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 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.