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Astrophysical fluid dynamics models gases and plasmas in stars, accretion discs, stellar winds and other astronomical flows. Its distinctive ingredients can include self-gravity, rotation, compressibility, radiative cooling and magnetic fields. ^[ref]

A useful scale comparison is the Mach number $\mathrm{Ma}=U/c_s$, with flow speed $U$ and sound speed $c_s$. Large compressions can produce shock waves; self-gravity can instead amplify density disturbances through Jeans instability. Conducting plasma adds magnetohydrodynamics. A continuum approximation must be justified: a very tenuous medium can require a kinetic description instead of a simple collisional fluid.

Gordon I. Ogilvie develops the fluid dynamics of astrophysical systems. His lecture notes connect conservation laws, rotation and magnetic fields with simplified models of stars and disks, while his accretion-disk teaching shows how boundary conditions constrain steady transport.^[ref][ref]

A local stress prescription alone does not fix an entire disk solution: the mass supply, angular-momentum flux and conditions at its boundaries must be specified as well.

Ideal magnetohydrodynamics describes a conducting fluid whose electromagnetic evolution neglects electrical resistivity. Its defining electric-field relation isso the electric field vanishes in the local nonrelativistic fluid rest frame. Together with Faraday's law this gives the ideal magnetohydrodynamic induction equationThe Lorentz force density couples the field back to fluid momentum. ^[ref]

For sufficiently smooth fields, the induction equation implies magnetic flux freezing: flux through a material surface moving with the fluid is conserved. Stretching the fluid can therefore amplify the magnetic field while preserving its connectivity. Magnetic tension supports Alfvén waves, providing a direct dynamical consequence of the coupling.

A large magnetic Reynolds number $\mathrm{Rm}=UL/\eta$ motivates neglecting diffusion on the chosen scales. Thin current sheets can still make nonideal terms important locally. The ideal induction condition alone does not set fluid viscosity; mechanical dissipation and magnetic diffusion are different approximations.

Magnetohydrodynamics couples an electrically conducting fluid to a magnetic field. The electric current density $\mathbf{J}$ produces a Lorentz force density $\mathbf{J}\times \mathbf{B}$, and fluid motion changes the magnetic field through induction.

For constant magnetic diffusivity $\eta$, a common nonrelativistic model includeswith $\nabla \cdot \mathbf{B}=0$. Ideal magnetohydrodynamics neglects the diffusion term; resistive magnetohydrodynamics retains it. The ratio of induction $UB/L$ to diffusion $\eta B/L^2$ is the magnetic Reynolds number $\mathrm{Rm}=UL/\eta$.

Magnetic tension supplies a restoring force for an Alfvén wave, with speed $v_A=B/\sqrt{{\mu }_0\rho }$ in a homogeneous medium. Magnetic pressure and ordinary pressure can both oppose compression. These effects explain why conducting-fluid waves and equilibria differ from those of an unmagnetized gas. ^[ref]

A finite-amplitude compression can steepen because different parts of the wave travel at different characteristic speeds. For a right-moving simple wave in a perfect gas, one may write $dx/dt=c_0+(\gamma +1)u/2$ in terms of the local velocity perturbation $u$. Faster compressive characteristics catch slower ones. When the smooth inviscid description would become multivalued, a physical compression shock replaces it. A rarefaction normally spreads rather than forming an entropy-violating expansion shock.^[ref]

Across a locally planar shock, use velocities relative to its moving front. The Rankine-Hugoniot conditions for a perfect gas conserve mass, momentum and total energy flux:where $h=\gamma p/[(\gamma -1)\rho ]$ is the specific enthalpy. For upstream normal Mach number $M_1>1$,Thus a normal compression shock increases pressure, density and temperature, reduces the normal velocity, and changes a supersonic upstream flow to subsonic downstream flow in the shock frame. In the strong-shock limit, compression approaches $(\gamma +1)/(\gamma -1)$, which is four for a monatomic gas. A shock conserves total energy but generates entropy; an adiabatic shock is not an isentropic compression. For a calorically perfect gas, stagnation temperature is conserved while stagnation pressure falls.^[ref]

Real dissipative processes resolve the narrow front; the jump conditions do not require its detailed internal profile. In a continuum-gas experiment a shock tube can produce a moving shock by releasing a pressure discontinuity. pressure probes separated along the tube measure arrival speed and the pressure jump; distinguishing the shock from the following contact surface is essential. A contact surface can have a density/entropy jump with continuous pressure and normal velocity, whereas a shock carries nonzero mass flux through its front.