Weakly supercritical dissipative structures on curved surfaces

Cellular, low amplitude structures appearing at cylindrical and spherical fronts of gaseous combustion and laser evaporation are described. In the case of a spherical front all these structures are found to be unstable. When the cylindrical front of gaseous combustion is expanded, we must expect the quasi one-dimensional structure homogeneous with respect to the ignorable coordinate to be replaced by a parquet-like pattern of rectangular cells, and later to reach a non-stationary regime. On the cylindrical front of laser evaporation the quasi one-dimensional structure of maximum amplitude is globally stable. The best known hydrodynamic example of a kinetic problem connected with the formation of dissipative structures i.e. thermodynamically nonequilibrium stationary structures appearing as a result of the development of aperiodic instability in a spatially homogeneous state, are Benard cells /1,2/. New problems of this kind are connected with the instability of plane fronts of laser evaporation of condensed material, and of gaseous combustion /3-5/. The instability is aperiodic and appears at finite values of the wave number of the perturbation representing curvature of a plane front. The development of the instability leads to the formation of a stationary, periodically curved front /3/. The purpose of this paper is to investigate such structures and their stability on cylindrical and spherical surfaces, and this corresponds to the problem of the propagation of a cylindrical or spherical flame through a gas, and of the laser evaporation of a spherical sample. Problems dealing with dissipative structures on curved surfaces are also of interest in biophysics, where a spherical surface models a cell membrane, while the cylindrical surface models the axon /6/.