Capillary Floating and the Billiard Ball Problem

Authors: Eugene Gutkin

Publish Date: 2011/08/16

Volume: 14, Issue: 2, Pages: 363-382

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Abstract

In a study of capillary floating Finn J Math Fluid Mech 11443–458 2009 described a procedure for determining crosssections of noncircular infinite convex cylinders that float horizontally on a liquid surface in every orientation with contact angle π/2 Finn’s procedure yielded incomplete results for other contact angles he raised the question as to whether an analogous construction would be feasible in that case In the note Finn J Math Fluid Mech 11464–465 2009 pointed out a connection with an independent problem on billiard caustics citing the unpublished work Gutkin in Proceedings of the Workshop on Dynamics and Related Questions PennState University 1993 of the present author Here we present a solution of the billiard problem in full detail thus settling Finn’s question in a surprising way In particular we show that such floating cylinders exist if and only if the contact angle lies in a certain explicitly described countably dense set Moreover for each element γ in this set we exhibit a family of convex noncircular cylinders that float in every orientation with contact angle γ Our discussion contains other material of independent interest for the billiard ball problemThis article is published under an open access license Please check the Copyright Information section for details of this license and what reuse is permitted If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and reuse information please contact the Rights and Permissions team

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