Until a few years ago the shear viscosity of a metal was not a topic of interest. However, that has changed, largely stimulated by some calculations based on string theory techniques! The history is described here.
In particular, these calculations suggest that for a quantum critical metal the ratio of the shear viscosity to the entropy density has a universal lower bound, hbar/(4 pi k_B).
Calculations for graphene suggest the ratio is close to the lower bound leading it to be dubbed "a nearly perfect fluid".
Recent experiments on fermionic cold atoms find the ratio is several times the universal minimum.
The quark-gluon plasma is close to the minimum.
Somehow this "minimum viscosity" (which in simple kinetic theory scales with the relaxation time) is related to a minimum conductivity, and thus the somewhat elusive and poorly defined Mott-Ioffe-Regel limit, which bad metals comfortably violate.
The exact relationship between viscosity [a hydrodynamic concept] and conductivity is a rather subtle one I don't understand. Some of the issues for strongly correlated systems are discussed by Andreev, Kivelson, and Spivak.
I welcome clarifications.
I am only aware of a few model calculations of the shear viscosity of metals, starting from model Hamiltonians.
In 1958 Steinberg did it for the Sommerfeld model, including electron-phonon scattering.
Calculations for ferromagnetic spin fluctuations (paramagnon model) are reviewed by Beal-Monod.
It would be nice to see some calculations of the shear viscosity for the bad metal state of a Hubbard model using a technique such as Dynamical Mean-Field Theory.
In July I am going to a workshop in Korea on "Bad metals and Mott criticality" and am sure these issues will be discussed extensively there.
But, in the mean time I would love to generate some discussion on this issue.