Coiling of an elastic filament on a spherical bubble

When a thin elastic filament is placed on top of a spherical bubble, it buckles under the bubble's Laplace pressure. When the elasto-capillary length scale, defined as the length at which bending force is balanced by capillary force, is much smaller than the bubble size the filament is confined to the bubble surface. In this scale-separated regime we ask what shape does a filament take? We find in our experiments that the geodesic minimises bending energy up to a critical filament length for a fixed bubble size beyond which the weight of the filament affects its shape. As we increase the ratio of filament length to bubble radius it shifts from a geodesic to take a path approximated by a latitude at a given polar angle. The filament then starts coiling around the latitude at this fixed polar angle. We construct a geometric model that describes our experimental observations and obtain a two-parameter coiling phase-diagram with geometry described by the ratio of filament length to bubble size and the ratio of bending force to filament weight.