting it dry finally for a couple of minutes before spreading the liquid. An electrical circuit is printed on the bottom side of the slides, with a pattern consisting in two squares connected by a line, as shown in Fig. 1a).

The squares are connected to a dc voltage generator and the resulting electrical current is measured. The central line, of width 100 µm, acts as a heat dissipator, generating a thermal field. We denote O the centre of the line and Ox and Oy the axes respectively perpendicular and parallel to the line. We make the approximation that the induced thermal field does not depend on y, which is justified in the vicinity of the centre of the line. We define the thermal excess θ(x) = T (x) -T 0 where T 0 is the room temperature and T (x) the thermal field. The dissipated power P along the line is a control parameter that sets the maximal temperature rise, θ max = θ(0). In our experiments P ∼ 1 mW and θ max ∼ 1 K. Infra-red measurements have evidenced that the horizontal extent of the thermal gradient w is a few millimeters.

The thermal gradient drives a flow in the liquid films away from the central line, inducing a thinning of the films along that line. A typical surface of a film once a very thin film has been formed at the central region is shown in Fig. 1b). The profile was measured via an implementation of the Free-surface synthetic Schlieren method [START_REF] Moisy | A synthetic schlieren method for the measurement of the topography of a liquid interface[END_REF], which we will not detail here since we focus on measurements at smaller thicknesses, which are described in the following. The region we focus on is that at the centre (where the heating line lies), which has been emptied of liquid so that the thickness is below the micrometre scale in an area spanning several millimetres. We emphasise that, as expected, the measured profile only very weakly depends on y, which justifies the 1D profile we will consider in the following.

Experiments were performed with different initial film thicknesses, h 0 and electrical powers P . The initial thickness h 0 is measured by optical profilometry just after spreading the films.

FIG. 1. a) Top view of the glass substrate. On its bottom side, a microprinted circuit connected to a voltage generator creates the thermal gradient. b) Typical thickness surface h(t, x) at an intermediate stage of the dynamics, reconstructed surface via Free-surface synthetic Schlieren method [START_REF] Moisy | A synthetic schlieren method for the measurement of the topography of a liquid interface[END_REF]. The thermocapillary effect promotes the apparition of a very thin (h < 1 µm) layer that extends over several millimetres.

B. Investigated liquids

The investigated liquids are silicone oil (ABCR AB112154), n-hexadecane (Sigma-Aldrich) and a ramified isomer of hexadecane: 2,2,4,4,6,8,8-heptamethylnonane (HMN, Sigma-Aldrich). Their useful physical properties are listed in table I; in particular, they have very close values of coefficient of surface tension variation, γ θ = -∂ T γ| T =T 0 . In contrast, we emphasise that their melting temperatures are very different.

Since the temperature increase imposed to produce the thermocapillary effect remains modest, the variations of viscosity and density are small. An estimation from the thermal dependencies of these properties for the three liquids investigated [START_REF] Roberts | Physical properties of low-molecular weight polydimethylsiloxane fluids[END_REF][START_REF] Dubey | Temperature and composition dependence of the densities, viscosities, and speeds of sound of binary liquid mixtures of 1-butanol with hexadecane and squalane[END_REF][START_REF] Luning Prak | Density, viscosity, speed of sound, bulk modulus, surface tension, and flash point of binary mixtures of n -dodecane with 2,2,4,6,6-pentamethylheptane or 2[END_REF] gives a variation on viscosity and density of around 2 % and 0.1 %, respectively, for a thermal rise of θ max = 1 K.

Similarly, the variation of refractive index of alkanes [START_REF] Camin | Physical properties of n-hexadecane, ndecylcyclopentane, n-decylcyclohexane, 1-hexadecene and n-decylbenzene[END_REF] and silicone oil [START_REF] Van Raalte | Measurement of the refractive index of silicone oils for the UKAEA[END_REF] caused by this temperature change is at most 0.01 %. Therefore, the variations of viscosity, density and refractive index produced by heating the liquid will be neglected in the following.

η (mPa s) ρ (kg m -3 ) γ θ (N m -1 K -1 ) T melt (°C) ref.
Silicone oil 19 950 6.5 × 10 -5 < -40 [20], [START_REF] Ricci | Density and surface tension of dioctylphthalate, silicone oil and their solutions[END_REF] n-C 16 2.2 733 8.7 × 10 -5 18 [START_REF] Korosi | Density and surface tension of 83 organic liquids[END_REF], [START_REF] Dubey | Temperature and composition dependence of the densities, viscosities, and speeds of sound of binary liquid mixtures of 1-butanol with hexadecane and squalane[END_REF], [START_REF] Ocko | Surface freezing in chain molecules: Normal alkanes[END_REF] HMN 3.2 781 8.7 × 10 -5 -4 [START_REF] Korosi | Density and surface tension of 83 organic liquids[END_REF], [START_REF] Luning Prak | Density, viscosity, speed of sound, bulk modulus, surface tension, and flash point of binary mixtures of n -dodecane with 2,2,4,6,6-pentamethylheptane or 2[END_REF], [24] TABLE I. Investigated liquids with their main properties: viscosity, η, density, ρ, thermal coefficient of the surface tension, γ θ = -∂ T γ| T =T 0 , and melting temperature, T melt .

C. Thickness measurements

The thickness of the film at the central region, where it will be below the micrometre scale,

is measured with 3-colour interferometry [START_REF] Schilling | Absolute interfacial distance measurements by dual-wavelength reflection interference contrast microscopy[END_REF][START_REF] De Ruiter | Air cushioning in droplet impact. i. dynamics of thin films studied by dual wavelength reflection interference microscopy[END_REF]. The measurement setup is sketched in Fig. 2a). The cell is shined from below using a RGB-LED of peak wavelengths λ R = 635 nm, λ G = 518 nm, and λ B = 463 nm. Images are taken from above with a CMOS monochrome camera. Images were taken typically each 10 min for silicone oil during a couple of days, and each 10 s during some hours for alkanes.

At any given measuring time, three images, one for each colour is taken. This leads to interference patterns such as the one shown in Fig. 2b). From such an image, an interference profile I C (x) is obtained at each time and for each colour C = R, G, B by averaging the intensity profiles along the y-axis within a region spaning around 1 mm, which corresponds to the rectangle in Fig. 2b). To obtain the thickness h(x) from I C (x), one considers the theoretical linearised expression of transmission interference through a thin film of thickness h,

I th C (h) ∝ 1 -A cos 4πnh λ C ( 1
)
where n is the refractive index of the liquid and A is a Fresnel coefficient depending on the refractive indexes of the media considered, in our experiments A ≃ 0.012. In the computations, the interference profiles I C (h) are convolved with a Gaussian function mimicking the broad-band emission of the LED source for each channel.

Working on the intensity space I = (I R , I G , I B ), as shown in Fig. 2c), it is possible to use h as a fitting parameter of the theoretical curve for the data points, resulting in an absolute-thickness curve h(x) at the time of measurement. This procedure allows to obtain thicknesses below the first-order fringe, i.e. smaller than 100 nm, thus in the range where intermolecular forces are to be expected. In our current implementation, overall thickness profiles ranging from the tens of nanometres to the micron scale can be measured with a nanometric accuracy. The floor sensitivity of the measurement is limited by experimental noise, which mainly originates from the camera sensor; as a result, thicknesses below 10 nm cannot be determined. Due to the opacity of the heating line, the film thickness cannot be measured for |x| ≲ 0.1 mm.

D. Numerical integration

Numerical integration of the thin film equation discussed below is done using the pdepe numerical solver of Matlab [27]. Since the equation is fourth order on the thickness gradient, the second derivative u(t, x) ≡ ∂ 2 x h is considered as an independent field and a simple closure expression u-∂ x {∂ x h} = 0 is simultaneously solved. The initial condition is a film of homogeneous thickness, h(0, x) = h 0 , u(0, x) = 0. For ease of convergence, the x spacing is logarithmic, allowing to better describe the evolution close to the centre as well as capturing the long range evolution. The numerical integration typically lasted some hours.

III. THEORETICAL BACKGROUND

As schemed in figure 3, we consider a homogeneous layer of incompressible liquid of constant viscosity η and density ρ with initial thickness h 0 , that is submitted to a symmetric temperature excess θ(x). The excess temperature remains small: θ max = θ(0) ≪ T 0 , with T 0 the room temperature and all the properties of the liquid are assumed to be uniform. The resulting dynamics is assumed to be invariant along the y-axis. In addition, since we use poorly volatile fluids we consider that evaporation is negligible.

A. Hypotheses and dimensionless numbers

The order of magnitude of the thermocapillary stress is γ θ θ max /w ∼ 0.1 Pa. The resulting velocity in a liquid film of thickness 10 µm and viscosity 10 -2 Pa s is v ∼ 100 µm s -1 . Since the liquids considered in the experiments have a thermal diffusivity κ ∼ 10 -7 m 2 s -1 [28,[START_REF] Vélez | Temperature-dependent thermal properties of solid/liquid phase change even-numbered n-alkanes: n-Hexadecane, n-octadecane and neicosane[END_REF], the thermal Péclet number Pe = h 0 v/κ ∼ 10 -2 is small enough to consider thermal convection negligible. Therefore, the transport of heat from the substrate to the liquid is purely diffusive.

