ECTS Credits: 6

I - Advanced fluid dynamics
Christophe Gissinger (Ecole Normale Superieure, Paris)

These lectures aim to bridge the gap between classical introductory lectures on fluid mechanics and the more advanced problems addressed in academic research. The first part of the course will be devoted to fundamental aspects, including the interplay between statistical physics and fluid dynamics. In particular, we will briefly discuss the theoretical process leading to the Navier-Stokes equations from the Boltzmann equation of the classical kinetic theory of gases. This approach highlights the relation between transport coefficients (such as heat conduction) and microscopic data and will lead us to describe some features of the theory of thermal conduction and diffusion in fluids. Similarly, a part of the course will focus on the fundamentals of compressible fluid motions and turbulent convection, which usually remain on the fringes of introductory courses. We will see that there exists a strong analogy between gas dynamics and shallow-depth interfacial waves, leading to interesting results on shock waves and solitary waves. An introduction to magnetohydrodynamics and astrophysical fluid dynamics will also be given.

• Fundamentals: - Introduction to fluid dynamics and conservation laws - Inviscid irrotational flows - Boundary layers and rotating fluids
• Statistical Physics: - Boltzman equation - Passage from Boltzman to Euler and Hilbert’s problem - Chapman-Enskog expansion
• Thermal effects: - The Boussinesq approximation - Heat transport and scaling laws for turbulent convection
• Compressible flows: - Gas dynamics and self-similar compressible flows - Supersonic shock waves
• Surface and Interfacial waves: - Free surface wave theory - Shallowdepth problem and the Korteweg-de Vries equation - Solitary waves and Solitons
• Complex fluids - Magnetohydrodynamics - Plasma physics