Lyapunov exponent for small particles in smooth one-dimensional flows

This paper discusses the Lyapunov exponent for small particles in a spatially and temporally smooth flow in one dimension. Using a plausible model for the statistics of the velocity gradient in the vicinity of a particle, the Lyapunov exponent is obtained as a series expansion in the Stokes number, St, which is a dimensionless measure of the importance of inertial effects. The approach described here can be extended to calculations of the Lyapunov exponents and of the correlation dimension for inertial particles in higher dimensions. It is also shown that there is correction to this theory which arises because the particles do not sample the velocity field ergodically. Using this non-ergodic correction, it is found that (contrary to expectations) the first order term in the expansion does not vanish.