A Scalar Version of the Caflisch‐Luke Paradox

Consider an infinite cloud of hard spheres sedimenting in a Stokes flow in the whole space ℝd. 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

• Publication year

  2019

• Venue

  Communications on Pure and Applied Mathematics

• Publication date

  2019-07-18

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1002/cpa.21970arXiv 1907.08182
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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