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

Table of contents
- Solution ii

Solution

The advection-diffusion equation is

c_{t} + u (y) c_{x} = D (c_{xx} + c_{yy}), u (y) = \frac{U}{h} y .

Write

c = \overset{c}{ˉ} (x, t) + c^{'} (x, y, t), \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

\overset{c}{ˉ}

varies slowly along the cell. The leading fluctuation balance is

u (y) \overset{c}{ˉ}_{x} = D c_{yy}^{'} .

Its scaling is

\frac{U c ˉ}{L} \sim \frac{D c ^{'}}{h ^{2}}, \frac{c ^{'}}{c ˉ} \sim \frac{U h ^{2}}{D L} = Pe_{h} \frac{h}{L} ≪ 1 .

This final inequality is precisely the transverse-equilibration condition from part i.

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