A Scalar Version of the Caflisch‐Luke Paradox

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