Implications of a viscosity bound on black hole accretion

Motivated by the viscosity bound in gauge/gravity duality, we consider the ratio of shear viscosity (η) to entropy density (s  ) in black hole accretion flows. We use both an ideal gas equation of state and the QCD equation of state obtained from lattice for the fluid accreting onto a Kerr black hole. The QCD equation of state is considered since the temperature of accreting matter is expected to approach ¹⁰¹² K${10}^{12}\text{K}$ in certain hot flows. We find that in both the cases η/s$\eta /s$ is small only for primordial black holes and several orders of magnitude larger than any known fluid for stellar and supermassive black holes. We show that a lower bound on the mass of primordial black holes leads to a lower bound on η/s$\eta /s$ and vice versa. Finally we speculate that the Shakura–Sunyaev viscosity parameter should decrease with increasing density and/or temperatures.