= Crocco form of the unsteady Euler equation
{c}
{title2=$\partial_t\mathbf u+\boldsymbol\omega\times\mathbf u+\nabla\varepsilon=T\nabla s$}
For a fixed-composition inviscid fluid, <specific enthalpy> satisfies $dw=Tds+dp/\rho$. Combining this identity with the <vorticity cross-product identity> gives the displayed equation, with $\boldsymbol\omega=\nabla\times\mathbf u$ and $\varepsilon=u^2/2+\Phi+w$. It exposes entropy gradients as the thermodynamic source of a mismatch between <Bernoulli function> gradients and the vorticity term. In a steady <adiabatic process>, its projection along a <streamline> makes $\varepsilon$ constant there.
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