Fluid-particle separation in a random flow described by the telegraph model

Abstract

We study the statistics of the relative separation between two fluid particles in a spatially smooth and temporally random flow. The Lagrangian strain is modelled by a telegraph noise, which is a stationary random Markov process that can only take two values with known transition probabilities. The simplicity of the model enables us to write closed equations for the inter-particle distance in the presence of a finite-correlated noise. In 1D, we are able to find analytically the long-time growth rates of the distance moments and the senior Lyapunov exponent, which consistently turns out to be negative. We also find the exact expression for the Cram\'er function and show that it satisfies the fluctuation relation (for the probability of positive and negative entropy production) despite the time irreversibility of the strain statistics. For the 2D incompressible isotropic case, we obtain the Lyapunov exponent (positive) and the asymptotic growth rates of the moments in two opposite limits of fast and slow strain. The quasi-deterministic limit (of slow strain) turns out to be singular, while a perfect agreement is found with the already-known delta-correlated case.