Oscillatory channel flux of a linear Maxwell fluid ($Q=-\frac{2\Delta ph}{i\omega \rho }\left(1-\frac{\tanh z}{z}\right)$)

For a no-slip channel with half-width $h$, a harmonic pressure gradient $\Delta p{e}^{i\omega t}$ and complex viscosity $\hat{\mu }=G_0\tau /(1+i\omega \tau )$, put $z^2=i\omega \rho h^2/\hat{\mu }$. Solving $\hat{\mu }{U}^{\prime \prime }-i\omega \rho U=\Delta p$ 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 $-2\Delta ph/(i\omega \rho )$, although shear waves can persist in the local profile. If instead $\omega \tau \gg 1$ while $\rho {\omega }^2{h}^2/G_0\ll 1$, the negligible-inertia elastic limit is $Q\simeq -2i\omega \Delta p{h}^3/(3G_0)$. The last two limits impose different scale orderings.