Phase transitions in the distribution of inelastically colliding inertial particles

TL;DR

This paper analytically investigates phase transitions in the localization of inelastically colliding inertial particles near turbulence minima, revealing a phase diagram with a collapse region and supporting the theory with numerical simulations.

Contribution

The study derives an analytical phase diagram for particle localization transitions considering inelastic collisions and confirms it through numerical simulations.

Findings

Identification of a localization-delocalization transition signaled by Lyapunov exponent change.

Discovery of an inelastic collapse region where particles always localize.

Numerical validation of the phase diagram and flow correlation effects.

Abstract

It was recently suggested that the sign of particle drift in inhomogeneous temperature or turbulence depends on the particle inertia: weakly inertial particles localize near minima of temperature or turbulence intensity (effects known as thermophoresis and turbophoresis), while strongly inertial particles fly away from minima in an unbounded space. The problem of a particle near minima of turbulence intensity is related to that of two particles in a random flow, so that the localization-delocalization transition in the former corresponds to the path-coalescence transition in the latter. The transition is signaled by the sign change of the Lyapunov exponent that characterizes the mean rate of particle approach to the minimum (which could be wall or another particle). Here we solve analytically this problem for inelastic collisions and derive the phase diagram for the transition in the inertia-inelasticity plane. An important feature of the phase diagram is the region of inelastic collapse: if the restitution coefficient of particle velocity is smaller than some critical value, then the particle is localized for any inertia. We present direct numerical simulations which support the theory and in addition reveal the dependence of the transition of the flow correlation time, characterized by the Stokes number.