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Boundary element methods for particles and microswimmers in a linear viscoelastic fluid

Published online by Cambridge University Press:  13 October 2017

Kenta Ishimoto Open the ORCID record for Kenta Ishimoto [Opens in a new window]  and
Eamonn A. Gaffney
Kenta Ishimoto*
Affiliation:
The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Eamonn A. Gaffney
Affiliation:
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
*
†Email address for correspondence: ishimoto@kurims.kyoto-u.ac.jp
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Abstract

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Boundary integral methods Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 831 , 25 November 2017 , pp. 228 - 251
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Copyright
© 2017 Cambridge University Press 

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