 Solution

ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2015/1/b/solution

Substitute the given Stokes streamfunction representation into the divergence in spherical coordinates. The mixed derivatives cancel:

\nabla \cdot u = \frac{1}{r ^{2}} \partial_{r} (\frac{Ψ _{θ}}{sin θ}) + \frac{1}{r sin θ} \partial_{θ} (- \frac{Ψ _{r}}{r}) = \frac{Ψ _{r θ} - Ψ _{θ r}}{r ^{2} sin θ} = 0.

Thus incompressibility is automatic for a smooth Stokes streamfunction. Since this is an axisymmetric flow with no azimuthal velocity, the only possibly nonzero component of its vorticity is

(\nabla \times u)_{φ} = \frac{1}{r} [\partial_{r} (- \frac{Ψ _{r}}{sin θ}) - \partial_{θ} (\frac{Ψ _{θ}}{r ^{2} sin θ})] = - \frac{1}{r ^{3} sin θ} [r^{2} Ψ_{rr} + sin θ \partial_{θ} (\frac{Ψ _{θ}}{sin θ})] .

L Ψ = r^{2} Ψ_{rr} + Ψ_{θθ} - cot θ Ψ_{θ} .

The stated equation is

L Ψ = 0

, and therefore

\nabla \cdot u = 0, \nabla \times u = 0 .

The irrotational flow has a velocity potential locally, and globally on a regular simply connected domain. Regularity of the physical velocity field supplies the limiting interpretation on the polar axis, where the coordinate expressions contain

sin θ

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