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

Published online by Cambridge University Press:  13 December 2017

Hisasi Tani Open the ORCID record for Hisasi Tani [Opens in a new window]  and
Youhei Fujitani
Hisasi Tani*
Affiliation:
Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, Kawasaki 214-8571, Kanagawa, Japan School of Fundamental Science and Technology, Keio University, Yokohama 223-8522, Kanagawa, Japan
Youhei Fujitani
Affiliation:
School of Fundamental Science and Technology, Keio University, Yokohama 223-8522, Kanagawa, Japan
*
†Email address for correspondence: hisasitani@gmail.com
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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.

JFM classification

Drops and Bubbles: Drops Biological Fluid Dynamics: Membranes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 910 - 931
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Copyright
© 2017 Cambridge University Press 

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