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A reduced model for a phoretic swimmer

Published online by Cambridge University Press:  18 November 2022

A. Farutin ,
M.S. Rizvi Open the ORCID record for M.S. Rizvi [Opens in a new window] ,
W.-F. Hu ,
T.S. Lin Open the ORCID record for T.S. Lin [Opens in a new window] ,
S. Rafaï  and
C. Misbah Open the ORCID record for C. Misbah [Opens in a new window]
A. Farutin
Affiliation:
University of Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
M.S. Rizvi
Affiliation:
University of Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
W.-F. Hu
Affiliation:
Department of Mathematics, National Central University, Taoyuan 32001, Taiwan
T.S. Lin
Affiliation:
Department of Applied Mathematics, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan
S. Rafaï
Affiliation:
University of Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
C. Misbah*
Affiliation:
University of Grenoble Alpes, CNRS, LIPhy, F-38000 Grenoble, France
*
†Email address for correspondence: chaouqi.misbah@univ-grenoble-alpes.fr
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Abstract

We consider a two-dimensional (2-D) model of an autophoretic particle. Beyond a certain emission/absorption rate (characterized by a dimensionless Péclet number, $Pe$) the particle is known to undergo a bifurcation from a non-motile to a motile state, with different trajectories, going from a straight to a chaotic motion by increasing $Pe$. From the full model, we derive a reduced closed model which involves only two time-dependent complex amplitudes $C_1(t)$ and $C_2(t)$ corresponding to the first two Fourier modes of the solute concentration field. It consists of two coupled nonlinear ordinary differential equations for $C_1$ and $C_2$ and presents several advantages: (i) the straight and circular motions can be handled fully analytically; (ii) complex motions such as chaos can be analysed numerically very efficiently in comparison with the numerically expensive full model involving partial differential equations; (iii) the reduced model has a universal form dictated only by symmetries (meaning that the form of the equations does not depend on a given phoretic model); (iv) the model can be extended to higher Fourier modes. The derivation method is exemplified for a 2-D model, for simplicity, but can easily be extended to three dimensions, not only for the presently selected phoretic model, but also for any model in which chemical activity triggers locomotion. A typical example can be found, for example, in the field of cell motility involving acto-myosin kinetics. This strategy offers an interesting way to cope with swimmers on the basis of ordinary differential equations, allowing for analytical tractability and efficient numerical treatment.

JFM classification

Biological Fluid Dynamics: Micro-organism dynamics Biological Fluid Dynamics: Swimming/flying Complex Fluids: Active matter
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 952 , 10 December 2022 , A6
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

‡

Present address: Department of Biomedical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, 502285, Telangana, India.

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