Introduction to continuum physics

A.Y. 2020/2021
6
Max ECTS
42
Overall hours
SSD
FIS/06
Language
Italian
Learning objectives
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.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
The course will be held remotely due to persisting limitations related to the COVID-19 emergency situation. Lectures will be delivered sinchronously according to the official schedule of the Physics Course and using the virtual classrooms system of the Physics Department implemented through the Zoom platform, thus enabling real-time interactions between students and teacher. Student reception and oral examinations will also be held remotely in a virtual room of the Physics Department. If deemed possible and appropriate, opportunities for live lectures and discussions will be arranged.
Course syllabus
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' problem. 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).
Teaching methods
Traditional lectures, supplemented by topical seminars.
Teaching Resources
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.
FIS/06 - PHYSICS OF THE EARTH AND OF THE CIRCUMTERRESTRIAL MEDIUM - University credits: 6
Lessons: 42 hours
Professor: Maero Giancarlo
Professor(s)