= Oscillatory channel flux of a linear Maxwell fluid
{title2=$Q=-\frac{2\Delta p h}{i\omega\rho}\left(1-\frac{\tanh z}{z}\right)$}
For a no-slip channel with half-width $h$, a harmonic <pressure gradient> $\Delta p e^{i\omega t}$ and <complex viscosity> $\widehat\mu=G_0\tau/(1+i\omega\tau)$, put $z^2=i\omega\rho h^2/\widehat\mu$. Solving $\widehat\mu U''-i\omega\rho U=\Delta p$ gives $U=-\Delta p[1-\cosh(zy/h)/\cosh z]/(i\omega\rho)$; integration gives the displayed flux per unit width. At low frequency it approaches $-2\Delta p h^3/(3G_0\tau)$. At fixed positive $h,\tau,G_0,\rho$, the high-frequency leading flux is $-2\Delta p h/(i\omega\rho)$, although shear waves can persist in the local profile. If instead $\omega\tau\gg1$ while $\rho\omega^2h^2/G_0\ll1$, the negligible-inertia elastic limit is $Q\simeq-2i\omega\Delta p h^3/(3G_0)$. The last two limits impose different scale orderings.
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