Quasipatterns in a parametrically forced horizontal fluid film

Abstract : We shake harmonically a thin horizontal viscous fluid layer (frequency forcing Ω, only one harmonic), to reproduce the Faraday experiment and using the system derived in [31] invariant under horizontal rotations. When the physical parameters are suitably chosen, there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency Ω/2, and the mode with frequency Ω. Moreover, at criticality the corresponding wave lengths kc and k′c are such that if we define the family of 2q equally spaced (horizontal) wave vectors kj on the circle of radius kc , then kj + kl = k′n, with |kj| = |kl| = kc , |k′n| = k′c . It results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequencyΩ/2) having a spatially quasiperiodic pattern if q ≥ 4. Restricting our study to solutions invariant under rotations of angle 2π/q, gives a kernel of dimension 4. In the spirit of Rucklidge and Silber (2009) [29] we derive formally amplitude equations for perturbations possessing this symmetry. Then we give simple necessary conditions on coefficients, for obtaining the bifurcation of (formally) stable time-periodic (frequency Ω/2) quasipatterns. In particular, we obtain a solution such that a time shift by half the period, is equivalent to a rotation of angle π/q of the pattern.
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Médéric Argentina, Gérard Iooss. Quasipatterns in a parametrically forced horizontal fluid film. Physica D: Nonlinear Phenomena, Elsevier, 2012, 〈10.1016/j.physd.2012.04.011〉. 〈hal-01265170〉



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