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 Incoming links: Local spin in an azimuthal power-law flow

Past exam of the mathematical methods II course of the University of Oxford / 2019 / 8 / c / i / Solution 2026-10-25

Write

r^{2} = x^{2} + y^{2}

and work away from the

z

axis, where

r > 0

. The vector field is independent of

z

and has zero vertical component, so only the vertical curl can be nonzero:

(\nabla \times F)_{z} = \frac{\partial}{\partial x} [x (x^{2} + y^{2})^{- n}] - \frac{\partial}{\partial y} [- y (x^{2} + y^{2})^{- n}] = 2 r^{- 2 n} - 2 n (x^{2} + y^{2}) r^{- 2 n - 2} .

Therefore

\nabla \times F = 2 (1 - n) r^{- 2 n} \overset{z}{^}, r > 0.

This local spin in an azimuthal power-law flow depends on the radial variation of the tangential speed, not merely on the sign of that speed.

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