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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 1

The steady incompressible <Stokes flow> equations are
$$
-\nabla p+\mu\nabla^2\mathbf u=0,
\qquad
\nabla\mathbin{\cdot}\mathbf u=0.
$$
Taking the <curl> eliminates the pressure gradient and commutes with the Laplacian, so for the <vorticity> $\boldsymbol\omega=\nabla\times\mathbf u$,
$$
\boxed{\nabla^2\boldsymbol\omega=0}.
$$
This is the vorticity part of <Harmonic pressure and vorticity in Stokes flow>.

For a two-dimensional incompressible flow, choose the <stream function> convention
$$
u_x=\psi_y,
\qquad
u_y=-\psi_x.
$$
Then
$$
\omega_z=\partial_xu_y-\partial_yu_x=-\nabla^2\psi.
$$
Applying the harmonic-vorticity equation gives
$$
\boxed{\nabla^4\psi=0},
$$
the <Biharmonic stream function for planar Stokes flow> equation.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 39C / a / Solution

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