Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 38C / b

Table of contents
- Solution b

Solution

Let

U

be the streamwise speed and

δ

the layer width. Continuity gives

v \sim U δ / x

, while inertia--viscosity balance gives

\frac{U ^{2}}{x} \sim \frac{νU}{δ ^{2}} .

The imposed stress gives

ρ νU / δ \sim S x^{- 1/2}

. Eliminating

U

yields

δ \sim (\frac{ρ ν ^{2}}{S})^{1/3} x^{1/2},

and

U \sim (\frac{S ^{2}}{ρ ^{2} ν})^{1/3},

which is independent of

x

.

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