 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4/37a/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Axisymmetric incompressibility is

\partial_{r} (r u_{r}) + r \partial_{z} u_{z} = 0

. Multiply the radial momentum equation by

r

and use this to put advection in conservative form:

\partial_{r} (r u_{r}^{2}) + r \partial_{z} (u_{r} u_{z}) = ν r \partial_{z}^{2} u_{r} .

Integrating through the layer gives

\frac{d F}{d r} = r [ν \partial_{z} u_{r} - u_{r} u_{z}]_{- \infty}^{\infty} = 0.

The boundary terms vanish because radial velocity and its shear decay into the surrounding fluid; an entrainment velocity need not itself vanish as long as its product with

u_{r}

does. Hence

F = \int_{- \infty}^{\infty} r u_{r}^{2} d z is independent of r .

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