Dynamics Of A Circular Array Of Falling Liquid Columns

Transcript

EPJ manuscript No. (will be inserted by the editor) Dynamics of a circular array of falling liquid columns Dynamics of an array of liquid columns P. Brunet1,3 and L. Limat1,2 1 2 3 Laboratoire PMMH-ESPCI - 10, rue Vauquelin 75005 Paris France F´ed´eration de Recherche Mati`ere et Syst`emes Complexes Present adress: Royal Institute of Technology (KTH) - Department of Mechanics 10044 Stockholm (Sweden) Received: date / Revised version: date Abstract. We report an experimental study of an array of falling liquid columns, hanging below an overflowing circular dish fed at constant flow-rate. This system constitutes a one-dimensional pattern which shows a host of dynamical regimes as well as spatio-temporal chaos, depending on flow-rate, initial positions and number of columns, and liquid properties. In this paper, we present an extensive quantitative study of the ordered, predictable states, and we show stability diagrams obtained with different liquid viscosities. Some mechanisms of destabilization of these regimes are also presented. PACS. 05.45.-a Nonlinear dynamics and nonlinear dynamical systems – 47.20.Lz Secondary instability – 47.20.Ma Interfacial instability 1 Introduction Pattern-forming instabilities are known to exhibit rich and fascinating dynamical behaviors [1,2]. They are observed in everyday’s life in various forms such as arrays of clouds, ripples in underwater sand sheared by the tide, or snow flakes. Since the last decades, they have been studied intensively because some of their properties are associated to generic concepts of symmetry-losses and transition toward disorder. They are also known to be involved in some transition scenarios toward fluid turbulence in channel flows: recently, a set of oblique and traveling waves have been evidenced in plane Couette flow [3,4] and in Poiseuille flow [5]. The destabilization of a patterned structure by creations of local or global traveling domains is seemingly a generic feature of pattern-forming instabilities. As a sub-class of these systems, one-dimensional destabilizing fronts have focussed lots of interests because of their relative simpleness and the easiness one can identify and measure their dynamical properties. Directional viscous fingering [6–11], directional solidification [12–15], arrays of ferrofluid pikes under oscillating magnetic field [16] or a thin layer of liquid submitted to a temperature gradient [17] are examples of such one-dimensional fronts. Numerically, the damped Kuramoto-Sivashinsky equation [18–21] exhibits striking analogous features. The ’circular fountain’ presented here and previously studied in [22–26], is another example of such destabilizing interfacial front. It consists in an array of liquid columns formed below an overflowing circular dish represented on fig. 1-a. Basically, it can be seen as a combination between the Rayleigh-Taylor instability [27,28], that occur for instance in an overhung layer of fluid and creating ultimately a network of pending drops [29], and a permanent supply of liquid at given flow-rate Q. The liquid supply insures that the gravity flow counterbalances the Rayleigh-Plateau instability [27]. The latter may cause the pinch-off of columns at their top, turning them into dripping sites, if the flow-rate is not high enough. Pioneering studies on arrays of columns formed below an overflowing cylinder were worked out by Pritchard [30]: it was there already remarked that such a network could exhibit unstable and complex dynamics. The problems of unstable phenomena in overflowing liquids has historically brought interests for practical applications, such as surface coating or atomization into drops. Afterwards, it became obvious that, in the regime of columns, liquids overflowing from overhangs could be studied as benchmarks for capturing the rich dynamics of destabilizing interfacial fronts [31– 33]. The geometry of the circular dish appeared then as the most suitable: it disables any edge effects that can lead to unwanted perturbations or can hinder the longlasting motion of columns. For instance below the circular overhang, domains of drifting columns can propagate endlessly, like in an infinite medium [22,24]. In such pattern-forming systems, the direct resolution of equations is far from obvious, and generally tells little about the complex states, as it is focused on the first stages of the primary instability. Instead, these systems have been described through more phenomenological models, involving local or global symmetry-breaking that occur from a basic state reference, generally, a static spatiallyperiodic pattern: see [34–40] amongst many other studies, and [1] for an exhaustive review. The determination 2 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns (a) (b) Fig. 1. The circular fountain experiment. (a) Side view of the array of liquid columns (dish diameter d=10 cm, silicon oil η = 100 cP). (b) The array viewed from above, suitable to extract grey levels in order to build spatiotemporal diagrams. of relevant parameters in those models needs inputs from benchmark experiments, particularly in the further stages, far beyond secondary thresholds [39,40]. One of the main interests of this experiment is not only to survey on the hydrodynamics of this liquid merry-goround (this likeness is particularly striking in the regime where all the columns drift at the same speed [22,24]), but also to suggest general phenomena in pattern-forming interfaces. In our system, the easy control of parameters enables a broad exploration of the rich dynamics, which can be compared to similar systems, although the latter involve really different physical mechanisms. The scope of our study is to present a set of results on the circular network of columns, obtained at different viscosities. Mainly, it concerns the presentation of stability diagrams assembling the limits of different possible states in the parameters space, exhaustive measurements of these dynamical regimes and finally their different break-up scenarios. The natural control parameter is the flow-rate per unit length Γ , which is equal to the flow-rate in cm3 /s divided by the perimeter of the circle along which columns travel. Γ = Q πd (1) In the pattern of columns, the order of magnitude of Γ is between 0.05 and 0.5 cm2 /s. The spacing between static columns is around one centimeter, but the spacing can be up to 2.2 cm between two drifting columns. Figure 1-b offers a view from above, showing a zone of dilation at the right side of the fountain, and related to a local drift suggested by an arrow. However, the selected state is not only determined by the value of Γ , but also by a set of initial conditions, like in most of out-of-equilibrium systems constantly fed with energy. These initial conditions can easily tuned by the experimentalist [22,24,25], and it is one of the advantages of this system. As also pointed out in previous papers, the viscosity is a crucial parameter: a higher viscosity seemingly increases the richness of the dynamics and the number of the different available states. For instance, there is a limit for η, around 90 cP, above which a regime of spatio-temporal chaos is observed. No such regime could be observed in the first studies conducted at low viscosities [22,23]. This chaotic state has been compared to disordered behaviors in other systems [6,15–17,41,42]. Most of them involve the so-called generic scenario of spatiotemporal intermittency (STI), i.e. a coexistence of laminar domains and turbulent patches. Chaotic regimes in the pattern of columns, and presumably in other destabilizing fronts, are somehow different as they involve spatiotemporal singularities in the phase space, called ’defects’. These defects result from complex interactions between laminar domains, particularly propagative structures and oscillating patches [26]. In most situations, they are clearly the signature of the break-up of states that have been driven out of their domain of stability. Let us finally mention a two-dimensional extension of this system [43,44], where the network of columns is selforganized around a hexagonal structure. Local departures from this basic hexagonal state leads to dynamical states, such as localized oscillations or a pair of columns, propagative solitons or oscillating arrays of columns. A quantitative study of turbulent states in 2D has been carried out in [45]. The paper is organized as follows: we firstly give insights of some existing models dedicated to such cellular structures (section 2). Then we describe the experimental setup (section 3). Section 4 offers a general overview of the dynamics of the pattern. Measurements related to different states are reported in section 5. Section 6 is dedicated to different break-up scenarios of various regimes. Section 7 concludes the paper. 2 Summary of existing models and remaining questions on 1D cellular patterns 2.1 Describing a cellular pattern with a symmetry-based model: the example of parity-breaking drifting cells Even if the direct resolution of the complete set of equations has led to remarkable achievements in some patternforming instabilities, these studies are mostly restricted to the very first stages of the primary instability. Their extension to secondary instabilities would constitute a heavy mathematical work, that furthermore would not bring out P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns U(x) Static columns x Oscillations Vd Vg λ1 λ0 Propagative domain of drifting columns Vg Fig. 2. Sketch of a one-dimensional cellular pattern by a function U (x, t), in some typical dynamical states. From top to bottom, a periodic function represents static states; a timedependent period-doubled function represents out-of-phase oscillations; a combination of modes represents a propagative domain of broken-parity drifting cells. the general characteristics one wishes to emphasize. Instead, it is common to reduce the problem to a more phenomenological approach, by seeking for simpler models, although rich enough to exhibit the complex behaviors one expects. The framework of time-dependent amplitude equations, such as the complex Ginzburg-Landau equation (CGLE) [46] or the Kuramoto-Sivashinsky equation (KSE) [21], related to considerations of possible broken symmetries in the system, constitutes a general and mathematically tractable approach. For one-dimensional patterns with primary static, spatially periodic state, Coullet and Iooss have extracted the possible dynamical states from symmetry arguments [34]. In this approach, the pattern is represented as a onedimensional function U (x, t), of one space variable x and time t, that obeys simple general laws. The function U (x, t) is generally the sum of a periodic function U0 (x + φ) chosen as a reference, and another function u(x + φ, t) which is the sum of all possible modes. The quantity φ is the spatial phase of the pattern. U (x, t) = U0 (x + φ) + u(x + φ, t) X u(x + φ, t) = Aα (x + φα , t) mα (x + φα , t) (2) (3) α is the index of unstable modes mα that can be timedependent or not. The quantity Aα is the amplitude of this mode. It has been shown that ten different generic modes exist, corresponding to ten different broken symmetries. Figure 2 illustrates how such a function can represent a pattern of cells, in specific states. For example, the well-known propagating domains of drifting cells [8,10,12,14,15,22–25,35] are associated to a left-right symmetry breaking. Figure 2 (bottom sketch) gives kinetic and geometrical definitions related to this state: λ0 and λ1 stand the wavelengths respectively se- 3 lected outside and inside a propagative domain; Vd and Vg stand respectively for the drift velocity of cells (phase speed) and