Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 16C / c

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- Solution c

Solution

Take the curl of the Navier-Stokes equation. The pressure term disappears, curl commutes with the Laplacian, and the incompressible vector identity

\nabla \times ((u \cdot \nabla) u) = (u \cdot \nabla) ω - (ω \cdot \nabla) u

gives

\frac{\partial ω}{\partial t} + (u \cdot \nabla) ω = (ω \cdot \nabla) u + ν \nabla^{2} ω .

Therefore the material derivative form is

\frac{D ω}{D t} = (ω \cdot \nabla) u + ν \nabla^{2} ω .

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