Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 2 / a / i

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- Solution i

Solution

For a unidirectional,

z

-independent velocity

u = u_{z} (r, θ) e_{z},

incompressibility holds automatically. The axial Stokes flow equation is

0 = - p_{z} + μ \nabla_{⊥}^{2} u_{z} .

No external pressure gradient is imposed, so

p_{z} = 0

and

\nabla_{⊥}^{2} u_{z} = 0 .

Thus the axial velocity is a harmonic function in the disk.

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