# A10: Fluids and Waves - Material for the year 2019-2020

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16 lectures

This course introduces students to the mathematical theory of inviscid fluids. The theory provides insight into physical phenomena such as flight, vortex motion, and water waves. The course also explains important concepts such as conservation laws and dispersive waves and, thus, serves as an introduction to the mathematical modelling of continuous media.

Incompressible flow. Convective derivative, streamlines and particle paths. Euler's equations of motion for an inviscid fluid. Bernoulli's Theorem. Vorticity, circulation and Kelvin's Theorem. The vorticity equation and vortex motion.

Irrotational incompressible flow; velocity potential. Two-dimensional flow, stream function and complex potential. Line sources and vortices. Method of images, circle theorem and Blasius's Theorem.

Uniform flow past a circular cylinder. Circulation, lift. Use of conformal mapping to determine flow past a flat wing. Water waves, including effects of finite depth and surface tension. Dispersion, simple introduction to group velocity.

D. J. Acheson, *Elementary Fluid Dynamics* (OUP, 1997). Chapters 1, 3.1-3.5, 4.1-4.8, 4.10-4.12, 5.1, 5.2, 5.6, 5.7.

R. P. Feynman, R.B. Leighton, M. Sands, *The Feynman Lectures on Physics*, volume II, (Addison Wesley 1964) Chapter 40 (http://www.feynmanlectures.caltech.edu/II_40.html)

M. van Dyke, *An Album of Fluid Motion* (Parabolic Press, 1982).