Motion of a disk embedded in a nearly inviscid Langmuir film. Part 1. Translation

Ehud Yariv, Rodolfo Brandão, Michael Siegel, Howard A. Stone

Research output: Contribution to journalArticlepeer-review

Abstract

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number,. A semianalytic solution of the dual integral equations governing the flow at arbitrary was devised by Hughes et al. (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit, it produces the value for the dimensionless translational drag, which is larger than the classical -value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction, being the Euler-Mascheroni constant.

Original languageEnglish (US)
Article numberA30
JournalJournal of Fluid Mechanics
Volume977
DOIs
StatePublished - Dec 18 2023

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

Keywords

  • thin films

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