  Most black holes possess accretion disks. Models of such disks inform
observations and constrain the properties of the black holes and their
surrounding medium. Here, we study isothermal shocks in a thin black hole
accretion flow. Modelling infinitesimal molecular viscosity allows the use of
multiple-scales matched asymptotic methods. We thus derive the first explicit
calculations of isothermal shock stability. We find that the inner shock is
always unstable, and the outer shock is always stable. The growth/decay rates
of perturbations depend only on an effective potential and the
incoming--outgoing flow difference at the shock location. We give a
prescription of accretion regimes in terms of angular momentum and black hole
radius. Accounting for angular momentum dissipation implies unstable outer
shocks in much of parameter space, even for realistic viscous Reynolds numbers
of the order $\approx 10^{20}$.
