Elastic properties of a cellular dissipative structure

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 [1], the velocity V <Subscript> g </Subscript> of a constant width domain is linked to the angular frequency <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\omega$</EquationSource> </InlineEquation> of oscillations and to the spacing between columns <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\lambda_0$</EquationSource> </InlineEquation> by the relationship <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$V_g=\alpha \lambda_0 \omega$</EquationSource> </InlineEquation>. We show by a simple geometrical argument that <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\alpha$</EquationSource> </InlineEquation> should be close to <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$1/ \pi$</EquationSource> </InlineEquation> instead of the initial value <InlineEquation ID="Equ6"> <EquationSource Format="TEX">$\alpha=1/2$</EquationSource> </InlineEquation> deduced from their analogy with phonons. This fact is in quantitative agreement with our data, with a slight deviation increasing with flow rate. Copyright Springer-Verlag Berlin/Heidelberg 2003