Stirring a square cup results in a strange pattern. What's the math behind it?
Had the cup been circular we would see vortex shaped upper surface deeper at the center than at edges with polar circular symmetry as there is no radial component. When boundary is a regular polygon of $n$ sides we can see standing wave interference patterns caused by vibration frequency due to radial and circumferential component streamline flow interfering to form $n$ spiral arms.
An attempt at modeling should include the two interactions.
EDIT1:
If such a cup is placed and rotated sufficiently fast on a turntable I suspect it may give rise to some similar pattern, that is if viewing at all is possible.
The shape which is formed must have $90 \deg$ rotational symmetry equivalent to the shape of the cup.
The shape will depend chaotically lots of factors (viscosity, exact shape, speed at which it is stirred, shape of the spoon which is stirring the liquid) hence it is almost impossible to mathematically show that the shape is as observed. I doubt whether the question as it stands can be answered at all.