for the wall velocity of the domain (group velocity). The mode mα (x + φα , t) is an anti-symmetrical function of x, that is added to the reference U0 in the bulk of a domain [35]. The amplitude of the mode Aα and the phase φα are supposed to vary slowly with the space and time variables. From symmetry arguments, one can deduce coupled partial differential equations for Aα and φα . When one only gets restricted to this specific antisymmetric mode - the assumption is valid inside a propagating domain - the following equation is found for the space-time evolution of A [35]: At = (µ + φx )A + Axx + γAAx − δA3 + . . . φt = ζA + φxx + . . . (4) (5) These equations have been restricted to their lowest orders for the powers of A and φ and their mutual combination. This assumption is done providing that the system lies sufficiently close to the bifurcation threshold, so that amplitudes keep small. Generally, µ appears as the natural control parameter of the system (presumably identifiable here to the flow-rate per unit length Γ ), and it turns out that the threshold of the bifurcation depends on the phase gradient. In this minimal model also, the coefficients δ and ζ are a priori independent on the phase gradient φx , but more subtle relationships could be found out in the array of columns [24]. The quantity φx is also the difference between the local wavenumber and the reference wavenumber k0 : φx = k1 − k0 = 2π( 1 1 − ) λ1 λ0 (6) The time derivative of the phase φt is related to the drift velocity of the cell, by the following relationship: φt = V d k (7) Assuming that the phase gradient and the asymmetry A are constant inside a propagative domain - which is equivalent to say that all the drifting cells are identical and at equal distance to their nearest neighbors, all the space derivatives of A and φx can be removed from (4) and (5). Thus, it is found that A and φt are proportional: φt = ζA. The solution for A at equilibrium (At =0) is: A2 = (µ + φx )/δ (8) Thus, the quantity to be measured is (Vd k1 )2 , that is predicted to vary linearly with the quantity (µ + φx ). In the printer’s instability experiments, the drift velocity was found to vary as the square-root of the speed of the internal cylinder vi [8,10,11] and then the identification of µ with vi was obvious. It was even possible to directly measure the amplitude of the antisymmetry from the shape of the interface [11]. In directional solidification µ was identified with the combination of the wavelength and the 4 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns pulling velocity: λV 2 . In the pattern of columns, the identification of µ to the flow-rate Γ was also straightforward from measurements. From (6) and (8), one obtains: (Vd k1 )2 = ζ2 (Γ + (k1 − k0 )) δ (9) The possible control of the wavelength in the pattern of columns, enables the determination of some coefficients in the model by Goldstein et al. as it has been achieved in a previous work [24]. For example, the coefficient  can be determined by measuring the variations of the flow-rate threshold with the wavelength. Also, the variation of the slope in (9) can provide expressions for ζ and δ, and it turns out that some higher-order terms have to be added in (4) and (5). Another unexpected feature is that this model is still valid far from threshold of the secondary bifurcation [24], even though it was built under the assumptions of close-to-threshold conditions. The comparison between the model of eqs.(4) and (5) and measurements on drifting states, is presented in more details in section V. Let us mention another model, based on nonlinear interactions between the basic mode k0 and its first spatial harmonic 2k0 (often called ’k − 2k model’) described for instance in [36]. This model predicts a broken parity of the cells, leading to their drift, if there is a phase mismatch between the basic mode and its first harmonic. This model is particularly attractive for interfacial fronts, where the birth and the frustrated growth of cells between the primary cells of the pattern, can be observed, resulting indeed from a insufficiently damped harmonic. Its relevance has also been emphasized in the KSE equation [21]. 2.2 Possible extension of usual models far from threshold One of the remaining issues in such systems is how far the different patterns can be compared to each others, when one examines their specific behaviors far from threshold. The model by Coullet and Iooss [34] offers a framework for secondary instabilities close to threshold. However, this model is not adapted to predict certain far-from-threshold behaviors. An extension of this model has recently been built by Gil [39]. It introduces a possible phase mismatch between the primary and the secondary mode as a new variable, and allows discontinuities in the amplitude and phase of the modes. This model has provided promising results, typically obtained in numerical and experimental systems that are driven far from secondary thresholds, such as (A) oscillating wakes at the trailing edge of localized propagative domains, (B) phase mismatch and amplitude holes in extended oscillating regimes and (C) turbulent oscillating patches seemingly associated to STI. Situation (A) is observed in the printer’s instability [7], in directional solidification [14] and the pattern of columns [25], although with specific relationships between the propagation speed and the pulsation of oscillations that does not seem to be reproduced by the model. Situation (B) has been observed in directional solidification [15] as well as in the array of columns [47] (see also later in this paper). Situation (C) is more likely to be related to STI in the Rayleigh-B´enard convection [41], where turbulent patches are characterized by existence of cells that become smoother and loose their initial shape. Spatiotemporal complexity appears differently in our system, as columns still keep the same shape they have in laminar states. Instead, the complexity lies only in the motions of the cells. The complex motions are sustained by multiple interactions between propagative and oscillating inclusions that give rise to defects, i.e. singularities in the space phase variable. To our knowledge, such a behavior could not be reproduced in the model by Gil, but rather in the CGLE [46] and KSE equations [21]. All these models, dedicated to pattern-forming instabilities [21,46,48–50], include a large number of tunable parameters, which are not directly identifiable to physical quantities. In order to capture the minimal set of required parameters, they need some inputs from experimental benchmarks such as the system presented here. In the following, we will relate the specific behaviors of the pattern of columns, trying systematically to compare to other experiments and numerical models when possible. 3 Experimental setup Silicon oil of viscosity η, surface tension σ and density ρ at 20o C is injected at the dish center through a hollow vertical tube. The different properties of the liquid are listed on table 1. Except for the flax oil, the only quantity that varies significantly is η The flow is measured by a float flow-meter (Brooks Full View GT 1024) and is kept constant by a gear pump (Ismatec BVP Z) followed by a cylindrical half-filled chamber, that damps the remaining pulsations (radius=20 cm, height=15 cm). The imposed flow-rate Q ranges from 2 to 30 cm3 /s. The oil temperature is regulated with a thermal bath at 20oC with a few percent accuracy. Plexiglas circular dishes with different external diameter (d) have been used. The flow-rate per unit length Γ is determined with an accuracy of ± 0.005 cm2 /s. Reported data are obtained with two dishes of diameter d=10 cm and 16.7 cm. The accuracy of the dish horizontality is crucial to obtain reproducible measurements. It is tuned with a three-feet table supporting the setup, by simply checking the uniformity of amplitudes of oscillating columns when the system undergoes a transition to an oscillatory state, as described in more details elsewhere [24]. The pattern of columns is observed from above by a CCD video camera, and lighten by a circular neon tube put in the periphery of the dish and slightly below it. Columns appear as series of U-spots (see fig. 1-b). Spatiotemporal diagrams are built by recording grey levels along the circle which intercepts the columns’ center. Experimentally, the diameter of this circle was found to be independent on the flow-rate [23,24] and respectively equal to D=9.54 cm and D=16.20 cm for the dishes of diameter d=10.00 cm and d=16.70 cm. Images were digitized using P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns NIH Image 1.62 on a Macintosh computer. To achieve reliable image processing, it is of first importance that the background color of pictures acquired from above be as homogeneous as possible and that the edge of columns be visibly sharp, in order to have well-controlled diagrams. Several pieces of black papers cover the surround between the dish and the neon tube to prevent unwanted light reflections. Special care is also devoted to protect the system from any sources of perturbations. The dish is surrounded by a transparent plexiglas cylinder of internal radius 9 cm, to protect the system against any air motions around the experiment. Moreover, in order to isolate the dish from vibrations induced by the thermostatic bath or the gearing pump, these devices are put on foam beds of few centimeters height each. Oil η ρ γ 3 (cP) (g/cm ) (dyn/cm) PDMS 47V20 20 0.95 20.6 PDMS 47V50 50 0.96 20.7 PDMS 47V100 100 0.97 20.7 PDMS 47V200 200 0.97 20.7 Flax Oil 50 0.92 26.0 Table 1. Physical properties of the liquids. 4 Dynamical regimes: a qualitative overview 4.1 Different flow structures in the circular fountain The pattern of columns appears in a range of flow-rate Γ between 0.05 cm2 /s and 0.7 cm2 /s. At smaller flowrate, an array of static dripping sites is observed, whereas at higher flow-rate an annular liquid sheet replaces the pattern of columns. The study of such an object has been published elsewhere [51]. The transitions between these different flow regimes show hysteresis: the flow-rate above which dripping sites turn into columns is slightly larger than the flow-rate below which columns break into drops at their top. Similarly, the annular sheet replaces the columns above a Γ between 0.6 cm2 /s and 0.8 cm2 /s (the value depends on viscosity), whereas such a liquid bell can be maintained at very low flow-rate once formed: if the flow-rate is carefully decreased and if the sheet is protected from surrounding perturbations, one can maintain the bell at Γ of order 0.01 cm2 /s. Thus, rainbow patterns appear on the surface, witnessing that the liquid thickness is locally equal to a few light wavelengths. As a consequence of these hysteresis, mixed states are observed, including both dripping sites and columns or columns and a local triangular-shaped sheet, in the same 5 Fig. 3. Transition from dripping sites to columns regime, by increasing the flow-rate (η=100 cP). In the beginning, one notices that dripping sites appear as dashed tracks. Then, columns appear as continuous tracks. way as it was already observed in an overflowing cylinder experiment by Pritchard [30] and Giorgiutti et al. [31, 55]. The network of dripping sites remains static and has a homogeneous wavelength, which has been measured as equal to the one of the Rayleigh-Taylor (RT) instability of a thin layer below an overhang [29]: λdrops = λRT √ rσ = 2π 2 ρg (10) With silicon oil, this wavelength is 1.30 cm, independently on the viscosity. Thus, the network of dripping sites is very similar to the case of a simple RT instability. The natural space between columns at rest, is slightly smaller than the RT wavelength. When the flow-rate is increased in order to progressively transit from dripping sites to columns, a transient state is noticed, during which columns simply come up at locations of the former dripping sites. However, the pattern of columns can not withstand a homogeneous, static state at this wavelength, and some columns are then created between others. This necessary adjustment causes a decrease of the mean wavelength λm . The spatiotemporal diagram of fig. 3 illustrates this behavior. The reached state includes three small propagating domains. One of the fundamental difference between dripping sites and columns holds in that the latter are coupled to each others. In other terms, any column of the pattern feels the motion of its nearest neighbors, contrary to dripping sites. To illustrate this assessment, let us relate the following: when one introduces a needle in a dripping site and tries to drag it by capillary effects, the resulting motion of the dripping site does not induce any motion of the neighboring sites, which remains static. This is to be related with that no dynamical regime involving dripping sites could be noticed. Whereas in the pattern of columns, any column motion is strongly related to its neighbors. This feature explains collective motions associated to dynamical regimes. Having situated the pattern of columns in regards to other flow regimes, we now focus to the pattern of columns, reminding some of its general features. In the following, the term ’regime’ will refer to dynamical states of the pat- 6 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns tern of columns, in opposition to the ’flow regimes’ previously used to qualify dripping sites, columns or sheets. 