Capillary dynamics of coupled spherical-cap droplets

Abstract

Centre-of-mass motions of two coupled spherical-cap droplets are considered. A model with surface tension and inertia that accounts for finite-amplitude deformations is derived in closed form. Total droplet volume λ and half-length L of the tube that connects the droplets are the control parameters. The model dynamics reside in the phase-plane. For lens-like droplets λ < 1, and for any L there is a single steady state about which the droplets vibrate with limit-cycle behaviour. For λ>1, the symmetric state loses stability (saddle point) and new antisymmetric steady states arise about which limit-cycle oscillations occur. These mirror states – big-droplet up or big-droplet down – are also stable. In addition, there are large finite-amplitude ‘looping’ oscillations corresponding to limit cycles that enclose both steady states in the phase-plane. All three kinds of oscillations are documented in an experiment that sets the system into motion by ‘kicking’ one of the droplets with a prescribed pressure-pulse. Model predictions for frequencies are consistent with observations. Small-amplitude predictions are placed in the wider context of constrained Rayleigh vibrations. A model extension to account for the small but non-negligible influence of viscosity is also presented.

Finite-amplitude dynamics of coupled cylindrical menisci

“…The dynamics of the spherical-cap system have been previously studied and are qualitatively similar to the dynamics reported above [1]. They are quantitatively different, however, because of the presence of the additional principal curvature.…”

“…Furthermore, according to these stability studies and the behavior of the analogous coupled spherical-cap problem [1] -supported by experiment -one expects switching. Study [1] also compares an inviscid model to experiments at Re = 240 and finds qualitative agreement of finiteamplitude predictions with observation and that this agreement becomes quantitative when weak dissipation is added to the model.…”

“…Studies on droplet deformation show that the effect of inertia is small with small Weber number [3], and viscosity can be neglected for large Laplace number [4]. The extent to which these conditions are satisfied for the spherical cap droplet oscillations is discussed in Theisen et al [1]. When inertia and surface tension dominate, the motions of centers-ofmass can be tracked ignoring details of internal flow.…”

“…The energy landscape has either a single-or double-welled potential depending on the extent of volume overfill. Total liquid volume is a bifurcation parameter, as in the corresponding problem for coupled spherical-cap droplets [1]. Unlike the spherical-cap problem, however, axial disturbances can also destabilize, depending on overfill.…”

High speed adaptive liquid microlens array

“…For substantially higher driving frequencies hydrodynamic coupling between adjacent microlenses in the array leads to the excitation of collective modes, which destroy the phase synchronization between the microlenses [20,21] and limit the operation range of the present device (supplementary information Media 6). One possible strategy to extend the operation range to even higher frequencies is to place suitably designed obstacles that selectively suppress the hydrodynamic coupling between adjacent microlenses.…”