= Solution
For an inviscid fluid of constant <mass density> $\rho$ with no body force, the <Euler equations for an inviscid fluid> are
$$
\partial_tu+(u\cdot\nabla)u=-\frac1\rho\nabla p.
$$
The vector identity
$$
(u\cdot\nabla)u=\nabla\!\left(\frac12|u|^2\right)-u\times(\nabla\times u)
$$
and <irrotational flow> give $(u\cdot\nabla)u=\nabla(|u|^2/2)$. Writing $u=\nabla\phi$ in terms of a <velocity potential>, we obtain
$$
\nabla\!\left(\partial_t\phi+\frac12|\nabla\phi|^2+\frac p\rho\right)=0.
$$
Thus the bracket depends only on time:
$$
\boxed{\partial_t\phi+\frac12|\nabla\phi|^2+\frac p\rho=C(t)}.
$$
Since adding a function of time to $\phi$ leaves $u=\nabla\phi$ unchanged, that function may be chosen to absorb $C(t)$. This is the <Unsteady Bernoulli equation>.
Back to article page
