Observation of two coupled Faraday waves in a vertically vibrating Hele-Shaw cell with one of them oscillating horizontally
OObservation of two coupled Faraday waves in avertically vibrating Hele-Shaw cell with one of themoscillating horizontally
Xiaochen Li, Xiaoming Li and Shijun Liao ∗ State Key Laboratory of Ocean EngineeringSchool of Naval Architecture,Ocean and Civil EngineeringShanghai Jiaotong University, Shanghai 200240, China
Abstract
A system of two-dimensional, two coupled Faraday interfacialwaves is experimentally observed at the two interfaces of the three layersof fluids (air, pure ethanol and silicon oil) in a sealed Hele-Shaw cell withperiodic vertical vibration. The upper and lower Faraday waves coexist: theupper vibrates vertically, but the crests of the lower one oscillate horizontallywith unchanged wave height and a frequency equal to the half of the forcingone of the vertically vibrating basin, while the troughs of the lower one alwaysstay in the same place (relative to the basin). Besides, they are stronglycoupled: the wave height of the lower Faraday wave is either a linear function(in the case of a fixed forcing frequency) or a parabolic function (in the caseof a fixed acceleration amplitude) of that of the upper, with the same wavelength. In addition, the upper Faraday wave temporarily loses its smoothnessat around t = T / and t = 3 T / , where T denotes the wave period, andthus has fundamental difference from the traditional one. To the best ofour knowledge, this system of the two coupled Faraday waves has never beenreported. Key Words
Faraday waves; multiple layers of fluids; experimental observa-tion
1. Introduction
The Faraday waves in a vertically oscillating basin were first discoveredby Faraday [1] and then analyzed by Benjamin and Ursell [2], who foundthat these standing waves vertically vibrate with a frequency equal to halfof the forcing one of the basin. These waves can organize in different forms, ∗ Corresponding author. Email address: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] A ug uch as stripes, squares, hexagons [3], and even stars [4]. The Faraday insta-bility in viscous fluids was also experimentally investigated by Bouchgl andAniss [5]. Thereafter, the motion of the interface between two fluids withdifferent ratios of density by means of forcing vertical oscillation became ahot topic. Some extreme steep interfacial waves which oscillate verticallyat the interface of two inviscid fluids were numerically simulated and theirstability was investigated by Mercer and Roberts [6]. The parametric insta-bility analysis for the interface of two viscous fluids was studied by Kumarand Tuckerman [7], who found that the effect of large viscosity on the wave-length selection is substantial. The two-dimensional Faraday waves of twoinviscid fluids were numerically studied by Wright et al. [8], the results werealso compared with the fully nonlinear numerical simulation by Takagi andMatusumoto [9]. The instability of Faraday interfacial waves between twoweakly viscous layers in a rectangular domain was studied by Hill [10]. Thespatiotemporal Fourier spectrum of Faraday waves on the interface of twoliquids in a three-dimensional closed cell were measured by Kityk [11]. Theexperimental study on Faraday waves in domains with flexible boundaries isimplemented by Pucci et al. [12, 13] in the instability of floating fluid drops.The walking and orbiting droplets were observed on the surface of a liquidat a sufficiently high acceleration by Couder et al. [14]. The linear Faradaystability of a two-layer liquid film with a free upper surface was investigatednumerically by Potosky and Bestehorn [15]. The diffuse interface betweentwo miscible liquids subject to vertical vibration was studied by means ofexperiments and numerical simulation [16], and a time-dependent densitygradient is established from the moment when the two layers were placed to-gether [17]. By singular perturbation theory, the interfacial wave modes in atwo-layer liquid-filled cylindrical vessel were found to become more complex,as the density ratio increases from the upper to the lower layer [18].In this letter we experimentally investigate the system of the two coupledinterfacial Faraday waves at the interfaces of air and two immiscible liquidsin a sealed Hele-Shaw cell with periodic vertical vibration. The upper liquidis pure ethanol (with the density ρ = 791kg/m and the viscosity µ =0.0011Pa s) and the lower is silicon oil (methyl-silicone-I, with the density ρ = 970 kg/m and the viscosity µ = 0.35 Pa s). Above the two immiscibleliquids is the air. So, there exist three layers of different fluids and twointerfaces. One is the interface between the air and the pure ethanol, calledthe upper interface. The other is the interface between the pure ethanol andthe silicon oil, called the lower interface.2 . Experimental setup The experimental setup is as follows. A Hele-Shaw cell (made of PMMA)with 300 mm length, 2mm width and 60mm depth is filled with two immisci-ble fluids: the upper is pure ethanol (4mm in depth) and the lower is siliconeoil (8mm in depth). For the sake of observation convenience, a very smallamount of phenol red is added in pure ethanol. The cell is fixed on a hor-izontal shaker and guided with a vertical sinusoidal vibration. The forcingfrequency (denoted by f ) and the acceleration amplitude (denoted by A ) ofthe shaker are output by a closed-loop control system. A high-speed camerais positioned perpendicular to the front of the cell to record the evolution ofthe upper interface (between the air and pure ethanol) and the lower inter-face (between the two liquids). The temperature is nearly 20 ◦ C. Consideringvolatility of pure ethanol, the cell is sealed (i.e. the depth of the air is 48mm) and the pure ethanol is replaced every twenty minutes.