4.2 General description of the dynamics As already reported in previous articles [22,24–26], the pattern of columns exhibits different dynamical regimes, depending on both the flow-rate per unit length Γ , and the initial conditions. Indeed Γ does not constitute the only quantity that rules the state of the system. The asymptotically reached state is strongly influenced by the initial conditions, i.e. the number of columns and the position and velocity of each column. These conditions can somehow be controlled by the experimentalist using needles to force local motions and spacing, as explained in more details in previous papers [22–25]. Some of these quantities are dependent on each others: for instance, the velocity of a column is strongly related to the space between it and its neighbors. Practically, with some experience and a little accuracy, it is possible to add or suppress columns at will, at specific locations on the pattern. The detailed phenomenological description of this manipulation is related elsewhere [22, 23], but can be summarized here in two main cases: - one can approach two neighbor columns to each other, in a quasi-static way. In that case, a column can be locally suppressed without any initial speed. This technique is used to change the number of columns and keep the pattern static, or to turn a static state to an oscillating one. As the same way, one can slowly move away two columns from each other, and provoke the birth of a new one. - one uses a needle to drag a column at a certain speed, towards one or several neighboring columns. Then, columns can be suppressed and several others have an initial speed that equals the speed of the needle. This technique is rather used to create propagative domains. Providing that the drag speed is included in a certain range, the drag speed and the drift velocity selected asymptotically by the system are the same. This technique enables a selection of the size of the domain and of the drift velocity, independently from each other. For example, it is possible to create a domain of drifting cells that extends to the whole dish [22,24], if a large number of columns are suppressed in that way. Depending on the speed one uses to drag columns with the needle, a different wavelength is obtained inside the final domain: the larger the speed is, the larger the obtained wavelength. Then, this enables to control the wavelength in a range allowed by the system, see also in the next paragraph. Examples of obtained states are shown on fig. 4 as spatiotemporal diagrams. (a), (b) and (c) represent successively a localized domain (LD) propagating endlessly along the dish, a globally extended drifting state (GD) and an extended oscillating state (OSC). These two regimes are ubiquitous in similar systems, like the printer’s instability [8,10] or directional solidification [12–15]. In the case of a propagative domain, the wavelength outside the domain is selected to a certain value λ0 , somehow by a robust process: indeed λ0 is independent on the flow-rate (a) (b) (c) Fig. 4. Overview of the main dynamical regimes. (a) Localized propagative domain (LD) followed by transient oscillations. One defines the quantities Vd (drift velocity), Vg (wall ’group’ velocity) and the wavelengths inside and outside the domain λ1 and λ0 . (b) Global drifting state (GD). (c) Extended oscillating regime (OSC). Γ , the size of the domain, the velocities Vg and Vg and the wavelength inside the domain λ1 . Only fluid parameters like the viscosity or the surface tension seem to influence it. Such a wavelength selection has also been noticed in other systems, like the printer’s instability or directional solidification [14,8]. Besides these now classical secondary instabilities, let us mention occurrences of an up to now unreported regime, namely the oscillating-drifting state (OSD). It was just briefly presented in a previous paper [26]. This state can not be obtained by the will of the experimentalist, but rather appears spontaneously after long chaotic transients. It is formed by successive small propagative domains of equal size, and it is then closely related to local drift. Furthermore, this state has the remarkable property to lead to a tripling of the spatial wavelength. In other terms, the motion of a given column is the same each third column as suggested on fig. 5. This state has similarities with some observations reported in directional solidification of small layers of eutectics [15], namely ’T - xλO’ states as mentioned in that paper: T stood for ’Tilt’, O stood for ’Oscillations’ and xλ suggested that this state induces a change in the spatial period, in the same way as the extended oscillatory regime was leading to a doubling of the spatial period. In our case, x=3. Besides these latter regimes that should be qualified as ’ordered’, ’predictable’ or ’laminar’, in the sense that the dynamics of columns show predictable motion, the P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 7 Fig. 5. Spatial tri-periodicity in the oscillating-drifting state (OSD) (d=10 cm, N =24). system exhibits chaotic regimes in a certain domain of parameters. In these regimes, chaos has both a temporal and spatial significance (see fig. 6), thus it is usually qualified as spatiotemporal chaos (STC). The columns have chaotic motion and the number of columns N does not keep constant. It means on the one hand that the trajectories of neighboring columns erratically meet, so that two columns can merge spontaneously, and on the other hand that a column can spontaneously split into two others. Then, by analogy of streamlines in usual channel flows, this type of regime is often qualified as ’non-laminar’ or ’turbulent’, in the sense that trajectories can cross. Of course, the word ’turbulent’ does not refer to the motion of elementary fluid particles which stay laminar, but rather to the space phase variable φ describing the positions of columns. The term ’phase turbulence’ is then commonly used. Each event such as the merging of two columns, or the splitting of a column, represents a singularity in the phase space. These events are then also called ’defects’. A recent study has shown that they contribute to sustain chaotic motion of columns, and that their space-time density constitutes a good measurement of disorder [26]. In the pattern of liquid columns, these defects are produced by interactions between dynamical regimes, for example oscillations and domains of drifting columns. No chaotic motion can be observed if there is no such defects in the structure. In other terms, the number of columns is necessarily a fluctuating quantity in a chaotic state. The conditions for obtaining the different regimes are detailed in the next paragraph, in the form of stability diagrams. 4.3 Stability diagrams Here we show the regimes obtained when one covers the parameters space through three directions: the flow-rate Γ , the mean wavelength λm and the viscosity η. Γ is controllable from 0.005 cm2 /s, up to 1 cm2 /s. The mean wavelength λm is controlled through the number of columns N , as λm = πD/N , and thus it takes discrete values. Its usual range of value is between 0.95 cm and 2.5 cm. Different viscosities have been used, see table 1. The resulting diagrams are depicted on figs. 7, 8 and 9 respectively for the 20 cP, 100 cP and 200 cP silicon oil. From these diagrams, several comments can be drawn up: Fig. 6. Regime of spatiotemporal chaos (STC). - At 20 cP, the diagram is quite simple: a large domain of ST, and narrow domains for LD and GD. At 50 cP, the diagram is qualitatively the same, with larger domains for LD and GD and a smaller domain for ST. Let us mention that for even lower viscosities, the system lies mainly in a static state. Some preliminary experiments conducted at 10 cP showed a very narrow range for dynamical regimes, and practically it was very difficult to keep stable those latter for reliable measurements. - The most complex dynamics are obtained for larger viscosities: at 100 and 200 cP, STC regimes are observed. Furthermore at these viscosities, the range of ST states is narrow, whereas those of LD and GD are much extended. At 200 cP, the new OSD state is observed. - The STC regime appears only at higher viscosities: a minimal viscosity of η around 100 cP seems to be necessary, as no permanent chaotic states could be obtained at a viscosity of 90 cP. Under those conditions, it turns out that there exists a domain of parameters (Γ ,λm ) where the pattern can not find an asymptotically stable, predictable state. Thus in this domain, the pattern of columns behaves chaotically, as shown in fig. 6. Crossing the boundaries of this domain from an initially predictable state, one observes various break-up scenarios, that will be shown in the last section of the paper. The main results obtained in the STC regime have already been reported in a previous article [26]. - Arrowed lines represent the transition from a state to another one. When the arrow is at both ends, it means that the transition is reversible, this latter case always concerns transitions at constant λm . For example, the arrow starts at the boundary of a GD state, and ends in LD or STC domains. It means that the latter GD state has been broken and has turned to another state, generally with a weaker λm . - The OSC domains have different shapes depending on the viscosity. At 20 cP, it is quite large and localized at low flow-rates, thus the system stabilizes when increasing Γ . At 200 cP, extended oscillations appear in a very narrow P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns range of flow-rate although they appear for a larger range of λm . At 100 cP, it is split in two small domains, lying at higher flow-rates, as a first step to STC. The upper domain can be reached by starting at low Γ with the suitable wavelength, and by increasing very suddenly Γ in order to skip the unstable area in between. - Some domains of parameters allow the existence of several states, for instance LD+GD, LD+OSC, LD+ST, ... It does not mean that two different states