Several years ago, I was a trainee in a physics lab where I was supposed to study atomisation in sprays (ensemble of liquid drops). As we did observe that the drops tended to adopt a spherical shape over time, I thought that maybe some kind of curvature diffusion was occurring during this process. What I would like to know is whether the Ricci flow can model the evolution of the shape of the drops, and if some reference in mathematical physics on this subject is available. Feel free to tell me if this question should be asked on physics.stackexchange.com instead.
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asked Oct 25, 2016 at 12:39
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1 Answer
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For shapes of liquid drops, it is probably not driven by Ricci flow.
Fluid interfaces with surface tension is better modeled by mean curvature, going back to Young and Laplace; and there is a lot of current work on structure of interfaces when coupled with fluid equations on either side.
What you are looking for is probably described as a free boundary problem where the boundary evolves by mean curvature flow, but with a source term coming from the outward pressure of the drop, coupled to (possibly some simplified version of the) fluid equations in the interior which will help guarantee that the droplet doesn't shrink to zero. (Notice that finite time extinction would also be a problem with Ricci flow.) In the incompressible case, for example, your fluid equations will guarantee that the interior volume remains constant.
answered Oct 25, 2016 at 13:43