Relativistic Hydrodynamics

When the velocity of fluids is high enough, it is important to take into account the relativistic effects, which yields the relativistic hydrodynamics. Recently, there have been considerable progress in relativistic hydrodynamics in a lot of different fields of physics, such as nuclear physics, high energy physics and plasma physics.In the non-relativistic limit, we can use 3-dimensional space coordinates which are sufficient to describe the motion of fluids, while in the relativistic limits, we should formu-late hydrodynamics in 4-dimensional space-time since space and time are fixed together by Lorentz transformation in relativity. By generating hydrodynamics from 3D to 4D space-time, we need to employ the tensor algebra in order to preserve Lorentz invariance of this formulism. Under this principle, we can derive the equations of hydrodynamics based on the conservation of energy momentum tensor, we established momentum tensor plus some physical intuition and as a basis to derive the hydrodynamic equations.However the energy momentum tensor is only given formally, and the problem is how to formulate concrete ex-pressions to describe the movement of the fluids. A natural idea is to expand it with Taylor series,and the principle is the entropy theory and conservation laws of energy momentum.Energy momentum tensor can expand to different orders with the results of different equations. When expands to zeroth order, it is non-viscosity Euler equation. Ignoring vis-cosity, there are two very important models, which are Bjiorken model and Landau model, both can be used to describe some behaviors in relativistic heavy Ion Collisions. Further more, when the energy momentum tensor is expanded to oneth order, it is necessary to consider the effects of the viscosity. Expanding to first order corresponds to the relativistic Navier-Stokes equations. When we transit it to low speed, Navier-Stokes equations have a wide application and fully meets the needs of engineering precision, but still complicated. Further more, when the energy momentum tensor is expanded to second order, we obtain Israel-Stewart theory, which has a very complex mathematical structure, and has made some progress in describing collective behaviors in nuclear-nuclear collisions.With the hydrodynamic equations at hand, a very important issue is to solve them. Relativistic hydrodynamics equations are nonlinear, partial differential equations, which are difficult to find analytical solutions. Recently, the most popular way is to solve fluid dynamics through numerical methods. Nevertheless analytical solutions is still important and can reflect the physical nature of the flow and can be viewed as the basis of numerical simulation study. In addition, hydrodynamic self-similarity is also an important method since it can convert multi-dimensional to low-dimensional.