 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/39c/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

The steady incompressible Stokes flow equations are

- \nabla p + μ \nabla^{2} u = 0, \nabla \cdot u = 0.

Taking the curl eliminates the pressure gradient and commutes with the Laplacian, so for the vorticity

ω = \nabla \times u

\nabla^{2} ω = 0 .

For a two-dimensional incompressible flow, choose the stream function convention

u_{x} = ψ_{y}, u_{y} = - ψ_{x} .

Then

ω_{z} = \partial_{x} u_{y} - \partial_{y} u_{x} = - \nabla^{2} ψ .

\nabla^{4} ψ = 0,

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