Fluid Solution - Special Cases

Special Cases

Several special cases of fluid solutions are noteworthy:

• A perfect fluid has vanishing viscous shear and vanishing heat flux:

,

• A dust is a pressureless perfect fluid:

,

• A radiation fluid is a perfect fluid with :

The last two are often used as cosmological models for (respectively) matter-dominated and radiation-dominated epochs. Notice that while in general it requires ten functions to specify a fluid, a perfect fluid requires only two, and dusts and radiation fluids each require only one function. It is much easier to find such solutions than it is to find a general fluid solution.

Among the perfect fluids other than dusts or radiation fluids, by far the most important special case is that of the static spherically symmetric perfect fluid solutions. These can always be matched to a Schwarzschild vacuum across a spherical surface, so they can be used as interior solutions in a stellar model. In such models, the sphere where the fluid interior is matched to the vacuum exterior is the surface of the star, and the pressure must vanish in the limit as the radius approaches . However, the density can be nonzero in the limit from below, while of course it is zero in the limit from above. In recent years, several surprisingly simple schemes have been given for obtaining all these solutions.