In addition, the thermal field is established in a time h 2 0 /κ ∼ 10 -2 s, much faster than the timescale of hours involved in the dynamics. Hence, we assume that the thermal field is established instantaneously and that it is constant throughout the film's thickness. The Reynolds number Re = h 0 v/η ∼ 10 -7 is also very small.

In the literature, a dimensionless number has been defined to measure the effect of the thermocapillary stress compared with that of hydrostatic pressure, R = 3∆γ/2ρgh 2 0 , where ∆γ = γ θ θ max [START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF]. We typically work with R ∼ 10, a larger thermocapillary effect than in Ref. [START_REF] Burelbach | Steady thermocapillary flows of thin liquid layers. II. Experiment[END_REF] so that the thickness at the centre will always reach the region where intermolecular effects are relevant [START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF].

B. Thin-film equation

In experiments, the thermal field decays over a typical length w ∼ 2 mm much larger than the initial thickness, h 0 ≪ w, allowing to consider a small gradient approximation. As a result, it is possible to obtain a lubrication approximation of the Navier-Stokes equations leading to a partial differential equation for the thickness profile h(t, x) belonging to the family of thin film equations [START_REF] Oron | Long-scale evolution of thin liquid films[END_REF].

For the system defined by the conditions above and schematized in Fig. 3, the thin film equation reads [START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF] where the pressure contributions considered on the second term of the l.h.s. are the capillary pressure, the hydrostatic pressure and Π(h) is a disjoining pressure term accounting for the intermolecular interactions. The third term represents the thermocapillary force driving film thinning. Capillarity and gravity oppose film thinning whereas disjoining pressure may oppose or amplify thinning, according to its sign. The relative importance of the pressure terms varies as the film thins down and this variation has been investigated theoretically [START_REF] Burelbach | Steady thermocapillary flows of thin liquid layers. II. Experiment[END_REF]. If the initial thickness of the film is larger than the range of molecular interactions, then in a first stage, the film dimples and both gravity and capillarity contribute to slow down its evolution. The central region of the film further empties and a central area of very small thickness is formed. In this region of small thickness and small curvature, both capillary and hydrostatic pressures are negligible. As a result, film thinning results from an interplay of thermocapillary stress and disjoining pressure. If the intermolecular forces are attractive then spontaneous rupture of the film occurs. In the case of repulsive intermolecular forces, a fully equilibrated stationary state can be reached and has been observed in experiments [START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF][START_REF] Clavaud | Modification of the Fluctuation Dynamics of Ultrathin Wetting Films[END_REF]; it corresponds to a balance of the thermocapillary stress and disjoining pressure. In such a state, it would be possible, in principle, for the disjoining pressure to be measured if the thickness of the ultra-thin liquid film at the central region is measured and the imposed thermal field is known. However, most experimental works have not aimed at measuring disjoining pressure and rather focused on the question of the film rupture [START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF][START_REF] Burelbach | Steady thermocapillary flows of thin liquid layers. II. Experiment[END_REF]. In addition, an accurate determination of the thermal field is required to measure the disjoining pressure, which is a challenging task. In the geometry we investigate, the thermal gradients extend over millimetric lengths and we observe a very slow dynamics of the central region, which thins down during a time scale of days. In practice, this implies that the full equilibrium cannot be experimentally reached since other effects, such as sample contamination or liquid evaporation, appear at these very long time scales. Nevertheless, the thinning dynamics itself provides information on the intermolecular forces. Therefore, rather than trying to reach a stationary state, we take profit of the thinning dynamics to probe the intermolecular interactions at stake.

∂ t h + 1 3η ∂ x h 3 ∂ x γ 0 ∂ 2 x h -ρgh + Π(h) - γ θ 2η ∂ x h 2 ∂ x θ(x) = 0, (2)

C. Pure thermocapillary regime

Before focusing on the effects of disjoining pressure, we first examine the preceding stage, in which the central area of small thickness has formed but its thickness is large enough for disjoining pressure to be neglected, i.e. larger than roughly 100 nm. In this stage, the capillary and gravitational pressures can as well be neglected in the description of the evolution, consistently with former works [START_REF] Tan | Steady thermocapillary flows of thin liquid layers. I. Theory[END_REF][START_REF] Burelbach | Steady thermocapillary flows of thin liquid layers. II. Experiment[END_REF]. Film thinning is then entirely driven by thermocapillarity, which no effect opposes. Equation ( 2) then becomes

∂ t h - γ θ 2η ∂ x h 2 ∂ x θ(x) = 0. (3)
Equation ( 3) can be solved by separation of variables, giving

h(t, x) = h 0 1 + (t/t * ) |∂ 2 x θ(0)| 2 |∂ x θ(x)| x 0 ds |∂ s θ(s)| , (4)
where t * = 2η/(γ θ |∂ 2 x θ(0)|h 0 ). Since t * is at most a few seconds and the timescale of the experiments is hours, the limit t ≫ t * is quickly reached. Equation (4) then simplifies and the product t × h(t, x) can be expressed as

t × h(t, x) = η γ θ 1 |∂ x θ(x)| x 0 ds |∂ s θ(s)| . (5)
The product t × h(t, x) is therefore independent of time and solely a function of the spatial coordinate, x. Since in addition it only depends on the liquid properties and not on the initial film thickness, for a given liquid, t × h(t, x) curves are expected to collapse onto a master curve that is a function of x only, whatever the initial conditions of the experiment. Inversely, at any given location x, the thickness has a simple dependency on time: h(t, x) ∝ t -1 . This temporal scaling has been reported in experiments performed by heating a substrate with a laser beam [START_REF] Wedershoven | Infrared laser induced rupture of thin liquid films on stationary substrates[END_REF]. The variations with time of the thickness of the oil film spread on the substrate were found to follow h(t, 0) ∝ t -1 for a wide range of laser power and beam waists under half a micrometre. It confirms the relevancy of a regime in which both capillarity and gravity are negligible. In the former work, the role of capillarity was found to be apparent only for very small beam waists at and below the micrometre scale, where the curvature of the film is sufficiently large. We demonstrate in the following that, in our experiments, capillarity and gravity are always negligible in the central area.

IV. EXPERIMENTAL RESULTS

A. Master evolution at intermediate thicknesses

In figure 4, we show the result of a typical measurement of the evolution at the central area, extending roughly a few millimeters away from the central line, similar as the one in figure 1b). The evolution is displayed, between 1 and 6 hours after turning the thermal gradient on. Since the evolution is symmetric with respect to the origin of the x-axis, the thickness of only one side of the film is reported. The experiment uses a film of silicone oil with initial thickness h 0 = 36(2) µm and the dissipated power is P = 0.81(1) mW.

The film thickness increases very slowly from the centre to the edge: it reaches a few micrometers only over a distance of a few millimeters. The spatial variations of t × h(t, x) are also displayed in the figures: remarkably, all curves nicely collapse onto a master curve.

According to the analysis of section III, the collapse indicates that the effects of both capillary and hydrostatic pressures are negligible and that the dynamics is driven by thermocapillarity only, following the simple time dependency given by equation [START_REF] Dai | Directional interfacial motion of liquids: Fundamentals, evaluations, and manipulation strategies[END_REF].

Following equation ( 5), the obtained master curve depends on the liquid properties, which are known, and on the thermal gradient driving the dynamics. The latter can hence be determined from the experimental master curve by inverting the relation given by ( 5) to get

∂ x θ(x) = - 2η γ θ 1 ⟨t × h(x, t)⟩ 2 exp x 0 ⟨t × h(s, t)⟩ exp ds. ( 6
)
where ⟨t × h(x, t)⟩ exp is the experimentally determined master curve from the collapse of profiles as in figure 4b), and is only a function of x. Figure 5 shows the thermal gradient ∂ x θ(x) obtained in this way. Since we do not have data up to x = 0, we extrapolate ⟨t × h(x, t)⟩ exp by a constant for x ≤ 0.1 mm in order to numerically compute the integral in equation ( 6). This extrapolation satisfies the symmetry requirement ∂ x h(t, 0) = 0 and we have checked that it results in a negligible correction for larger values of x. A heuristic expression for ∂ x θ(x) can further be determined from the temperature gradient found from the master curve. Once integrated, it corresponds to a thermal field given by

θ(x) = θ max sech x δ δ/w (7)
where δ and w are two characteristic lengths. The length δ accounts for the finite length of the zone of maximum temperature. This length results from both the finite size of the central line of the circuit dissipating heat and the diffusion of heat through the thickness of the substrate. Since the line width and substrate thickness are both of 100 µm, δ is expected to be close to this value. The length w is the decay length of the thermal field at distances from the central line larger than δ and smaller than roughly 1 cm. We emphasise that at a longer range, the thermal excess is expected to go to zero asymptotically, with a characteristic 6). The dashed line represents a fit to the experimentally determined gradient following the derivative of the thermal excess given by equation ( 7)

with the parameter values given in the text. Inset: integrated thermal excess inferred from the experimentally determined gradient. The dashed line represents equation [START_REF] Pimputkar | Transient thermocapillary flow in thin liquid layers[END_REF].

decay length one order of magnitude larger than w. However, since we are interested in the central area of the film, for which submicron thicknesses are reached, equation ( 7) provides a satisfactory description of the thermal gradient in this area and further detail on the thermal field is not needed. In the particular case of Fig. 6, a fit of equation ( 7) to the experimentally determined gradient yields the following values: θ max = 0.37 K, δ = 110 µm and w = 2.3 mm. These values are consistent with independent measurements of the thermal field at the bare substrate measured with an IR camera. We emphasise that the modest value of θ max confirms that the liquids can be considered as having constant physical properties (i.e. viscosity, optical index and density), although they are heated.

