Perturbation to free surface in bispherical coordinates

cathalcummins

Hi there!


I'm working on a physics problem where there is a liquid droplet (not necessarily spherical) on a plane. Transforming from cylindrical coordinates to bispherical:

[tex](r,\phi,z)\mapsto (\xi,\eta,\varphi;a)[/tex]

such that

[tex]r={\frac {a\sin \left( \eta \right) }{\cosh
\left( \eta \right) +\cos \left( \xi \right) }};\phantom{space}z={\frac {a\sinh \left( \xi \right) }{\cosh \left( \eta \right) +\cos
\left( \xi \right) }}; \phantom{space}\phi=\varphi[/tex]

thus the [itex]z-[/itex]axis is given by [itex]\eta=0[/itex], the contact curve is [itex]\eta=\infty[/itex] and the base of the drop is [itex]\xi=0[/itex]. The spherical cap free surface is then described by [itex]\xi=\alpha[/itex].

I'm looking to describe deformations to the sphere's surface in this coordinate system: has anyone tried to describe surfaces other than planes, spheres, apples or lemons? Or, can someone point out why this would be a bad choice of coordinate system to perform deformations to free surfaces in?

Thanks!