TY - JOUR

T1 - Motion of a disk embedded in a nearly inviscid Langmuir film. Part 1. Translation

AU - Yariv, Ehud

AU - Brandão, Rodolfo

AU - Siegel, Michael

AU - Stone, Howard A.

N1 - Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press.

PY - 2023/12/18

Y1 - 2023/12/18

N2 - 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.

AB - 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.

KW - thin films

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U2 - 10.1017/jfm.2023.954

DO - 10.1017/jfm.2023.954

M3 - Article

AN - SCOPUS:85180978303

SN - 0022-1120

VL - 977

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

M1 - A30

ER -