In summary, the master evolution followed by the film thickness within intermediate times allows the determination of the thermal gradient. The empirical expression that is obtained can be used in numerical resolutions to test the consistency of the results, as described in the next section.

B. Comparison of experimental data with numerical resolution

Since the thermal field is determined, equation ( 2) that includes capillary and gravity terms can be solved, allowing the comparison of experimental data with numerical results obtained without any assumption. We emphasise that, in the following analysis, we do not consider the disjoining pressure term since we focus here on large thickness values, for which intermolecular forces are negligible. In Fig. 6, we can see that the numerical integration faithfully represents the measured local dynamics, even for thicknesses well below the micron scale. Therefore, the determination of the thermal gradient based on the assumption of negligible effects of capillarity and gravity is fully validated. As expected, at thicknesses below 100 nm, we observe that the numerical profiles deviate from experimental data. In this thickness range, both capillarity and gravity can still be neglected and the dynamics results from thermocapillary and intermolecular effects. We have not tried to quantitatively describe the disjoining pressure term in silicone oil. Instead, we focus on a qualitative effect resulting from intermolecular forces that we have evidenced.

C. Transition to the ultra-thin film

We now examine the long term regime, when the thickness is on the order of tens of nanometres, and when the effect of intermolecular forces becomes important. In this regime, the reduced thermocapillary evolution ( 4) is not valid any more, and the curves t × h(t, x)

are expected to deviate from the master curve.

Figure 7 shows the long-term regime of the evolution when the thickness enters the range in which intermolecular forces are relevant, for the same experiment as Fig. 6. The corresponding t × h(t, x) curves clearly lie above the master curve, which demonstrates that the thinning induced by thermocapillarity has been slowed down. This deceleration can be explained by the presence of an effective repulsive intermolecular interaction that opposes the thermocapillary effect. A repulsion between the interfaces of the silicone oil film is expected since silicone oil totally wets the glass surface [START_REF] De Gennes | Gouttes, bulles, perles et ondes[END_REF]. Therefore, the thinning slows down and the final stationary state would correspond to an equilibrium of disjoining pressure and thermocapillary stress.

We have performed similar experiments with alkanes, and panels c) and d) of figure 7 display the thickness profiles measured in a film of n-hexadecane. Because of the smaller viscosity of n-hexadecane, the time scales of the experiment are much shorter. The curves at earlier times are nevertheless qualitatively similar to those of silicone oil and collapse as well onto a master curve. In contrast, at later times, the experimental profiles obtained with n-hexadecane reveal an acceleration of film thinning since they lie below the master curve. The acceleration is associated with an unstable behavior: the thickness profiles of n-hexadecane clearly collapse and form a foot receding from the centre. We were however unable to evidence the presence or absence of an ultra-thin film in the central region, since, if it exists, its thickness lies below the sensitivity level of the measurement (10 nm). We can nevertheless state that, in the range of investigated thicknesses, the disjoining pressure for n-hexadecane is attractive since it accelerates thinning instead of opposing it, in contrast to silicone oil.

This result is in contradiction with reports in the literature of hexadecane fully wetting glass surfaces [START_REF] Shiri | Thermal marangoni flow impacts the shape of single component volatile droplets on thin, completely wetting substrates[END_REF], hence suggesting that the interactions between the interfaces of n-hexadecane films should be repulsive. The attractive interactions evidence rather evoke a partial or pseudo-partial wetting of n-hexadecane on glass. In the case of partial wetting, the spreading parameter is negative and the equilibrium situation is a drop sitting on the substrate with a finite angle of the contact line; in the case of pseudo-partial wetting, the spreading parameter is postive and the equilibrium situation is also a drop with a finite angle but that sits on a film of nanometric thickness. The latter situation results from non-monotonic variations of the disjoining pressure with thickness [START_REF] De Gennes | Gouttes, bulles, perles et ondes[END_REF]. Since glass is a material of high surface energy, it is likely that the spreading parameter is positive, favoring the hypothesis of pseudo-partial wetting.

We emphasise that we observe the same unexpected behavior with the ramified alkane HMN as with the linear n-hexadecane, and hence that it does not result from crystallisation effects. Actually, it is well known that surface freezing occurs in alkanes at temperatures up to 3 K above their solidification temperature [START_REF] Wu | Surface crystallization of liquid normal-alkanes[END_REF], which affect the wetting behavior of n-hexadecane films [START_REF] Lazar | Spreading and two-dimensional mobility of longchain alkanes at solid/gas interfaces[END_REF][START_REF] Lu | Stability of the parallel layer during alkane spreading and the domain structures of the standing-up layer[END_REF]. However, HMN has a crystallisation temperature which is well below room temperature, and remains liquid at room temperature even when present on sub-nanometric layers [START_REF] Gosvami | Squeeze-out of branched alkanes on graphite[END_REF]. As a result, surface freezing cannot be invoked to explain the observed behaviors that are common to n-hexadecane and HMN. We rather suggest that these alkanes only pseudo-partially wet glass. In this case, the distinction from total wetting is difficult in macroscopic experiments if the equilibrium angles are small. In addition, times to attain an equilibrium state may reach the scale of the week [START_REF] Pérez | Spreading dynamics of polydimethylsiloxane drops: Crossover from laplace to van der waals spreading[END_REF]. These difficulties are likely to explain why n-hexadecane has been suggested to totally wet glass, and show that static experiments at the macroscopic scale may not be sufficient to determine wetting behaviors properly.

Additional insight is provided by numerical integration of the thin-film equation with a given disjoining pressure. In the apolar liquids and the range of film thicknesses we investigate, intermolecular forces are expected to be non-retarded van der Waals interactions Π(h) = -A H /(6πh 3 ), where A H is the Hamaker constant [START_REF] Churaev | Surface forces in wetting films[END_REF]. A repulsive interaction then corresponds to a negative Hamaker constant and an attraction to a positive Hamaker constant. Using typical values for the thermal gradient, we have solved equation (2) with disjoining pressure terms of respectively attractive and repulsive nature. Because of numerical instabilities, full spatial variations as in figure 7 cannot be obtained in the case of attractive intermolecular forces. We nonetheless show in figure [START_REF] Yeo | Marangoni instability of a thin liquid film resting on a locally heated horizontal wall[END_REF] the dynamics at the centre, h(t, 0), for attractive (A H > 0) and repulsive (A H < 0) interactions, as well as for zero disjoining pressure. Since the dynamics at the centre cannot be measured in the ex-periment, the dynamics measured at a distance of 0.2 mm from the centre h(t, x = 0.2 mm) is displayed for experiments with silicone oil and both n-hexadecane and HMN. In order to compare the dynamics measured in different experimental conditions with liquids that have different viscosities, we have represented the dimensionless thickness a function of the ratio of the time and a thermocapillary timescale, t 0 = 2ηw 2 /(∆γh 0 ). As expected, the case A H = 0 results in a pure-thermocapillary evolution h(t, 0) ∝ t -1 . In contrast, for non-zero interactions, a deviation from t -1 variations appears for thicknesses in the range of tens on nanometres. For A H < 0 the thinning is slowed down, whereas for A H > 0 it is abruptly accelerated, leading to film rupture. as in a) but with profiles at the centre h(t, 0) from the numerical integration of equation ( 2) with physical properties of n-hexadecane, when the intermolecular forces generate an effective repulsion (A H < 0) vs. attraction (A H > 0) of the interfaces. In the case where there is no disjoining pressure (e.g. A H = 0), the evolution follows equation (4), i.e. t × h(t, 0) ∝ const. for t ≫ t 0 .

Changing the sign of the disjoning pressure in the equation therefore satisfactorily ac-counts for the different behaviors observed. This result confirms that attractive intermolecular forces, at least in the thickness range investigated, are at stake between the interfaces of films of the considered alkanes. Beyond this qualitative finding, the very similar shape of the experimental and computed curves suggest that the expression for disjoining pressure we use is close to the real one. However, we have found that the dynamics in silicone oil could not be quantitatively described by the simple law we have used: clearly in figure [START_REF] Yeo | Marangoni instability of a thin liquid film resting on a locally heated horizontal wall[END_REF], the experimental thinning is faster than in the computation. Consistently, the full thickness profiles, as the ones reported in figure 6 could not be described, even when changing the value of the Hamaker constant. This discrepancy may result from the fact that the used silicone oil is not monodisperse but rather constituted by a mixture of macromolecules of different lengths. Mixtures effects have actually been predicted to change the exponent of the thickness-dependency of molecular interactions [START_REF] Derjaguin | On the question of determining the concept of disjoining pressure and its role in the equilibrium and flow of thin films[END_REF]. These complex effects are out of the scope of the present work and we have not tried to further determine the disjoining pressure at stake. However, since silicone oil is widely used as a model wetting liquid, it could be an interesting aim for future research.

V. CONCLUSION

In conclusion, we have demonstrated that the dynamics of thin films induced by a thermal gradient can serve as a tool for investigating intermolecular forces. We predict and experimentally evidence a regime in which thermocapillarity fully drives thinning. The thermal gradient can thus be determined from thickness profile measurements and, in turn, be used to compute the expected dynamics when intermolecular forces are at stake. We demonstrate that thickness measurements allow a clear discrimination between the cases of attractive and repulsive interactions between the interfaces of the liquid films. The experiments reveal an unexpected behavior of both linear and ramified alkanes when film thicknesses reach a few tens of nanometres, and thus show that accurate probes of liquid films at very small scales are still needed.

