Stability of permeative flows in 1 dimensionally ordered systems
Jacques Prost, Yves Pomeau, E. Guyon · Journal de Physique II · 1991
Layered structures are met in dissipative systems, such as Rayleigh Bénard rolls, as well as in liquid crystalline phases (smectics and cholesterics). We present here a general description, in the framework of phase dynamics, of the stability of these structures when submitted to an external force field (flow, electric field) acting perpendicular to the roll axis for various boundary conditions. The one-dimensional equilibrium solution with fixed boundary conditions leads to an effect, discovered experimentally by Pocheau and Croquette on Rayleigh-Bérnard rolls in the presence of a transverse flow, and involving the coexistence of compressed and dilated rolls; this effect has a known counterpart in cholesterics. Using the same boundary conditions, we generalize the well known undulation instability obtained under a dilative stress to the case of the action of a transverse force both from the point of view of linear stability and in the highly nonlinear limit. The possibility of observing fractal structures is indicated. For mixed boundary conditions, it is possible to have a sustained time dependent behavior involving the nucleation of new layers as also observed in the above mentioned experiments.