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Directional viscous-fingering experiments are reported which extend previous studies by Rabaud, Michalland, and Couder. With the external cylinder rotation speed Ve fixed at a small constant value, the counter-rotation speed of the inner cylinder V 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

We present a mechanism of limitation for the possible wave numbers above an instability threshold. This mechanism is experimentally investigated in the interfacial instability of directional viscous fingering in a finite system. It is shown experimentally to be controlled by the divergence of a phase-diffusion constant. Theoretically, this limitation on the low value of the accessible wave numbers is a consequence of the interaction between the fundamental and the first harmonic modes. The analysis of coupled amplitude equations demonstrates theoretically the existence of a divergence of a phase-diffusion constant when approaching the threshold