For this question I'm struggling to gather exactly what PDE's I should be solving here. I'm also struggling to understand what the boundary conditions would look like for such a question, which is the reason why I don't know how to start. Thanks in advance for any help.
You only need to express the Laplacian in cylindrical polar coordinates $(\sigma, \phi)$ [what a strange notation!] to have:
$$\rho \frac{\partial w}{\partial t} = G + \mu \left[ \frac{1}{\sigma} \frac{\partial}{\partial \sigma} \left( \sigma \frac{\partial w}{\partial \sigma}\right) + \frac{1}{\sigma^2} \frac{\partial^2 w}{\partial \phi^2} \right] $$
where we have used the fact that the flow is fully developed and therefore does not depend on $z$ (that is, $w = w(\sigma, \phi)$).
To solve for the axial velocity you must impose a no-slip condition at the walls, meaning that $w = 0$ at $\sigma = a$, $-\beta < \phi < \beta$ and at $\phi = \pm \beta$, $\sigma < a$. As for the center of the cylinder $\sigma = 0$, you will have to get rid of the singular part of the solution so as to keep the velocity bounded at all times.
Hope this helps!