FIG. 2

 2 FIG. 2. a) Sketch of the measurement setup. A RGB Panel illuminates from below the experimental cell, consisting of a glass substrate over which a liquid film is deposited via spin-coating. A monochrome camera on top takes pictures at specified intervals. b) Typical interferometric profile of a silicone oil film obtained after an initial transient period. Here λ R = 635 nm (red) and the height different between intensity maxima is ∆h = 113 nm. The black rectangle indicates the region of interest for measuring the film dynamics, the interference intensity I R (x) is obtained by performing an average along the vertical direction (and similarly for green, I G and blue, I B ). c) Grey dots: interferometric data in intensity space (I R , I G , I B ) for the film at a given time. Black line: theoretical intensity curve, with the thickness h as parameter. The coloured points and lines are projections into the orthogonal planes for better visualization. Fitting the data points to the theoretical curve gives an absolute measurement of the thickness h at the corresponding x coordinate. The squares corresponds to h = 80 nm, the stars to h = 300 nm.

FIG. 3 .

 3 FIG.3. Sketch of the film dynamics with main variables: a thin liquid film of density ρ and viscosity η, with initial thickness h 0 , is deposited on top of a glass substrate which is heated locally, w being the decay length of the thermal field.

FIG. 4

 4 FIG. 4. a) Experimental profiles of an experiment with silicone oil with h 0 = 36(2) µm and P = 0.81(1) mW. The time between profiles is 10 min. The inset shows the same profiles in linear scale. b) The products t × h(t, x) for the experimental profiles of panel a) at corresponding times collapse onto a master curve, which indicates that the local dynamics is solely described by the thermocapillary effect. The dashed line is the average of the experimental curves that is used to determine the thermal gradient.

FIG. 5 .

 5 FIG. 5. Derivative of the thermal field ∂ x θ(x) obtained experimentally from the collapse of curves shown in panel b) of Fig. 4 using equation (6). The dashed line represents a fit to the experi-

FIG. 6

 6 FIG.6. a) Same experimental thickness profiles as in figure4. The solid lines were obtained by numerical integration of equation (2), with the thermal gradient of figure5and a zero disjoining pressure. b) Zoomed view of the curves at small thicknesses showing the discrepancy between numerical and experimental data at this small scale (for better visualization, only one out of two curves is shown).

FIG. 7

 7 FIG. 7. a) and b): same plots as in Fig. 4 (dimmed curves), with the addition of profiles at later times t = 13, 23, 33, 43 h. The deviation of these last profiles from the master curve in panel b) reveals the effect of intermolecular forces. c) Experimental profiles h(t, x), and d) curves t × h(t, x) for an experiment using n-hexadecane (n-C 16 ), with h 0 = 13.0(2) µm and P = 0.95(9) mW. The time between profiles is 1 min. Observe that the time arrow on panel d) points in the opposite direction as the corresponding arrow on panel b).

FIG. 8

 8 FIG. 8. a) Comparison of the thickness evolution near the centre, h c (t) = h(t, x = 0.2 mm), for the experiment with silicone oil and with n-hexadecane of Fig. 7. The time has been scaled by a thermocapillary timescale t 0 = 2ηw 2 /(∆γh 0 ) and the thickness by h 0 . Silicone oil: h 0 = 36(2) µm, θ max = 0.37 K, w = 2.3 mm (t 0 ≃ 211 s). n-hexadecane: h 0 = 13.0(2) µm, θ max = 0.44 K, w = 2.2 mm (t 0 ≃ 42 s). HMN: h 0 = 19.5(6) µm, θ max = 0.54 K, w = 2.2 mm (t 0 ≃ 30 s). b) Same

ACKNOWLEDGMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No 754387.


Cyril Tissot

Mathias Rouan

Théo Petitjean

Valerie Bonnardot

Jeanne Thibault

Hervé Quénol

Simulating climate change impact on viticultural systems

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Each activity is represented by an autonomous agent able to react and adapt its reaction to the variability of environmental constraints.

Agents are distinguished according to their objectives: "Supervisors" Agent plays an overseeing role in the model, "Winegrower" Agents aim to grow grapes and produce wine that meets precise specifications according to their end-product goals and "Vine" Agents are grape production entities. The relationships between these three types of agents determine the production strategies adopted by the winegrowers.

Simulating vine phenology

At the local scale, for example at the Layon pilot site, simulation results show a significant increase in the number of actions, especially for organic plots. This result is largely related to the increase in the number of fungicide treatments. In organic production, the use of contact products requires a high spray frequency during periods of pathogenic risk (a few days between each spay depending on rainfall frequency). In conventional production, by using systemic product, wine growers reduce significantly this frequency (usually at least 15 days between spraying). These results show that with a changing climate, The risk of disease is higher in most of the pilot sites. In some cases, the increased earliness of the grapevine partly compensates this risk by reducing the exposure period.

Implemented during the LIFE ADVICLIM program, SEVE model (Simulating Environmental impacts on Viticultural Ecosystems), present a generic modeling environment for simulating grapevine growth and berry ripening under different conditions and constraints (slope, aspect, soil type, climate variability...) as well as production strategies and adaptation rules according to climate change scenarios.

Real

Entités élémentaires

Agents, Objets

Entities Agent

Virtual reality

In SEVE model grapevine growth dynamics uses simple algorithms based on bioclimatic indices (Winkler, GFV and Huglin) specific to grapevine behaviour. These indices allows to study climate variability and change in relation with local environment characteristics and type of grapevine variety.

These results provide adaptation trajectories among a set of possible trajectories.

Other adaptation strategies can be considered. One of the principal challenges for winegrowers will be to manage increasing temperatures in order to delay grapevine phenology. A too early onset of ripening will lead to unbalanced grape compositions. Increased early phenology can also lead to an increased risk of frost during bud-break and flowering period. Linking the results of the SEVE model with the climate projections of RCP 4.5 and 8.5 shows that the risk of spring frost may be significant in some plots.

In SEVE model, winegrower agents are created from three production profiles: conventional (traditional viticulture), integrated (limitation of pesticides, fertilizers and weeding) and organic (strong limitation of pesticides, mechanical weeding... ). Surveys conducted at vineyard scale are used to define potential practice schedules for each production profiles. The main practice constraints are also integrated into the model. Climate data are used to calculate the available days for each agronomic action. According to these data, its production profile, vine phenological cycle and agro-climatic characteristics (soil type, slope, aspect, water balance ...), each wine grower agent performs specific agronomic actions.

  Simulating climate change impact on viticultural systems TERCLIM I 2 nd ClimWine Symposium I XIV th International Terroir Congress I 3-8 July 2022 I Bordeaux, France

  variability (Layon pilot site) Cyril Tissot, Mathias Rouan, Théo Petitjean, Valérie Bonnardot, Jeanne Thibault, Hervé Quénol UMR 6554 CNRS LETG Highlighting potential adaptation strategies according to propective scenarios (Layon pilote site example)

strategy Vegetative Rest Vegetative Growth Reproductive Cycle Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Early Shoot Growth Budburst Shoot Growth Ceases Leaf Fall Flowering Fruit set Veraison Harvesting

								Environmental	Practice	Climate
								constraints	schedule	conditions
								Plot
								Agent	Phenological cycle	Potential practice
									calendar
	Layon							T in °C
								16
								15
							Choice of non local
							grape variety	14
		Adaptation of winemaking practices
								13
	Technical operations management
	2029	2039	2049	2059	2069	2079	2089	2099


I. INTRODUCTION

The interaction of a flame with acoustics is a fundamental problem in combustion. It is traditionally studied to understand and prevent the formation of thermo-acoustic instabilities. Acoustic waves are also useful to analyze how flames respond to time dependent flows. In particular, in combustion models where the flame is described as a discontinuity of the flow, the local speed of propagation of the flame front w is related to the speed of the upcoming flow u 1 by a kinematic relation of the form [START_REF] Clavin | Flamelet library for turbulent wrinkled flames[END_REF][START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF][START_REF] Bechtold | The dependence of the Markstein length on stoichiometry[END_REF][START_REF] Clavin | Curved and stretched flames: the two markstein numbers[END_REF]:

