Drag coefficient of a liquid domain with distinct viscosity in a fluid membrane

Hisasi Tani, Youhei Fujitani

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

Original languageEnglish
Pages (from-to)910-931
Number of pages22
JournalJournal of Fluid Mechanics
Volume836
DOIs
Publication statusPublished - 2018 Feb 10

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drag coefficients
Drag coefficient
Viscosity
viscosity
membranes
Membranes
Fluids
fluids
Liquids
unity
liquids
infinity
gradients
power series
continuity
velocity distribution
dipoles
coefficients

Keywords

  • drops
  • membranes

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering

Cite this

Drag coefficient of a liquid domain with distinct viscosity in a fluid membrane. / Tani, Hisasi; Fujitani, Youhei.

In: Journal of Fluid Mechanics, Vol. 836, 10.02.2018, p. 910-931.

Research output: Contribution to journalArticle

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AB - We calculate the drag coefficient of a circular liquid domain in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. As the ratio of the former viscosity to the latter increases to infinity, the drag coefficient of a liquid domain should approach that of the disk of the same size in the same ambient viscosities. This approach has not yet been shown explicitly, however. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. This means that we need to assume dipole singularity along the domain perimeter in solving the governing equations unless the ratio is unity. In the present study, we take this singularity into account and obtain the drag coefficient of a liquid domain as a power series with respect to a dimensionless parameter, which equals zero when the ratio is unity and approaches unity when the ratio tends to infinity. As the parameter increases to unity, the sum of the series is numerically shown to approach the drag coefficient of the disk.

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