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

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 Pe vanishes. This indeterminacy, which has no counterpart in three dimensions, is remedied by introducing a non-zero value of 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 Da. As Pe!0 the dominance of diffusion breaks down at distances that scale inversely with 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 Pe. Another scenario involving weak advection takes place under strong reactions, where Pe and Da are large and comparable. In that limit, the breakdown of diffusive dominance occurs at distances that scale as Da2=Pe.