(u 1 -w) • n = u L -L I u L R + L II n • ∇u 1 • n, (1)
where n is the normal vector to the flame front (directed towards the burnt gases), u L is the laminar flame speed, R is the local radius of curvature of the flame front, n • ∇u 1 • n is the normal-normal component of the rate-of-strain tensor and L I and L II are the Markstein lengths. These two parameters are key quantities in combustion models. They express the role of transport phenomena happening in the thickness of the flame and allow them to be taken into account while still considering the flame as a flow discontinuity. Their expression is traditionally obtained through asymptotic analysis of the internal structure of the flame under the assumption that the typical spatial and time scales of the incident flow are large compared to the transit time τ L and thickness d L of the flame, respectively [START_REF] Pelcé | Influence of hydrodynamics and diffusion upon the stability limits of laminar premixed flames[END_REF][START_REF] Matalon | Flames as gasdynamic discontinuities[END_REF][START_REF] Clavin | The influence of the temperature dependence of diffusivities on the dynamics of flame fronts[END_REF]. However, when the time scales associated with the flow are no longer large in comparison to the transit time of the flame, its typical response to curvature and strain is likely to be modified. This question was first addressed theoretically by Joulin [START_REF] Joulin | On the response of premixed flames to time-dependent stretch and curvature[END_REF] in the framework of a constant density model. He demonstrated that the flame response to stretch is indeed frequency dependent by performing an asymptotic analysis of the internal structure of the flame and considering a curvature and normal-normal component of the rate of strain tensor that vary periodically in time at a frequency ω a . More precisely, when ω a is increased, L II monotonically decreases until the flame response to strain completely disappears at high forcing frequency. This is due to a memory effect that makes the flame insensitive to hydrodynamic stretch when the latter occurs on a too small time scale (compared with τ L ). The flame does not have time to see the flow variations occurring at times scales that are small compared to the transit time. The diffusive mechanisms are no longer effective on these short time scales for the same reason. Therefore, the Lewis number component of L I vanishes at high frequency for which all the flames behave as if Le = 1. These analytical results were qualitatively confirmed later in the more realistic variable density framework [START_REF] Clavin | High-frequency response of premixed flames to weak stretch and curvature: a variable-density analysis[END_REF]. From a numerical or experimental standpoint, there are only a few studies that have addressed the frequency dependence of Markstein lengths. Indeed, due to the fact that their value depends on the position of the adopted surface of reference within the flame, their direct measurement is made very complicated [START_REF] Davis | Determination of Markstein numbers in counterflow premixed flames[END_REF][START_REF] Davis | Markstein numbers in counterflow, methane-and propane-air flames: a computational study[END_REF]. An alternative is to perform indirect measurement of their value by analyzing the dynamics of a flame submitted to a periodic flow. As first suggested by Markstein [START_REF] Markstein | Interaction of Flow Pulsations and Flame Propagation[END_REF], when advected by a periodic flow, the flame front is described by a parametric oscillator equation. As such, parametric restabilization and parametric resonance can be observed. Searby and Rochwerger [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF] have proposed a refined version of Markstein's theory that allows the prediction of the parametric restabilization and parametric resonance thresholds in terms of the amplitude of the acoustic wave and the wavenumber of the perturbation. In their study they have compared these theoretical thresholds to experimental measurements obtained for hydrocarbon flames propagating downwards in a tube; these flames were forced by an acoustic wave that was generated by a speaker positioned at the bottom of the tube. Their measurements seem to corroborate Joulin's theory: at high frequency, all the flames seem to respond in the same way, independently of their chemical composition. Similar observations were obtained in [START_REF] Denet | Numerical Study of Premixed Flames Parametric Acoustic Instability[END_REF] with threshold measurements obtained from DNS.

In the present paper, we propose to supplement these previous studies by presenting new experimental measurements of these threshold values on flames propagating in a Hele-Shaw burner. This type of burner has been extensively exploited in recent studies for its potential applications in micro combustion devices and the easy visualization of flame dynamics that it offers [START_REF] Alexeev | Experimental Study on Cellular Premixed Propane Flames in a Narrow Gap between Parallel Plates[END_REF][START_REF] Hye | Unsteady propagation of premixed methane/propane flames in a mesoscale disk burner of variable-gaps[END_REF][START_REF] Al Sarraf | Darrieus-Landau instability and Markstein numbers of premixed flames in a Hele-Shaw cell[END_REF][START_REF] Almarcha | Interface dynamics, pole trajectories, and cell size statistics[END_REF][START_REF] Veiga-López | Experimental analysis of oscillatory premixed flames in a Hele-Shaw cell propagating towards a closed end[END_REF][START_REF] Veiga-López | Unexpected propagation of ultra-lean hydrogen flames in narrow gaps[END_REF][START_REF] Radisson | Nonlinear dynamics of premixed flames: from deterministic stages to stochastic influence[END_REF]. Until recently, it was still not clear if the same parametric behavior could be obtained in such a device due to the confined geometry associated with this type of burner. The viscous losses at the walls of the burner were shown to inhibit the occurrence of the standard thermo-acoustic instability [START_REF] Veiga-López | Experimental analysis of oscillatory premixed flames in a Hele-Shaw cell propagating towards a closed end[END_REF][START_REF] Kumar Dubey | Exploring a critical diameter for thermoacoustic instability of downward propagating flames in tubes[END_REF]. In a recent study, we demonstrated that parametric restabilization and parametric resonance can be observed in this kind of apparatus and we uncovered a new type of coupling between the flame propagation and the structural modes of the burner [START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF]. Here we propose to exploit this coupling mechanism to perform measurements of restabilization and parametric thresholds. For this purpose, one of the walls of the burner is forced with a shaker in order to excite one of its eigen modes. As demonstrated in [START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF], this in turn produces an oscillating flow in the burner cavity that interacts with the flame propagation in the same way as in the tube of the Searby and Rochwerger experiment. Our apparatus is more versatile than that in the Searby and Rochwerger experiment for two reasons : i) due to its particular geometry the dynamics of the flame are quasi-2D and easier to visualize. This allows us to define precisely the instant when a threshold is crossed. Moreover, it is possible to design very large Hele-Shaw burners and determine accurately the restabilization threshold ( k * I , u * I in the following) corresponding to large wavelengths. This was more difficult with their tube experiments and so they focused only on the parametric instability threshold ( k * II , u * II in the following). In this paper we will see that studying this restabilization threshold provides interesting information on the flame response. ii) the eigen mode of the structure can be easily varied by changing the material or geometrical properties of the walls of the Hele-Shaw burner, which allows us to study the flame response to a large range of frequency with the same apparatus. The present paper is organized as follows: in §II we introduce the experimental apparatus and the procedure employed to perform these measurements. In §III we briefly present the Searby and Rochwerger theory and the main properties of the parametric oscillator equation describing the flame dynamics. Our experimental results are then presented and compared to the theory in §IV and we finally discuss these results, emphasizing particularly the discrepancy between our measurements and the theory. The eigenmodes of one of the plates are forced using a shaker and the resulting vibrations are recorded using two accelerometers positioned on the plate (see [START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF] for details). B. The oscillations of the plates generate an oscillating flow in the burner cavity. Here we plot the vertical component of the flow speed ua (on the horizontal axis) in terms of the vertical position in the Hele-shaw burner y (on the vertical axis where the zero corresponds to the top end of the burner). Moving downwards in the burner, we observe a succession of areas with a low amplitude oscillating flow (nodes) and areas with a large amplitude oscillating flow (antinodes). These nodes and antinodes of the flow speed correspond to nodes and antinodes of the plate vibrations [START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF]. C. When propagating downwards, the flame undergoes a periodic oscillating flow with different acoustic amplitude depending on the position in the tube, and exhibits parametric restabilization and/or parametric destabilization.

II. PARAMETRIC FORCING OF A PREMIXED FLAME

The Hele-Shaw burner used in this study is composed of two borosilicate glass plates (Young modulus E = 69GP a) separated by a 5mm gap. The two plates are 1500mm in height and 500mm in width. The burner cavity is closed on the sides and at the bottom and open at the top. The oscillating flow is generated by enforcing a structural mode of the burner with a shaker attached to one of the walls (see Fig. 1A). The glass plate of the Hele-Shaw burner can be switched between h = 5mm and h = 19mm thicknesses. This modifies the bending stiffness of the plate and allows for expansion of the range of frequencies we can study. For each experiment reported hereafter, the shaker frequency is adjusted to an eigenmode of the plate in order to generate large bending oscillations (see [START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF] for details). The deformations of the wall induce volume variation in the burner cavity. This in turn generates vertical flow oscillations with variable amplitude depending on the vertical position in the burner. A typical profile of the amplitude of the vertical flow oscillations u a is presented on Fig. 1B. The bending mode shape leads to non-homogeneous amplitude of the oscillating velocity with a succession of low velocity amplitude regions (nodes) and high velocity amplitude regions (antinodes). When a flame, ignited at the top of the burner, propagates downwards through this oscillating flow, its dynamics are strongly influenced. A typical space time diagram of the obtained flame is presented on Fig. 1. We notice that the flame is wrinkled when crossing low flow velocity regions, but it is flattened when passing through regions where the amplitude of the oscillating flow is large. This is a typical observation of a parametric flattening of a Darrieus-Landau (DL) unstable flame. These dynamics are correctly described by the following Searby Rochwerger theory.

