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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 3

The definitions through the Stokes stream function give
$$
{1\over r}\partial_r(ru)+\partial_zw
=-{1\over r}\partial_r\partial_zΨ+{1\over r}\partial_z\partial_rΨ=0,
$$
so <incompressible flow>[incompressibility] is automatic.

Balancing inertia $A^2/r^5$ with viscosity $νU/δ^2$ gives
$$
δ(r)\sim(νr^3/A)^{1/2}.
$$
The streamfunction scale is $rUδ\sim(Aνr)^{1/2}$, motivating
$$
\boxed{Ψ=(Aνr)^{1/2}F(η),\qquad η=z(A/(νr^3))^{1/2}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 38A / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 38A / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 38A / b / Solution

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