Navier-Stokes equation is non-relativistic, what is relativistic Navier-Stokes equation through Einstein notation?


Are you referring to the exact relativistic equivalent to Navier-Stokes equation or a more general Dissipative Relativistic Hydrodynamics Equation?

The "relativistic equivalent to Navier-Stokes equation" would be something like this:

There would be an energy momentum tensor with the following form:

$T_{\mu\nu} = (e+p)u_\mu u_\nu - p g_{\mu\nu} + \tau_{\mu\nu}$

with the viscous stress tensor given by:

$\tau_{\mu\nu}= -\eta \left[(\partial_\mu u_\nu+\partial_\nu u_\mu) - u_\mu u^\alpha \partial_\alpha u_\nu - u_\nu u^\alpha\partial_\alpha u_\mu\right] - (\zeta - \frac{2}{3}\eta)\left[\partial_\alpha u^\alpha\right](g_{\mu\nu}-u_\mu u_\nu) $

And the Conservation of Energy and Momentum would lead to:

$\partial^\mu T_{\mu\nu}=0$

The idea is to try to transpose the expressions you use in usual non-relativistic NS equation and fix the terms via positivity of entropy production. The relativistic Euler equation can be obtained setting $\tau_{\mu\nu}=0$, although some effort must be made to recover the Euler equation, it can be done.

The reference that I used was L. D. Landau and E.M. Lifshitz, Fluid Mechanics, chap. XV

I'm asking this because this is not the only formulation of dissipative relativistic hydrodynamics, and it`s not necessarily the best one.


I'm using $g_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$, $u^\mu$ refers to the 4-velocity of the fluid, $e$ and $p$ refer to the proper energy density and the pressure, $\zeta$ and $\eta$ would refer to the bulk and shear viscosity.

Edit: Sorry, more typo corrections

  • $\begingroup$ Is $\partial_a(\rho u^a) = 0$ implied by $\partial_aT^a{}_b=0$? Or do you need to provide it separately? $\endgroup$ – Jackozee Hakkiuz Jul 3 '18 at 5:30

protected by Qmechanic Feb 20 '14 at 13:58

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.