Spontaneous autophoretic motion of isotropic disks

It is theoretically known that a chemically isotropic active sphere in an unbounded solution undergoes symmetry breaking when the intrinsic Péclet number Pe exceeds a finite critical value (Michelin et al., Phys. Fluids, 2013), marking a transition from a stationary state to spontaneous motion. In two dimensions, a linear stability analysis in a large bounded domain reveals that the critical 𝑃𝑒 value diminishes as the domain size increases (Hu et al., Phys. Rev. Lett., 2019). Motivated by those findings, we consider an unbounded domain from the outset, addressing the two-dimensional problem of steady self-propulsion with a focus on the limit of small Pe. This singular limit is handled using matched asymptotic expansions, decomposing the fluid domain into a particle-scale region, where the solute transport is primarily diffusive, and a remote region, where advection and diffusion are comparable. We derive a closed-form approximation for the particle speed, which reveals that in two dimensions the spontaneous motion of isotropic particles can occur for any nonzero Pe, in contrast to the three-dimensional case.