Asymptotic dynamics of inertial particles with memory

TL;DR

This paper proves that inertial particles with memory in fluid flows exhibit universal algebraic decay to a limit, extending understanding of their long-term dynamics and providing new mathematical bounds and solution existence results.

Contribution

It establishes the universal algebraic decay property for the Maxey--Riley equation with memory and provides the first proof of global solution existence and uniqueness.

Findings

Particle velocity decays algebraically to a fluid velocity limit

Derived a sharp upper bound for particle velocity

Proved existence and uniqueness of solutions to the fractional-order system

Abstract

Recent experimental and numerical observations have shown the significance of the Basset--Boussinesq memory term on the dynamics of small spherical rigid particles (or inertial particles) suspended in an ambient fluid flow. These observations suggest an algebraic decay to an asymptotic state, as opposed to the exponential convergence in the absence of the memory term. Here, we prove that the observed algebraic decay is a universal property of the Maxey--Riley equation. Specifically, the particle velocity decays algebraically in time to a limit that is $\mathcal O(\epsilon)$-close to the fluid velocity, where $0<\epsilon\ll 1$ is proportional to the square of the ratio of the particle radius to the fluid characteristic length-scale. These results follows from a sharp analytic upper bound that we derive for the particle velocity. For completeness, we also present a first proof of existence and uniqueness of global solutions to the Maxey--Riley equation, a nonlinear system of fractional-order differential equations.