- Fundamentals of hydrodynamics: continuity and Euler equations, equation of state, conservation of mass/momentum/energy, viscosity and thermal conduction, Lagrangian vs Eulerian views, vorticity equation, Kelvin circulation theorem, rotating frames, Reynolds number, radiative heating and cooling. - Fundamentals of magneto-hydrodynamics: MHD equations, induction equation, magnetic pressure and tension, magnetic flux freezing, magnetic fields amplification. - Hydrostatic equilibrium: polytropic and isothermal spheres, polytropic and isothermal slabs, application: Chandrasekhar upper mass limit for white dwarfs - Spherical steady flows: Parker solar wind and Bondi spherical accretion - Waves: sound waves, water waves, group velocity, analogy between shallow water theory and gas dynamics, MHD waves (Alfvén waves. fast and slow waves), internal gravity waves. Effects of rotation on waves, epicyclic frequency and Lindblad resonances, density waves in discs - Shocks: steepening of sound waves and the formation of shocks, 1D jump conditions at discontinuity (Rankine-Hugoniot conditions), Mach number - Spherical blast waves: strong explosion in uniform atmosphere, Sedov-Taylor self-similar solution, application: supernovae - Accretion discs: Eddington limit, basic equations of viscous thin disc evolution, steady-state viscous thin discs, angular momentum transport, Shakura-Sunyaev alpha-disc model, emitted spectrum, vertical and radial balance, time-dependent accretion - Instabilities: Kelvin-Helmholtz instability, Thermal instability, Rayleigh instability, rotational instability and Rayleigh criterion, magneto-rotational instability, Jeans instability, Toomre instability - Gravity: Legendre expansion, Gauss’s Law, Poisson equation, Tidal forces, Virial theorem. Tides, the equilibrium tide, stability of satellites in orbit around the Earth - Elements of turbulence, Kolmogorov’s theory