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In a directional viscous fingering experiment, a phase diffusion behavior of the regular array of cells is demonstrated. At constant wave number, the evolution of the phase diffusion coefficient D with the parameter of the instability \ensuremath{\epsilon} is reported. This coefficient D is found to decrease at low \ensuremath{\epsilon} values in agreement with the presence of an Eckhaus instability, but to increase strongly at large \ensuremath{\epsilon} values. This unusual increase of D is also present in the analytical and numerical study of two coupled amplitude equations for the modes k and 2k. The phase diffusion coefficient is found to diverge at the threshold of the parity-breaking bifurcation.

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

The transition to chaos in diverse physical systems far from thermodynamic equilibrium is one of the challenging problems of modern physics. The most commonly studied models in hydrodynamics are the Rayleigh-Benard thermal convection system and the Taylor-Couette instabilityl, 2, both of which have been intensively investigated during the last two decades. The transition to chaos has been characterized and various scenarios have been discovered experimentally and in numerical simulations of those systems3. These two systems possess several symmetries, the breaking of which gives rise to new patterns. However, the real world is far from these simple cases and an effort is underway to study more complicated systems such as thermal convection in superposed layers of immiscible fluids4, the horizontal Taylor-Couette system with a partially filled gap5, the flow in curved channel6 or the boundary layer flow over a concave wall2.

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