Successive bifurcations in directional viscous fingering

Directional viscous-fingering experiments are reported which extend previous studies by Rabaud, Michalland, and Couder [Phys. Rev. Lett. 64, 184 (1990)]. With the external cylinder rotation speed ${\mathit{V}}_{\mathit{e}}$ fixed at a small constant value, the counter-rotation speed of the inner cylinder ${\mathit{V}}_{\mathit{i}}$, which is then the single control parameter of the experiment, was increased or decreased in small steps. Beyond the primary planar-cellular bifurcation of the air-oil interface, a secondary bifurcation was observed to a state with uniform space-filling traveling cells, followed by a spatial period-doubling bifurcation. Close to these bifurcations, we also observed transient states with solitary traveling cells, traveling domains of tilted cells, and mixed states of coexisting large and small traveling cells. These states resemble observations in recent directional-solidification experiments on liquid crystals and eutectic alloys. We discuss these results in the context of recent theoretical descriptions of parity-breaking tilt bifurcations and spatial period-doubling bifurcations based on k-2k mode-interaction models.