3. Experimental results
When the Hele-Shaw cell vibrates vertically with the forcing frequency f =18 Hz and the acceleration amplitude A =17 m/s , we observed a sys-tem of two coupled Faraday waves. For details, please see figure 1 and thecorresponding movie. At the upper interface, there exists a standing wavethat oscillates vertically in a similar way like a traditional Faraday wave (butwith a few fundamental differences mentioned later), call the upper Faradaywave. At the lower interface, there exists a standing wave whose crests oscil-late horizontally with an unchanged height and a frequency equal to half ofthe forcing frequency of the vertically vibrating basin, called the lower Fara-day wave. To the best of our knowledge, such kind of horizontally oscillatingFaraday waves have never been reported. The upper and lower Faraday wavescoexist and are strongly coupled, with the same period (denoted by T ) andthe same wave length (denoted by L ). At t = 0, the crest of the upper Fara-day wave reaches its maximum height, below which there are two adjoiningcrests of the lower Faraday wave that are in the shortest distance (denoted by δ min ). Thereafter, the crest of the upper Faraday wave falls vertically untilit becomes a trough at t = T /
2, while the above-mentioned two adjoiningcrests of the lower Faraday wave depart horizontally from each other withalmost unchanged height, and their distance (denoted by δ ) increases untilit reaches the maximum (denoted by δ max ) at t = T /
2. As the time further3 igure 1: (Colour online) The system of two coupled Faraday waves in the case of theforcing frequency f =18 Hz and the acceleration amplitude A = 17 m/s , where T denotesthe wave period. For more details, please see the corresponding movie. t = 3 T /
4, and then becomes a crest againthat reaches its maximum at t = T . In the same time, the two adjoiningcrests of the lower Faraday wave horizontally approach each other with theunchanged height until δ decreases to δ min at t = T . Note that all troughs ofthe lower Faraday wave are still (relative to the Hele-Shaw cell) on the samehorizontal line, and the distance of any two adjoining troughs is equal to thehalf of the wave length L of the upper Faraday wave. However, unlike theupper Faraday wave which has a symmetry about crest, the lower Faradaywave loses its symmetry about the crest, although both of the upper andlower Faraday waves retain the symmetry about the trough. Besides, unlikethe traditional Faraday wave, the upper Faraday wave temporarily loses itssmoothness at around t = T / t = 3 T /
4. It implies that the upper andlower Faraday waves strongly interact each other. In addition, it is foundthat, using the same forcing frequency f = 18 Hz and the same accelerationamplitude A =17 m/s , we can not observe any Faraday waves if there existsonly the 8mm silicone oil (with the air) in the sealed Hele-Shaw cell, or if weincrease the depth of pure ethanol up to 10mm. This phenomenon stronglysuggests that the lower horizontally oscillating Faraday wave is excited bythe upper vertically vibrating Faraday wave via the viscous friction on theinterface between the two immiscible liquids.The schematic illustration is as shown in figure 2(a), where H and H denote the wave height of the upper and lower Faraday waves, L denotestheir wave length, δ is the distance between the two adjoining crests of thelower Faraday wave, respectively. In the case of f =18 Hz and A = 17 m/s ,the time-dependent variation of δ is as shown in figure 2(b), which can befitted by a simple formula δ ( t ) = 12 ( δ max + δ min ) −
12 ( δ max − δ min ) cos( πf t ) (1)with δ max = 14 .
08 mm and δ min = 7 .
26 mm in a good agreement with themeasured data, and by the fitted formula δ ( t ) = 12 ( δ max + δ min ) −
12 ( δ max − δ min ) cos( πf t ) + B sin( πf t ) (2)with B = 0 . f fixed at 18Hz, the system of the two coupled Faraday waves canbe observed within a region of the acceleration amplitude 12.5 m/s ≤ A ≤ a) Figure 2: (Colour online) (a) Schematic illustration of the upper and lower Faraday waves. H and H denote the wave height of the upper and lower Faraday waves, L is their wavelength, δ is the distance of the two crests of the lower Faraday wave, respectively. (b)Variation of δ ( t ) in case of f =18 Hz and A = 17 m/s .Figure 3: (Colour online) Existence window of A versus f for the couple two Faradaywaves. Solid line: the lower threshold; Dashed line: the upper threshold.
