 Stokes' second problem

Stokes' second problem describes the oscillatory shear flow generated by a tangentially oscillating flat plate in a semi-infinite viscous fluid flow. The velocity satisfies $u_t=\nu u_{xx}$, with $u(0,t)=U_0\cos \omega t$ and decay away from the plate. For $\omega >0$, the displayed time-periodic solution follows by taking the real part of $U_0{e}^{i\omega t-(1+i)x/\delta }$. The length $\delta$ is the penetration depth: both amplitude attenuation and phase lag grow with $x/\delta$. Boundary data alone also allow homogeneous transients; the periodic solution describes the established oscillation after those transients decay.