Physique non-linéaire et phénomènes hors équilibre
Nonlinear Physics and out of equilibrium phenomena

Nonlinear Fluid Dynamics: transport and interfacial dynamics
Christophe Gissinger (Ecole Normale Superieure, Paris)

This course aims to provide training at the interface between classical introductory courses in fluid mechanics and the more advanced problems addressed in research. It focuses mainly on the nonlinear aspects of fluid dynamics and provides an opportunity to discuss several aspects of advanced hydrodynamics, such as turbulent transport phenomena, shock waves, fluid interfaces, hydrodynamic solitons, and the Boltzmann equation. The course incorporates examples drawn from recent advances in theoretical and experimental research, as well as industrial and astrophysical applications.

1. Hydrodynamic limit of Boltzmann equation: - Boltzmann equation - Chapman-Enskog transformation - Molecular transport - Onsager-Casimir relations

The first part of the course is devoted to fundamental aspects. Traditionally, fluid mechanics is approached within the framework of continuum mechanics, where the equations are obtained by applying Newton's second law to each fluid particle. In this course, I present an alternative approach where the Navier-Stokes equations are derived directly from the Boltzmann equation. This approach highlights the relationship between transport coefficients (such as heat conduction or molecular viscosity) and microscopic phenomena and allows us to discuss the phenomenological aspects of transport in fluid mechanics. Recent theoretical research efforts in this area will be discussed.



2. Turbulent transport: - Turbulent transport and mixing - Boussinesq approximation and turbulent convection - Heat transport and scaling laws - Experimental aspects

The introduction to molecular transport in Chapter 1 will then enable us to deal with modern aspects of heat transfer in fluid mechanics, with particular emphasis on the heat transport properties of turbulence, a very active field of research. I will discuss in detail recent theories of turbulent convection, its transport properties, and the scaling laws obtained in recent years. Some examples will be given of the various experimental advances in this field, some industrial applications, astrophysics (star formation), and climate science.



3. Interface dynamics and shock waves: - Nonlinear theory of surface waves - Nonlinear shallow-water approximation. - Korteweg-de-Vries equations, solitary waves and hydrodynamic solitons - Formal analogy with compressible gas dynamics, supersonic shock waves

The final part of the course focuses on wave phenomena and interfaces with strong non-linearities. This enables us to present various perturbative methods in fluid mechanics and to propose a unified approach to surface wave dynamics. We will also explore the analogy between gas dynamics and shallow interfacial waves, offering interesting insights into shock waves, solitary waves, and supersonic fluid acceleration. Applications include supersonic flight, tsunami and tidal bore propagation, and compressible turbulence in astrophysics.



Prerequisites:
Basic hydrodynamics (Eulerian or Lagrangian description, Navier-Stokes equation), basic Statistical physics


Bibliography:
1. Landau L.D., Lifschitz E.M.- Vol. 6 - Fluid Mechanics
2. R.S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press


Timing: The Course is offered in the second part (december-february) of the M2 year.
It consists of 8 Lectures on Fridays from 1.45pm to 5.45pm at Sorbonne University, room to be announced
First lecture on Friday 6th December

ECTS Credits:3

Hours: 32 hours.