Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 3 / b / Solution

In steady inviscid two-dimensional flow the spanwise vorticity obeys

u \cdot \nabla ω = [\nabla (ρ^{'} / ρ_{0}) \times g]_{y} .

Integrating this equation over the front-frame control volume converts the left side to vorticity flux through its upstream and downstream faces. For a sharp interface, the baroclinic source integrates to the circulation generated by the hydrostatic pressure jump,

g^{'} h

. With plug flow downstream, the resulting balance is

\frac{1}{2} u_{2}^{2} = g^{'} h, u_{2} = 2 g^{'} h .

Volume conservation in the front frame gives

u_{1} H = u_{2} (H - h)

. The undisturbed indoor air is stationary in the laboratory, so

u_{1}

is the front speed:

U_{f} = \frac{H - h}{H} 2 g^{'} h .

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