= Solution
For an <inviscid flow> of constant density $\rho$, driven by a conservative body force $-\nabla\Phi$, the <Euler equation> is
$$
\partial_tu+(u\cdot\nabla)u=-\nabla(p/\rho)-\nabla\Phi.
$$
The vector identity $(u\cdot\nabla)u=\nabla(|u|^2/2)-u\times(\nabla\times u)$ and irrotationality give the <Unsteady Bernoulli equation>. Specifically, wherever a single-valued potential $u=\nabla\phi$ is available,
$$
\nabla\left(\phi_t+\frac12|u|^2+\frac p\rho+\Phi\right)=0.
$$
On a connected potential-flow region this means
$$
\boxed{\phi_t+\frac12|u|^2+\frac p\rho+\Phi=C(t).}
$$
The integration function is spatially constant but need not be constant in time; it can be removed by changing the potential's time-dependent gauge. A simply connected irrotational region provides the requisite global potential. For a barotropic fluid the <pressure> term is replaced by $\int dp/\rho(p)$; here the liquid density is constant.
For the tube, let $U(t)$ be its approximately uniform axial <velocity>. The large reservoir's surface <velocity> is negligible and gravity is neglected as stipulated. Entrance-region inertia contributes an effective length of order $a$, negligible against $L$. The potential difference between the outlet and the reservoir is consequently $LU$ to leading order. Subtract Bernoulli at the reservoir surface, where <pressure> is $P+p_a$, from its value at the outlet, where <pressure> is $p_a$ and speed is $U$. This gives
$$
L\dot U+\frac12U^2=\frac P\rho,\qquad U(0)=0.
$$
The inertial term cannot be dropped during startup. With $U_\infty=\sqrt{2P/\rho}$, separate variables:
$$
\frac{d(U/U_\infty)}{1-(U/U_\infty)^2}=\frac{U_\infty}{2L}\,dt.
$$
Hence the <inviscid startup in a pressure-driven tube> gives
$$
\boxed{Q(t)=\pi a^2\sqrt{\frac{2P}{\rho}}
\tanh\left(\frac tL\sqrt{\frac P{2\rho}}\right).}
$$
Initially $U\sim Pt/(\rho L)$, the acceleration of the stationary liquid column. At long times $U\to\sqrt{2P/\rho}$, the inviscid pressure-to-kinetic-energy limit. The result applies while the tube remains full and reservoir dimensions are large enough that surface motion and the neglected entrance inertia remain small.
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