Scaling Theory of Singularities

Jens Eggers

Department of Physics, Universitaet Essen, 45117 Essen, Germany

Abstract: Owing to the nonlinear character of the equations of motion describing fluid motion, sudden discontinuities in the pressure, the density, or the surface shape of a fluid often occur. Examples are shock waves, the separation of a fluid drop, bubbles rising in a viscous fluid, or drops in strong electric fields. Singularities are the crucial events in the evolution of a flow, and signal the emergence of new structures. Physically, smaller and smaller structures are produced near the singularity, so it lacks a characteristic length scale. Singular solutions thus must be invariant under a change of scale, and have a simple scaling form. Moreover, singularities turn out to be universal, so the same structure is found in a broad variety of circumstances. This talk explores the mathematics and the physical consequences of these two fundamental properties.