Motion of a disk embedded in a nearly-inviscid Langmuir film

The translation 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 semi-analytic solution of the dual integral equations governing the flow at arbitrary Boussinesq numbers was devised by Hughes, Pailthorpe & White (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit, it produces the value 8 for the dimensionless translational drag, which is 50% larger than the classical 16/3-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 naive 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. Utilizing the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the requisite drag correction.