ECTS Credits: 6
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 -
Astrophysical fluid dynamics
Jean-Marcel Rax (Univ. Paris-Saclay, Ecole Polytechnique)
Besides ordinary temperature usual (i) solid state, (ii) liquid state and
(iii) gaseous state, at very low and very high temperatures new exotic states
appear : (iv) quantum fluids and (v) ionized gases. These states display
a variety of specific and new physical phenomena : (i) at low temperature
quantum coherence, correlation and indiscernibility lead to superfluidity, superconductivity and Bose-Einstein condensation ; (ii) at high temperature
ionization provides a significant fraction of free charges responsible for instabilities, nonlinear, chaotic and turbulent behaviors characteristic of the
“plasma state”. This set of lectures provides an introduction to the basic
tools, main results and advanced methods of Plasma Physics.
• Plasma Physics : History, orders of magnitudes, Bogoliubov’s hierarchy, fluid and kinetic reductions. Electric screening : Langmuir
frequency, Maxwell time, Debye length, hybrid frequencies. Magnetic
screening : London and Kelvin lengths. Alfven and Bohm velocities.
• Charged particles dynamics : adiabaticity and stochasticity. Alfven
and Ehrenfest adiabatic invariants. Electric and magnetic drifts. Ponderomotive force. Wave/particle interactions : Chirikov criterion and
chaotic dynamics. Quasilinear kinetic equation for Landau resonance.
• Structure and self-organization : Debye and Child-Langmuir sheath,
Chapman-Ferraro boundary layer. Brillouin and basic MHD flow.
Flux conservation : Alfven’s theorem. Magnetic topology : magnetic
helicity conservation. Grad-Shafranov and basic MHD equilibrium.
• Waves and instabilities : Fluid and kinetic dispersions : resonance
and cut-off. Landau and cyclotron absorption. Bernstein’s modes.
Fluid instabilities : drift, interchange and kink. Alfvenic and geometric
coupling. Introduction to gyrokinetic. Wave/wave interactions and
parametric instabilities.
• Relaxation an dissipation : Coulomb collisions : Landau equation.
Electron impact ionization: Thomson’s cross section. Friction and
momentum transfer. Thermalization and energy transfer. Relaxation :
Landau equation Green’s functions. Fluids moments, coupled fluxes
and entropy production.