Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 38B / iii

Table of contents
- Solution iii

Solution

Let the supplied volume per unit width be

V (t)

. The governing balance gives the scales

\frac{h}{t} \sim C \frac{h ^{3}}{x}, V \sim h x .

Eliminating

x

yields

V (t) \sim C h^{3} t .

For the head height to approach a nonzero constant

h_{\infty}

, the total supplied volume must therefore grow linearly:

\frac{d V}{d t} \sim C h_{\infty}^{3} = \frac{g sin α}{3 ν} h_{\infty}^{3} = constant .

This is the constant-head supply for a gravity-driven thin film.

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