Stability of a helicoidal surface inside a cylinder with pinned diameters

A mathematical analysis is presented of the stability of a soap film with uniform surface tension when stretched between two diameters on the inside of a circular cylinder. The stability boundary is found as a critical twist angle θ between the two diameters, as a function of the aspect ratio l of the cylinder. Numerical and asymptotic results agree well with previous numerical simulations and experiments by Cox & Jones (J.\ Engr Math, 2014, 86, 1-7). Their hypothesis that the stability boundary for the multiple-vane case is identical to the single film case is confirmed. It is also shown that two distinct instability mechanisms operate. For moderate and small θ/l, the instability is driven by the decrease in area caused by the film moving to an off-diameter position. But for larger θ/l (more twisted films), the decrease in area is dominated by an internal rearrangement of the surface. The latter mechanism is more relevant to Plateau borders in foams, and our results indicate that straight Plateau borders should be stable at any length provided the total twist is less than π/√2.