Complex Wave Dynamics on Thin Films

Introduction and History.Formulation and Linear Orr-Sommerfeld Theory. Navier-Stokes Equation with interfacial conditions. Linear stability of the trivial solution to two- and three-dimensional perturbations. Longwave expansion for surface waves. Unusual case of zero surface tension. Surface waves: the limit of R → ∞. Numerical solution of the Orr-Sommerfeld equations.Hierarchy of Model Equations.Kuramoto-Sivashinsky (KS), KdV and related weakly nonlinear equations. Lubrication theory to derive Benney's longwave equation. Depth-averaged integral equations. Combination of Galerkin-Petrov method with weighted residuals. Validity of the equations. Spatial and temporal primary instability of the Shkadov model.Experiments and Numerical Simulation.Experiments on falling-film wave dynamics. Numerical formulation. Numerical simulation of noise-driven wave transitions. Pulse formation and coarsening. Periodic and Solitary Wave Families. Main properties of weakly nonlinear waves in an active/dissipative medium. Phase space of stationary KS equation. Solitary waves and Shilnikov theorem. Bifurcations of spatially periodic travelling waves and their stability. Normal Form analysis for the Kawahara equation. Nonlinear waves far from criticality - the Shkadov model. Stationary waves of the boundary layer equation and Shkadov model. Navier-Stokes equation of motion - the effects of surface tension. Floquet Theory and Selection of periodic Waves Stability and selection of stationary waves. Stable intervals from a Coherent Structure Theory. Evolution towards solitary waves.Spectral Theory for gKS Solitary Pulses.Pulse spectra. Some numerical recipes to construct eigenfunctions and obtain spectra. Stability of gKS pulses. Attenuation of radiation wave packet by stable pulses. Resonance pole-a discrete culmination of the continuous spectrum. Resonance pole description of mass drainage. Suppression of wave packets by a periodic train of pulses.Spectral Theory and Drainage Dynamics of Realistic Pulses.Role of drainage in pulse coalescence. Spectrum of the solitary pulse. Quasi-jump decay dynamics. Essential and resonance pole spectra of the pulses.Pulse Interaction Theory.Coherent Structure theory due to translational zero mode Repulsive pulse interaction of the gKS pulses. Coupled drainage and binary interaction dynamics of pulses for the Shkadov model.Coarsening Theory for Naturally Excited Waves.Spatial evolution, linear filtering and excitation of low-frequency band. A theory for the characteristic modulation frequency. Universal coarsening rate based on ▵. Noise-driven wave dynamics.Transverse Instability.Coupled oblique waves and triad resonance. Transverse breakup of equilibrium 2D-waves. Scallop waves. Stability of nonlinear localized patterns.Hydraulic Shocks.Governing equations. Numerical simulation. Coherent wave structures and self-similarity. Self-similar coarsening dynamics. Summary and discussion.Drop Formation on a Coated Vertical Fiber.Pulse coalescence dynamics. Equilibrium subcritical pulses and stability. Growth dynamics of supercritical pulses. Discussion.