Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 38D / a / Solution

The incompressibility equation turns the advective term into a divergence:

u u_{x} + v u_{y} = \partial_{x} (u^{2}) + \partial_{y} (uv),

because

v_{y} = - u_{x}

. For a plume in an otherwise stationary environment, the boundary-layer pressure equals the ambient pressure and has no imposed

x

-gradient. Integrating the vertical momentum equation across

y

therefore gives

\frac{d}{d x} \int_{- \infty}^{\infty} ρ u^{2} d y + [ρ uv]_{- \infty}^{\infty} = μ [u_{y}]_{- \infty}^{\infty} + b (x) .

Both boundary terms vanish as the velocity and shear decay away from the plume. Hence the integral momentum-flux balance for a two-dimensional plume is

\frac{d}{d x} \int_{- \infty}^{\infty} ρ u^{2} d y = b (x) .

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