Hydrodynamics

Fluids are all around us all the time at all scales. To understand fluid mechanics is to understand the mechanics of what surrounds us: air and water in particular. As such, hydrodynamics is part of the physicist's basic knowledge.

The UE Hydrodynamics is an introduction to incompressible perfect (Euler) and viscous Newtonian (Navier-Stokes) fluid mechanics. The classical flows are presented, as well as the notion of boundary layer, instability and turbulence. Emphasis is placed more on physical ideas than on advanced mathematical or numerical resolution methods.

To master the basic concepts and flows of fluid dynamics in order to be able to approach more advanced courses, either in numerical solution of hydrodynamic equations, or in complex fluids in M2, or in magneto-hydrodynamics (Fluid Dynamics in Astrophysics and Cosmology in M2) and applications in the field of biophysics, colloid physics, or coastal engineering, to name but a few examples

Newtonian Dynamics 1&2 in L

Recommended Prerequisites:

The UE Hydrodynamics, Elasticity and Hydrodynamics, and Mathematics for Physics of L are useful but not mandatory. In general, a solid background in physics is necessary to approach this course.

CCI 100 %

Hydrodynamic description. Eulerian and Lagrangian points of view. Conservation of matter equation, incompressibility. Perfect fluid: Euler equation, momentum balance and stress tensor of the ideal fluid, boundary conditions, hydrostatics, Bernoulli theorem and applications, conservation of vorticity, potential flow, examples of ideal flows.

Viscous stress tensor, viscosity, Newtonian fluid, boundary layer. Navier-Stokes equation and boundary conditions. Euler ↔ Navier-Stokes transition, role of Reynolds number, examples and orders of magnitude. Simple classical examples of stationary and non-stationary viscous flows: plane and cylindrical Couette flow, flow in a pipe, Poiseuille formula. Viscous fluid set in motion by tangential stress and relationship to diffusion. Highly viscous fluid, flow and Stokes formula.

Laminar boundary layer, Prandtl theory, boundary layer shape. Boundary layer separation and its role in turbulence. Notions on the turbulent boundary layer.

Instability: the principle of calculating stability conditions. Empirical conditions and Reynolds number, unstable flows. Examples: Von Karman's path, Couette instability in rotating cylinders, Kelvin Helhmotz instability, instability of flow in a pipe (Reynolds) and between 2 parallel planes, Rayleigh Taylor instability ... the march towards turbulence.

Turbulence: statistical description of turbulent fluid, correlations, Reynolds tensor, role and interpretation. Application example: logarithmic velocity profile near an obstacle. Kolmogorov cascade and energy dissipation in a fluid.

Gravity waves: the boundary condition at the interface between two fluids (with and without surface tension). Linear boundary, waves in deep water and finite depth, dispersion relation, interpretation and role of phase velocity and group velocity. Stokes waves (non linear periodic solution). Applications.

The course is illustrated with numerous applications from physics and everyday life, as well as environmental.