coexist at the same time, but rather that two different states are possible in that range of parameters, the choice for one or another depending on initial conditions, i.e. position and speed of columns. - At 100 and 200 cP, the threshold for transition to STC is almost the same for both viscosities: Γc = 0.34 cm2 /s. It is worth giving a significance for Γc , as one can clearly observe that several ordered states still exist at much higher flow-rates, for instance GD at large λm or ST at small λm . These states correspond to specific situations that are never spontaneously obtained, but are rather carefully ’prepared’ by the experimentalist. The threshold Γc is meaningful for the other usual conditions, in fact most of the conditions. Thus, when one starts from an initially chaotic state, the latter will turn laminar after a finite time, if the flow-rate is decreased below Γc . Also, from random initial conditions (practically when the dish starts to overflow), the system converges towards an ordered state if Γ < Γc (after a more or less short transient), whereas it stays endlessly chaotic Γ > Γc [26]. The presence of domains of coexistence in the diagrams questions the choice of λm for one of the axis. In fact, the case of domains of drifting cells can be approached through another point of view, in order to emphasize the wavelength selection process inside and outside a domain. The stability domain of drifting states deserves a closeup, replacing λm by the phase gradient φx defined in eq. (6). Figure 10 represents the phase gradient for various domains, versus flow-rate. The dotted line represents the stability limit of GD states. Except at very high flow-rate, the wavelength of standing columns outside domains is equal to the reference one: λ0 = 1.08 cm with the silicon oil of η= 100 cP. For most of the LD states, the phase gradient is included in the boundaries of GD states. There exists anyway a few measurements at small λ1 which show abnormally small phase gradients and lie out of the GD domain. Otherwise, the range of existence in (φx ,Γ ) is larger for GD than for LD. Global states are constrained to stay at the same wavelength, whereas LD states can adjust their wavelength dynamically. This adjustment apparently occurs before the limit values for GD are reached as shown in fig. 10. This suggests that the wavelength selection inside a domain is somehow more subtle than that was reported in previous studies [23]. The wavelength λ1 is primarily chosen in a large range of values, depending on how the domain is generated. Then, the chosen value will remain until a significant change is applied, so that the system can not hold the initial λ1 anymore and adjusts itself to 0.6 0.5 Γ (cm2/s) 8 Limit of GD states 0.4 0.3 Large λ1 0.2 Small λ 1 0.1 0 -3.2 -2.8 DROPS -2.4 k1-k0 (cm- 1) -2 -1.6 Fig. 10. Range of existence of domains of drifting cells, in gradient of the space phase φx versus flow-rate (η = 100 cP). Open symbols stand for LD states, filled symbols and the dotted line bound the stability domain of GD states. another value. The change can be an increase of Γ or the creation of another domain. 5 Measurements 5.1 Static states As stated in the previous section, states of static columns (ST) are obtained at large number of columns N , small λm . As shown in stability diagrams, the range of λm they appear in, is almost independent on Γ providing that the flow-rate keeps moderate. Then, the existence domain becomes narrower at higher flow-rates. This is apparently the simplest state as it follows from the primary instability, before eventually secondary bifurcations should occur. At the very first stages of its appearance, slow motions have been transiently noticed. These motions, which correspond to corrections toward homogeneity in the space phase, do not last more than one minute. However, they seem to play a role in the further appearance of a global slow drift of the pattern. Indeed, it has been noticed that a state of slow drift could appear from a static state, if Γ overcomes a certain threshold. This slow drift can be considered as a the outward sign of a Goldstone mode, i.e. induced by an invariance of the properties of the system in regards to a translation of the space phase. A possible way to kill occurrences of the slow drift, is to let the pattern turn perfectly spatially homogeneous, by waiting for the phase diffusion to be achieved. To proceed so, the flow-rate has to be kept below the threshold value during a while: typically one minute after the primary stage of overflowing. Under this condition, the pattern will stay frozen even at Γ higher than the usual threshold. However in this situation, a slow drift can be turned on anyway, providing that a suitable perturbation is brought to the pattern: for instance, by dragging columns very slowly with a needle, and thus by initiating the expected motion. In that sense, the bifurcation towards slow drift can somehow be considered as sub- P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 9 1 ANNULAR SHEETS Γ(cm2/s) 0,8 ST 0,6 OSC + ST 0,4 ST ST + LD 0,2 LD GD DROPS 0 0,8 λ0 1 1,4 1,6 1,8 λm(cm) 2 2,2 2.2 2.4 Fig. 7. Stability diagram at η= 20 cP. 0.6 Annular sheets 0.5 ST STC GD 0.3 OSC LD 0.1 0 0.8 LD + GD (+ LD) 0.2 ST Γ (cm2/s) 0.4 GD LD + ST DROPS 1 λ0 1.2 1.4 1.6 λm(cm) 1.8 Fig. 8. Stability diagram at η= 100 cP. 2 10 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 0,6 Annular sheets Γ (cm2/s) 0,5 0,4 STC OSC 0,3 GD OSD +STC STC 0,2 ST LD + OSD 0,1 LD 0 0,8 λ0 GD LD + GD DROPS 1,2 1,6 2 λm(cm) 2,4 Fig. 9. Stability diagram at η= 200 cP. Γ 0.6 ST Γ (cm2/ s ) POSSIBLE SLOW DRIFT STC GD To dynamic states 0.3 OSC ST 0.1 0 0.8 LD + GD (+ LD) 0.2 LD Threshold for the slow drift GD LD + ST DROPS 1 λ0 1.2 1.4 1.6 λm( c m ) 1.8 2 2.2 2.4 λmin 1,2 λ (cm) 0.5 0.4 λRT 1,3 Annular sheets λmoy λ0 λ max Fig. 11. Schematic close-up around the range of existence of static states (100 cP). critical. Measurements of this mode are presented later in the section. A close-up to static and quasi-static (with slow drift) states is presented on fig. 11. The reference wavelength λ0 , selected outside a local propagative domain (fig. 4-a), is also almost independent on Γ . However both the range of ST and λ0 depend on viscosity η. This dependency is reproduced on fig. 12. Some comments can be extracted from fig. 12: - λ0 is always lower than the theoretical RT wavelength λRT , and decreases as η is increased. At small viscosity, λ0 approaches λRT . - The range of stability in λm of the ST states becomes narrower at large viscosity, as those states become less and less prevalent. However, the minimal wavelength λmin does not vary, and nearly equals the minimal wavelength at which there is a positive growth-rate for the RT q σ = √12 λRT [29]. Thus any instability, i.e. λmin = 2π ρg 1,1 1 λinf 0,9 0 50 100 150 η (cP) 200 250 Fig. 12. Range of wavelength where static states are stable, versus viscosity. Black circles represent the reference wavelength λ0 . attempt to add a column when the system lies at λmin provokes the merging of two columns somewhere else in the pattern: so the system keeps itself above λmin . At the other end of the stability domain (fig. 11), the ST states bifurcate towards dynamical ones (OSC or LD) when it crosses the stability limit. The maximal wavelength is generally smaller than λRT , except at lower viscosities when it becomes equal to this limit value, see fig. 12. Although they are apparently simple, the study of ST states have revealed meaningful information about the further study of dynamical states. For example in the following, measurements of the thickness of liquid between static columns are presented. The dependancy of this thickness P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 0.2 h (cm) on viscosity, flow-rate and local wavelength is barely accessible from hydrodynamic arguments, although this dependancy seem crucial to explain some of the behaviors in dynamical regimes. Liquid thickness between two static columns h ∼ λ2 Γ 1/2 λ =1.15 λ =1.11 λ =1.07 λ =1.03 λ =1.00 λ =0.97 cm cm cm cm cm cm 0.1 0.05 0 0.1 0.2 (a) 0.2 (11) The influence of viscosity has also been explored: given Γ and λ, the larger the viscosity, the thicker the arch. For example, with a viscosity twice larger (η=200 cP) the thickness is measured 1.2 to 1.5 times larger. Otherwise, the thickness h follows a different behavior from what is predicted by the lubrication theory, which would rather lead to a law such as: h ∼ Γ 1/3 . Whereas the liquid thickness at the edge of the dish has been measured to behave as: e ∼ Γ 1/3 η 1/3 , then in agreement with the lubrication theory. Thus the thickness h increases with the square-root of flow-rate, and becomes larger for larger λ and η. These measurements are particularly useful to access some scaling laws of dynamical regimes, like the semi-empirical determination of the pulsation of oscillations and the drift velocity. Indeed, it would be more puzzling to extract a reliable hydrodynamic quantity in the two latter states, although some qualitative attempts were also conducted likewise [47]. 0.15 h / λ2 (cm- 1) We report measurements of the film thickness between two columns in a static state, at the saddle point of the arch. Thus, this corresponds to a local minimal thickness of liquid that is let by the flow under the overhang. Figure 13-a illustrates this thickness h for η=100 cP versus Γ , for various wavelengths λ. Similar results have been found using different viscosities. It is shown that the plots are correctly fitted by power-laws, the coefficients of which vary between 0.4 and 0.5. Figure 13-b proposes a rescalling by dividing by λ2 . The following empirical relation provides a good approximation of the thickness: 11 0.15 0.4 0.5 0.6 cm cm cm cm cm cm 0.1 0.05 0 (b) λ =1.15 λ =1.11 λ =1.07 λ =1.03 λ =1.00 λ =0.97 0.3 Γ (cm2/s) 0.1 0.2 0.3 Γ (cm2/s) 0.4 0.5 Fig. 13. (a) Liquid thickness between two columns, at the saddle point (η=100 cP). Dashed lines stand for power law fits. (b) Plot of the quantity h/λ2 collapses points on a single curve. The empirical relationship ω ∼ Γ 1/2 can be obtained from a coarse argument, based on the seek for characteristic length and time. This is illustrated on figure 14-c, and supported by direct observations of a transiently growing drop between two oscillating columns at their maximal 5.2 Measurements of oscillatory states spacing. Trying to roughly evaluates the volume of such a transient drop, one obtains V = λh2πlc , multiplying Out-of-phase oscillations appear when the local wavelength the three characteristic lengths along the azimuthal (λ), is slightly larger than λ0 . This situation can occur (a) the vertical (h) and the radial direction (2πl ), recalling c p globally on a homogeneous state with a suitable number of columns (fig. 4-c), or (b) locally in the trailing edge that lc is the capillary length equal to γ/ρg. Between of a propagating domain of drifting columns (fig. 4-a). As two columns, the injected quantity of liquid is Γ λ. The suggested in stability diagrams (figs. 7, 8 and 9), situation characteristic time for the transient drop to be filled is : (a) occurs in a narrow range of parameters. Figure 14-a 2πlc λh τR ' shows measurements of angular frequency ω versus Γ , for Γλ various viscosities from 10 cP to 200 cP, obtained for both which evaluates the angular frequency to: situations (a) and (b). No difference in ω can be noticed between measurements obtained in both situations, even if Γ in situation (a) the global wavelength is generally slightly ω' l ch larger than λ0 . Considering values of h that are presented in section At any given viscosity, data fit well with a square-root law: ω = CΓ 1/2 . Figure 14-b shows the evolution of C 4, the order of magnitude for ω is found around 10 rad/s, corresponding to the experimentally found values. with the viscosity. 