Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 54 / 1 / c / Solution

The instantaneous kinematic viscosity law is $\bar{\nu }(r,\Sigma )$, with no explicit constitutive memory. Expand its transported quantity at the possibly evolving background:The viscous transport response exponent satisfies ${\partial }_{\Sigma }(\bar{\nu }\Sigma ){|}_0=q{\bar{\nu }}_0$. Subtract the background equation and retain only linear terms to obtainThis remains a linear equation with space- and time-dependent background coefficients; neither a steady background nor constant $q$ is required for this step. The sign of the response coefficient is the negative-diffusion criterion for viscous disk instability.