= Inviscid startup in a pressure-driven tube
{title2=$U(t)=\sqrt{2P/\rho}\tanh\left(\frac tL\sqrt{\frac P{2\rho}}\right)$}
For a slender tube of length $L$ supplied from a large reservoir with excess pressure $P$, negligible gravity and entrance inertia, the <Unsteady Bernoulli equation> gives $L\dot U+U^2/2=P/\rho$. A stationary initial liquid column has the displayed speed and <volume flux> $\pi a^2U$ for tube radius $a$. The inertial term controls the early linear acceleration; the kinetic-energy term sets the terminal inviscid speed. This is a leading slender-tube model while the reservoir remains large and the tube full.
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