Stokes' second problem ($u(x,t)=U_0{e}^{-x/\delta }\cos (\omega t-x/\delta ),\delta =\sqrt{2\nu /\omega }$)

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 $U_0{e}^{i\omega t-(1+i)x/\delta }$. 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.

The viscous penetration depth is the distance over which a time-periodic shear velocity driven by an oscillating boundary decays by $1/e$ in a semi-infinite Newtonian fluid. The momentum diffusion equation gives $u=U_0{e}^{-z/\delta }\cos (\omega t-z/\delta )$. Both attenuation and phase lag are controlled by $z/\delta$. The corresponding fluid-displacement amplitude has the same spatial decay, because dividing each velocity amplitude by the common angular frequency does not change their ratio.