Weakly nonlinear nonaxisymmetric oscillations of capillary bridges at small viscosity

Abstract

Weakly nonlinear nonaxisymmetric oscillations of a capillary bridge are considered in the limit of small viscosity. The supporting disks of the liquid bridge are subjected to small amplitude mechanical vibrations with a frequency that is cióse to a natural frequency. A set of equations is derived for accounting the slow dynamics of the capillary bridge. These equations describe the coupled evolution of two counter-rotating capillary waves and an associated streaming flow. Our derivation shows that the effect of the streaming flow on the capillary waves cannot be a priori ignored because it arises at the same order as the leading (cubic) nonlinearity. The system obtained is simplified, then analyzed both analytically and numerically to provide qualitative predictions of both the relevant large time dynamics and the role of the streaming flow. The case of parametric forcing at a frequency near twice a natural frequency is also considered.

Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

“…which holds for any real vector w and any complex vector u such that V • u = 0 and V x u = 0 [6], to obtain equation (2.4), with g k ¡ as in (2.38) and p s given by Property (a) is a direct consequence of the slowly varying nature of the streaming flow, and property (c) is obvious. Properties (b) and (d) are well known in two dimensions [7], [57], [8], [58] and have been checked [59] for general, not necessarily plañe, solid and free boundaries in three dimensions.…”

“…The techniques developed in this paper show that this is indeed the case. Similar coupling arises in vibrating liquid bridges [6], [29], and may well play a role in the dynamics of acoustically driven drops and bubbles. In particular the current description of self-propulsion of acoustically driven bubbles relies on the excitation of speciflc mode interactions but remains entirely inviscid [30], [31], [32].…”

“…The mechanisms responsible for driving the streaming flow (also called "acoustic streaming" or "viscous mean flow") are well known and go back to the work of Schlichting [7] and Longuet-Higgins [8] (see [9] for a review). However, their importance for the dynamics of Faraday waves under experimentally relevant conditions has been recognized only relatively recently [5], [6]. In this paper we focus on three-dimensional containers of small aspect ratio, Le., systems in which the frequency of the vibration selects a wavenumber of the instability that is comparable to the size of the container.…”

“…However, this limit is singular and must be treated with care. This is because viscous oscillatory boundary layers attached to the container and the free surface are capable of driving streaming flows that in turn interact with the waves responsible for them [4], [5], [6]. This interaction arises already at leading (Le., cubic) order and as a result has a strong effect on the stability of the waves.…”

“…The resulting oscillation is a standing wave and so is completely determined up to an overall phase; however, it is this phase that is coupled to the streaming flow and that can exhibit nontrivial dynamics (see Section 4.2). This special property of the system is a consequence of rotational invariance of the system, and is in direct contrast to (counter-rotating) waves excited directly by lateral vibration where the wave amplitudes couple to the streaming flow as well [4], [6]. In the present paper we show that even with vertical vibration the full coupling between the streaming flow and the wave amplitude and phase is restored when (i) the (circular) cross section of the Faraday container is slightly perturbed so that invariance under rotation is lost, or (ii) when a second pair a counter-rotating modes is present.…”