12 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 14 2,5 ω (rad/s) 10 8 6 2 ν = 10 mm / s ν = 20 mm2 / s 4 2 ν = 50 mm / s 2 ν = 100 mm2 / s 2 1,5 12 10 ω (rad/s) ρg)1/2 (cm3/s) ω λ2 (γγ /ρ 12 Silicon oil V50 Flax oil 1 0,5 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 2 Γ (cm /s) Γλ h ω (rad/s) (a) ν = 200 mm / s 0 C 4 2 0 0 0,1 0,2 0,1 0,3 2 0,2 0,3 0,4 Γ (cm2/s) Γ (cm /s) 0,4 0,5 0,5 Fig. 15. Dependency of ω on surface tension. The quantity ωλ2 lc , suggested in the text, provides a good rescalling. Insert : Raw data. 25 20 15 (b) 6 0 0 2 0 8 0 50 100 150 ν (mm2 / s ) 200 250 (c)) not with a power-third law. Thus, even if (12) gives a rough relationship between ω and the physical properties of the system, some more subtle dependency remains to be found. 2 π lc Fig. 14. (a) Angular frequency of oscillations, versus flowrate for different viscosities. Dashed curves stand for fits by a square-root function of flow-rate : ω = CΓ 1/2 . (b) Value of C versus viscosity. (c) Sketch of the dimensional argument, leading to (12). In section 4, it was reported that h varied with the viscosity, the flow-rate and √ the wavelength, as the following empirical law : h ∼ Γ λ2 . The dependency in viscosity was not systematically explored, but the tendency for a slight decrease with viscosity suggests that h varies seemingly like ∼ η 1/3 , like the thickness of a liquid layer flowing of a solid surface at low Reynolds number. A simplified law for h can then be the following : √ h ∼ Γ η 1/3 λ2 One obtains for ω : ω∼ √ Γ η 1/3 λ2 l (12) c However, the relationship between ω and the viscosity is not as simple as this. The values for C obtained from the fit of ω, and plotted in fig. 14-a, suggest indeed that a higher viscosity coarsely leads to a decrease of ω, but Checking the dependency on surface tension γ: others measurements carried out with flax oil, the characteristics is which are given on table 1, have been compared to the previous ones, particularly with the V50 silicon oil, of comparable viscosity. Figure 15-a shows two series of measurements for both oils. Since raw values of ω are quite different for the two liquids (see insert), to plot the quantity ωλ2 lc , which has the dimension of a flow-rate, suggested by the previous analysis, makes data collapsing. This quantity should only depend on viscosity, which is coherent with an angular frequency varying as (γ)−1/2 , according to eq. (12). To summarize, the oscillating regime shows non-obvious dependancy on the different parameters and physical quantities. As a consequence of dynamics ruled by the complex free surface below the overhang, simple hydrodynamical arguments show their limitation for a complete description of phenomena. Otherwise, a more precise description of these regimes can benefit from more phenomenological approaches [32,25]. These approaches were inspired from that oscillations and drifts seemingly originate from a common mechanism. The occurrence of an oscillating wake left behind a propagative domain constitutes an illustration of this statement, as presented later in a specific paragraph at the very end of section 5. P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 13 (a) (b) Fig. 16. Phase and amplitude inhomogeneities in an extended oscillating regime with a odd-number of columns (η=100 cP, Γ =0.26 cm2 /s, d=10 cm, 27 cols.). (a) Progressive shift of phase along the pattern. (b) Sharp jump of phase coinciding with an amplitude hole. Fig. 18. Drifting amplitude hole in an oscillating state. 26 col., Γ =0.139 cm2 /s, η=100 cP. Duration 64 s. The close-up shows variations of the amplitude of oscillations. Nevertheless, an amplitude hole can exist with an even number of columns. In this case, its occurrence can be coupled to a global slow drift, as illustrated in 18, and the amplitude hole seemingly exists in order to correct the phase mismatch resulting from the slow drift. This object presents similarities with the one simulated by the model of Gil [39]. Fig. 17. Almost homogeneous amplitude in an extended oscillating regime, without any phase jump. 26 col. Γ =0.18 cm2 /s, η=100 cP. Duration 2 s. 5.3 Measurements of global drifting states We present measurements of phase velocities of GD states, at different viscosities (η=20, 50, 100 and 200 cP). As suggested in stability diagrams, such states can be obtained On phase inhomogeneities inside extended oscillating at several wavelengths. The wavelength being simply equal regimes to λ = 2πd n , it is possible to tune it to several fixed values in order to study separately the influence of flow-rate, By doing a close-up to extended oscillating regimes, one viscosity and wavelength on Vd . Measurements are shown observes that two consecutive columns never oscillate per- on fig. 19-a,b, where Vd is plotted versus Γ . They concern fectly out-of-phase. This fact comes from two main rea- both dishes of diameter d=10 and 16.7 cm. For sake of sons. Firstly, this regime has fair chances to appear for a clarity, a single graph only shows two values of viscosities. odd number of columns. In that situation, it is straightIt is first noticeable that, given the flow-rate and the forward that a homogeneous phase profile will not match viscosity values, the choice of the wavelength characterizes the periodic boundary conditions. For example, if d=10 completely the dynamics. The wavelength λ can vary from cm, η=100 cP and N =27. This is illustrated in figs. 16. 1.55 to 2.21 cm; when one fixes the number of columns, one The black line stands for an isophase in regards to the always obtains the same value for Vd . Measurements conperiod-doubling mode. Figures (a) and (b) show two dif- cern several values of λ, only some of them being reported ferent kind of phase mismatch in such a situation. Figure for sake of clarity: each value corresponds to a branch 16-b shows a sharp phase jump that coincides with an of plots. Larger velocities are obtained for larger λ. From amplitude hole. This behavior has also been observed in these measurements, it also appears that a larger viscosity experiments of directional solidification [15] and in simu- increases the velocity. For most wavelengths/viscosities, lations of coupled amplitude/phase model introducing a Vd shows sharp increase at low flow-rate, whereas it satuphase shift [39]. rates or under certain conditions even decreases at highest With the same experimental conditions (d=10 cm, η=100 flow-rates. cP), it is possible to obtain an extended oscillating regime Detailed measurements for a single viscosity η=100 cP for an even number of columns (N =26). In that case, the are shown on fig. 20-a,b. Figure 20-b represents the square isophase line behaves evenly, see fig. 17. Only smooth vari- of the quantity (Vd k), which is also the square of the time derivative of the spatial phase (φt ), see section 2. This ations along the isophase can be noticed. 14 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 3 2,4 2,2 λ 2,5 Vd (cm/s) Vd (cm/s) 2 1,8 1,4 η = 20 cP η = 50 cP 1,2 0 (a) 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 2 Γ (cm /s) 0 60 3 50 λ (Vd k)2 (s- 2) 2,5 Vd (cm/s) λ =2.21 λ =2.03 λ =1.89 λ =1.76 λ =1.64 2 1,5 1 η = 100 cP η = 200 cP 0,5 0 0,1 0,2 0,3 2 0,4 Γ (cm /s) 0,5 0,1 40 λ =2.21 λ =2.04 λ =1.89 λ =1.76 λ =1.64 0,15 0,2 Γ (cm2/s) cm cm cm cm cm 0,25 cm cm cm cm cm 0,3 0,35 λ 30 20 10 0 0,6 (b) Fig. 19. Phase velocity in a global drifting state, for various wavelengths λ. (a) η=20 and 50 cP. (b) η=100 and 200 cP. quantity shows a linear dependance with flow-rate, in a significant range of flow-rate above threshold. The threshold itself is defined as the value of flow-rate extrapolated at Vd =0. Such a dependance is predicted by the model of Goldstein et al [35] for a supercritical bifurcation to a parity-broken state. However our measurements show that the slope and threshold depend on λ, which was not included in the model. From this set of measurements, it is possible to determine coefficients of the model of eqs. (4) and (5). Defining a and Γc respectively as the slope in fig. 20-b and the flow-rate threshold, one can write an empirical law as follows: φ2t = (Vd k)2 = a(φx )(Γ − Γc (φx )) 0,05 (a) 3,5 (b) 1 0,5 1 0 2 1,5 1,6 0,8 λ (13) The quantity φx , defined in eq. (6), represents the spatial derivative of the phase. It is negative inside a domain of drifting, dilated cells. It contains implicitly the dependance on λ. It turns out that, for all utilized viscosities, a and Γc show linear variations with φx . Figure 21-a,b illustrates this fact for η=100 cP. The following laws come up: 0 0,05 0,1 Γ (cm2/s) 0,15 0,2 Fig. 20. Measurements for η=100 cP, by series of constant wavelength. (a) Phase speed Vd . (b) Square of the time derivative of the spatial phase φ2t = (Vd k)2 . Γc (φx ) = Γc0 − φx a(φx ) = αφx + β (14) (15) Measurements at four different viscosities allow to draw a tendency of the evolution of α and β with η, see fig. 22-a: β decreases significantly as the viscosity increases, whereas α just shows a slight increase around a value of -90. The coefficients  and Γc0 appearing on the threshold law are both decreasing with viscosity as power laws, seefig. 22-b. The detailed comparison of these measurements with the model of Goldstein et al. [35] is reported in a previous paper [24]. As briefly presented in section 2, this model can describe the dynamics of drifting cells, in both localized domains and global states, providing that higher order terms, introducing the wavelength dependence, be added. The conclusion of the study of [24] was that the equations (4) and (5) of the initial model have to be completed as follows: At = (µ + φx )A + γAx A + Axx − (β + αφx )A3 + ... (16) P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 160 380 a (cm2/s) β) α.φ x+β φx) = (α a(φ 120 360 340 15 80 320 40 300 280 0 260 -40 1.8 2 (a) 2.2 2.4 2.6 k0-k (cm- 1) 2.8 3 α β -80 -120 0.022 0 50 100 150 η (cP) (a) 200 250 Γc (cm2/s) 0.1 0.018 0.014 Slope=-0.41 0.01 0.01 1.8 (b) 2 2.2 2.4 2.6 k0-k (cm- 1) 2.8 Γc 0- ε.φx Γc =Γ 3 Γ0 c ε Fig. 21. Constitutive parameters of the predictive law for the phase dynamics, obtained from coupled amplitude/phase equations (η = 100 cP). (a) Coefficient a, versus k0 − k1 = −φx and (b) Threshold Γc . 0.001 10 Slope=-0.94 100 η (cP) (b) 1000 Fig. 22. Coefficients  and α versus viscosity. φt = (β + αφx )A + Dφ φxx + ... (17) These equations include higher order terms that couple the amplitude and the phase gradient. The coefficients ζ and δ of (9) are linear functions of φx , and this dependancy matches with the initial symmetries of the problem (see also section 2). 5.4 Measurements on localized propagative domains of drifting cells Figures 23-a,b,c present measurements of wavelengths λ0 and λ1 , group (Vg ) and phase (Vd ) velocities for various acquisitions series, on domains of various sizes, with a viscosity of η=100 cP. Other similar measurements have been performed as well for different viscosities, and have revealed similar tendencies. As in measurements on global drifting states, Vd increases with flow-rate. In LD states, so does the group velocity Vg . Measurements have been obtained from different localized domains: these domains have been built with different sizes and different internal wavelengths λ1 . As noticed in section 4, on fig. 10, λ1 can take various values for the same flow-rate. The same kind of dispersion is observed in Vd and Vg values. However, it is confirmed that λ0 is almost constant within the studied range of flow-rate, and thus constitutes a suitable reference wavelength. In order to draw deeper relationships between λ0 , λ1 , Vg and Vd , it is worth reminding the kinematic relationship of the forward or backward fronts of a localized domain [14]: Vd λ1 =1+ λ0 Vg (18) Or written differently: Vd k 1 = V g φx (19) This suggests that if one wants to unify all the measurements, one needs to check the dependency between Vd and λ1 . It is worth noticing that λ1 can not be controlled inside a localized domain, contrary to the case of global drifting states. However, it stays bounded in a flow-ratedependent range, as emphasized in section 4 and as shown in fig. 10: this range is surrounded by the stability limits of global drifting states. Considering eq. (18), it comes up 16 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 2.2 7 λ0 λ1 6 λ0, λ1 (cm) 2 φx) (Vd .k)2 / a(φ 1.8 1.6 1.4 1.2 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2 Γ (cm /s) (a) Vg (cm/s) 4 2 1 0.1 0.2 0.3 0.4 Γ (cm2/s) (b) 0.5 3 2.5 V d (cm/s) 2 1.5 1 0.5 (c) 0 3 2 GD states LD states 1 0 2 4 6 8 Γc 0 Γ-Γ Γc 0+ εφx)/Γ (Γ 10 12 Fig. 24. Collapse of all measurements of Vd on a single master curve. 