Phoretic self-propulsion of a slightly inhomogeneous disc

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 Damkohler number D alpha and the intrinsic Peclet number 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 D alpha-dependent critical value of Pe. We also address the respective limits of small and large D alpha for fixed Pe and arbitrary inhomogeneity, and illuminate their linkage to the limit of weak inhomogeneity.

Automated summary

This study addresses the well-posed problem of a homogeneous phoretic swimmer with circular cross-section in two dimensions. By introducing a slight inhomogeneity, diffusio-osmosis is used to animate the phoretic motion. The singular problem associated with the scale disparity is analyzed using matched asymptotic expansions, which provide an implicit equation for the particle velocity. The velocity exhibits non-trivial dependence on the sign of the slip coefficient.