Patterns and quasi-patterns in the Faraday experiment

Abstract

Parametric excitation of surface waves via forced vertical oscillation of a container filled with fluid (the Faraday instability) is investigated experimentally in a small-depth large-aspect-ratio system, with a viscous fluid and with two simultaneous forcing frequencies. The asymptotic pattern observed just above the threshold for the first instability of the flat surface is found to depend strongly on the frequency ratio and the amplitudes and phases of the two sinusoidal components of the driving acceleration. Parallel lines, squares, and hexagons are observed. With viscosity 100 cS, these stable standing-wave patterns do not exhibit strong sidewall effects, and are found in containers of various shapes including an irregular shape. A ‘quasi-pattern’ of twelvefold symmetry, analogous to a two-dimensional quasi-crystal, is observed for some even/odd frequency ratios. Many of the experimental phenomena can be modelled via cubic-order amplitude equations derived from symmetry arguments.

Existence of Quasipattern Solutions of the Swift–Hohenberg Equation

“…To our knowledge, hereafter is the first proof of existence of such quasipatterns of a PDE. Quasipatterns were discovered in nonlinear pattern-forming systems in the Faraday wave experiment [BCM92,EF94], in which a layer of fluid is subjected to vertical oscillations. Since their discovery, they have also been found in nonlinear optical systems, shaken convection and in liquid crystals (see references in [AG12]) .…”

Pattern formation in large domains

“…A notable example of this is quasi-patterns, which are most readily found in Faraday wave experiments in which a tray of liquid is subjected to vertical vibrations with two commensurate forcing frequencies (Edwards & Fauve 1994). A recent survey of experimental results can be found in Arbell & Fineberg (2002), and one experimental example of a quasi-pattern is shown in¯gure 2a.…”

Footprint geometry and sessile drop resonance

“…For both, typical observables are mode shapes [11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], resonance frequencies [11,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32], and the evolution of surface waves with forcing amplitude [18,23,[29][30][31][33][34][35]. The free surface waves oscillate at the forcing frequency (harmonically) when the forcing acceleration is low.…”