= Stokes' second problem
{c}
{title2=$u(x,t)=U_0e^{-x/\delta}\cos(\omega t-x/\delta),\quad\delta=\sqrt{2\nu/\omega}$}
<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 $U_0e^{i\omega t-(1+i)x/\delta}$. 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.
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