Astrophysical fluid dynamics
Astrophysical fluid dynamics is a modern branch of astronomy involving fluid mechanics which deals with the motion of fluids, like the gases which the stars are made up of or any fluid which is found in outer space. The subject covers the fundamentals of mechanics of fluids using various equations, ranging from the continuity equation, Navier Stokes to Euler's equations of collisional fluids and the like. It is an extensive study of the physical realms of the astral bodies and their movements in space. A thorough understanding of this subject requires detailed knowledge of the equations  governing fluid mechanics. Most of the applications of astrophysical fluid dynamics include dynamics of stellar systems, accretion disks, Astrophysical jets, Newtonian fluids, and the fluid dynamics of galaxies. Introduction. Astrophysical fluid dynamics deals with the application of fluid dynamics and its equations in the movement of the fluids in space. The applications are entirely different from what we usually study as all of this happens in vacuum with zero gravity. Most of the Interstellar Medium is not at rest, but is in supersonic motion under the action of supernova explosions, stellar winds and radiation fields and the time dependent gravitational  field due to spiral density waves in the stellar disc of the galaxy. Since supersonic motions almost always involve shock waves, these play a crucial role. The galaxy also contains a dynamically significant magnetic field which means that the dynamics is governed by the equations of compressible magnetohydrodynamics. In many cases the electrical conductivity is large enough for the ideal magnetohydrodynamics to be a good approximation, but this is not true in star forming regions where the gas density is high and the degree of ionization is low. One of the most interesting problems is that of star formation. It  is known that stars form out of the Interstellar Medium and that this mostly occurs in Giant Molecular Clouds such as the Rosette Nebula for example. It has been known for a long time that an interstellar cloud can collapse due to its self-gravity if it is large enough, but in the ordinary interstellar medium, this can only happen if the cloud has a mass of several thousand solar masses - much larger than that of any star. There must therefore be some process that fragments the cloud into smaller high density clouds whose masses are in the same range  as that of stars. Self-gravity cannot do this, but it turns out that there are processes that do this if the magnetic pressure is much larger than the thermal pressure, as it is in Giant Molecular Clouds. These processes rely on the interaction of magnetohydrodynamic waves with a thermal instability. A magnetohydrodynamic wave in a medium in which the magnetic pressure is much larger than the thermal pressure can produce dense regions, but they cannot by themselves make the density high enough for self-gravity to act. However, the gas in star forming regions is heated by  cosmic rays and is cooled by radiative processes. The net result is that gas in a thermal equilibrium state in which heating balances cooling can exist in three different phases at the same pressure: a warm phase with a low density, an unstable phase with intermediate density and a cold phase at low temperature. An increase in pressure, due to a supernova or a spiral density wave can flip the gas from the warm phase into the unstable phase and a Magnetohydrodynamic wave can then produce dense fragments in the cold phase whose self-gravity is strong enough for them  to collapse to form stars . In this process, we can study the dynamics of the cosmic gas and understand the formation of stars. This is just one example. Even Magnetohydrodynamics has its basis on the fundamentals of astrophysical fluid dynamics. Basic concepts. Concepts of Fluid Dynamics. The equations of Fluid Dynamics are tools in developing an understanding of the phenomena in Astrophysical Fluid Dynamics. The important equations with their applications are as mentioned below. Conservation of Mass The continuity equation applies the principle of conservation of mass to fluid flow. Consider a fluid flowing through a fixed volume tank having  one inlet and one outlet as shown below. If the flow is steady i.e. no accumulation of fluid within the tank, then the rate of fluid flow at entry must be equal to the rate of fluid flow at exit for mass conservation. If, at entry (or exit) having a cross-sectional area A (m), a fluid parcel travels a distance dL in time dt, then the volume flow rate (V, m/s) is given by: "V = (A . dL)/∆t" but since dL/∆t is the fluid velocity (v, m/s) we can write: "Q = V x A" The mass  flow rate (m, kg/s) is given by the product of density and volume flow rate "i.e m = ρ.Q = ρ .V.A" Between two points in flowing fluid for mass conservation we can write: m1=m2 If the fluid is incompressible i.e. ρ = ρ then: "V""A" "= V""A" But, We shall apply this theorem for Astrophysicsical Fluid Dynamics in supersonic Flow regime which will require us to consider a Compressible flow condition where density is not constant. An application for fluid dynamics in astrophysics is the Neutron stars, which are ancient remnants of stars that have  reached the end of their evolutionary journey through space and time. These interesting objects are born from once-large stars that grew to four to eight times the size of our own sun before exploding in catastrophic supernovae. After such an explosion blows a star's outer layers into space, the core remains—but it no longer produces nuclear fusion. With no outward pressure from fusion to counterbalance gravity's inward pull, the star condenses and collapses in upon itself. Despite their small diameters—about 12.5 miles (20 kilometers)—neutron stars boast nearly 1.5 times the mass of our sun, and  are thus incredibly dense. Just a sugar cube of neutron star matter would weigh about one hundred million tons on Earth. A neutron star's almost incomprehensible density causes protons and electrons to combine into neutrons—the process that gives such stars their name. The composition of their cores is unknown, but they may consist of a neutron superfluid or some unknown state of matter. Neutron stars pack an extremely strong gravitational pull, much greater than Earth's. This gravitational strength is particularly impressive because of the stars' small size. When they are formed, neutron stars rotate in space. As  they compress and shrink, this spinning speeds up because of the conservation of angular momentum—the same principle that causes a spinning skater to speed up when she pulls in her arms. These stars gradually slow down over the eons, but those bodies that are still spinning rapidly may emit radiation that from Earth appears to blink on and off as the star spins, like the beam of light from a turning lighthouse. This "pulsing" appearance gives some neutron stars the name pulsars. After spinning for several million years pulsars are drained of their energy and become normal neutron stars . Few of the known existing neutron stars are pulsars. Only about 1,000 pulsars are known to exist, though there may be hundreds of millions of old neutron stars in the galaxy. The staggering pressures that exist at the core of neutron stars may be like those that existed at the time of the big bang, but these states cannot be simulated on Earth. EMG (Estakhr's Material Geodesic) Equations. The Estakhr's Material-Geodesic (EMG) equations have applications in astrophysical fluid dynamics. They were introduced for the first time by the American Physical Society in 2013. These equations are  a relativistic version of the Navier-Stokes equations.