The operator:
$$\frac{\partial^4}{\partial x^4}+2\frac{\partial^4}{\partial x^2 \, \partial y^2}+\frac{\partial^4}{\partial y^4}$$
... appears in the equation of motion of an oscillating rigid elastic plate (see e.g. Chladni patterns).
To try and evaluate the equation for the circular case, conversion to polar coordinates $(r,\theta)$ would be an obvious choice. Conversion of the much simpler Laplacian to polar coordinates is well-known and shows how tedious that conversion is.
Has anyone here made this conversion or has a reference to a reputable paper that mentions it?