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Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant

Published online by Cambridge University Press:  26 April 2006

S. W. Joo ,
A. F. Messiter  and
W. W. Schultz
S. W. Joo
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, Universitv of Michigan. Ann Arbor, MI 48109–2125, USA
A. F. Messiter
Affiliation:
Department of Aerospace Engineering, University of Michigan, Ann Arbor. MI 48109, USA
W. W. Schultz
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, Universitv of Michigan. Ann Arbor, MI 48109–2125, USA
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Abstract

A formal derivation of evolution equations is given for viscous gravity waves and viscous capillary—gravity waves with surfactants in water of infinite depth. Multiple scales are used to describe the slow modulation of a wave packet, and matched asymptotic expansions are introduced to represent the viscous boundary layer at the free surface. The resulting dissipative nonlinear Schrödinger equations show that the largest terms in the damping coefficients are unaltered from previous linear results up to third order in the amplitude expansions. The modulational instability of infinite wavetrains of small but finite amplitude is studied numerically. The results show the effect of viscosity and surfactants on the Benjamin—Feir instability and subsequent nonlinear evolution. In an inviscid limit for capillary—gravity waves, a small-amplitude recurrence is observed that is not directly related to the Benjamin—Feir instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 135 - 158
Copyright
© 1991 Cambridge University Press

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