= Weak peristaltic pumping of a collapsible tube
{title2=$Q_2=\frac{c(n+1)}{2\alpha_0}\frac{P_e^2}{\widetilde P_0'^2+(R_0c/(k\alpha_0))^2}$}
For negligible inertia, periodic forcing $p_e=\epsilon P_e\sin kX$, resistance $R\propto\alpha^{-n}$ and prescribed mean area $\alpha_0$, write $Q=-c\alpha_0+\epsilon^2Q_2+\cdots$. First-order area obeys $\widetilde P'_0\alpha_1'+R_0c\alpha_1/\alpha_0=-kP_e\cos kX$. Its mean square is $P_e^2/[2(\widetilde P_0'^2+(R_0c/(k\alpha_0))^2)]$. Averaging the second-order friction balance gives $Q_2=c(n+1)\langle\alpha_1^2\rangle/\alpha_0$, proving the displayed forward pumping rate. Periodicity entails zero mean <pressure> drop; the mean-area convention fixes the expansion's otherwise free constant.
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