Ceniceros, Hector D. and Hou, Thomas Y. (1999) Dynamic generation of capillary waves. Physics of Fluids, 11 (5). pp. 1042-1050. ISSN 1070-6631. http://resolver.caltech.edu/CaltechAUTHORS:CENpof99a
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We investigate the dynamic generation of capillary waves in two-dimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the forward front of steep water waves. Although various experimental and analytical studies have contributed to the understanding of this physical phenomenon, the precise mechanism that generates the dynamic formation of capillary waves is still not well understood. Using a numerically stable and spectrally accurate boundary integral method, we perform a systematic study of the time evolution of breaking waves in the presence of surface tension. We find that the capillary waves originate near the crest in a neighborhood, where both the curvature and its derivative are maximum. For fixed but small surface tension, the maximum of curvature increases in time and the interface develops an oscillatory train of capillary waves in the forward front of the crest. Our numerical experiments also show that, as time increases, the interface tends to a possible formation of trapped bubbles through self-intersection. On the other hand, for a fixed time, as the surface tension coefficient tau is reduced, both the capillary wavelength and its amplitude decrease nonlinearly. The interface solutions approach the tau = 0 profile. At the onset of the capillaries, the derivative of the convection is comparable to that of the gravity term in the dynamic boundary condition and the surface tension becomes appreciable with respect to these two terms. We find that, based on the tau = 0 wave, it is possible to estimate a threshold value tau0 such that if tau <= tau0 then no capillary waves arise. On the other hand, for tau sufficiently large, breaking is inhibited and pure capillary motion is observed. The limiting behavior is very similar to that in the classical KdV equation. We also investigate the effect of viscosity on the generation of capillary waves. We find that the capillary waves still persist as long as the viscosity is not significantly greater than surface tension.
|Additional Information:||©1999 American Institute of Physics. (Received 28 April 1998; accepted 27 January 1999) We would like to thank Professor Bjorn Engquist, Professor Phillip Saffman, and Professor Ted Wu for many helpful discussions regarding this work. We would also like to thank the referees for their suggestions and comments. The research was supported in part by an Office of Naval Research grant No. N00014-96-1-0438 and by a National Science Foundation grant No. DMS-9704976.|
|Subject Keywords:||capillary waves; water waves; surface tension; wave equations|
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|Deposited On:||03 Mar 2007|
|Last Modified:||26 Dec 2012 09:32|
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