09224 - Continuum Mechanics

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Good health and well-being Quality education Clean water and sanitation Climate Action

Academic Year 2021/2022

Learning outcomes

At the end of the course the students know some models of fluidodynamics, elasticity, and magnetofluidodynamics. He can understand and solve problems of stability and wave propagation, and acquires consciousness of the physical relevance of the problems dealt with.

Course contents

Description of the Dispersive Wave tool for linear stability analysis on first and second order PDEs occurring throughout mathematics: linearization around a stationary and homogeneous, quiescent or not, equilibrium state. Introduction to partial differential equations: Cauchy problem, characteristic curves and surfaces, classification and propagation of singularities. Short account of wave propagation theory, with examples of hyperbolic models.Tensorial algebra and analysis. Introduction to continuum bodies: kinematics, dynamics and thermodynamics of fluids and elastic media. General balance laws, in integral and local forms, even in the presence of a singular surface relative to scalar/vectorial unknown fields.Local balance equations and constitutive equations. Derivation of the  Rankine-Hugoniot equations. The Conservation Principles of the Continuum Mechanics and their local divergence and convective forms. The Lagrangian formulation.Description of some hyperbolic and parabolic mathematical models of fluids: the Euler model for barotropic ideal fluids and the Navier-Stokes model for viscous/dissipative fluids. Reynolds number. Existence of stationary classical solutions. Uniqueness and stability in L^2. The Navier model for linear and homogeneous elastic solids: hyperbolic waves towards dispersive waves. Gravitational instability problems  and Jeans Criteria, even in the presence of the cosmological constant, accounting for universe expansion. Collapse formation and gravito-sonic waves. Mathematical models fo chemotactic mechanisms towards gravitational instability in self-gravitating interstellar gas clouds. Analysis of viscous and magnetic effects on collapse formation.The model of ideal magnetofluidodynamics and its hyperbolicity. Magneto-rotational instabilities: non-gravitational transversal Alfven waves and their properties and gravitational longitudinal fast and slow magneto-sonic waves. Gradient catastrophe and shock waves: Euler's and Lundquist's models.

Convective properties of fluids: the Benard problem.

Mathematical modelings for non-isothermal galactic coronae in the presence of ideal and non ideal magnetic effects: the effect of a thermal relaxation time for a delayed response of the heat flux vector.


M.E. Gurtin An introduction to continuum mechanics, Academic Press 2003
F. John: Partial Differential Equations, Springer
B. Straughan: The energy method, stability and nonlinear convection, Springer 2007
M. Renardy, R.C. Rogers: An Introduction to Partial Differential Equations, Springer 2004
T. Ruggeri: Introduzione alla Termomeccanica dei Continui; Monduzzi 2007
E.R. Priest, T.G. Forbes: Magnetic Reconnection: MHD Theory and Applications; Cambridge University Press 2000

J. Binney, S. Tremaine: Galactic Dynamics, Princeton University Press, Princeton 2008

I Sih Liu: Continuum Mechanics (primer), 2006

Lecture notes and papers of the teacher, published on virtuale.

Teaching methods

The course consists of class-room lectures, where the basic elements of Continuum Mechanics are introduced, with a particular care for the mathematical models of Astrophysics.

Assessment methods

The assessmnet method consists in an oral exam which aims to assess the knowledge of the various topics of the course.

Office hours

See the website of Franca Franchi