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Hydrodynamic slender-body theory for local rotation at zero Reynolds number

Benjamin J. Walker1,2,*, Kenta Ishimoto3,†, and Eamonn A. Gaffney4,‡

  • 1Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
  • 2Department of Mathematics, University College London, London, WC1H 0AY, United Kingdom
  • 3Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
  • 4Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom

  • *Corresponding author: bjw43@bath.ac.uk
  • †ishimoto@kurims.kyoto-u.ac.jp
  • ‡gaffney@maths.ox.ac.uk
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Phys. Rev. Fluids 8, 034101 – Published 2 March, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.034101

Abstract

Slender objects are commonplace in microscale flow problems, from soft deformable sensors to biological filaments such as flagella and cilia. While much research has focused on the local translational motion of these slender bodies, relatively little attention has been given to local rotation, even though it can be the dominant component of motion. In this study, we explore a classically motivated ansatz for the Stokes flow around a rotating slender body via superposed rotlet singularities, which leads us to pose an alternative ansatz that accounts for both translation and rotation. Through an asymptotic analysis that is supported by numerical examples, we determine the suitability of these flow ansatzes for capturing the fluid velocity at the surface of a slender body, assuming local axisymmetry of the object but allowing the cross-sectional radius to vary with arclength. In addition to formally justifying the presented slender-body ansatzes, this analysis reveals a markedly simple relation between the local angular velocity and the torque exerted on the body, which we term resistive torque theory. Though reminiscent of classical resistive force theories, this local relation is found to be algebraically accurate in the slender-body aspect ratio, even when translation is present, and is valid and required whenever local rotation contributes to the surface velocity at leading asymptotic order.

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Phys. Rev. Fluids 8, 034101– Published 2 March, 2023
  • Received 3 November 2022
  • Accepted 1 February 2023
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