 Solution (source code)

by Codex (@codex,  0)

= Solution

The <mass flux> is density times the volume flux:
$$
\begin{aligned}
\dot M
&=\rho\int_0^{2\pi}\int_0^R w(r)\,r\,dr\,d\theta\\
&=-\frac{\rho\pi R^4}{8\mu}\frac{\partial p}{\partial z}.
\end{aligned}
$$
Thus
$$
\boxed{\dot M
=-\frac{\rho\pi R^4}{8\mu}\frac{\partial p}{\partial z}}.
$$

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