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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 2

The wall speed sets
$$
\boxed{U(x)=\alpha x}.
$$
Using $x$ as the local streamwise length gives
$$
\operatorname{Re}_x=\frac{U(x)x}{\nu}
=\frac{\alpha x^2}{\nu},
$$
which is large when $x\gg\sqrt{\nu/\alpha}$. The <boundary-layer scaling>
$$
\frac{U^2}{x}\sim\frac{\nu U}{\delta^2}
$$
then gives
$$
\delta(x)\sim\sqrt{\frac{\nu x}{U(x)}}
=\boxed{\sqrt{\frac{\nu}{\alpha}}},
$$
so the stretching-sheet layer has constant thickness to leading order.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 39C / b / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 39C / b / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 39C / b / i / Solution

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