64 m/s . When A < , no interfacial waves were observed at all.When A >
24 m/s , the interfacial waves at the upper and lower interfaceare disordered. In the cases of f = 23 Hz and f =15 Hz, such kind oftwo coupled Faraday waves are always observed, but with different upperand lower thresholds of A . It is found that there exists the correspondingupper and lower thresholds of A for a given forcing frequency f , as shown infigure 3. Note that the lower threshold of A increases with the frequency f very slowly, but the upper threshold rises rapidly.In the case of the fixed acceleration amplitude A = 15 m/s with thedifferent forcing frequency f in the region of 15 Hz ≤ f ≤
23 Hz, the waveheight H of the lower Faraday wave is a parabolic function of the waveheight H of the upper one, but δ max has a linear relationship with the wavelength L , respectively, as shown in figure 4(a,b). In the case of the fixedforcing frequency f =18 Hz with the different acceleration amplitude A inthe region of 14 m/s ≤ A ≤
22 m/s , the wave heights H of the lowerFaraday wave has a linear relationship with the wave height H of the upperone, but δ max is a parabolic function of the wave length L , respectively, asshown in figure 4(c,d). These phenomena reveal the close relationship andstrong coupling between the upper and lower Faraday waves.Note that the upper Faraday wave looks like the traditional Faradaywave at the interface of two immiscible fluids only (such as water and air).For the sake of comparison, we measured the traditional Faraday wave atthe interface of the air and pure ethanol with the same depth of 4mm (but without silicon oil) in the same Hele-Shaw cell [19, 20]. In the case of thefixed acceleration amplitude A =15 m/s with the different forcing frequency f in the region of 15 Hz ≤ f ≤
23 Hz, both of the wave height H andwave length L of the upper Faraday wave decreases with the increase ofthe frequency f , as shown in figure 5(a,b). Besides, it is found that thewave height H of the upper Faraday wave is always smaller than the waveheight H of the traditional one, but the wave length L of the upper Faradaywave is almost the same as that of the traditional one, respectively. Thisis easy to understand, since the upper layer liquid (pure ethanol) transferssome kinetic energy to the lower one (silicon oil) via viscous friction at theirinterface. In the case of the fixed frequency f =18 Hz with the differentacceleration amplitude A in the region 14 m/s ≤ A ≤
22 m/s , both ofthe wave height H and wave length L of the upper Faraday wave increaseswith the increase of A , as shown in figure 5(c,d). However, it is interestingthat the traditional Faraday wave does not exist when A >
17 m/s , but7 igure 4: (Colour online) Relations between H and H , L and δ max of the coupled twoFaraday waves. (a) and (b): in case of A =15 m/s and 15 Hz ≤ f ≤
23 Hz; (c) and (d):in the case of f =18 Hz and 14 m/s ≤ A ≤
22 m/s ; Dashed-line: the fitting formulas. igure 5: (Colour online) Comparison of the upper Faraday wave and the traditionalFaraday wave of pure ethanol with the same physical parameters (but without the layerof silicon oil below), where H and L denote wave height and wave length of the upperFaraday wave, H and L denote those of the traditional one. (a) and (b): in case of A =15 m/s and 15 Hz ≤ f ≤
23 Hz; (c) and (d): in the case of f =18 Hz and 14 m/s ≤ A ≤
22 m/s . A up to 22 m/s . It means that, for a given forcing frequency f , the upper threshold of A for the coexistence of the two coupled Faradaywaves is larger than that for only one traditional Faraday wave. It indicatesthat the system of the two coupled Faraday waves is stable even for a largeacceleration amplitude A that corresponds to a high nonlinearity. Note that,for a given frequency f , more kinetic energy is needed to excite the systemof the two coupled Faraday waves than the only one traditional Faradaywave. In addition, the wave length L of the upper Faraday wave is almostthe same as L of the traditional one, although its wave height H is alwayssmaller than H . Furthermore, unlike the traditional Faraday wave, theupper Faraday wave temporarily loses its smoothness at around t = T /
T /
4, as shown in figure 1. All of these indicate that the upper Faradaywave is fundamentally different from the traditional one, although both ofthem are vertically vibrating waves. Finally, it should be emphasized thatthe upper and lower Faraday waves coexist and are strongly coupled: neitherof them can exist along.Note that Potosky and Bestehorn [15] numerically investigated the linearinstability of Faraday waves of the three-layer fluids (air and two immisci-ble liquids) in a three-dimensional domain. However, they only gained thecoupled two Faraday waves that vibrate vertically. Unlike their theoreticalinvestigation, our physical experiments are related to the two-dimensional
Faraday waves by means of physical parameters quite different from theirs:the ratio of the viscosity µ /µ ≈ .
4. Conclusions
In conclusion, we experimentally observed a system of the two-dimensional,two coupled Faraday waves at two interfaces of three layers of fluids (air, pureethanol and silicon oil) in a sealed Hele-Shaw cell with periodic vertical vi-bration. The upper Faraday wave vibrates vertically, and the lower oscillateshorizontally. They coexist and are strongly coupled. This system of two cou-pled Faraday waves has never been reported, to the best of our knowledge.So, it fleshes out the picture of Faraday waves as a type of vertical standingwaves. They also bring us some new challenges in theoretical analysis andnumerical simulations. 10 cknowledgement
This work is partly supported by National Natural Science Foundation ofChina (Approval No. 11272209 and 11432009).