III. SEARBY ROCHWERGER THEORY

A. The equivalent parametric oscillator

This theory, following the pioneering idea of Markstein [START_REF] Markstein | Interaction of Flow Pulsations and Flame Propagation[END_REF], considers that the coupling with the flame is essentially controlled by the acceleration. This is contrary to other models where the coupling is caused by the pressure and leads to a pulsating acceleration term added to the gravity in the equation introduced in [START_REF] Clavin | The influence of the temperature dependence of diffusivities on the dynamics of flame fronts[END_REF] (eq. ( 37)). When all the lengths are rescaled by the flame thickness d L and the time is rescaled by the transit time τ L = d L /u L , the following non-dimensional oscillator equation for the vertical coordinate of the front is obtained

∂ 2 Φ ∂ t 2 + 2c ∂ Φ ∂ t + a + a 1 cos( ω a t) Φ = 0, ( 2
)
where Φ is the amplitude of the mode k, t is the non-dimensional time and ω a is the non-dimensional pulsation of the acoustic oscillation. c , a and a 1 are given by :

c = k 1 + k(M a -J )/ (1 -γ) (2 -γ) + γ k (M a -J /γ) (3)
a = γ F r 2 k - γ 1 -γ k 2 1 + 1 -γ F r 2 M a - J γ + γ 1 -γ k 3 λ 2 + (2 + γ) M a γ - 2J γ + (2P r -1) H (2 -γ) + γ k (M a -J /γ) (4)
a 1 = γ k u a ω a 1 -k (M a -J /γ) (2 -γ) + γ k (M a -J /γ) (5)
where γ = (ρ u -ρ b )/ρ u depends on the density in the burned and unburned gases, λ 2 is the non-dimensional thermal conductivity evaluated at burnt gases temperature, M a = L I /d L = L II /d L is the Markstein number (which is unique in the case of simple chemistry and when the front position is defined as the position of the reacting sheet as done in [START_REF] Clavin | The influence of the temperature dependence of diffusivities on the dynamics of flame fronts[END_REF]), F r = u L / √ gd L is the Froude number associated with flame propagation, P r = ν/D th (with ν the momentum diffusivity and D th the thermal diffusivity) is the Prandtl number evaluated at fresh gases temperature and u a = u a /u L is the reduced acoustic velocity. The integrals H and J are given by :

H = 1 0 λ 2 -λ(θ)dθ (6) J = γ 1 -γ 1 0 λ(θ) 1 + θγ/(1 -γ) dθ (7)
where λ (θ) is the thermal conductivity evaluated at reduced temperature θ normalized by the thermal conductivity λ 1 in fresh gas, and λ 2 is the conductivity in burned gas.

Setting :

z = 1 2 ω a t, (8)
Y(z) = φ( t) exp(c t), (9)
a = 4 a -c 2 ω 2 a , (10)
q = - 2a 1 ω 2 a , (11)
equation ( 2) is reduced to the so called Mathieu equation :

d 2 Y dz 2 + ( a -2q cos(2z)) Y = 0 (12)
Floquet theory predicts solutions in the form :

Y(z) = exp(µz)ψ(z) + exp(-µz)ψ(-z) ( 13
)
Where ψ is a 2π-periodic function and µ is the Floquet exponent (also called characteristic exponent). Thanks to [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF], the stability condition of Mathieu equation solutions is made explicit : If Re(µ) > 0, Y(z) grows exponentially with z (unstable), otherwise the solution is stable (Re(µ) ≤ 0). Typical stability diagrams for Y(z) can be seen in [START_REF] William | Theory and application of Mathieu functions[END_REF] for example s???.

The stability condition for Φ(t) is then obtained from ( 8) and [START_REF] Clavin | High-frequency response of premixed flames to weak stretch and curvature: a variable-density analysis[END_REF]. The flame front is stable if Re(µ) ≤ 2c/ ω a and unstable if Re(µ) > 2c/ ω a . As a and q depend on the wavenumber of the perturbation k, on the physico-chemical parameters of the flame and on the forcing parameters ( ω a , u a ), their physical meaning is difficult to understand. Therefore, in the following, the stability diagrams for Φ will be shown in the ( k, u a ) plan. Typical diagrams are plotted on Fig. 2 for different values of ω a , F r, M a. Here we use the same parameters as those used in [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF] Fig. 1, which are restated here:

P r = 0.689 γ = 0.822 λ 2 = 3.19 H = 0.96 J = 3.33 (14)
Other parameters are given on the plots. To obtain these diagrams, for each set of parameters ( ω a , F r, M a), the corresponding values for a, q and κ = 2c/ ω a are computed. The Floquet exponent µ( a, q) is then obtained numerically [START_REF] Alhargan | A complete method for the computations of Mathieu characteristic numbers of integer orders[END_REF] and the stability condition Re(µ( a, q)) ≤ κ( a, q) is verified. The process is repeated for each set of parameters ( k, u a ). With the above parameters, the obtained diagrams agree well with those obtained in [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF]. This validates our method to compute the characteristic exponents. On diagram B and C, a third unstable zone, not visible in [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF], is observed (see also [START_REF] Yanez | The acoustic-parametric instability for hydrogen-air mixtures[END_REF][START_REF] Ajit | Range of "complete" instability of flat flames propagating downward in the acoustic field in combustion tube: Lewis number effect[END_REF]). It corresponds to the second region of unstable solutions visible on the typical Mathieu stability diagram (see [START_REF] William | Theory and application of Mathieu functions[END_REF]). This second parametric resonance is never observed (at the parametric threshold) in our experiments, as the acoustic threshold associated with it is always higher than that associated with the first resonance.

We now focus on the first two regions. The first one corresponds to the zone where the flame is Darrieus-Landau unstable. For u a = 0 (no forcing), this region corresponds to the range of unstable wavenumbers given by the dispersion relation obtained in [START_REF] Clavin | The influence of the temperature dependence of diffusivities on the dynamics of flame fronts[END_REF]. As u a is increased, the range of unstable wavenumbers narrows until this zone completely disappears for u a = u * I . At this threshold intensity, there is only one DL unstable wavenumber k * I . The second region II , a typical parametric instability is observed with patterns pulsating at half the forcing frequency ωa. On frame G. the evolution of a parametrically excited front is represented over one full accoustic period (from top to bottom). We note that at the end of the period the minima and maxima of the front have exchanged positions. After another acoustic period, the minima and maxima would get back to their original position (The period of the pattern is twice the acoustic period). On the other hand, if for a given set of parameters, u * II < u * I , there is no parametric restabilization zone (see B. and C.). Instead DL instability and parametric instability may coexist for a certain range of ua (see H. where one can note the DL (large wavelength) and the parametric (small wavelength) instabilities).

corresponds to the first parametric resonance of the oscillator equation associated with the flame dynamics. This zone appears for a non zero acoustic intensity u a = u * II at a wave number k * II . As the forcing intensity is increased, this zone becomes larger, leading to a wider range of unstable wavenumbers. As can be seen on Fig. 2, the thresholds u * I and u * II depend on the flame parameters and on the forcing frequency. If for a given set of parameters, u * II > u * I (for example, in Fig. 2A and2D), three different scenarios may be observed, depending on the forcing intensity u a : (i) If u a < u * I , the flame is Darrieus-Landau unstable (see Fig. 2E). (ii) If u * I < u a < u * II , the DL instability disappears and the flame is flattened (see Fig. 2F). We will call this zone the restabilization zone. (iii) Finally, if u a > u * II , a parametric instability is observe and the flame front exhibits a cellular aspect with structures oscillating at a frequency twice smaller than the forcing frequency (see Fig. 2G). Otherwise, if u * II < u * I there is no restabilization zone (see Fig. 2B and2C). The DL instability and the parametric one may coexist in the case where u * II < u a < u * I (see Fig. 2H).

By computing stability diagrams, as the ones presented in Fig. 2, the theoretical thresholds of the parametric flattening (acoustic velocity u * I with wavenumber k * I ) and parametric instability (acoustic velocity u * II with wavenumber k * II ) can be obtained for several sets of parameters. forcing frequency. When M a increases, both the stabilization and the parametric thresholds drift toward larger wavelengths. This is due to the fact that for larger M a, the small wavelengths are more damped by diffusive effects. This tendency is weaker for k * I than for k * II . On the other hand, two opposite trends are observed for u * I and u * II . When M a is increased, the restabilization (respectively the parametric) threshold is decreased (respectively increased). The restabilization zone that occurs when u * II > u * I is therefore larger at high M a and tends to be completely suppressed when M a is decreased at low frequency(see Fig. 3A). This trend is due to the dependency of the damping coefficient c in the oscillator equation ( 2) on the Markstein number. When M a decreases, the damping is reduced too, and the parametric zone goes down (for zero damping (c = 0) u * II → 0). At high frequency (see Fig. 3B) , the variation of u * II with M a is strong whereas the u * I is weakly dependent on M a. The wavelength at the parametric instability threshold is smaller than that of the DL, provided that the forcing frequency is sufficiently large (see Fig. 3D). However, at low forcing frequency, and especially at a low Markstein number, the DL wavelength may be smaller than that of the parametric instability (see Fig. 3C).

