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Two-dimensional phoretic swimmers: the singular weak-advection limits

Published online by Cambridge University Press:  10 March 2017

Ehud Yariv Open the ORCID record for Ehud Yariv [Opens in a new window]
Ehud Yariv*
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: udi@technion.ac.il
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Abstract

Because of the associated far-field logarithmic divergence, the transport problem governing two-dimensional phoretic self-propulsion lacks a steady solution when the Péclet number $\mathit{Pe}$ vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of $\mathit{Pe}$, however small. We consider that problem employing a first-order kinetic model of solute absorption, where the ratio of the characteristic magnitudes of reaction and diffusion is quantified by the Damköhler number $\mathit{Da}$. As $\mathit{Pe}\rightarrow 0$ the dominance of diffusion breaks down at distances that scale inversely with $\mathit{Pe}$; at these distances, the leading-order transport represents a two-dimensional point source in a uniform stream. Asymptotic matching between the latter region and the diffusion-dominated near-particle region provides the leading-order particle velocity as an implicit function of $\log \mathit{Pe}$. Another scenario involving weak advection takes place under strong reactions, where $\mathit{Pe}$ and $\mathit{Da}$ are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as $\mathit{Da}^{2}/\mathit{Pe}$.

JFM classification

Complex Fluids: Colloids Complex Fluids: Complex Fluids Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 816 , 10 April 2017 , R3
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Copyright
© 2017 Cambridge University Press 

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