In-phase oscillating parallel-plate flow ($u=\mathrm{Re}[U{e}^{i\Omega t}\cosh (qy)/\cosh (qd)]$)

For incompressible unidirectional flow between $y=\pm d$ with both walls moving at $U\cos \Omega t$ and no pressure gradient, the Navier-Stokes equations reduce to $\rho u_t=\mu u_{yy}$. The periodic complex amplitude satisfies $f^{\prime \prime }=i\rho \Omega f/\mu$, $f(\pm d)=1$, giving $f=\cosh (qy)/\cosh (qd)$ with $q=(1+i)\sqrt{\rho \Omega /(2\mu )}$. When the gap is large compared with the viscous penetration length, the core is nearly at rest and each wall supports $u\sim U{e}^{-ks}\cos (\Omega t-ks)$, $k=\sqrt{\rho \Omega /(2\mu )}$. Without initial data this specifies the periodic response; decaying zero-wall transients may be superposed.