# Euler equation: momentum conservation in terms of enthalpy?

So I stumbled upon this equation:

$\frac{\partial h v}{\partial t}= - \nabla(hv \cdot v)- \nabla P$

where $h$ is enthalpy, $v$ is fluid velocity, $t$ time, and $P$ pressure.

It seems to have the form of the momentum conservation equation in hydrodynamics. However I don't understand the use of enthalpy here. It makes sense to me to replace enthalpy in energy density but not in momentum. I can't find anything online about this.

The source of this equation is equations 2 and 3 in http://arxiv.org/pdf/1008.4806v1.pdf

• Can you provide the source for that equation? I've never seen enthalpy used there either. Commented Aug 8, 2015 at 17:49
• Hi. I added the source. Commented Aug 8, 2015 at 18:34
• It appears they are using it as an energy equation there. I've seen something similar in some relativistic situations (though there are densities & Lorentz-$\gamma$'s included as well in those cases) Commented Aug 8, 2015 at 19:02
• Why the conservation of "flux" tho, instead of just h. Like why not $\frac{\partial h}{\partial t}$ instead of $\frac{\partial h v}{\partial t}$, that's what is confusing me. Commented Aug 8, 2015 at 19:18
• In the aforementioned relativistic situations, it is $\partial_t\left(\rho\gamma^2hv\right)$. I am unsure how the authors use that, but they cite Landau & Lifshitz for their equation, so you may want to check that book. Commented Aug 8, 2015 at 19:23