The energy-momentum tensor for a perfect fluid is [itex]T^{ab}=(\rho_0+p)u^au^b-pg^{ab}[/itex] (using the +--- Minkowski metric).(adsbygoogle = window.adsbygoogle || []).push({});

Using the conservation law [itex]\partial_bT^{ab}=0[/itex], I'm coming up with [itex](\rho+\gamma^2p) [\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]= -{\nabla}p[/itex] instead of [itex]\rho[\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]= -{\nabla}p[/itex] (disregarding the body force part of the equation). Why is there a term for the part in brackets multiplied by [itex]\rho[/itex], but not for [itex]\gamma^2p[/itex]?

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# Deriving the Navier-Stokes equation from energy-momentum tensor

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