= In-phase oscillating parallel-plate flow
{title2=$u=\operatorname{Re}[Ue^{i\Omega t}\cosh(qy)/\cosh(qd)]$}
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 $f''=i\rho\Omega f/\mu$, $f(\pm d)=1$, 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 $u\sim Ue^{-ks}\cos(\Omega t-ks)$, $k=\sqrt{\rho\Omega/(2\mu)}$. Without initial data this specifies the periodic response; decaying zero-wall transients may be superposed.
Back to article page
