PHYSICAL REVIEW E 86, 066313 (2012) Faraday instability at foam-water interface A. Bronfort * and H. Caps GRASP, Physics Department B5, University of Li` ege, B-4000 Li` ege, Belgium (Received 17 August 2012; published 14 December 2012) A nearly two-dimensional foam is generated inside a Hele-shaw cell and left at rest on its liquid bath. The system is then vertically shaken and, above a well-defined acceleration threshold, surface waves appear at the foam-liquid interface. Those waves are shown to be subharmonic. The acceleration threshold is studied and compared to the common liquid-gas case, emphasizing the energy dissipation inside the foam. An empirical model is proposed for this energy loss, accounting for the foam characteristics such as the bubble size but also the excitation parameter, namely the linear velocity. DOI: 10.1103/PhysRevE.86.066313 PACS number(s): 47.20.−k, 47.57.Bc, 47.35.−i I. INTRODUCTION Aqueous foams can be seen as dispersions of gas bubbles inside a surrounding fluid composed of water and surfactant. The rheology of those complex fluids depends on their liquid content ϕ [1]. When the amount of liquid, with respect to that of gas, is large (typically ϕ  10%) the foam is identified as wet. Adjacent bubbles are free to move relatively to each other but some energy is dissipated due to the viscosity of the motion of the interstitial fluid [2]. In the case of dry foams (ϕ  10%) the bubble interaction is more elastic and energy is dissipated by the geometrical reorganization of the liquid films separating the bubbles [2,3]. Up to now, a lot of experimental [4,5] and theoretical [6,7] studies have been conducted on the rheology of aqueous foams, both in two-dimensional and three-dimensional configurations. Most of those works focus on the bulk properties of the foam, under external constraints. To our knowledge, the interface between the foam and its liquid bath did not receive such attention. However, the interface between two fluids is defined in terms of an interfacial tension, and depends on the viscosity, the density, and the chemistry of both fluids. In the present case, the foam and its liquid bath can be seen as two nonmiscible fluids, with some kind of mixing length defined by the height the water that can imbibe the foam thanks to capillary rise. In order to probe the mechanical properties of an interface, hydrodynamic instabilities appear as useful tools. One of those instabilities is the Faraday instability [8], emerging when a free surface experiences vertical oscillations above acceleration  c . For acceleration values in the vicinity of  c , the free surface is covered by a set of standing waves which can be ordered into different geometries [8–13]. The Faraday waves are parametrically excited and oscillate at half the forcing frequency [8,19]. The dispersion relationship in the inviscid and spatially infinite free fluid surface hypothesis reads ω m (k) =  g 0 k + σ ρ k 3  tanh(kh)  1/2 , (1) where the ω m (k) are the natural frequency of each of the eigenmode k of the surface, σ is the fluid surface tension, ρ its density, g 0 is Earth’s gravitational acceleration, and h is the depth of the liquid pool [22]. * abronfort@ulg.ac.be The threshold for instability is observed to depend on the fluid viscosity [14,15] as well as on capillary effects [16–18]. In case of miscible fluids, an instability can also be observed [20] but experiments are limited by the mixing. In the present paper, we propose to study the stability of a foam-liquid interface under vertical oscillations. Control parameters such as the bubble diameter D, the amplitude A, and the frequency f of oscillations will be investigated. After a description of the experimental setup, we focus on the critical acceleration  c . Attention is then put on the damping of the waves due to the foam. A phenomenological model is eventually proposed and validated. II. EXPERIMENTAL SETUP All the experiments presented below have been carried out inside Hele-Shaw (HS) cells made out of polycarbonate plates (130 × 100 × 3 mm 3 ). Half of the cells were filled with an aqueous surfactant solution composed of 94% bi-distilled water, 1% commercial dishwashing soap, and 5% glycerol. Using milli-fluidic T junctions [21], bubbles are blown into this liquid pool and compose the foam. The cell does not have a top wall so the foam is left in contact with air for all the following experiments. The bubble diameter D ranges in [1; 4] mm. The amount of foam inside the HS cell is defined as the height H reached by the bubbles above the liquid surface. The depth of the liquid pool is assumed as infinite. The cell is then vertically fixed onto an electromagnetic shaker (as shown in Fig. 1) providing oscillations which are defined by their frequency f ∈ [5; 50] Hz and their amplitude A. The latter is deduced from the measurement of the acceleration  ≡ A(2πf ) 2 ∈ [0.5; 3]g 0 . The frequency ranges in [18; 28] Hz for experiments with foam. The lower limit is due to our shaker acceleration range and the upper one is set by the features of Faraday waves. Indeed at high frequencies the wavelength is typically of the order of the bubble diameter or smaller. The HS cells are backlighted and images of the foam are recorded using a high-speed video camera configured at 1000 frames per second (fps). Before each experiment the freshly generated foam was left at rest for a few minutes in order to reach its gravitational equilibrium. The regular replacement of the foam was so that the foam neither ages nor evolves over the experiment duration time. 066313-1 1539-3755/2012/86(6)/066313(5) ©2012 American Physical Society