Theoretical modeling of walker interactions with slowly varying topography

A droplet may bounce indefinitely on the surface of an oscillating fluid bath. Beyond a critical forcing acceleration, the bouncing state is destabilized by the underlying wavefield, causing the droplet to be propelled horizontally. Many interesting phenomena arise when these walking droplets interact with abrupt changes in bottom topography. The model of Faria (2017) permits a theoretical treatment of such problems. We here present the results of an integrated theoretical and experimental study into the motion of a walker on a bath whose depth varies slowly in space. In this limit, it is possible to derive an asymptotic correction to the integral stroboscopic wave model derived by Oza et al. (2013). We compare the theoretical predictions of this model with experiments of droplets walking on a bath with a bottom varying linearly; firstly in a single direction, so that the bottom forms a linear slope; and secondly in the radial direction, so that the bottom forms an inverted cone. The latter configuration gives rise to circular orbits, which destabilize into precessing orbits and more complex motion as the forcing acceleration is increased progressively.