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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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References

Bearman, P. W., Downie, M. J., Graham, J. M. R. & Obasaju, E. 1985 Forces on cylinders in viscous oscillatory flows at low Keulegan—Carpenter numbers. J. Fluid Mech. 154, 337.Google Scholar
Chaplin, J. R. 1984a Mass transport around a horizontal cylinder beneath waves. J. Fluid Mech. 140, 175.Google Scholar
Chaplin, J. R. 1984b Nonlinear forces on a horizontal cylinder beneath waves. J. Fluid Mech. 147, 449.Google Scholar
Chorin, A. J. 1973 Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785.Google Scholar
Chorin, A. J. 1978 Vortex sheet approximation of boundary layers. J. Comput. Phys. 27, 428.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Macmillan.
Ogilvie, T. F. 1963 First and second order forces on a cylinder submerged under a free surface. J. Fluid Mech. 16, 451.Google Scholar
Riley, N. 1971 Stirring of a viscous fluid. Z. Angew. Math. Phys. 29, 439.Google Scholar
Sarpkaya, T. 1986 Force on a cylinder in viscous oscillatory at low Keulegan—Carpenter number. J. Fluid Mech. 165, 61.Google Scholar
Smith, P. A. 1986 Computation of viscous flows by the vortex method. Ph.D. thesis, University of Manchester.
Smith, P. A. & Stansby, P. K. 1988 Impulsively started flow around a circular cylinder by the vortex method. J. Fluid Mech. 194, 45.Google Scholar
Smith, P. A. & Stansby, P. K. 1989 An efficient surface algorithm for random-particle simulation of vorticity and heat transport. J. Comput. Phys. 81, 349.Google Scholar
Smith, P. A. & Stansby, P. K. 1991 Viscous oscillatory flows around cylindrical bodies at low Keulegan—Carpenter numbers using the vortex method. J. Fluids Structures (to appear).Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of a pendulum. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Wang, C. Y. 1968 On high frequency oscillating viscous flows. J. Fluid Mech. 32, 55.Google Scholar