Thresholds dependence on forcing frequency

The same kind of analysis is now performed to study the dependency of the threshold values on forcing frequency. The Markstein number is now fixed (three different values are reported) and the reduced forcing frequency is varied over one order of magnitude. This kind of parametric exploration has already been presented in [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF] Fig. 4 and in [START_REF] Baillot | Parametric Response of a Conical Flame to Acoustic Waves[END_REF] Fig. 11 for the parametric threshold (u * II , k * II ). Their results are recalled now for the purpose of the following discussion. We also present the restabilization threshold dependency on the forcing frequency; the results are plotted on Fig. 4. As shown, the restabilization threshold ( k * I , u * I ) is frequency dependent only at small frequency values. This property is consistent with the theory developed by Bychkov in [START_REF] Bychkov | Analytical scalings for flame interaction with sound waves[END_REF], which predicts no frequency dependency for the DL zone (this analysis is asymptotically exact in the limit ω a k √ a). On the other hand, the threshold of the parametric zone exhibits a strong linear dependence on ω a . The variation is even more pronounced when the Markstein number is high. As the theory predicts a linear increasing parametric threshold u * II and a constant threshold u * I , all flames should exhibit a restabilization zone at high frequency (i.e a range of forcing for which the flame is completely flattened). As we shall see in the following, this is not what we observe in our experiments. As we are interested in measuring the flattening and the parametric thresholds experimentally, we want to analyze flames that admit a restabilisation zone (i.e u * I < u * II ) so that these thresholds can be measured from a flat front. Lean heavy hydrocarbon-air flames typically have a large Markstein number and therefore admit a restabilisation zone on a larger range of frequencies. Here, we perform lean propane-air flame experiments with three different equivalence ratios. The experiment is performed several times for each flame, adjusting the shaker intensity between each run. This is done in order to find the appropriate value to reach the parametric instability threshold in the very vicinity of the antinode of the mode. This setup allows the generation of flow oscillations that go from u a ≈ 0 (nodes) to u a = u * II (antinodes). In this way, the flame experiences all of the interesting u a range during its propagation. Here, we perform the threshold measurements for different flames at different forcing frequencies. The results are then compared to theoretical values. To compute these theoretical thresholds, the laminar flame speeds are taken from Bosschaart and de Goey [START_REF] Bosschaart | The laminar burning velocity of flames propagating in mixtures of hydrocarbons and air measured with the heat flux method[END_REF], the expansion ratios are taken from a numerical resolution of the flame structure using the Cantera software [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes[END_REF] with a complex chemical-kinetic SanDiego mechanism [START_REF]San Diego Mechanism: Chemical-Kinetic Mechanisms for Combustion Applications[END_REF] and the Markstein numbers are taken from Searby and Quinard [START_REF] Searby | Direct and indirect measurements of Markstein numbers of premixed flames[END_REF]. The resulting set of parameters is summarized in table I and here:

IV. EXPERIMENTAL RESULTS

D th = 2 × 10 -5 m 2 .s -1 , g = 9.
81m.s -2 , P r = 0.9, ( 16)

d L = D th /u L , τ L = D th /u 2 L , ω a = 2πτ L f shaker , F r = u L / gd L . ( 17
)
The first thing to observe as a qualitative comparison between theory and experiments is the existence or absence of a stabilization zone (i.e u * I < u * II ). For each type of experiment, this conclusion has been summarized in the last two ϕ f shaker ωa uL γ M a F r λ2 J H Rest Rest th 0.8 63Hz 0.096 0.28m.s -1 0.854 4.28 10.58 2.62 1.62 0.69 y y 0.8 116Hz 0.186 0.28m.s -1 0.854 4.28 10.58 2.62 1.62 0.69 y y 0.8 156Hz 0.250 0.28m.s -1 0.854 4.28 10.58 2.62 1.62 0.69 y y 0.8 400Hz 0.641 0.28m.s -1 0.854 4.28 10.58 2.62 1.62 0.69 y y 0.9 63Hz 0.058 0.36m.s -1 0.863 4.21 15.42 2.7 1.7 0.72 n n 0.9 400Hz 0.388 0.36m.s -1 0.863 4.21 15.42 2.7 1.7 0.72 y y 1.0 63Hz 0.047 0.40m.s -1 0.868 3.69 18.06 2.75 1.75 0.74 n n 1.0 400Hz 0.314 0.40m.s -1 0.868 3.69 18.06 2.75 1.75 0.74 n y Table I. Parameters corresponding to each experiments. The equivalence ratio ϕ is prescribed using Bronkhorst mass flow regulators. The forcing frequency f shaker is prescribed by adjusting the shaker frequency. The non-dimensional angular frequency ωa is computed as ωa = 2πf shaker τL. The laminar flame speed uL is taken from [START_REF] Bosschaart | The laminar burning velocity of flames propagating in mixtures of hydrocarbons and air measured with the heat flux method[END_REF]. The expansion parameter γ is computed using Cantera [START_REF] Goodwin | Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes[END_REF]. The Markstein number M a is taken from [START_REF] Searby | Direct and indirect measurements of Markstein numbers of premixed flames[END_REF]. Finally, J and H are computed following their theoretical expression recalled in the main text.

columns of the above table. The columns are noted "y" if u * I < u * II and "n" in the opposite case. The comparison between the "Rest" and the "Rest th" columns, gives the agreement between experiments and theory, respectively. The agreement is satisfying for all the flames we analyzed except for the case where ϕ = 1.0 with forcing frequency f shaker = 400Hz.

The comparison between experimental and theoretical thresholds is visible on Fig. 5 : it shows the u * and the k * values for three different equivalence ratios at fixed forcing frequency f shaker = 400Hz. Despite the fact that the value for f shaker is fixed, it is important to note that the reduced forcing frequency ω a is not. For the restabilization threshold, we observe that both the k * I and the u * I values remain nearly constant when the equivalence ratio is increased (Fig. 5A and5C). This is in good agreement with the trend predicted by theory (red curve). Looking now at the actual values, we remark that the measured u * I values are substantially lower than the ones predicted by theory. On the other hand, the measured k * I values are in relatively good agreement. For the parametric threshold, our measurements indicate that u * II increases with the equivalence ratio (Fig. 5B) while k * II (Fig. 5D) decreases. This is not the trend predicted by theory, where u * II is expected to decrease with the equivalence ratio and k * II is expected to remain nearly constant for these flames at this forcing frequency. These data and additional ones at other forcing frequencies f shaker are also reported on Fig. 4 in term of the nondimensional forcing frequency ω a . As mentioned in the previous section, the theoretical thresholds on this figure have been computed for 3 different Markstein numbers and with all the other parameters corresponding to a Propane-air flame with equivalence ratio ϕ = 0.8. As M a = 4.28 corresponds to the Marsktein value associated with this type of flame (see table I), we theoretically expect the experimental data associated with equivalence ratio ϕ = 0.8 to collapse on the theoretical red curve M a = 4.28. The two other equivalence ratios studied in the present experiments are not expected to collapse on any of the theoretical curves as they not only have a different Markstein number but also different F r, J and H. However, we shall see that analyzing the trend of the evolution of their thresholds with ω a provides interesting information.

Among the flames studied here, complete restabilization is observed only for the flame with an equivalence ratio ϕ = 0.8 (at all the forcing frequencies studied here) and for the flame ϕ = 0.9 at low frequency. For the case ϕ = 0.8, the restabilization threshold remains nearly constant when ω a is increased both in terms of acoustic speed ( u * I ) and wavenumber ( k * I ). This trend is in good agreement with the theoretical results that predict a constant restabilization threshold except in a thin boundary layer at low forcing frequency where both u * I and k * I decrease drastically (Fig. 4A and4C). We note that this boundary layer is not observed in our experimental data. Comparing the actual values of these thresholds with the theoretical ones, we note that the agreement is satisfying for the wavenumbers (Fig. 4C) but not for the acoustic forcing speed (Fig. 4A), where the measured thresholds are substantially lower than the expected ones.

Looking now at the results obtained for the parametric instability threshold (Fig. 4B and4D), we observe that for the three equivalence ratios studied here the measured values for u * II remain nearly constant when ω a is increased while the theory predicts a linearly increasing threshold value if M a remains constant (red curves Fig. 4B). For the wavenumbers, we observe that k * II increases with the forcing frequency for the three equivalence ratios analyzed here (Fig. 4D). This tendency is in relatively good agreement with what the theory predicts, although the experimental data seem to increase faster than the theoretical.

V. DISCUSSION

The experimental measurements presented in the previous section display major discrepancies with the theoretical predictions given by the Searby & Rochwerger theory. In particular, for the restabilization threshold, the agreement on the wavenumber values k * I is satisfying (Fig. 4C and5C). However, while the experimental values for u * I (Fig. 4A and5A) show the correct trend (except at very low frequency), the measured values are substantially lower than the ones predicted by theory. This large discrepancy cannot be explained by a misestimation of the actual M a and/or ω a associated with the experimental flame, as this threshold is not very dependent on these two quantities (see parametric study carried out in §III B). Moreover, as pointed out earlier, while some discrepancies are observed on u * I , the agreement on k * I is satisfying. Here we argue that heat losses might be responsible for the discrepancy observed between experiments and theory on the restabilization threshold. Indeed, (2) was obtained under an adiabatic hypothesis. In our apparatus, the flame is confined between two walls and the heat losses play an important role in the dynamics [START_REF] Basile Radisson | Nonlinear dynamics of flame fronts with large-scale stabilizing effects[END_REF]. Heat losses were shown to inhibit the standard Darrieus-Landau instability [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF]. With the present type of burner, they were shown to affect the entire range of unstable wave numbers quite uniformly. Mathematically, this corresponds to the addition of a negative growth rate to the entire range of unstable wavelengths [START_REF] Basile Radisson | Nonlinear dynamics of flame fronts with large-scale stabilizing effects[END_REF]. A flame in this apparatus is therefore less unstable than its adiabatic counterpart and is easier to restabilize (i.e it requires a smaller forcing amplitude u a to observe restabilization). This explains the smaller values of u * I measured in our experiments. Meanwhile, as these heat losses are affecting all the wavenumbers uniformly, a good agreement on k * I is expected with the adiabatic theory. This is what we observe in our experiments (Fig. 3C and4C). Finally, the steep decrease of both u * I and k * I predicted by theory and not observed with our experimental data (Fig. 4A and4C) can also be explained by heat losses. Heat losses tend to decrease the actual u L and therefore, by using the adiabatic u L from [START_REF] Bosschaart | The laminar burning velocity of flames propagating in mixtures of hydrocarbons and air measured with the heat flux method[END_REF] to non dimensionalize the forcing frequency, we are underestimating the actual ω a associated with our experimental flames. This means that the experimental data on Fig. 4A and 4C should be shifted horizontally towards higher ω a values and are probably already in the frequency range where u * I and k * I are frequency independent. To validate this hypothesis, in future works, it might be interesting to search for this boundary layer predicted by theory at low frequency by forcing these flames with lower frequencies. This could be achieved in a Hele-Shaw burner that uses walls with a lower fundamental frequency (for example using plastic walls instead of glass walls).

