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A New Approach to Free Surface Euler Flows with Capillarity
Author(s) -
Crowdy Darren G.
Publication year - 2000
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00141
Subject(s) - surface tension , connection (principal bundle) , euler's formula , capillary action , bubble , capillary wave , free surface , surface (topology) , mathematics , calculus (dental) , classical mechanics , mathematical analysis , mechanics , physics , geometry , thermodynamics , medicine , dentistry
This article attempts to elucidate the underlying mathematical connection between the well‐known exact solutions for the deep water capillary wave problem [ G.D. Crapper , J. Fluid Mech. , 2:532–540 (1957)] and the recent discovery of a very special polar decomposition of solutions for a steadily translating bubble with surface tension [ S. Tanveer , Proc.Roy. Soc. A , 452:1397–1410 (1996)]. This is achieved by describing a new and unified mathematical approach to the two separate physical problems. Using the new approach, Crapper's capillary wave solutions are retrieved in a novel and simplified fashion, while additional analytical insight into the nature of solutions for a steadily‐translating bubble is obtained. The new approach is quite general and can also be used to obtain new exact results to other related free surface problems.