On solitary waves running down an inclined plane
We study the existence and the role of solitary waves in the instability of a fluid layer flowing down an inclined plane. The approach presented is fully nonlinear. Solitary waves steady in a moving frame are described by homoclinic trajectories of an associated ordinary differential equation. They are searched numerically for a given value of viscosity and surface tension. Several kinds of solitary waves can exist, characterized by their number n of humps. We investigate the stability of these waves by integrating the initial-value problem directly. Solitary waves with more than 1 hump did not appear in the simulation, and moreover a catastrophic behaviour took place for too large a Reynolds number (R [greater, similar] R*1) or too large an amplitude, suggesting a finite-time singularity. The long-term evolution is shown to be a very slow relaxation to a steady state in a moving frame. The relation to the experimental observation of localized wavetrains is also discussed.(Published Online April 20 2006)
(Received August 16 1982)
(Revised May 12 1983)
p1 Permanent address: DPh-G/PRSM, Orme des Merisiers 91191 Gif-sur-Yvette, France.