Concerning the parametric threshold ( u * II , k * II ), the theory does not even describe the correct trend observed in our experimental measurements. In particular, u * II is expected to increase linearly with the forcing frequency. In our experiments we have observed that the thresholds remain nearly constant when the frequency is increased. The theoretical red curves are computed for a constant M a number, but the actual Markstein value associated with our experimental flames is apparently changing when the forcing frequency is increased. According to Joulin's theory, the Lewis component is disappearing when the acoustic time is no longer small compared to the transit time. For these lean propane air flames, this corresponds to Markstein values that decrease when the frequency is increased. This is exactly what we observe with our measurements. For the speed u * II , the threshold remains nearly constant instead of growing linearly because M a is decreasing with frequency. In practice it corresponds to a flame starting at a darker red curve (high Markstein value) and moving towards the lighter red curves (lower Markstein values). This is corroborated by the k * II measurements that display a steeper slope than the theoretical curves for the same reason: the flames are evolving from the darker red curves to the lighter ones when the frequency is increased (see Fig. 4B and4D). This would also explain the incorrect trend observed when the dimensional forcing frequency f shaker is fixed and the equivalence ratio ϕ is varied. For this lean propane-air flames both M a and ω a are decreasing when the equivalence ratio is increased from ϕ = 0.8 to ϕ = 1.0. At fixed M a, based on theory, u * II is expected to increase with ω a (see Fig. 4B). At fixed frequency, u * II is expected to increase with M a. By decreasing M a and ω a , we therefore have two different effects that are supposed to make u * II decrease, which explain the theoretical trend observed on Fig. 5B. However, if M a is indeed decreasing with ω a as expected by Joulin's theory, it means that we are overestimating M a by taking the low frequency value obtained in [START_REF] Davis | Determination of Markstein numbers in counterflow premixed flames[END_REF][START_REF] Davis | Markstein numbers in counterflow, methane-and propane-air flames: a computational study[END_REF] and therefore overestimating u * II . At fixed f shaker , ω a is higher at ϕ = 0.8 than at higher equivalence ratio and we are therefore overestimating M a by a larger amount. This leads in turn to a larger overestimation of u * II than at a higher equivalence ratio, as observed on Fig. 5.

Finally, we note that our experiments are compared to linear stability analysis although no total restabilization zone could be observed in some cases ((ϕ = 0.9, f shaker = 63Hz), (ϕ = 1.0, f shaker = 63Hz) and (ϕ = 1.0, f shaker = 400Hz)). In those cases, where the parametric instability does not appear from a flat front but from a Darrieus-Landau wrinkled one (see Fig. 2H), the oscillator model described by [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF] is no longer valid and non-linear effects could have a non-negligible influence on the threshold values (see [START_REF] Rego | On nonlinear model equations for the response of premixed flames to acoustic like accelerations[END_REF][START_REF] Assier | Linear and weakly nonlinear instability of a premixed curved flame under the influence of its spontaneous acoustic field[END_REF]).

VI. CONCLUSION

In this paper, the flame interaction with a periodic time dependent flow has been studied experimentally using the flame/structure coupling uncovered in [START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF]. The measured parametric restabilization and destabilization thresholds have been compared to those given by the theory introduced in Searby and Rochwerger [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF]. Both the u * I and u * II values from our experiments are much smaller than those predicted by theory. However, the agreement on k * I and k * II is surprisingly good. This trend has been attributed to thermal losses which lower the DL growth rate quite uniformly over the range of unstable wavenumbers. In addition, our measurements suggest a nearly constant parametric threshold when the forcing frequency ω a is increased. This is in contradiction with theory, which predicts a linear increase of u * II with ω a . This disagreement may be explained by a decrease of the Markstein number when the frequency increases, as predicted in [START_REF] Joulin | Influence of Momentum and Heat Losses on the Large-Scale Stability of Quasi-2D Premixed Flames[END_REF] (for flames with Le > 1) and already observed experimentally and numerically in previous studies [START_REF] Searby | A parametric acoustic instability in premixed flames[END_REF][START_REF] Denet | Numerical Study of Premixed Flames Parametric Acoustic Instability[END_REF][START_REF] Chen | Stretch effects on the burning velocity of turbulent premixed hydrogen/air flames[END_REF]. Our results constitute a confirmation of the results obtained in these former studies in a completely different kind of apparatus and suggest that frequency dependent Markstein numbers should be used in combustion models for time dependent flows.

VII. ACKNOWLEDGEMENTS

Figure 1 .

 1 Figure 1. Experimental apparatus. A. The flame propagates downwards in a Hele-Shaw burner composed of two borosilicate glass plates oriented vertically and separated by a 5 mm gap. The burner cavity is closed on the sides and at the bottom and open at the top.The eigenmodes of one of the plates are forced using a shaker and the resulting vibrations are recorded using two accelerometers positioned on the plate (see[START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF] for details). B. The oscillations of the plates generate an oscillating flow in the burner cavity. Here we plot the vertical component of the flow speed ua (on the horizontal axis) in terms of the vertical position in the Hele-shaw burner y (on the vertical axis where the zero corresponds to the top end of the burner). Moving downwards in the burner, we observe a succession of areas with a low amplitude oscillating flow (nodes) and areas with a large amplitude oscillating flow (antinodes). These nodes and antinodes of the flow speed correspond to nodes and antinodes of the plate vibrations[START_REF] Basile Radisson | Coupling of vibro-acoustic waves with premixed flame[END_REF]. C. When propagating downwards, the flame undergoes a periodic oscillating flow with different acoustic amplitude depending on the position in the tube, and exhibits parametric restabilization and/or parametric destabilization.

Figure 2 .

 2 Figure 2. Stability diagrams for different set of parameters and their corresponding regimes of propagation. For a given set of parameters where u * I < u * II (see A. and D. for examples), three different scenarios of propagation can be observed depending on the acoustic amplitude ua : (i) if ua < u *I , the flame is DL unstable and exhibits characteristic wrinkling with cusps pointing towards the burnt gases (see D.), (ii) if u * I < ua < u * II , all the wavenumbers are stable and parametric flattening is observed (see F.), (iii) if ua > u * II , a typical parametric instability is observed with patterns pulsating at half the forcing frequency ωa. On frame G. the evolution of a parametrically excited front is represented over one full accoustic period (from top to bottom). We note that at the end of the period the minima and maxima of the front have exchanged positions. After another acoustic period, the minima and maxima would get back to their original position (The period of the pattern is twice the acoustic period). On the other hand, if for a given set of parameters, u * II < u * I , there is no parametric restabilization zone (see B. and C.). Instead DL instability and parametric instability may coexist for a certain range of ua (see H. where one can note the DL (large wavelength) and the parametric (small wavelength) instabilities).

B.F 1 .

 1 Parametric oscillator properties Here, we focus on the dependence of the theoretical threshold values ( k * I , u * I ) and ( k * II , u * II ), on the forcing frequency ω a and on the value of the Markstein number. The thresholds are computed numerically using the same procedure as the one used to obtain the diagram of Fig. 2. The Markstein number M a and the reduced forcing frequency ω a are varied. Other parameters are fixed with the following values (these values correspond to lean propane-air flames ϕ = 0.8.): Thresholds dependency on Markstein number First, the parameters given in (15) and the forcing frequency ω a are fixed and the M a value is varied from M a = 2 to M a = 5.5. For each M a value, the bottom point of the parametric zone and the top point of the Darrieus-Landau zone are evaluated numerically. The obtained results are shown on Fig. 3C and 3D for two different values of the

Figure 3 .

 3 Figure 3. Dependence of the restabilization/parametric thresholds to M a. A. u * I and u * II against M a at small forcing frequency. B. u * I and u * II against M a at high forcing frequency. C. k * I and k * II against M a at small forcing frequency. D. k * I and k * II

Figure 4 .

 4 Figure 4. Comparison between experimental and theoretical threshold values with respect to the non-dimensional forcing frequency ωa. Theoretical data for different Markstein number values are reported in shades of red and experimental data for different equivalence ratios ϕ are reported in shades of grey. A. Comparison of the restabilization threshold u * I . B. Comparison of the restabilization threshold u * II . C. Comparison of the last unstable reduced wavenumber k * I before restabilization. D. Comparison of the first unstable reduced wavenumber k * II above the parametric threshold.

Figure 5 .

 5 Figure 5. Comparison of experimental thresholds (gray) with those predicted by Searby & Rochwerger theory (red). For the three cases presented here the forcing frequency is f shaker = 400Hz. A. Comparison of the restabilization threshold for propane-air flame with different equivalence ratios ϕ. B. Same for parametric thresholds. C. Comparison of the last unstable wavenumber at the restabilization threshold. D. First unstable wavenumber at the parametric threshold.

We thank Daniel Mazzoni for his implication in the experimental facility. We thank the Excellence Initiative of Aix-Marseille University -A*MIDEX, and Labex MEC, for funding.