{
"id": "1907.08182",
"version": "v1",
"published": "2019-07-18T17:43:31.000Z",
"updated": "2019-07-18T17:43:31.000Z",
"title": "A scalar version of the Caflish-Luke paradox",
"authors": [
"Antoine Gloria"
],
"comment": "36 pages",
"categories": [
"math.AP",
"math-ph",
"math.MP",
"math.PR"
],
"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 Caflish 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 Caflish-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 Caflish and Luke, and opening a way to the systematic study of such problems.",
"revisions": [
{
"version": "v1",
"updated": "2019-07-18T17:43:31.000Z"
}
],
"analyses": {
"keywords": [
"caflish-luke paradox",
"scalar version",
"hard spheres",
"dilute point process",
"effective velocity"
],
"note": {
"typesetting": "TeX",
"pages": 36,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}