Past exam of the fluids course of the University of Oxford / 2021 / 1 / b

The divergence in polar coordinates and the given streamfunction convention yieldThus the incompressibility condition holds wherever the mixed partial derivatives commute. The vertical vorticity isNeglecting both inertial terms in the Navier-Stokes equation, for constant dynamic viscosity and mass density, gives the Stokes equation:Its curl eliminates the pressure gradient and the uniform gravitational body force, so $\mu {\nabla }^2\zeta =0$. Hence the biharmonic streamfunction satisfiesThis conclusion requires the quasi-steady Stokes approximation. If local acceleration were retained, the corresponding planar vorticity equation would instead be ${\partial }_t{\nabla }^2\psi =\nu {\nabla }^4\psi$.