Non-wetting impact of a sphere onto a bath and its application to bouncing droplets

Abstract

We present a fully predictive model for the impact of a smooth, convex and perfectly hydrophobic solid onto the free surface of an incompressible fluid bath of infinite depth in a regime where surface tension is important. During impact, we impose natural kinematic constraints along the portion of the fluid interface that is pressed by the solid. This provides a mechanism for the generation of linear surface waves and simultaneously yields the pressure applied on the impacting masses. The model compares remarkably well with data of the impact of spheres and bouncing droplet experiments, and is completely free of any of impact parametrisation.

Free rings of bouncing droplets: stability and dynamics

“…Figure 10(b) reveals that our theoretical predictions slightly underpredict the stable ring radii, r 0 , with the discrepancy between theory and experiment being larger for smaller rings, but no greater than 15 %. We expect that this discrepancy may arise from our stroboscopic wave model failing to capture the influence of the radially propagating wave fronts generated at droplet impact (Eddi et al 2011b;Damiano et al 2016), that are captured by more detailed wave models (Milewski et al 2015;Durey & Milewski 2017;Galeano-Rios et al 2017;Galeano-Rios, Milewski & Vanden-Broeck 2019). As shown by Galeano-Rios et al (2018), these fronts influence droplet-droplet interactions when the drops are both in contact with the bath for a significant fraction of the Faraday period, T F , and are in close proximity.…”

“…Coalescence is prevented by a thin air layer between the droplet and the bath that is sustained during the drop's impact (Couder et al 2005a). The dependence of the drop's bouncing mode on the droplet radius R, vibrational frequency f and vibrational acceleration γ has been studied in detail (Protière, Boudaoud & Couder 2006;Eddi et al 2008;Moláček & Bush 2013a,b;Wind-Willassen et al 2013;Galeano-Rios, Milewski & Vanden-Broeck 2017). As γ is increased progressively beyond γ B , the droplet undergoes a period-doubling transition and eventually becomes resonant with the Faraday wave period, T F , leading to a significant increase in the wave energy.…”

Ratcheting droplet pairs

“…7(b), if pair A in the n = 1.5 mode smoothly transitioned from a (2, 2) to a (2, 1) mode instead of passing through the (8, 7) and (4, 3) modes deduced from the simulations, the erroneous abrupt changes in the ratcheting speed not seen in experiments might have been averted. We note that the shortcomings of the current simulations might be eliminated through the application of the most recent model of Galeano-Rios, Milewski, and Vanden-Broeck, 6 which more accurately models the dropletbath interaction and has been shown to predict bouncing modes more accurately.…”

“…The equations of the model are formulated in the frame of reference of the shaker, which requires the use of a time dependent gravity field, G(t) = g (1 + γ sin ωt). A thorough derivation of these equations is presented in Galeano-Rioset et al 6 Periodic boundary conditions are imposed in x and y, allowing the use of spectral methods. Spectral decomposition also diagonalizes the Dirichlet-to-Neumann map, defined as φ(x, y, 0, t) → φ z (x, y, 0, t), which is necessary to reduce Eqs.…”

“…As γ is further increased, the bouncing mode changes a) C.A.Galeano.Rios@bath.ac.uk b) couchman@mit.edu c) bush@math.mit.edu according to a sequence that depends on drop size, which may include (2,2), (4,4), or (4, 3) modes. [3][4][5][6] Eventually, once the drop's bouncing amplitude is sufficiently large, it bounces at twice the period of the driving, in the (2,1) mode. The drop then achieves resonance with the most unstable wave mode of the bath, namely, the subharmonic Faraday waves excited by its impact.…”