  Stellar-mass binary black holes (BBHs) embedded in active galactic nucleus
(AGN) discs offer a distinct dynamical channel to produce black hole mergers
detected in gravitational waves by LIGO/Virgo. To understand their orbital
evolution through interactions with the disc gas, we perform a suite of 2D
high-resolution, local shearing box, viscous hydrodynamical simulations of
equal-mass binaries. We find that viscosity not only smooths the flow structure
around prograde circular binaries, but also greatly raises their accretion
rates. The torque associated with accretion may be overwhelmingly positive and
dominate over the gravitational torque at a high accretion rate. However, the
accreted angular momentum per unit mass decreases with increasing viscosity,
making it easier to shrink the binary orbit. In addition, retrograde binaries
still experience rapid orbital decay, and prograde eccentric binaries still
experience eccentricity damping. Our numerical experiments further show that
prograde binaries are more likely to be hardened if the physical sizes of the
accretors are sufficiently small such that the accretion rate is reduced. The
dependency of the binary accretion rate on the accretor size can be weaken
through boosted accretion either due to a high viscosity or a more
isothermal-like equation of state (EOS). Our results widen the explored
parameter space for the hydrodynamics of embedded BBHs and demonstrate that
their orbital evolution in AGN discs is a complex, multifaceted problem.
