The <advection-diffusion equation> is
$$
c_t+u(y)c_x=D(c_{xx}+c_{yy}),
\qquad
u(y)=\frac Uh y.
$$
Write
$$
c=\bar c(x,t)+c'(x,y,t),
\qquad
\overline{c'}=0,
$$
where the bar is the cross-gap average. In the long, late-time Taylor regime, $c'$ adjusts rapidly across the gap while $\bar c$ varies slowly along the cell. The leading fluctuation balance is
$$
\boxed{
u(y)\bar c_x=Dc'_{yy}}.
$$
Its scaling is
$$
\frac{U\bar c}{L}\sim\frac{Dc'}{h^2},
\qquad
\boxed{
\frac{c'}{\bar c}
\sim\frac{Uh^2}{DL}
=\operatorname{Pe}_h\frac hL\ll1}.
$$
This final inequality is precisely the transverse-equilibration condition from part i.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-334/2/b/ii/solution