3 0 4 0 5 0 0 5 0.05 0.1 0.15 0.2 Γ (cm2/s) 0.25 0.3 0.35 Fig. 23. Measurements on propagative domains (η=100 cP). (a) Wavelengths versus flow-rate. (b) Group velocity versus flow-rate. (c) Phase velocity versus flow-rate. that one can explain the dispersion for Vd and Vg by the fact that λ1 is adjusted in a certain range of values by the system itself (so behave k1 and φx as well). It turns out that, by using the same scaling-laws as for global domains, it is possible to make measurements of Vd and Vg collapsing on the same master curve. Figure 24 shows measurements of Vd for both local and global drifting states, corresponding to figs. 20 and 23-c, and rescaled with the quantities defined by eq. (14) and (15). The scaling law holds very well, even far from threshold where the dependance of (Vd k1 )2 on Γ is not linear any more. The same scaling is possible with Vg , using eq. (19): the collapse of Vg measurements on a single curve is obtained by plotting the quantity Vg φx (αφx + β)−1/2 versus Γ − Γc (φx ), see figures 25-a and b. From these scalings, it is possible to draw some short conclusions. Firstly, it appears that localized domains and global states have the same kinematic properties, the only difference being that the system chooses itself the wavelength inside a localized domain. Secondly, if the pattern includes several domains, they will necessary have the same λ1 and Vd , as they have to propagate with the same speed Vg , so that no domain can go ahead another one. Thus, if a domain is created in a pattern that already contains a domain, the new domain properties will be selected by the first one. 5.5 Measurements on oscillating-drifting state As briefly mentioned in section 4, the OSD state can be obtained for quite specific conditions. Firstly, a high viscosity is needed: it was mainly observed at η= 200 cP, see fig. 9. It could be created and kept stable for η=100 cP as well, although in a very narrow range of parameters: this range of appearance was too narrow and not reproducible enough to trace a specific domain for it anyway. It is called ’oscillating-drifting’, as it consists in successive small, equally sized domains which propagate at the same speed. As each small domain lets a short oscillatory wake at its trailing edge, the global motion of a column is a mix between oscillations and drifts. Such a state appears spontaneously after long chaotic transients: at η=200 cP, there exists a range of flow-rate P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 17 α φx + β)-1/2 Vg φx (α 0,6 0,5 0,4 0,3 0,2 0,1 0 0 (a) 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 Γ - Γc (φ φx) α φx + β)-1/2 Vg φx (α 1 0,1 0,01 (b) Slope : 0.5 Fig. 26. Convergence of an initially chaotic state towards a particular state mixing oscillations and drift. Γ =0.26 cm2 /s, η=200 cP. The total duration is around 70s. (a) 0,1 Γ - Γc (φ φx) 1 Fig. 25. Collapse of all measurements of Vg on a single master curve (a) Linear axes. (b) Logarithmic axes. at which the transiently chaotic system can not but reach the OSD state. Firstly one lets the system stand in a permanent STC regime, then one slightly decreases Γ until it reaches a slightly smaller value than the critical flow-rate Γc separating ordered and permanent disordered states (see stability diagram on fig. 9, and ref. [26]). Figure 26 illustrates how a transient chaotic state can suddenly turn into an oscillating-drifting state. Afterwards, it is possible to increase or decrease Γ in a certain range without breaking this state, and perform measurements of its properties. In that sense, the domain of existence is much less narrow than the domain of parameters for the state to be created. Figure 27-a includes definitions of three characteristic velocities: the group velocity Vg is the same as for a classical propagative domain; but one can define two drifting speed: the maximal drifting speed Vdmax , corresponding to the usual speed of a column inside a domain, and the mean (b) Fig. 27. (a) Oscillation-drifting state, extended on the whole pattern (η=200 cP, Γ =0.242 cm2 /s, d=10 cm, N =24, duration=10.5 s). (b) Co-existence with a local domain of drifting cells. η=200 cP, Γ =0.34 cm2 /s, d=10 cm, N = 21, duration=13 s. drifting speed Vdmean . The small domains that constitute this state, can co-exist with another domain of larger size (figure 27-b), but propagating at the same speed. As the same way as an oscillatory state could exhibit a quasi-homogeneous period-doubling only for an even number of columns, the oscillating-drifting state is homogeneous only is the number of columns is divisible by 3. With the dish of diameter d=10 cm for instance, it ap- 18 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 0,8 pears for N =23, 24 and 25; the state is homogeneous and shows perfect tri-periodicity only for N =24. Measurements for the three velocities defined on figure 27-a are plotted on figs. 28 -(a,b and c), for three values of mean wavelength. As expected, the velocities increase with flow-rate and with the mean wavelength. The group velocity is well fitted by a power-law of exponent one half. Figure 28-d is the ratio between Vdmax and Vdmean . This ratio is close to 3. λ m=1.30 cm λ m=1.25 cm λ m=1.20 cm V d_mean (cm/s) 0,7 0,6 0,5 0,4 5.6 Slow drift of quasi-static regimes 2 (Vd k) = α(Γ − Γc ) 0,3 0 0,05 0,1 (a) V d_max(cm/s) 2,5 0,15 0,2 0,25 0,3 0,35 0,2 0,25 0,3 0,35 Γ (cm2/s) λ m=1.30 cm λ m=1.25 cm λ m=1.20 cm 2 1,5 1 0 0,05 0,1 (b) 4 Vg (cm/s) 0,15 Γ (cm2/s) λ m=1.30 cm λ m=1.25 cm λ m=1.20 cm 3 2 1 0 0 0,05 (c) 0,1 0,15 0,2 0,25 Γ (cm2/s) 0,3 0,35 4 3,5 Vd_max / Vd_mean As previously stated, this regime consists in a slow drift of the whole pattern, that can be perceived during long observations. The typical order of magnitude for the drifting speed Vd is 0.01 cm/s. This regime generally appears when the flow-rate exceeds a certain threshold. The direction of the slow drift is arbitrarily chosen by the system: seemingly, this direction is fixed by imperfections in initial conditions, i.e. by initial tiny departures from spatial homogeneity. Thus, in most conditions, the system keeps its initial direction. An example is depicted on fig. 29-a. It is remarkable that slow undulations are superimposed to the drift. These undulations exist as well without drift, at lower flow-rates. The existence of these states is also bounded to a upper threshold in flow-rate. Above the threshold, they cease to exist and turn to other dynamical states, see stability diagrams figs. 7 to 9. Close to this limit, the drift can exhibit spontaneous changes of direction as shown on figure 29-b. These unstable phenomena occur in a narrow range of parameters, and constitute the first step to the destabilization towards OSC or STC regimes. The quantitative study of such a regime needs long acquisitions (up to 20 minutes), with a much longer time step between each line of spatiotemporal diagrams than for usual dynamical regimes. Measurements of the drift velocity Vdslow are reported on figs. 30-a et b for two viscosities 100 and 200 cP. The velocity is a growing function of λ, Γ and η, as does the drift velocity of LD and GD states. Otherwise, the threshold for appearance of slow-drift Γc is smaller at higher λm and higher η. If one analyses the data more carefully, it turns out that the velocity follows a square-root law with the flowrate, similarly as the LD and GD states. Figure 31-a plots the quantity (Vd k)2 versus Γ , for values close to threshold. These results suggest that the bifurcation to a slow drift is supercritical. The quantity Vd k is then compared to the measured pulsation of undulations ω. It turns out that the ratio Vωd k is close to one, with noticeable discrepancies at low flow-rate. This suggests that the undulations and the drift are related, in the same way as non-dispersive propagative waves. To summarize, the slow drift velocity obeys the simple empirical relationship: 3 2,5 2 1,5 1 0,5 (20) 0 0,05 (d) 0,1 0,15 0,2 0,25 Γ (cm2/s) 0,3 0,35 Fig. 28. Velocity measurements in an oscillating-drifting state, for different number of columns (η=200 cP, d=10 cm). (a) Mean phase speed. (b) Maximal phase speed. (c) Group ve. locity. (d) Ratio VVdmax dmoy P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 19 0.04 Vd slow (cm/s) λ =1.15 λ =1.11 λ =1.07 λ =1.03 λ =1.00 0.03 cm cm cm cm cm 0.02 0.01 0 0 (a) 0.1 (a) λ =1.07 λ =1.03 λ =1.00 λ =0.96 0.3 0.4 0.5 0.15 0.2 0.25 cm cm cm cm Vd slow (cm/s) 0.03 0.2 Γ (cm2/s) 0.02 0.01 (b) Fig. 29. Diagrams of slow-drift states (η=200 cP, d=10 cm). (a) Γ =0.089 cm2 /s, 28 cols. (λ=1.067 cm). Duration 640 s. (b) Changes of direction (Γ =0.266 cm2 /s, 32 cols. (λ=0.933 cm). Duration 1280 s.) 0 0 (b) 0.05 0.1 2 Γ (cm /s) Fig. 30. Measurements of slow-drift velocity versus flow-rate, for different λ (d=10 cm). (a) η=100 cP. (b) η=200 cP. with α and Γc which are functions of λ and η. It is then remarkable that the slow drift and the ’usual’ drift, due to parity-breaking of cells, show up striking similarities in the laws relating the drift velocity to the flowrate, see eqs. (13) and (20), although the velocities differ by two order of magnitude, and keeping in mind that these two states originate from distinct mechanisms. Consequently, a same set of equations as the one by Goldstein et al. [35] can be adapted and modified like eqs. (17) and (17). The order parameter would not be the asymmetry of the cells, but more simply the drift velocity Vdslow . the pattern of columns: they could here be recorded during more than one day, before the acquisition was stopped. There are still different points about this regime that remain to be clarified: for instance why flow-rate, wavelength and viscosity increase the speed ? Furthermore, why does this state appear above a certain threshold ? What determines this threshold value ? Why spontaneous changes of direction can be observed in a narrow range of parameters ? Such slow dynamical phenomena have also been observed in the printer’s instability [9]. As this experiment has rigid boundary conditions, one can not expect an endlessly drift of cells in one direction. Instead, slow in-phase undulations of cell were observed, rather like the example on fig. 29-b. Otherwise in the printer’s instability, slow motions stop after some hours [9], which was not observed on Oscillations and drift as two distinct outward signs of the same mechanism ? In this paragraph, we highlight that oscillations and drifts of columns have a common origin, as deduced from direct observations, and we discuss consequences of such observations. 