Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T17:51:54.306Z Has data issue: false hasContentIssue false

Linearized planing-surface theory with surface tension. Part II: Detachment with discontinuous slope

Published online by Cambridge University Press:  17 February 2009

E. O. Tuck
Affiliation:
Department of Applied Mathematics, University of Adelaide, Box 498, G.P.O., Adelaide, South Australia 5001
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In Part I of this series, surface tension was included in the classical two-dimensional planing-surface problem, and the usual smooth-detachment trailing-edge condition enforced. However, the results exhibited a paradox, in that the classical results were not approached in the limit as the surface tension approached zero. This paradox is resolved here by abandoning the smooth-detachment condition, that is, by allowing a jump discontinuity in slope between the planing surface and the free surface at the trailing edge. A unique solution is obtainable for any input planing surface at fixed wetted length if one allows such jumps at both leading and trailing edges. If, as is the case in practice, the wetted length is allowed to vary, uniqueness may be restored by requiring either, but not both, of these slope discontinuities to vanish. The results of Part I correspond to the seemingly more-natural choice of making the trailing-edge detachment continuous, but it appears that the correct choice is to require the leading-edge attachment to be continuous.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Ackerberg, R. C., “The effects of capillarity on free-streamline separation”, J. Fluid Mech. 70 (1975), 333352.CrossRefGoogle Scholar
[2]Cumberbatch, E. and Norbury, J., “Capillary modification of the singularity at a free-streamline separation point”, Quart. J. Mech. Appl. Math. 32 (1979), 303312.CrossRefGoogle Scholar
[3]Milne, L. J. and Milne, M., “Insects of the water surface”, Scientific American (04 1978), 134142.CrossRefGoogle Scholar
[4]Oertel, R. P., The steady motion of a flat ship including an investigation of local flow near the bow (Ph.D. Thesis, University of Adelaide, 1974).Google Scholar
[5]Tuck, E. O., “Linearized planing-surface theory with surface tension. Part I: Smooth detachment”, J. Austral. Math. Soc. Ser. B (1982), 241258.CrossRefGoogle Scholar
[6]Broeck, J.-M. Vanden, “The influence of capillarity on cavitating flow past a flat plate”, Quart. J. Mech. Appl. Math. 34 (1981), (in press).Google Scholar
[7]Wehausen, J. V. and Laitone, E. V., “Surface waves”, in Handb. der Physik, Vol. 9, ed. Flugge, S. (Springer-Verlag, Berlin, 1960).Google Scholar