Introduction to Continuum Physics

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 continuous media. Lagrangian and Eulerian approach. Material derivative. Derivative of volume, surface, line integrals. Mass conservation and continuity equation.
Volume and surface forces. Stress tensor, pressure and deviatoric stress tensor, first law of motion for a continuous medium.
STATICS OF IDEAL FLUIDS
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
Momentum 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 under arbitrary and small perturbations. Tangential discontinuities, Kelvin-Helmholtz instability.
DYNAMICS OF CONTINUOUS MEDIA
Constitutive equations for elastic and viscous-elasto-plastic materials; objectivity principle or material frame-indifference. Equation for the incremental linear momentum for a pre-stressed continuum. Elasto-plastic continua: Navier equation and elastic waves.
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 in thermal conduction: 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. Free convection: Boussinesq approximation, Grashof and Rayleigh numbers, Rayleigh-Bénard instability.