20 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 0.05 λ λ λ λ λ λ λ λ λ 100 cP 0.04 (Vd k)2 (s- 2) 200 cP 0.03 distance of two consecutive columns that lie at their maximal distance. Inside propagative domains, a drop grows between columns, but it stays located closer to one of the columns: if the drop is close to the left column, the drift of columns is oriented to the right direction, as shown in fig. 32-b, and vice-versa. As already put into evidence in other experimental systems [8,10,11] and theoretically predicted [21,36,52–54], the resonant combination of the fundamental mode and its first harmonic leads to a parity-breaking bifurcation if the phase-mismatch between the two modes is non-zero. This approach has been denoted as k − 2k model. = 1.15 cm = 1.11 cm = 1.07 cm = 1.03 cm = 0.97 cm = 1.07 cm = 1.03 cm = 1.00 cm = 0.97 cm 0.02 0.01 0 0 0.05 (a) 0.1 0.15 Γ (cm2/s) 0.2 0.25 0.3 1.3 100 cP 200 cP 1.25 ω Vd .k/ω 1.2 1.15 1.1 1.05 1 0.95 0.9 0 (b) 0.1 0.2 0.3 Γ (cm2/s) 0.4 0.5 Fig. 31. (a) (Vd k)2 versus flow-rate, for two viscosities 100 and 200 cP. (b) The dimensionless quantity (Vd k)/ω versus flow-rate. Oscillations appear in coincidence with the cycle of growth and absorption of a transient drop on an arch between two consecutive columns. It occurs as soon as the local wavelength overcomes a certain threshold. This threshold is dependent on Γ , and it is slightly larger than λ0 . The growth of the transient drop can be related to the appearance and growth of a insufficiently damped second harmonic (a new column would like to develop between two neighbor ones), as already depicted in fig. 14, and this appears each second column. Then, arches between columns are alternatively thicker and thinner. The positions of those local thickening and thinning switch alternatively in time with the pulsation ω. Furthermore, by making a close-up on the shape of arches between two columns, we observe that a drop grows between arches in both drifting and oscillating states, see fig. 32. In oscillating regimes, drops tend to grow at equal The relationship between superimposition of k and 2k modes, and the deformation of arches in the pattern of columns is illustrated on fig. 32-c. The figure includes also the basic combination between the mode k and the mode 2k that lead to symmetry break-ups. To understand better the relationship between deformations of arches and motion of columns, one has to keep in mind that a drop growing on the arch will lead to an attraction force of the column towards the drop [55]. This force could be estimated at a qualitative level by an argument on surface energy. In case (1), the initially homogeneous pattern with a wavelength λ, is dilated to a larger wavelength λ1 when columns are suppressed by a needle. At the same time, the needle drags neighboring columns and initiates the drift. The dilation leads to the growth of the second harmonic, i.e. a drop grows between each column; but because of the drift motions, it does not grow in the middle of the columns. This leads to a parity-breaking of the arch, which in turn sustain the drift of columns (the drop attracts the column that lies closer). In case (2), the pattern is at a wavelength of λ0 or slightly larger, and the flow-rate is increased: the second harmonic starts to grow locally on the pattern. Because of remaining spatial inhomogeneities, a very first drop starts to grow where the local spacing is higher (the relevance of this scenario is supported by observations that extended oscillating states always start to appear locally, and finally get extended to the whole pattern). The drop takes the shape of an extra thickness on the arch. At a certain thickness, the columns get attracted by the drop, and move towards the center of the arch. As columns get closer to the growing drop, they absorb a part of its mass and the drop vanishes. In the meantime, it creates local dilations at neighbor arches, and drops start to grow there. Afterwards the motion of each column undergoes a cycle of oscillations, and each second arch shows an increase of thickness when columns around lie at their maximal distance: this leads to a period-doubling bifurcation. The resulting collective motions look like the motions of a spring-and-mass system. The seemingly common hydrodynamical origin for oscillations and drifting columns has important consequences, as it has recently been emphasized [7,25,32]. The particular case of transient oscillations following a propagative domain, reveals this common origin (see fig. 4-a and more visibly later in the paper on fig. 36-d). It is associated to elastic properties of the pattern, in the sense that propagative domains carry a certain amount of elastic energy P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 21 anism for the wavelength λ1 via the value of Vd would directly ensue from the condition (21), also taking into account eq. (19). Let us notice that at higher flow-rates, this selection is not any more supported by the system and a slight progressive departure from eq. (21) has been measured [25]: columns do not oscillate in perfect phaseopposition, and it breaks one of the constraints that leads to eq. (21). The question is now to check if usual models are able to reproduce such a behavior. An oscillating wake left behind propagative domains has been observed in numerical simulations by Gil [39,40], as well as in phonon-like models [7,32], although they could not reproduce the relationship (21). Indeed in [39,40], ω is an adjustable parameter selected at will, and not selected in the pattern of columns. In phonon-like models, the relationship (21) is quantitatively different, as the value of 1/π was to be replaced by one half: no period-doubling is imposed within oscillating patches. Otherwise, the equation KSE could reproduce out-of-phase oscillating patches in the wake of drifting cells [56], that could in turn amplify and cause the transition to spatiotemporal chaos. Thus, capturing or not the features presented above is an essential clue that the model is able to reproduce at least qualitatively contaminating processes involved in disordered states. (a) (b) 6 The breakup of dynamical regimes (c) Fig. 32. (a) Arches between oscillating columns. One remarks that the arch is more inflated where the local space is larger, witnessing a transient growth of a drop. (b) Asymmetric arch between two drifting columns. The arrow suggests the drifting direction. (c) Sketch of the argument that put in relation the growth of a second harmonic and the deformations of arches. due to the local dilation. At the trailing edge of domains, this energy is relaxed through oscillations which are more or less damped. If they amplify, they lead to the breakup of the pattern and transition toward disorder: then as stated previously in the paper, propagative domains and oscillations get involved in the contamination process of disorder. Let us now focus more precisely on the case of the oscillating wake: as oscillating columns are in phase opposition with their nearest neighbors, a kinematic construction leads to a relationship between the wall velocity of the domain Vg and the pulsation ω as: 1 Vg λ0 (21) π The pulsation ω of oscillations at the trailing edge of propagative domains, is the same as the one measured for the same flow-rate in extended oscillating regimes. Thus, this value is strongly selected by intrinsic properties of the system (whatever propagative domains appear or not), and the kinetic quantities of a propagative domain have to fit with the value of ω. For example, the selection mechω= In this section, we report a catalogue of the different breakup scenarios of ordered regimes. Such situations are generally provoked when the flow-rate is progressively increased or decreased, in order to cross the boundaries of stability domains described on diagrams figs. 7, 8 and 9. Starting from initial conditions at given flow-rate and number/positions of columns, one lets the system evolve and converge to an asymptotic state. In a second step, the flow-rate is modified. The break-up of the initial state is often associated to occurrences of one or several defects, i.e. changes in the number of columns. However, this is not always the case: a change of state can also be observed without any defect occurrences. The resulting final state can be either another laminar regime or a chaotic one. 6.1 Break-up of static and quasi-static regimes A static regime can break through various scenarios. For any initial mean spacing λm , a decrease of flow-rate leads to the dripping state. The dripping occurs at a constant flow-rate per column qc . At some values of λm , an increase of flow-rate can lead to a slow-drift superimposed to slow undulations, as shown in figs. 29. In most cases, the slowdrift is the first stage towards more dramatic transitions of the initial state: if the flow-rate is increased further, the pattern will show oscillations, propagative domains or chaos. However, it is difficult to figure out how the slow dynamics interact with other swift regimes and eventually influence the break-up scenarios. It acts as a slow phase 22 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns (a) (a) (b) Fig. 33. (a) Two coalescing columns after an increase of flowrate: η=100 cP, d=10 cm, Γ =0.45 cm2 /s, N = 32 and finally 31 columns. (b) Column nucleation after transient oscillations: η=100 cP, d=16.7 cm, Γ =0.16 cm2 /s, N = 45 and finally 46 columns. (b) diffusion, and as previously shown, it could be for instance related to a localization of oscillations (fig. 18). To summarize, by increasing flow-rate from an initial static or quasi-static state, the two following break-up scenarios are observed : - For λ ' λmin or slightly larger (shrunk structure), an increase of flow-rate can lead to the merging of two columns into a single one. This leads to an increase of λm , see figure 33-a and consequently to the increase of λmin with flow-rate, see also 11. This secondary instability is seemingly an Eckhaus instability, followed by an adjustment due to phase-diffusion. The Eckhaus instability itself is tuned by phase inhomogeneities. This instability is ubiquitous in other similar systems, like the printer’s instability [8], the directional solidification [13,15] and simulations of KSE [21]. - For λ > λ0 , a change in flow-rate (generally increasing) can lead to an oscillatory state. At a further stage, the system can generate a defect (birth of a column) and then it reaches a new stable state (figure 33-b). In some situations, it was even observed that the system could turn directly to LD or STC states, after a short oscillatory transient. 