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# Introduction to interfacial waves

## Overview

The course is an introduction to the rich field of interfacial waves. The first half of the course prepares the student for studying wave phenomena by introducing discrete mechanical analogues of wave phenomena in fluid systems. The basic principles of normal mode analysis are introduced through point-mass systems connected through springs. The exact solution to the (nonlinear) pendulum equation is used to introduce the notion of amplitude dependence on frequency of the oscillator. The Kapitza pendulum is introduced as a discrete analogue for Faraday waves. Basic perturbation techniques are then introduced for subsequent use. The second half of the course introduces basics of interfacial waves viz. shallow and deep-water approximations, phase and group velocity, frequency and amplitude dispersion etc.. Capillary as well as capillary-gravity waves in various base state geometries (rectilinear, spherical (drops and bubbles including volumetric oscillations of the latter), cylindrical (filaments) are taught and the corresponding dispersion relation derived. The Stokes travelling wave is derived using the Lindstedt-Poincare technique and the amplitude dependence in the dispersion relation is highlighted. Side-band instability of the Stokes wave is discussed. Fluid particle trajectories for linear water waves is derived and the Stokes drift expression is derived. Time permitting, the Kelvin ship wave pattern in deep water is derived using the method of stationary phase. Introductory ideas in resonant interactions among surface gravity waves are discussed. The fundamental aspects studied in the course will be related to various engineering applications continuously.INTENDED AUDIENCE :Chemical & Mechanical Engineering studentsPRE-REQUISITES :Introductory Transport Phenomena / Fluid MechanicsINDUSTRY SUPPORT :Industrial personnel working on two phase flows

## Syllabus

Week-1:Introduction to waves and oscillations, Normal modes of linear vibrating systems with finite degrees of freedom, Eigenmodes (shapes of oscillation) and frequencies
Week-2:Normal modes of a linear, N degree of freedom spring-mass system, continuum limit, linear wave equation and normal modes Week-3:Nonlinear pendulum: exact solution using elliptic integrals, amplitude dependence of frequency, intro. to perturbation methods: regular and singular, Lindstedt-Poincare technique
Week-4:Damped harmonic oscillator, Duffing oscillator, method of multiple scales Week-5:Parametric instability and the Kapitza Pendulum, Introduction to Floquet analysis; Capillary-gravity waves on a fluid interface: governing equations and boundary conditions, Normal mode analysis, Deep and shallow water approximations and dispersion relations. Week-6:Phase and group velocity, Cauchy-Poisson problem for surface waves in deep water: 2D rectilinear and cylindrical geometry, Standing and travelling waves, kinematic interpretation of group velocity; Waves on a fluid cylinder, Rayleigh-Plateau instability, oscillations of a hollow filament.
Week-7:Normal modes of a liquid drop and bubble, Normal modes of compound drops
Week-8:Wind waves and the Kelvin-Helmholtz instability, KH instability as a model for wind wave generation, surface waves in an uniform flow due to an oscillatory pressure source at the surface
Week-9:Stokes wave in deep water, nonlinear travelling wave of constant form, stability of Stokes wave (sideband instability), solitary waves, KdV equation and solitons
Week-10:Faraday instability on a fluid interface, subharmonic response, Floquet analysis, atomization from Faraday waves
Week-11:Particle trajectories in water waves, Stokes drift, long surface gravity waves on inviscid shear flows: Burns dispersion relation
Week-12:Shape and volume oscillations of bubbles, Minnaert frequency, Rayleigh-Plesset equation. (If time permits) Kelvin wave pattern of ship wake in deep water and method of stationary phase, Resonant interactions among water waves

### Taught by

Prof. Ratul Dasgupta

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