 Solution

For $\mathbf{u}=u(y,t){\mathbf{e}}_x$, incompressibility is automatic and the nonlinear advection term is zero. With no pressure gradient the streamwise momentum equation is $\boxed{\rho u_t=\mu u_{yy}}$. Seek its periodic response as $u=\mathrm{Re}[Uf(y)e^{i\Omega t}]$. The amplitude satisfies $f^{\prime \prime }=i\rho \Omega f/\mu$ with $f(\pm d)=1$. Since $[k(1+i){]}^2=i\rho \Omega /\mu$ for $k=\sqrt{\rho \Omega /(2\mu )}$, even symmetry givesOdd components vanish because the two boundary amplitudes are equal. This is the unique periodic solution. Without a specified initial profile, decaying zero-wall diffusion transients may be added, so the formula should not be read as a claim that every transient is absent.

As $kd\to \infty$, the wall motions penetrate only a distance $k^{-1}=\sqrt{2\mu /(\rho \Omega )}$. Near either wall, with inward distance $s=d-|y|$, the leading velocity isThe two oscillatory viscous layers are separated by an almost motionless core; the phase lag increases inward. This is in-phase oscillating parallel-plate flow.