Torque-driven superhydrophobic cylinders swim in circles
File(s)
Author(s)
Crowdy, Darren
Type
Journal Article
Abstract
A superhydrophobic cylinder is a circular cylinder
with a boundary pattern of alternating shear-free
menisci and solid portions. This paper derives
analytical solutions for the torque-driven Stokes flow
around a superhydrophobic cylinder that is free
to move, or “swim”, subject to the constraint that
it is free of net force. The mathematical approach
is a natural generalization of one used to solve
for transverse shear flow over a superhydrophobic
surface comprising a periodic array of no-shear slots
[Crowdy, Phys. Fluids, (2011)]. First, the torque-driven
Stokes flow around a superhydrophobic cylinder with
two unequal no-shear menisci and two equal solid
portions is solved in detail. It is then shown how
that analysis generalizes, in a natural way, to a
cylinder with M no-shear menisci between M solid
portions with full exposition given of the case of
M = 3 equal solid portions. A further generalization
to flow around a superhydrophobic cylinder with
an N-fold rotationally symmetric patterning with
M menisci in each 2π/N sector is made. Torque driven superhydrophobic cylinders with a no-shear
pattern devoid of any rotational symmetry are shown
to swim on circular trajectories with formulas for
the radius and cylinder angular velocity on these
trajectories derived here as functions of the no shear pattern geometry. The M = 1 case of a “Janus”
superhydrophobic cylinder having one meniscus and
one solid portion can be solved completely explicitly.
with a boundary pattern of alternating shear-free
menisci and solid portions. This paper derives
analytical solutions for the torque-driven Stokes flow
around a superhydrophobic cylinder that is free
to move, or “swim”, subject to the constraint that
it is free of net force. The mathematical approach
is a natural generalization of one used to solve
for transverse shear flow over a superhydrophobic
surface comprising a periodic array of no-shear slots
[Crowdy, Phys. Fluids, (2011)]. First, the torque-driven
Stokes flow around a superhydrophobic cylinder with
two unequal no-shear menisci and two equal solid
portions is solved in detail. It is then shown how
that analysis generalizes, in a natural way, to a
cylinder with M no-shear menisci between M solid
portions with full exposition given of the case of
M = 3 equal solid portions. A further generalization
to flow around a superhydrophobic cylinder with
an N-fold rotationally symmetric patterning with
M menisci in each 2π/N sector is made. Torque driven superhydrophobic cylinders with a no-shear
pattern devoid of any rotational symmetry are shown
to swim on circular trajectories with formulas for
the radius and cylinder angular velocity on these
trajectories derived here as functions of the no shear pattern geometry. The M = 1 case of a “Janus”
superhydrophobic cylinder having one meniscus and
one solid portion can be solved completely explicitly.
Date Issued
2024-10
Date Acceptance
2024-08-23
Citation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2024, 480 (2300)
ISSN
1364-5021
Publisher
The Royal Society
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
480
Issue
2300
Copyright Statement
© 2024 The Author(s).
Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
License URL
Identifier
https://royalsocietypublishing.org/doi/10.1098/rspa.2024.0149
Publication Status
Published
Article Number
20240149
Date Publish Online
2024-10-23