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Interactions of large amplitude solitary waves in viscous fluid conduits

Published online by Cambridge University Press:  11 June 2014

Nicholas K. Lowman ,
M. A. Hoefer  and
G. A. El
Nicholas K. Lowman
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
M. A. Hoefer*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
G. A. El
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
*
†Email address for correspondence: mahoefer@ncsu.edu
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Abstract

The free interface separating an exterior, viscous fluid from an intrusive conduit of buoyant, less viscous fluid is known to support strongly nonlinear solitary waves due to a balance between viscosity-induced dispersion and buoyancy-induced nonlinearity. The overtaking, pairwise interaction of weakly nonlinear solitary waves has been classified theoretically for the Korteweg–de Vries equation and experimentally in the context of shallow water waves, but a theoretical and experimental classification of strongly nonlinear solitary wave interactions is lacking. The interactions of large amplitude solitary waves in viscous fluid conduits, a model physical system for the study of one-dimensional, truly dissipationless, dispersive nonlinear waves, are classified. Using a combined numerical and experimental approach, three classes of nonlinear interaction behaviour are identified: purely bimodal, purely unimodal, and a mixed type. The magnitude of the dispersive radiation due to solitary wave interactions is quantified numerically and observed to be beyond the sensitivity of our experiments, suggesting that conduit solitary waves behave as ‘physical solitons’. Experimental data are shown to be in excellent agreement with numerical simulations of the reduced model. Experimental movies are available with the online version of the paper.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Waves/Free-surface Flows: Solitary waves Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 750 , 10 July 2014 , pp. 372 - 384
Copyright
© 2014 Cambridge University Press 

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Lowman et al. supplementary movie

Rescaled bimodal interaction

Download Lowman et al. supplementary movie(Video)
Video 3.5 MB

Lowman et al. supplementary movie

Bimodal interaction

Download Lowman et al. supplementary movie(Video)
Video 8.5 MB

Lowman et al. supplementary movie

Bimodal interaction, zoomed in and rescaled. Corresponds to the bimodal interaction in Fig. 4.

Download Lowman et al. supplementary movie(Video)
Video 2.4 MB

Lowman et al. supplementary movie

Bimodal interaction, zoomed in and unscaled. Corresponds to the bimodal interaction in Fig. 4.

Download Lowman et al. supplementary movie(Video)
Video 7.7 MB

Lowman et al. supplementary movie

Intermediate interaction, zoomed in and rescaled. Corresponds to the intermediate interaction in Fig. 4.

Download Lowman et al. supplementary movie(Video)
Video 1.1 MB

Lowman et al. supplementary movie

Intermediate interaction, zoomed in and unscaled. Corresponds to the intermediate interaction in Fig. 4.

Download Lowman et al. supplementary movie(Video)
Video 3.3 MB

Lowman et al. supplementary movie

Unimodal interaction, zoomed in and rescaled. Corresponds to the unimodal interaction in Fig. 4.

Download Lowman et al. supplementary movie(Video)
Video 1.3 MB

Lowman et al. supplementary movie

Unimodal interaction, zoomed in and unscaled. Corresponds to the unimodal interaction in Fig. 4.

Download Lowman et al. supplementary movie(Video)
Video 3.2 MB