cathalcummins
Nov30-11, 11:47 AM
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:
(r,\phi,z)\mapsto (\xi,\eta,\varphi;a)
such that
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
thus the z-axis is given by \eta=0, the contact curve is \eta=\infty and the base of the drop is \xi=0. The spherical cap free surface is then described by \xi=\alpha.
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!
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:
(r,\phi,z)\mapsto (\xi,\eta,\varphi;a)
such that
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
thus the z-axis is given by \eta=0, the contact curve is \eta=\infty and the base of the drop is \xi=0. The spherical cap free surface is then described by \xi=\alpha.
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!