For a unidirectional, $z$-independent velocity
$$
\mathbf u=u_z(r,\theta)\mathbf e_z,
$$
incompressibility holds automatically. The axial <Stokes flow> equation is
$$
0=-p_z+\mu\nabla_\perp^2u_z.
$$
No external pressure gradient is imposed, so $p_z=0$ and
$$
\boxed{\nabla_\perp^2u_z=0}.
$$
Thus the axial velocity is a <harmonic function> in the disk.

Solution

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

 Solution

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