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:

$$(r,\phi,z)\mapsto (\xi,\eta,\varphi;a)$$

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!

 

[jvicens]

Hi there!

Thank you for sharing your problem and the coordinate transformation you are using. I have not personally worked with this specific coordinate system for deformations of free surfaces, but I can offer some insights and potential considerations.

First, it is important to consider the physical significance and applicability of the coordinate system you are using. While bispherical coordinates can be useful for certain problems in physics, they may not be the most appropriate choice for studying deformations of free surfaces. This is because free surfaces are typically described in terms of Cartesian coordinates, where the z-axis is perpendicular to the surface and the x and y axes lie on the surface. This allows for a more intuitive understanding of the surface and its deformations.

Additionally, using bispherical coordinates may introduce unnecessary complexity and difficulty in analyzing the deformations of the free surface. The equations and transformations involved may be more complicated and may not have a direct physical interpretation. It may also be more challenging to visualize and understand the deformations in this coordinate system.

That being said, there may be certain situations where bispherical coordinates could be useful for studying deformations of free surfaces. For example, if the free surface has a specific shape or symmetry that can be described using these coordinates, then it may be a valid choice. However, it is important to carefully consider the advantages and disadvantages of using this coordinate system for your specific problem.

In terms of describing surfaces other than planes, spheres, apples, or lemons, there are many different coordinate systems that can be used. Some other common coordinate systems for describing free surfaces include cylindrical coordinates, polar coordinates, and spherical coordinates. It may be worth exploring these alternatives and seeing if they offer any advantages for your problem.

Overall, while bispherical coordinates may not be the most common or intuitive choice for studying deformations of free surfaces, they may still have some potential applications. I would suggest further exploring the physical significance and limitations of this coordinate system in relation to your problem, and considering other coordinate systems that may be more suitable. I hope this helps and good luck with your research!