# Stability of Soap Films

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.

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.

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.