# Relative velocities in bidisperse turbulent suspensions

## TL;DR

This paper studies how the relative velocities of particles of different sizes in turbulence are distributed, revealing power-law tails and their dependence on particle inertia, with implications for understanding particle interactions.

## Contribution

It introduces a statistical model for bidisperse turbulent suspensions and analyzes the distribution of relative velocities, highlighting non-analytic behavior of power-law exponents.

## Key findings

- Power-law tails occur when Stokes numbers are similar.
- Exponent depends non-analytically on mean Stokes number.
- Tail dominance persists even with larger Stokes-number differences.

## Abstract

We investigate the distribution of relative velocities between small heavy particles of different sizes in turbulence by analysing a statistical model for bidisperse turbulent suspensions, containing particles with two different Stokes numbers. This number, ${\rm St}$, is a measure of particle inertia which in turn depends on particle size. When the Stokes numbers are similar, the distribution exhibits power-law tails, just as in the case of equal ${\rm St}$. The power-law exponent is a non-analytic function of the mean Stokes number $\overline{\rm St}$, so that the exponent cannot be calculated in perturbation theory around the advective limit. When the Stokes-number difference is larger, the power law disappears, but the tails of the distribution still dominate the relative-velocity moments, if $\overline{\rm St}$ is large enough.