= Solution
Let $e$ be the specific internal energy, $s$ the <specific entropy>, $T$ the absolute <temperature>, and $w=e+p/\rho$ the <specific enthalpy>. The <first law of thermodynamics> for a simple fluid of fixed composition is $de=Tds-p\,d(1/\rho)$, so
$$
\boxed{dw=T\,ds+dp/\rho,\qquad \nabla p/\rho=\nabla w-T\nabla s.}
$$
The unmagnetized <Euler equations for an inviscid fluid> are $\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\nabla\Phi-\nabla p/\rho$. Define the <vorticity> $\boldsymbol\omega=\nabla\times\mathbf u$ and use the <vorticity cross-product identity> $(\mathbf u\cdot\nabla)\mathbf u=\boldsymbol\omega\times\mathbf u+\nabla(u^2/2)$. Substitution gives the <Crocco form of the unsteady Euler equation>:
$$
\boxed{\partial_t\mathbf u+\boldsymbol\omega\times\mathbf u
+\nabla\left(u^2/2+\Phi+w\right)=T\nabla s.}
$$
Adiabaticity means $Ds/Dt=0$ for smooth flow without dissipative heating. It does not require $s$ to be uniform between different fluid elements, so the entropy-gradient term generally remains. For a <perfect gas> with constant <adiabatic exponent> $\gamma$, one may write $w=\gamma p/[(\gamma-1)\rho]$, but the thermodynamic derivation does not require that particular <equation of state>.
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