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In this paper, we derive a viscous generalization of the Dysthe (1979) system from the weakly viscous generalization of the Euler equations introduced by Dias, Dyachenko, and Zakharov (2008). This viscous Dysthe system models the evolution of a weakly viscous, nearly monochromatic wave train on deep water. It contains a term which provides a mechanism for frequency downshifting in the absence of wind and wave breaking. The equation does not preserve the spectral mean. Numerical simulations demonstrate that the spectral mean typically decreases and that the spectral peak decreases for certain initial conditions. The linear stability analysis of the plane-wave solutions of the viscous Dysthe system demonstrates that waves with wave numbers closer to zero decay more slowly than waves with wave numbers further from zero. Comparisons between experimental data and numerical simulations of the NLS, dissipative NLS, Dysthe, and viscous Dysthe systems establish that the viscous Dysthe system accurately models data from experiments in which frequency downshifting was observed and experiments in which frequency downshift was not observed.
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This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large, localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications including tsunami and internal ocean wave modelling. This study is performed in the context of the viscous fluid conduit system-the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Due to buoyancy induced nonlinear self-steepening balanced by stress induced interfacial dispersion, the disturbance evolves into a slowly modulated wavetrain and further, into a sequence of solitary waves. An extension of Whitham modulation theory, termed the solitary wave resolution method, is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbances profile, the number of emergent solitary waves, and their amplitude distribution, quantifying an extension of the well-known soliton resolution conjecture from integrable systems to non-integrable systems that often provide a more accurate modelling of physical systems. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally. The number of observed solitary waves is consistently within 1-2 waves of the prediction, and the amplitude distribution shows remarkable agreement. Universal properties of solitary wave fission in other fluid dynamics problems are identified.
Frequency downshift (FD) in wave trains on deep water occurs when a measure of the frequency, typically the spectral peak or the spectral mean, decreases as the waves travel down a tank or across the ocean. Many FD models rely on wind or wave breaking. We consider seven models that do not include these effects and compare their predictions with four sets of experiments that also do not include these effects. The models are the (i) nonlinear Schrodinger equation (NLS), (ii) dissipative NLS equation (dNLS), (iii) Dysthe equation, (iv) viscous Dysthe equation (vDysthe), (v) Gordon equation (Gordon) (which has a free parameter), (vi) Islas-Schober equation (IS) (which has a free parameter), and (vii) a new model, the dissipative Gramstad-Trulsen (dGT) equation. The dGT equation has no free parameters and addresses some of the difficulties associated with the Dysthe and vDysthe equations. We compare a measure of overall error and the evolution of the spectral amplitudes, mean, and peak. We find: (i) The NLS and Dysthe equations do not accurately predict the measured spectral amplitudes. (ii) The Gordon equation, which is a successful model of FD in optics, does not accurately model FD in water waves, regardless of the choice of free parameter. (iii) The dNLS, vDysthe, dGT, and IS (with optimized free parameter) models all do a reasonable job predicting the measured spectral amplitudes, but none captures all spectral evolutions. (iv) The vDysthe, dGT, and IS (with optimized free parameter) models do the best at predicting the observed evolution of the spectral peak and the spectral mean. (v) The IS model, optimized over its free parameter, has the smallest overall error for three of the four experiments. The vDysthe equation has the smallest overall error in the other experiment.
A physical model of a three-dimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model is based on mechanics and thermodynamics of a single bubble coupled to the dynamics of a viscous fluid as a whole, and takes into account multiple physical effects, including gravity, viscosity, and surface tension. Dimensionle
We show experimentally that a stable wave propagating into a region characterized by an opposite current may become modulationaly unstable. Experiments have been performed in two independent wave tank facilities; both of them are equipped with a wavemaker and a pump for generating a current propagating in the opposite direction with respect to the waves. The experimental results support a recent conjecture based on a current-modified Nonlinear Schrodinger equation which establishes that rogue waves can be triggered by non-homogeneous current characterized by a negative horizontal velocity gradient.
The viscous drag on a slender rod by a wall is important to many biological and industrial systems. This drag critically depends on the separation between the rod and the wall and can be approximated asymptotically in specific regimes, namely far from, or very close to, the wall, but is typically determined numerically for general separations. In this note we determine an asymptotic representation of the local drag for a slender rod parallel to a wall which is valid for all separations. This is possible through matching the behaviour of a rod close to the wall and a rod far from the wall. We show that the leading order drag in both these regimes has been known since 1981 and that they can used to produce a composite representation of the drag which is valid for all separations. This is in contrast to a sphere above a wall, where no simple uniformly valid representation exists. We estimate the error on this composite representation as the separation increases, discuss how the results could be used as resistive-force theory and demonstrate their use on a two-hinged swimmer above a wall.
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