The course is meant to provide the basics of a macroscopic description of continuous media, and especially fluids, together with necessary tools such as tensor calculus.
Expected learning outcomes
After attending the course, the student will possess the following set of knowledge and skills: - Knowledge of macroscopic behaviour of matter treated as a continuum within a field theory - Use of tools such as tensor calculus and dimensionless numbers as well as analytical methods for the description of continuous media - Use of mechanics and thermodynamics concepts necessary to continuum dynamics - Knowledge of basic properties, laws and phenomena concerning ideal fluids - Knowledge of basic properties, laws and phenomena concerning real (viscous) fluids - Knowledge of basic properties, laws and phenomena concerning heat transport in fluids - Knowledge of application examples in geophysical, astrophysical and laboratory continuous media.
Lesson period: First semester
(In case of multiple editions, please check the period, as it may vary)
INTRODUCTION Memories from thermodynamics (thermodynamic potentials, Maxwell relations, specific heats, adiabatic processes). Tensors down to the bone (vectors and tensors; quotient rule, tensor subspaces, geometric decomposition of rank-2 tensors, alternating tensor, duality relations). General notions on fluids. Lagrangian and Eulerian approach. Material derivative. Derivative of volume, surface, line integrals. Mass conservation and continuity equation. STATICS OF IDEAL FLUIDS Volume and surface forces. Ideal fluid, Euler and entropy equations. Fluid statics: pressure, mechanical equilibrium, stability of the atmosphere. Incompressibility conditions. Statics of incompressible fluids. FLUID DYNAMICS OF IDEAL FLUIDS Impulse and energy flux. Bernoulli's theorem and applications. Kelvin's theorem, potential flow, Laplace equation. Gravity waves in ideal fluids, dispersion relations, applications. REAL (VISCOUS) FLUIDS Velocity gradient tensor, kinematic interpretation of its geometric decomposition. Cauchy's stress theorem and stress tensor, constitutive equations, Newtonian stress tensor. Navier-Stokes equation. Examples of viscous flows. Similarity laws and dimensionless numbers in the Navier-Stokes equation. Stokes' proble. Oscillatory motions in viscous fluids, damping of gravity waves, surface currents. STABILITY Stability under arbitrary and small perturbations. Tangential discontinuities, Kelvin-Helmholtz instability. HEAT EXCHANGE Heat equation for energy and entropy and second principle of thermodynamics, Clausius-Duhem inequality. Heat equation for incompressible fluids and solids (Fourier equation). Boundary conditions, transient and steady solutions: examples for the modelling of the Earth's crust. Green's function. Reversibility and irreversibility. Similarity law and dimensionless numbers in the heat equation. The ideal fluid as a limit of the viscous fluids at large global and local Reynolds numbers.
Prerequisites for admission
A solid knowledge of the calculus, algebra and geometry notions taught at the Physics Bachelor level is implied. The same holds for classical physics (mechanics and thermodynamics in particular; the course will also make use of fruitful analogies with the classical theory of electricity and magnetism).
Traditional lectures, supplemented by topical seminars.
Notes supplied by the teacher, available at the teacher's website 'Fluid Mechanics', L.D. Landau and E.M. Lisfshitz (Course of Theoretical Physics, Volume 6), Butterworth-Heinemann 'Elementi di fisica dei continui', G. Parravicini, CUSL 'Mathematics Applied to Continuum Mechanics', L.A. Segel, Dover Publications (or later SIAM edition) Supplementary material 'Fluid Mechanics', P.K. Kundu and I.M. Cohen (second or later edition), Academic Press 'Fluid Dynamics for Physicists', T.E. Faber, Cambridge University Press
Assessment methods and Criteria
The examination is based on a 45-60 minute discussion. The student must demonstrate an adequate mastery of the physical and mathematical contents of the course, with an accent on the modelling of physical phenomena and on the use of tools such as dimensionless numbers and approximations governed by the hierarchy of the terms concurring to the description of the phenomena under analysis.