 ## Course content:

• Properties and equations of fluids: Lagrangian and Eulerian views of fluid flow; equations of fluids (continuity equation, Euler’s equation, conservation of energy), equations of state (including polytropic models); application to astrophysical systems.
• Static solutions to fluid equations: hydrostatic equilibrium, application to modelling stellar structure.
• Forces acting on fluids: viscosity (Navier-Stokes equation), gravity and Poisson’s equation, magnetic fields, source terms (e.g., mass loading, heating, cooling), and their roles in astrophysical systems.
• Accretion and shocks: Bernoulli equation and Bondi accretion, continuous shocks, Rankine-Hugoniot conditions, astrophysical applications (e.g., active galactic nuclei, supernovae remnants, radio galaxies).
• Fluid instabilities: convection, sound waves in fluids, instabilities at fluid boundaries (Rayleigh-Taylor and Kelvin-Helmholtz instabilities), Jeans instability and star formation.
• Eulerian approaches to fluid simulation: finite difference methods, von Neumann stability analysis, Courant-Friedrichs-Lewy stability criterion, stiff equations and implicit methods, the shock tube problem and Godunov’s method, applications in astrophysics.
• Lagrangian approaches to fluid simulation: N-body codes, tree codes, force softening, smoothed particle hydrodynamics, applications in astrophysics.
• Limitations of astrophysical simulations: sub-grid physics, semi-analytic models, the effects of spatial and temporal resolution limits on the interpretation of simulation results.

## After completing this module students are expected to be able to:

• Explain and derive the fundamental equations that describe fluid motion.
• Explain the differences between Eulerian and Lagrangian approaches to fluid simulation, and the advantages and disadvantages of each.
• Understand the relationship between the fluid equations and the equations of stellar structure.
• Evaluate situations in which the behaviour of fluids in astrophysical systems is significantly affected by external forces, such as viscosity, magnetic fields, heating/cooling, or mass loading, and describe these in terms of modifications of the fluid equations.
• Apply the theory of Bondi accretion to astrophysical systems and calculate accretion rates.
• Identify instabilities and shocks in astrophysical observations or simulations, and describe and explain their causes and effects on the system in question.
• Evaluate the stability of various finite difference schemes.
• Describe and explain the implementation of various N-body approaches to fluid simulation.
• Discuss critically the limitations of numerical methods applied in astrophysical simulations.