Taylor dispersion in a linear porous-layer velocity profile ($D_{\mathrm{eff}}=D_L+U^2{H}^2/(30D_T)$)

For pore velocity $u(y)=2Uy/H$ on $0<y<H$, with reflecting transverse boundaries, solve $D_T{\chi }^{\prime \prime }=u-U$, ${\chi }^{\prime }(0)={\chi }^{\prime }(H)=0$, $\overline{\chi }=0$. The Taylor dispersion enhancement is $D_T\overline{({\chi }^{\prime }{)}^2}=U^2{H}^2/(30D_T)$. Here $U$ is the mean advective pore velocity, not the Darcy velocity, and $D_L$ is the original longitudinal dispersion. The averaged equation applies after transverse equilibration and on suitably long longitudinal scales.