A scalar version of the Caflisch-Luke paradox

TL;DR

This paper introduces a simplified scalar model of the Caflisch-Luke paradox, proving the existence of effective sedimentation velocity in certain dimensions and finite variance in higher dimensions, thus advancing understanding of sedimentation in random particle systems.

Contribution

It provides the first rigorous proof of the effective velocity's existence and finite variance in a scalar model, clarifying the paradox in sedimentation of random particle clouds.

Findings

Effective velocity is well-defined in dimensions d>2.

Variance of sedimentation speed is finite in dimensions d>4.

Supports formal predictions by Caflisch and Luke.

Abstract

Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space $\mathbb R^d$. Despite many contributions in fluid mechanics and applied mathematics, there is so far no rigorous definition of the associated effective sedimentation velocity. Calculations by Caflisch and Luke in dimension $d=3$ suggest that the effective velocity is well-defined for hard spheres distributed according to a weakly correlated and dilute point process, and that the variance of the sedimentation speed is infinite. This constitutes the Caflisch-Luke paradox. In this contribution, we consider a scalar version of this problem that displays the same difficulties in terms of interaction between the differential operator and the randomness, but is simpler in terms of PDE analysis. For a class of hardcore point processes we rigorously prove that the effective velocity is well-defined in dimensions $d>2$, and that the variance is finite in dimensions $d>4$, confirming the formal calculations by Caflisch and Luke, and opening a way to the systematic study of such problems.