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
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$\begingroup$ The relativistic version of Navier-Stokes equations is a quite different theory called Israel-Stewart hydrodynamics ("relativistic Navier-Stokes" are known as "Eckart" or "Landau" relativistic hydrodynamics, but this naive relativistic generalization of Navier-Stokes does not work, it is highly unstable and acausal, so the more advanced and complex formulation of Israel-Stewart is needed). $\endgroup$– QuilloCommented Oct 4, 2022 at 16:55
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2$\begingroup$ Possibly interesting: "Relativistic theories of dissipative fluids" J. Math. Phys. 36, 4226 (1995); doi.org/10.1063/1.530958 Robert Geroch $\endgroup$– robphyCommented Oct 5, 2022 at 23:53
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$\begingroup$ @robphy yes, this paper of Geroch is a seminal work! $\endgroup$– QuilloCommented Oct 6, 2022 at 0:06
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$\begingroup$ MHD is typically derived using Eulerian flows, which neglect viscosity, rather than NS equations. Including viscosity in such a scenario is generally called "Extended MHD" or something akin to that (with the inviscid case generally called "Ideal MHD"). $\endgroup$– Kyle KanosCommented Oct 27, 2022 at 20:48
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I expand my previous comment. Yes, there are relativistic versions of the Navier-Stokes equation: They extend to special (or general) relativity the usual Navier-Stokes equations coupled to heat conduction (i.e., energy diffusion, see this answer). 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).
Beyond relativistic Navier-Stokes: To fix the stability and causality problems of relativistic versions of Navier-Stokes hydrodynamics, we have to look for a more general framework for dissipative relativistic hydrodynamics. The various possibilities, their underlying assumptions and motivations are summarized in this review. A possible (and widely used) alternative to Eckart or Landau versions of the relativistic Navier-Stokes is the so-called "Israel-Stewart hydrodynamics" (later revised and upgraded in Derivation of fluid dynamics from kinetic theory with the 14-moment approximation). You can find an introduction to Israel-Stewart hydrodynamics 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 "When the entropy has no maximum: A new perspective on the instability of the first-order theories of dissipation": 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.
