Learning outcomes of the course unit
Investigation tools at mesoscopic level, entropy methods, foundation of thermo-fluid-dynamics starting from molecular gas dynamics, extension to other different applied sciences. Mathematical algorithms and methods for understanding and modelling complex phenomena.
Course contents summary
Kinetic theory, distribution function, Boltzmann equation. Collision operator, collision invariants, Maxwellian distributions. Entropy functionals and second law of thermodynamics. Hydrodynamic limit, Euler and Navier-Stokes equations. Extension to gas mixtures and to other kinetic approaches in the applied sciences.
Elements of kinetic theory of gases, macroscopic and microscopic quantities. Statistical mechanics, phase space, distribution function. Mean free path, collision dynamics and conservation laws. Macroscopic fields: density, velocity, pressure tensor, temperature, heat flow. Derivation of the Boltzmann equation, "streaming" and "scattering" operators and their properties. Weak form of the kinetic equation and transport equations of molecular properties. Collisional invariants and macroscopic conservation equations for mass, momentum and energy. Equilibrium and Maxwellian distribution configurations. Boltzmann H-functional, H-theorem and second principle of thermodynamics. Introduction to the linearized and the linear Boltzmann equation. Hydrodynamic limit, Euler and Navier-Stokes equations. Gas mixtures, global and species quantities, diffusion velocities. Collisional invariants, H-theorem, and thermodynamic equilibria. Kinetic approaches to other problems of applied sciences.
C. CERCIGNANI, Theory and applications of the Boltzmann equation, SPRINGER, New York.
S. CHAPMAN, T.G.COWLING, The mathematical theory of nonuniform gases, UNIVERSITY PRESS, Cambridge.
M. N. KOGAN, Rarefied gas dynamics, PLENUM PRESS, New York.
Y. SONE, Kinetic theory and fluid dynamics, SPRINGER, New York.
Assessment methods and criteria