Elastic properties of cellular dissipative structure

Abstract

Transition towards spatio-temporal chaos in one-dimensional interfacial patterns often involves two degrees of freedom: drift and out-of-phase oscillations of cells, respectively associated to parity breaking and vacillating-breathing secondary bifurcations. In this paper, the interaction between these two modes is investigated in the case of a single domain propagating along a circular array of liquid jets. As observed by Michalland and Rabaud for the printer's instability \cite{Rabaud92}, the velocity $V_g$ of a constant width domain is linked to the angular frequency $ω$ of oscillations and to the spacing between columns $λ_0$ by the relationship $ V_g = αλ_0 ω$. We show by a simple geometrical argument that $α$ should be close to $1/ π$ instead of the initial value $α= 1/2$ deduced from their analogy with phonons. This fact is in quantitative agreement with our data, with a slight deviation increasing with flow rate.