Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 2 / b / Solution

Let $F(\Sigma )=\nu (\Sigma )\Sigma$ and perturb the homogeneous state by $\delta \Sigma \propto e^{st+ikx}$. Linearization givesThus $d(\nu \Sigma )/d\Sigma <0$ produces the viscous instability of an accretion disk: a density enhancement transports angular momentum less effectively, loses material more slowly, and grows. The ring separates into denser narrow ringlets and lower-density gaps until nonlinear effects regularize the backward diffusion.