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Rate invariance and scallop theorem in viscosity gradients

Christian Esparza López1,2 and Eric Lauga1,*

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
  • 2Biological Physics Theory Unit, OIST Graduate University, Okinawa 904-0495, Japan

  • *e.lauga@damtp.cam.ac.uk
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Phys. Rev. Fluids 8, 063301 – Published 9 June, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.063301

Abstract

Purcell's scallop theorem states that a swimmer deforming in a fluid at low Reynolds number cannot undergo a net displacement if its sequence of shapes is symmetric in time. Motivated by heterogeneous biological environments, here we consider a fluid with broken translational symmetry where the viscosity of the medium varies spatially. We show that, in a prescribed and smooth but otherwise arbitrary viscosity field, the swimmer's displacement is independent of its deformation rate and thus the scallop theorem continues to hold. Transport of the viscosity by the flow would therefore be key to enabling reciprocal locomotion in heterogeneous media.

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Phys. Rev. Fluids 8, 063301– Published 9 June, 2023
  • Received 26 March 2022
  • Accepted 12 May 2023
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