On the analytical aspects of inertial particle motion

TL;DR

This paper provides a comprehensive theoretical analysis of the Maxey-Riley equation for inertial particle motion, establishing global existence, uniqueness, differentiability conditions, and sensitivity properties, with implications for fractional differential equations.

Contribution

It extends previous local results to global solutions, proves differentiability at initial time, and analyzes sensitivity, enhancing understanding of inertial particle dynamics modeled by fractional equations.

Findings

Solutions are globally well-defined in time.

Solutions are differentiable at initial conditions.

Inertial trajectories cannot intersect.

Abstract

In their seminal 1983 paper, M. Maxey and J. Riley introduced an equation for the motion of a sphere through a fluid. Since this equation features the Basset history integral, the popularity of this equation has broadened the use of a certain form of fractional differential equation to study inertial particle motion. In this paper, we give a comprehensive theoretical analysis of the Maxey-Riley equation. In particular, we build on previous local in time existence and uniqueness results to prove that solutions of the Maxey-Riley equation are global in time. In doing so, we also prove that the notion of a maximal solution extends to this equation. We furthermore prove conditions under which solutions are differentiable at the initial time. By considering the derivative of the solution with respect to the initial conditions, we perform a sensitivity analysis and demonstrate that two inertial trajectories can not meet, as well as provide a control on the growth of the distance between a pair of inertial particles. The properties we prove here for the Maxey-Riley equations are also possessed, mutatis mutandis, by a broader class of fractional differential equations of a similar form.