Unsteady Bernoulli equation

For time-dependent irrotational flow with $u=\nabla \varphi$, constant mass density, and no body force, the Euler equations for an inviscid fluid integrate toA time-dependent change of gauge in $\varphi$ may set $C(t)=0$.

An inviscid uniform tube plug driven by entrance excess pressure $KV$ obeys $\rho L\dot{u}=KV$ and $\dot{V}=-\pi a^{2}u$. From rest, the first zero of $V$ occurs after a quarter harmonic period, independent of its initial volume. The maximum speed is proportional to that initial volume. Entrance losses, a reservoir kinetic head or independently prescribed initial flow change this ideal model.