The transverse and longitudinal <Péclet numbers> are
$$
\boxed{\operatorname{Pe}_h=\frac{Uh}{D}},
\qquad
\boxed{\operatorname{Pe}_L=\frac{UL}{D}
=\frac Lh\operatorname{Pe}_h}.
$$
The <Taylor dispersion> regime requires transverse diffusion to act within a longitudinal advection time,
$$
\frac{h^2}{D}\ll\frac LU
\quad\Longleftrightarrow\quad
\operatorname{Pe}_h\ll\frac Lh,
$$
while longitudinal molecular diffusion is slow on that advection time,
$$
\operatorname{Pe}_L\gg1.
$$
Together,
$$
\boxed{
\frac hL\ll\operatorname{Pe}_h\ll\frac Lh},
\qquad
\boxed{
1\ll\operatorname{Pe}_L\ll\left(\frac Lh\right)^2}.
$$

Solution

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

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{title2=$\operatorname{Pe}=UL/D$}
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The Péclet number compares advection over length $L$ at speed $U$ with molecular diffusion of diffusivity $D$.


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