  Artificial phoretic particles swim using self-generated gradients in chemical
species (self-diffusiophoresis) or charges and currents (self-electrophoresis).
These particles can be used to study the physics of collective motion in active
matter and might have promising applications in bioengineering. In the case of
self-diffusiophoresis, the classical physical model relies on a steady solution
of the diffusion equation, from which chemical gradients, phoretic flows and
ultimately the swimming velocity, may be derived. Motivated by disk-shaped
particles in thin films and under confinement, we examine the extension to two
dimensions. Because the two-dimensional diffusion equation lacks a steady state
with the correct boundary conditions, Laplace transforms must be used to study
the long-time behavior of the problem and determine the swimming velocity. For
fixed chemical fluxes on the particle surface, we find that the swimming
velocity ultimately always decays logarithmically in time. In the case of
finite Peclet numbers, we solve the full advection-diffusion equation
numerically and show that this decay can be avoided by the particle moving to
regions of unconsumed reactant. Finite advection thus regularizes the
two-dimensional phoretic problem.
