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MA3D1 Content

Content: The lectures will provide a solid background in the mathematical description of fluid dynamics. They will cover the derivation of the conservation laws (mass, momentum,energy) that describe the dynamics of fluids and their application to a remarkable range of phenomena including water waves, sound propagation, atmospheric dynamics and aerodynamics. The focus will be on deriving approximate expressions using (usually) known mathematical techniques that yield analytic (as opposed to computational) solutions.

The module will cover the following topics:

  • Mathematical modelling of fluid flow. Specification of the flow by field variables; vorticity; stream function; strain tensor; stress tensor. Euler's equation. Navier-Stokes equation. Introduction of non-dimensional parameters
  • Additional conservation laws. Bernoulli's equations. Global conservation laws.
  • Vortex dynamics. Kelvin's circulation theorem. Helmholtz theorems. Cauchy-Lagrange theorem. 3D vorticity equation, vortex lines, vortex tubes and vortex stretching.
  • 2D flows. Flow in a pipe. Shear flows. Jet flow by similarity solution. Round vortices.
  • Irrotational 2D flows & classical aerofoil theory. Complex analysis methods in Fluid Dynamics. Blasius theorem. Zhukovskii lift theorem. Force on a cylinder and force on an aerofoil.
  • Rotating flows. Navier-Stokes in a rotating frame. Rossby number. Taylor-Proudman theorem. Geophysical flow.
  • Boundary layers. Prandtl's boundary layer theory. Ekman boundary layer in rotating fluids.
  • Waves. General theory of waves. Sound waves. Free-surface flows & surface tension. Gravity-capillary water waves.
  • Instabilities.  Rayleigh criterion. Orr-Sommerfeld equation. Kelvin-Helmholtz instability. Stability of parallel flows.
  • Turbulence. Richardson cascade and Kolmogorov spectrum. 2D turbulence and dual cascades. Reynolds Averaged Navier-Stokes equations.

Aims:
An important aim of the module is to provide an appreciation of the complexities and beauty of fluid motion. This will be highlighted in class using videos of the phenomena under consideration (usually available on YouTube).

Objectives: It is expected that by the end of this module students will be able to:
- be able to understand the derivation of the equations of fluid dynamics
- master a range of mathematical techniques that enable the approximate solution to the aforementioned equations
- be able to interpret the meanings of these solutions in 'real life' problems

Strongly recommended texts:
D.J. Acheson, Elementary Fluid Dynamics, OUP. (Excellent text with derivations, examples and solutions)
S. Nazarenko, Fluid Dynamics via Examples and Solutions, Taylor and Francis. (Great source of questions and detailed solutions.)

Further Reading:
A.R. Paterson, A First Course in Fluid Dynamics, CUP. (Easier than Acheson.)
L.D. Landau and E.M. Livshitz, Fluid Mechanics, OUP. (A classic for those with a deep interest in fluid dynamics in modern physics.)
D.J. Tritton, Physical Fluid Dynamics , Oxford Science Publs. (The emphasis is on the physical phenomena and less on the mathematics.)