Stability of Soap Films

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Cox & Jones (2014) investigated the stability of soap films trapped inside a rigid cylinder using experiments and numerical simulations. They set up a rectangular film trapped between two diameters and the curved cylinder walls of radius R. The distance H between the diameters and/or the relative twist θ between them was slowly increased. Initially the surface adopted the shape of a helicoid, but at larger values of the aspect ratio H/R and/or θ this became unstable. They found a critical aspect ratio, which decreased as θ increased.

Cox & Jones also looked at the case of multiple films meeting along the cylinder axis, and again found the stability boundary for. To good approximation, the stability boundary was found to be independent of the number of films present. Theoretical calculations were also present for case θ=0.


Illustrations of the setup with a twisted soap film inside a cylinder, for (a) a single film, and (b) multiple vanes (in this case N=3).

In a collaboration between myself and Simon Cox, we have extended the theoretical calculations in Cox & Jones (2014) to non-zero θ. The stability boundary we find is in good agreement with the earlier experiments and simulations. We find a new instability mechanism that allows the films to become unstable even in the absence of curved walls, provided θ>π/√2. We also prove that the multi-vane case does indeed have the same stability boundary as the single vane case.


The stability boundary in θ space, where θ is the twist angle and =L/H is the aspect ratio of the cylinder. The theoretical results (continuous line) are shown alongside numerical simulations of the full surface (circles). The dotted lines are the asymptotic solutions for large and small θ/.

In on-going work, we are continuing to investigate the effects of adding a line tension and/or a bending stiffness to the Plateau border in the above setup. It is expected that positive tensions and bending stiffnesses will act to stabilise the border, while either effect with a negative coefficient will inevitably lead to a short-wavelength instability. The most interesting (and realistic) case will be where there is a negative line tension and a positive bending stiffness. With the right parameters, this should destabilise longer-wavelength perturbations, but still be stable at short wavelengths.

Publications

Stability of a Helicoidal Surface inside a Cylinder with Pinned Diameters (pre-print)
Robert J. Whittaker & Simon Cox, 2015.
Quarterly Journal of Mechanics and Applied Mathematics, 68 (1), 23–52.
Stability of Twisted Plateau Border with Line Tension and Bending Stiffness
Robert J. Whittaker & Simon Cox.
In preparation.