Self-propulsion of chemically-active droplets along a wall

Chemically-active droplets exploit the non-linear coupling of the transport of a chemical solute produced at their surface to the Marangoni flows such solute generates, in order to swim at the micron scale. The onset of self-propulsion above a critical advective-to-diffusive transport ratio (Péclet number) is now well established in unbounded flows. Although it is an intrinsic feature of experimental systems, confinement by a rigid wall and its influence on the propulsion is systematically absent in active droplet models. Using non-axisymmetric bi-spherical coordinates, we therefore investigate the onset of self-propulsion of active droplets hovering near a rigid wall. To do so, we characterize the axisymmetric hovering state and analyse its stability with respect to horizontal (wall-parallel) swimming, for varying wall-to-drop distance and Péclet number.