Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 2 / b

The transverse and longitudinal Péclet numbers areThe Taylor dispersion regime requires transverse diffusion to act within a longitudinal advection time,while longitudinal molecular diffusion is slow on that advection time,Together,

The advection-diffusion equation isWritewhere the bar is the cross-gap average. In the long, late-time Taylor regime, $c^{\prime }$ adjusts rapidly across the gap while $\bar{c}$ varies slowly along the cell. The leading fluctuation balance isIts scaling isThis final inequality is precisely the transverse-equilibration condition from part i.

Put $c^{\prime }=\chi (y){\bar{c}}_x$. The balance and reflecting boundary conditions giveIntegration yieldsAveraging the full transport equation givesNowbecause $\overline{y^2}=h^2/3$ and $\overline{y^4}=h^4/5$. Thus the flow-induced Taylor dispersion coefficient and total effective diffusivity are