Is there a relativistic version of Navier-Stokes equations? Just as the title says, is there a relativistic version of Navier-Stokes equations?
In electromagnetic hydrodynamics it would be very useful to have relativistic version of Navier-Stokes equations, although I couldn't find one
 A: I expand my previous comment. Yes, there are relativistic versions of the Navier-Stokes equation. You can find them in several famous books, in particular:
Landau, Fluid Mechanics (volume 6 of the theoretical physics course). In the chapter dedicated to relativistic hydrodynamics, you find the famous relativistic version of the Navier-Stokes equation in the so-called "Landau frame". See, e.g., this answer.
Weinberg, Gravitation and Cosmology: here you can find the relativistic generalization of Navier-Stokes in the so-called "Eckart frame" (chapter 11).
The problem is that both these "naive" relativistic generalizations of Navier-Stokes do not work: the partial differential equations, despite being written in a covariant fashion, display instabilities and lead to non-causal propagation of signals (they display instabilities both at the computer simulation level, i.e. once discretized, as well as at the exact mathematical level!). This is a theorem based on the analysis of Hiscock and Lindblom (1985).
Therefore, we have to look for a more general (and complex) framework for (dissipative) relativistic hydrodynamics. A possible (and widely used) alternative to Eckart or Landay versions of the relativistic Navier-Stokes is the so-called "Israel-Stewart hydrodynamics". You can find an introduction in the recent book by Rezzolla and Zanotti: this formulation overcomes the instability problem and is causal (signals, like sound waves, propagate subliminally, namely with a speed less than the one of light), as shown in a seminal work by Hiscock and Lindblom (1983).
Why Eckart & Landau's approaches fail: A simple explanation of why Navier-Stokes does not work in special and/or general relativity is given in this article: in the formulations of Landau and Eckart the entropy function turns out not to have a maximum (the homogeneous equilibrium state is an unstable equilibrium point). Therefore, since the fluid wants to maximise entropy, the fluid explodes because entropy "wants" to grow indefinitely.
