Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

Hydrodynamic and hydromagnetic stability

An asymptotic modal system is derived for modelling nonlinear sloshing in a rectangular tank with similar width and breadth. The system couples nonlinearly nine modal functions describing the time evolution of the natural modes. Two primary modes are assumed to be dominant. The system is equivalent to the model by Faltinsen et al. (2000) for the two-dimensional case. It is validated for resonant sloshing in a square-base basin. Emphasis is on finite fluid depth but the behaviour with decreasing depth to intermediate depths is also discussed. The tank is forced in surge/sway/roll/pitch with frequency close to the lowest degenerate natural frequency. The theoretical part concentrates on periodic solutions of the modal system (steady-state wave motions) for longitudinal (along the walls) and diagonal (in the vertical diagonal plane) excitations. Three types of solutions are established for each case: (i) ‘planar’/‘diagonal’ resonant standing waves for longitudinal/diagonal forcing, (ii) ‘swirling’ waves moving along tank walls clockwise or counterclockwise and (iii) ‘square’-like resonant standing wave coupling in-phase oscillations of both the lowest modes. The frequency domains for stable and unstable waves (i)–(iii), the contribution of higher modes and the influence of decreasing fluid depth are studied in detail. The zones where either unstable steady regimes exist or there are two or more stable periodic solutions with similar amplitudes are found. New experimental results are presented and show generally good agreement with theoretical data on effective domains of steady-state sloshing. Three-dimensional sloshing regimes demonstrate a significant contribution of higher modes in steady-state and transient flows.

Resonant three-dimensional nonlinear sloshing in a square-base basin

The modal system describing nonlinear sloshing with inviscid flows in a rectangular rigid tank is revised to match both shallow fluid and secondary (internal) resonance asymptotics. The main goal is to examine nonlinear resonant waves for intermediate depth/breadth ratio 0.1 [lsim ] h/l [lsim ] 0.24 forced by surge/pitch excitation with frequency in the vicinity of the lowest natural frequency. The revised modal equations take full account of nonlinearities up to fourth-order polynomial terms in generalized coordinates and h/l and may be treated as a modal Boussinesq-type theory. The system is truncated with a high number of modes and shows good agreement with experimental data by Rognebakke (1998) for transient motions, where previous finite depth modal theories failed. However, difficulties may occur when experiments show significant energy dissipation associated with run-up at the walls and wave breaking. After reviewing published results on damping rates for lower and higher modes, the linear damping terms due to the linear laminar boundary layer near the tank's surface and viscosity in the fluid bulk are incorporated. This improves the simulation of transient motions. The steady-state response agrees well with experiments by Chester & Bones (1968) for shallow water, and Abramson et al. (1974), Olsen & Johnsen (1975) for intermediate fluid depths. When h/l [lsim ] 0.05, convergence problems associated with increasing the dimension of the modal system are reported.

Asymptotic modal approximation of nonlinear resonant sloshing in a rectangular tank with small fluid depth

The natural frequencies and damping rates of surface waves in a circular cylinder with pinned-end boundary conditions are calculated in terms of the gravitational Reynolds and Bond numbers, C−1 and B, and the slenderness of the cylinder Λ, in the limit C→0. We consider higher-order approximations that include the effect of viscous dissipation in the Stokes boundary layers and the bulk. A comparison with clean-surface experiments by Henderson & Miles (1994) shows a satisfactory agreement except for the first axisymmetric mode, which exhibits a 26% discrepancy. The much larger dramatic discrepancy of former theoretical predictions is hereby improved and explained.

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Surface-wave damping in a brimful circular cylinder

A weakly nonlinear analysis is presented of the small oscillations of nearly inviscid liquid bridges subjected to almost resonant axial vibrations of the disks. An amplitude equation is derived for the evolution of the complex amplitude of the oscillations that exhibits hysteresis and period doublings. We also analyse the steady streaming in the bulk forced by the oscillatory boundary layers near the disks; the boundary layer near the free surface produces no forcing terms. In particular some experimentally observed patterns are explained, and some new, non-observed ones are predicted. We conclude that the structure of this steady flow is not the appropriate one to counterbalance steady thermocapillary convection, but our results indicate how to get the desired counterbalancing effect.

Weakly nonlinear oscillations of nearly inviscid axisymmetric liquid bridges

Linear oscillations of axisymmetric capillary bridges are analyzed for large values of the modified Reynolds number C−1. There are two kinds of oscillating modes. For nearly inviscid modes (the flow being potential, except in boundary layers), it is seen that the damping rate −ΩR and the frequency ΩI are of the form ΩR=ω1C1/2+ω2C+O(C3/2) and ΩI=ω0+ω1C1/2+O(C3/2), where the coefficients ω0≳0, ω1<0, and ω2<0 depend on the aspect ratio of the bridge and the mode being excited. This result compares well with numerical results if C≲0.01, while the leading term in the expansion of the damping rate (that was already known) gives a bad approximation, except for unrealistically large values of the modified Reynolds number (C≲10−6). Viscous modes (involving a nonvanishing vorticity distribution everywhere in the liquid bridge), providing damping rates of the order of C, are also considered.

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Linear oscillations of weakly dissipative axisymmetric liquid bridges

The steady streaming flow due to vibration in capillary bridges is considered in the limiting case when both the capillary Reynolds number and the non-dimensional vibration frequency (based on the capillary time) are large. An asymptotic model is obtained that provides the streaming flow in the bulk, outside the thin oscillatory boundary layers near the disks and the interface. Numerical integration of this model shows that several symmetric and non-symmetric streaming flow patterns are obtained for varying values of the vibration parameters. As a by-product, the quantitative response of the liquid bridge to high-frequency axial vibrations of the disks is also obtained.

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On the steady streaming flow due to high-frequency vibration in nearly inviscid liquid bridges

Asymptotic formulas are derived for the effect of contamination on surface wave damping in a brimful circular cylinder; viscosity is assumed to be small and contamination is modelled through Marangoni elasticity with insoluble surfactant. It is seen that an appropriately chosen finite Marangoni elasticity provides an explanation for a significant amount of the unexplained additional damping rate in a well-known experiment by Henderson & Miles (1994); discrepancies are within 15%, significantly lower than those encountered by Henderson & Miles (1994) under the assumption of inextensible film.

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José A. Nicolás

Papers

Surface-wave damping in a brimful circular cylinder

Weakly nonlinear oscillations of nearly inviscid axisymmetric liquid bridges

Linear oscillations of weakly dissipative axisymmetric liquid bridges

On the steady streaming flow due to high-frequency vibration in nearly inviscid liquid bridges

A note on the effect of surface contamination in water wave damping