6.2 Break-up of drifting states Within the denomination ’drifting states’, are included both LD and GD. We firstly report break-up of GD states when crossing the lower limit of flow-rate. In that situation, two main scenarios are observed: - The break-up is caused by the rupture of one or several columns into dripping sites. Generally, the dripping site does not follow the drift motion of columns, which provokes a cascade of break-ups to consecutive sites (fig. Fig. 34. Break-up of drifting states, when decreasing flow-rate, η=100 cP. (a) Pinch-off of a column into drops, and cascade of generations of dripping sites: d=16.7 cm, N =29, Γ =0.042 cm2 /s. (b) Oscillatory instability as a first stage to break-up, with final dripping sites: d=10 cm, N =14, Γ =0.055 cm2 /s. 34-a). These originate from the hydrodynamical Rayleigh instability: indeed the break-up occurs for a fixed flow-rate per column qc , and it is found equal to the flow-rate per column below which static states turn locally into array of dripping sites. The value of qc is 0.07 cm3 /s for the 100 cP oil. - The break-up is caused by a phase-instability, witnessed by oscillations of the positions of columns (fig. 34b). This scenario arises when q is slightly higher than qc and concerns the most dilated GD states. At the ending stage, the Rayleigh instability also occurs. This phase instability shows that the parity-breaking mode involved in GD and LD is unstable near the threshold value of ν +φx , see eqs. (4) and (17), as it was predicted in some models [38,36]. Then we present the break-up scenarios when the upper flow-rate limit is reached: it is observed that the rupture of a drifting state can either result to another drifting state with a larger number of columns, either a static state, either a chaotic state. This last situation is only encountered at higher viscosities (100 or 200 cP). Figures 35-(a) to (c) give several examples of break-ups of global drift states, and figures 36-(a) to (c) show some examples of broken localized states. Amongst this host of break-up scenarios, some phenomena deserve further comments: P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 23 (a) (a) (b) (b) (c) Fig. 35. Examples of break-up of global drifting states at the upper flow-rate limit (η= 100 cP) (a) Final state: several domains following each others (d=16.7 cm, N = 25, Γ =0.28 cm2 /s). (b) Oscillations as first step of destabilization: d=10 cm, N = 17, Γ =0.36 cm2 /s. (c) Transition towards spatiotemporal chaos: d=10 cm, N =17, Γ =0.55 cm2 /s. - Oscillations of drifting columns, appearing at both lower (fig. 34-b) and upper (fig. 35-b) limits of existence in flow-rate, are of the same kind. Based on careful observations on the shape of the arch between two columns, it is noticeable that a transient drop develops between each pair of consecutive drifting columns. The mentioned growing drop is already observed in the classical non-oscillating drifting cells: it is the one that breaks the parity symmetry (c) (d) Fig. 36. Examples of break-up of propagative domains at the upper flow-rate limit, η = 100 cP except (c). (a) The final state consists in multiple domains: d=10 cm, N = 16, Γ =0.176 cm2 /s. (b) Transition towards spatiotemporal chaos: d=10 cm, N = 21, Γ =0.5 cm2 /s. (c) Transition to a static state: η= 20 cP, d=16.7 cm, N = 37, Γ =0.32 cm2 /s. (d) Break-up induced by amplification of oscillations following the trailing-edge of the domain: d=16.7 cm, N = 41, Γ =0.33 cm2 /s. 24 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns of the arch between two columns. During this oscillatory instability, the shape of this drop shows periodic cycles of growth and retraction. Then the drop is periodically absorbed by one of the columns, so that its growth is hindered like in the period-doubling oscillatory states. This cycle of growth and absorption has a well-defined characteristic time, of the order of one second. From a point of view of a phase instability, the growth of such transient drop may signify that, in this range of parameters, the system tries to catch a smaller wavelength than the initially imposed one. Finally, this oscillatory behavior, which is a first stage of rupture of global drifting states, has not to be confused with the oscillating-drifting state (OSD) shown in fig. 5, which rather consists in successive small propagative domains. - A global drifting state generally gets unstable beyond a critical flow-rate, higher than the upper limit for local drifting states, see also fig. 10. To explain this, one can argue that the break-up of local domains is generally initiated in the oscillating wake, the period-doubling oscillations being much less stable at high flow-rates. This is an important mechanism for creation of disorder [25, 26]. Then, in spite of a constrained wavelength for global states, these latter can be withstood further than their local domains. One noticeable exception is shown on fig. 36-b: the break-up seems to be initiated in the bulk of the domain such as the one on fig. 27-b. This is to be related to the fact that the oscillating wake has been turned into a domain of oscillating-drifting cells, which is much more stable. (a) (b) 6.3 Break-up of oscillatory states Extended oscillatory regimes, which do not include oscillations that follow the trailing edge of propagative domains, appear in a reduced range of parameters whatever the viscosity. Their stability is then particularly sensitive to any variation of flow-rate. If one decreases Γ from an initial oscillatory regime until the lower limit of existence, one observes that oscillations progressively fade out and that the pattern turns static. No change in N is noticed. This scenario is also observed when one increases Γ at low-viscosity (20 cP). However, at higher viscosities (50 to 200 cP), the increase of flow-rate leads to different behaviors: it generally breaks the state with generation of defects. The latter situation lead to a final state which is dependent on viscosity. At 50 cP, the final state is either S or LD; but at 100 or 200 cP, the final state is LD (fig. 37-a) or STC (fig. 37-b). Seldom, a LD is created without any defect (fig. 37-c). Thus, the three presented situations emphasize the non-trivial interactions between oscillations and propagative domains: it was previously shown in figs. 4-a and 36-d that out-of-phase oscillations follow propagative domains. Figures 37-(a) to (c) show in some sense the inverse process: oscillations can lead to small propagative domains when they get amplified. (c) Fig. 37. Break-up of an oscillatory state induced by an increase of flow-rate (η=100 cP). (a) A pair of domains propagating to opposite directions are generated by the break-up (d=16.7 cm, N = 46, Γ =0.27 cm2 /s). (b) Transition towards spatiotemporal chaos (d=10 cm, N = 27, Γ =0.29 cm2 /s). (c) A domain is created although no defect is shown (d=16.7 cm, N = 45, Γ =0.225 cm2 /s). 6.4 Collision between two propagative domains In this paragraph, we relate seemingly marginal experiments, whereas they are spectacular and tell about how propagative structures get involved in the production of disorder. It consists in making two propagative domains, which travel to opposite directions, collide each others. For these experiments, the largest dish (d=16.7 cm) has been used, in order to enable the development of two independent domains. To set up such a situation, one launches the two domains closely to each others, using the method P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns 25 of propagating domains are involved in the disorder creation process [26]. In that sense, propagative domains contribute indirectly to chaos, even though they are perfectly predictable. 7 Conclusion (a) (b) (c) Fig. 38. Collisions between two domains propagating to opposite directions (η=100 cP, Γ =0.276 cm2 /s). (a) Same size for the two domains: cancelation of drift and transient oscillations. (b) and (c) Different sizes: only the larger one survives, with a final smaller size. described in the experimental set-up, by moving a needle at constant speed on top of columns. Figures 38 show three examples of collisions: (a) two domains with same size cancel each others when they collide. If one of the domains is larger than the other one, it will continue its course, but its size is diminished by the size of the smaller one (b and c). It is worth noticing that such behavior has been observed in experiments of directional solidification [12] and in numerical simulations of generic models describing the parity-broken state in one-dimensional patterns [35]. Several features can be extracted from such observations, in order to get a better understanding of the nature of traveling domains. Firstly, although they are often referred in the literature as ’solitary waves’, they can not be qualified as solitons, the collisions of which are not destructive. Secondly, generated defects during collisions The study presented in this paper has focused to the ordered dynamics of the pattern of columns, through three main points. 1) A broad view of the different regimes has firstly been presented, with the overall pictures of their range of stability, at different viscosities. 2) Numerous measurements in a broad range of viscosity has shown general tendencies of oscillating and drifting states, varying η, Γ and λ in the maximal allowed ranges. The extended range of stability for dynamical regimes at high viscosities is a asset for such a quantitative study, and to our knowledge, our system is the only one that can enable it. 3) A host of break-up scenarios of various regimes has been shown, that present transitions from a well-defined laminar state to an another one, as well as different routes to spatio-temporal chaos. Most of these transitions are accompanied by defects. Amongst results reported in this study, some conclusive points of more general interest can be drawn. - The pattern of liquid columns exhibits a host of states. Particularly at higher viscosities, the richest dynamics is observed, with a large range of existence for the regime of spatio-temporal chaos. Also, a complex but predictable oscillating-drifting state, which possesses a striking property of spatial tri-periodicity, has been evidenced. Let us notice that to our knowledge, no available model or equation could reproduce such a state. - A generic set of equations has been found to describe regimes of parity-broken drifting cells. Especially, probing the dependency on λ has revealed relationships between the phase and group velocities, and the local phase gradients. This has enabled a more precise determination of coefficients and higher order terms of the initial model by Coullet and Iooss [34] based on broken symmetries. A rescaling based on this modified model has revealed that both localized and global states obey the same kinetics. - Although originating from distinct mechanisms, the regime of slow drift and the regime of parity-broken drifting cells have several features in common: the drifting speed of the cells increases with λ, Γ and η in both regimes. Also, their appearances need that Γ overcomes a certain threshold, that is wavelength-dependent. Many remaining questions concern the regime of slow drift: particularly how a process of phase diffusion can tune a homogeneous displacement of the pattern. - A strong coupling between propagating domains of drifting cells and oscillations has been evidenced within many situations. This suggests that both states may involve common origins, which is reinforced by possible reconstruction of local deformations in both cases, using a combination of the fundamental mode and its first harmonic. 26 P. Brunet, L. Limat: Dynamics of a circular array of falling liquid columns - Many features of the array of columns show similarities with other pattern forming unstable fronts. Considering morphology and space/time scales, the most resembling experimental system should be the directional viscous fingering [6–11]. Otherwise, some situations of directional solidification with lamellar eutectics [15], although dealing with smaller space and slower time scales, have shown striking similarities with our system, particularly on far-from-threshold behaviors: amplitude holes within an oscillating period-doubling state, oscillations of global drifting states (denoted as T − xλO in [15]) or a pair of propagating ’tilted’ domains launched at the break-up of an oscillating patch [14]. All of these systems are suitable to provide inputs in order to validate and improve existing models. As an alternative approach, some attempts to calculate the secondary instabilities of KS equation, from an initial cellular solution with a few tens of cells, have reproduced several typical behaviors resulting from nontrivial mode interactions: for instance, oscillating wakes behind propagative domains, phase jumps and amplitude holes in oscillating regimes, or oscillations prior to ruptures of global drifting states [56]. 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