= Oscillating shear dispersion in an insulating channel
{title2=$K=\kappa+U^2a/[4(a^2+\omega^2)],\quad a=\kappa\pi^2/L^2$}
For $u=U\cos(\omega t)\sin(\pi y/L)$ on $|y|<L/2$, positive <diffusivity> $\kappa$, nonzero frequency and insulating walls, a transversely uniform zeroth moment remains constant. Its first moment is proportional to $g(t)\sin(\pi y/L)$, where $g'+ag=U\cos(\omega t)$. The periodic solution is $g=U[a\cos(\omega t)+\omega\sin(\omega t)]/(a^2+\omega^2)$. The <longitudinal moment hierarchy for shear transport> then gives $m_2=2Ktm_0+O(1)$ after both transverse and periodic transients. The spatial average of $\sin^2(\pi y/L)$ is $1/2$ and the time average of $\cos^2(\omega t)$ is $1/2$, producing the factor $1/4$. At exactly zero frequency the steady enhancement is instead $U^2/(2a)$; averaging indefinitely over slow cycles does not commute with setting the frequency to zero.
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