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A Scalar Version of the Caflisch‐Luke Paradox
Author(s) -
Gloria Antoine
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21970
Subject(s) - randomness , mathematics , scalar (mathematics) , stokes flow , spheres , dimension (graph theory) , flow (mathematics) , mathematical analysis , calculus (dental) , pure mathematics , physics , geometry , statistics , astronomy , medicine , dentistry
Abstract Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space 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 . © 2020 Wiley Periodicals LLC