Phoretic self-propulsion of a slightly inhomogeneous disc

Abstract

In two dimensions, the problem governing a homogeneous phoretic swimmer of circular cross-section is ill-posed because of the logarithmic divergence associated with a purely diffusive solute transport. We address here the well-posed problem that is devised by introducing a slight inhomogeneity in the interfacial chemical activity. With the radial symmetry being perturbed, phoretic motion is animated by diffusio-osmosis. Solute advection, associated with that motion, becomes comparable to diffusion at large distances. The singular problem associated with that scale disparity is analysed using matched asymptotic expansions for arbitrary values of the Damköhler number $\textit {Da}$ and the intrinsic Péclet number $\textit {Pe}$ . Asymptotic matching provides an implicit equation for the particle velocity in terms of these two parameters. The velocity exhibits a non-trivial dependence upon the sign $M$ of the slip coefficient. For $M=-1$ , we observe the appearance of several solutions beyond a $\textit {Da}$ -dependent critical value of $\textit {Pe}$ . We also address the respective limits of small and large $\textit {Da}$ for fixed $\textit {Pe}$ and arbitrary inhomogeneity, and illuminate their linkage to the limit of weak inhomogeneity.