Theoretical and experimental study of the stability of a soap film spanning a flexible loop

Abstract

A variational model is used to study the stability of a soap film spanning a flexible loop. The film is modeled as a fluid surface endowed with constant tension and the loop is modeled as an elastic rod resistant to both bending and twist. The first and second energy variations of the underlying energy functional are derived, leading to governing equilibrium equations and stability criteria. The latter criteria are employed to explore the stability of flat circular configurations with respect to planar and transverse perturbations. Experiments are performed to study post-buckled configurations and to explore the validity of the theoretical predictions. Imparting a twist to the bounding loop destabilizes flat circular loops which are short-enough to be stable when twist-free. Nevertheless, when the circular disk is perturbed merely radially or merely transversely, the stability condition is indifferent to twist. By the equilibrium equations, the twisting angle of a planar equilibrium configuration should be distributed uniformly along the loop. The stability criteria guarantee that this uniformity is preserved after buckling. The stability with respect to increasing the surface tension of the film or the length of the loop is sensitive to the cross-sectional thickness of the bounding loop. However, stability with respect to increasing the total twist is indifferent to that thickness.