Elastic properties of cellular dissipative structure

TL;DR

This paper investigates the interaction of drift and oscillation modes in cellular dissipative structures, specifically in liquid jet arrays, revealing a geometric relationship between domain velocity, oscillation frequency, and spacing.

Contribution

It provides a simple geometrical explanation for the velocity-oscillation relationship, refining previous assumptions and aligning well with experimental data.

Findings

Velocity of propagating domain linked to oscillation frequency and spacing.

Derived that the proportionality constant should be close to 1/π.

Experimental data supports the refined geometric relationship.

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 $\omega$ of oscillations and to the spacing between columns $\lambda_0$ by the relationship $ V_g = \alpha \lambda_0 \omega$. We show by a simple geometrical argument that $\alpha$ should be close to $1/ \pi$ instead of the initial value $\alpha = 1/2$ deduced from their analogy with phonons. This fact is in quantitative agreement with our data, with a slight deviation increasing with flow rate.