Faraday instability threshold in large-aspect-ratio containers

Abstract

We consider the Floquet linear problem giving the threshold acceleration for the appearance of Faraday waves in large-aspect-ratio containers, without further restrictions on the valúes of the parameters. We classify all distinguished limits for varying valúes of the various parameters and simplify the exact problem in each limit. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar & Tuckerman (1994). Some comparisons are made with (a) the numerical solution of the original exact problem, (b) some ad hoc approximations in the literature, and (c) some experimental results.

Viscous Faraday waves in two-dimensional large-aspect-ratio containers

“…This is in contrast with other mean flows that appear in, for example, strictly inviscid water waves , Poiseuille flow (Davey, Hocking & Stewartson 1974), and Rayleigh-Benard convection (Zippelius & Siggia 1982). Most theoretical studies in the viscous limit (Beyer & Friedrich 1995;Müller et al 1997;Cerda & Tirapegui 1998;Mancebo & Vega 2002) are linear. Nonlinear terms have been considered in the viscous limit only by Chen & Viñals (1999), who in fact considered three-dimensional deep containers, but ignored both spatial modulation and the mean flow.…”

“…A plot of a c vs. d 2 for the indicated valúes of íM 3 and Sfd (which are independent of the forcing frequeney) is given in figure 2 (b). Assuming that d is not too small, which would require a large forcing amplitude (figure 2b, see also Mancebo & Vega 2002), the first instability is subharmonic and the eigenfunctions of (2.7)-(2.9) are such that which means that the mode is a standing wave (SW).…”

Standing wave description of nearly conservative, parametrically driven waves in extended systems

“…As above, see (10), we assume that both parametric forcing and dissipation be small, which require in particular that viscous effects be weak, namely that ε be small. This in turn requires that both kinematic viscosity be small and the forcing frequency be not too large (for as ω * → ∞, the denominator ω * * 2 → 0, see (45)); a complete quantitative analysis of the combined effects of viscosity and the forcing frequency in the Faraday instability threshold can be found in [14]. Also, we consider the limit of deep layer,…”

Weakly-nonlinear analysis of the Rayleigh–Taylor instability in a vertically vibrated, large aspect ratio container

“…In addition, we must avoid the Faraday (parametric) instability [7] which would give short waves (with a wavelength small as compared to depth) along the free surface. If in addition to (1.7a-c), it is satisfied that BC~2 <C ui z l 2 and C~2 -C w 1 / 2 , as we will assume hereafter, the Faraday instability is avoided if CI 2 LJ < 2.79..., see [8].…”