Shapes of a filament on the surface of a bubble.

The shape assumed by a slender elastic structure is a function both of the geometry of the space in which it exists as well as the forces it experiences. Here, we explore by experiments, a calculation, and numerical analysis, the morphological phase-space of a filament confined to the surface of a spherical bubble. The morphology is controlled by varying the bending stiffness and weight of the filament, and its length relative to the radius of the bubble. In the regime where the dominant considerations are the geometry of confinement and the elastic energy, we observe that the filament lies along a geodesic. When gravitational energy becomes significant, a bifurcation occurs with a part of the filament occupying a longitude and the rest occupying a curve that may be approximated by a latitude. Far from the transition, when the filament is long compared to the diameter, it coils around the selected latitude. A simple model that describes the shape as a composite of two arcs that lie along longitude and latitude, captures the transition. For better quantitative agreement with the subcritical nature of the bifurcation, we study the morphology by numerical energy minimization. Our observations and analysis of a filament on a sphere in terms of a morphological space spanned by a purely geometric parameter, and a parameter that compares elastic energy with body forces, may provide guidance for packing slender structures on more complex surfaces.