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A physicist and mathematician associated with Stokes flow, Stokes' second problem and Stokes theorem.

For the established time-periodic response, take a complex separated solution $v=X(x)e^{i\omega t}$ and set $u=\mathrm{Re}v$. The diffusion equation gives $X^{\prime \prime }=(i\omega /\nu )X$. For $\omega >0$, define the penetration depth $\delta =\sqrt{2\nu /\omega }$; the spatial roots are $\pm (1+i)/\delta$. Decay at infinity selectsIts real part is the solution of Stokes' second problem:The velocity amplitude decays exponentially, while the phase lags by $x/\delta$. Direct substitution verifies the diffusion equation, and the two boundary conditions follow immediately.

These data select a unique periodic response, but do not specify an arbitrary initial transient. With no initial velocity supplied, one may add any suitable solution of the homogeneous-boundary diffusion problem. The displayed solution is the usual established oscillatory flow after transients have decayed.

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.