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 cells^3,4. On the other hand electroconvection of liquid crystals^5,6, convection in binary fluid mixtures^7,8 and the Faraday experiment^9,10 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 structures^11–13. 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 experiments^3,4 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 previously^17a 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.