The Printer’s Instability: The Dynamical Regimes of Directional Viscous Fingering

Abstract

Important experimental and theoretical efforts have been devoted recently to the dynamical behaviours of extended systems (reviews can be found in References 1 and 2). In this spirit, experiments were done in the Rayleigh-Bénard instability aimed at the study of the onset of chaos in periodic structures with a large number of cells3,4. On the other hand electroconvection of liquid crystals5,6, convection in binary fluid mixtures7,8 and the Faraday experiment9,10 show a rich dynamics due to the presence of two independent control parameters. In particular, propagative modes are observed, sometimes confined to limited domains. Propagation also exists in Taylor-Couette flows where it creates spiral structures11–13. For the sake of simplicity and comparison to theoretical models, one-dimensionality has been sought in extended systems. This is obtained by confinement in the above-mentioned convection experiments3,4 Another way to reach one-dimensionality is to study a boundary. This boundary can be the limit between two convective rolls affected by the oscillatory instability14 or an interface between two-dimensionally confined media. In the latter case, if a front instability generates a large number of cells, their dynamics is one dimensional to a good approximation. Such situations are obtained in directional phase transition of liquid crystal15, eutectic crystallisation16 and directional viscous fingering l7a,b. We presented previously17a results on the instability onset of the front of a viscous fluid in a widening gap. The present article reports the development of our work on the nonlinear regimes of this front instability.KeywordsSolitary WaveDirectional SolidificationChaotic StateSpatiotemporal ChaosFront InstabilityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.