Drag on a partially immersed sphere at the capillary scale*

We study the drag on a centimetric sphere in a uniform flow in the presence of a free surface as a function of submergence depth. Through direct force measurements in a custom benchtop recirculating flume, we demonstrate that the drag can significantly exceed the corresponding drag in a single-phase flow and achieves a peak at submergence depths just prior to complete immersion. The additional drag in the partially immersed state is rationalized by considering hydrostatic effects associated with the asymmetric surface height profile induced by the obstacle in the flow direction which persists for flow speeds below the minimum capillary-gravity wave speed. At these scales, the sphere's wettability plays a pronounced role in determining the maximum possible drag and results in hysteretic behaviors near touchdown and complete immersion. The influence of flow speed, sphere size, and surface tension on the drag characteristics are additionally explored through a combination of experiments and numerical simulations.