  As the number of observed merging binary black holes (BHs) grows, accurate
models are required to disentangle multiple formation channels. In models with
isolated binaries, important uncertainties remain regarding the stability of
mass transfer (MT) and common-envelope (CE) evolution. To study some of these
uncertainties, we have computed simulations using MESA of a $30M_\odot$, low
metallicity ($Z_\odot/10$) star with a BH companion. We developed a
prescription to compute MT rates including possible outflows from outer
Lagrangian points, and a method to self-consistently determine the
core-envelope boundary in the case of CE evolution. We find that binaries
survive a CE only if unstable MT happens after the formation of a deep
convective envelope, resulting in a narrow range (0.2 dex) in period for
envelope ejection. All cases where interaction is initiated with a radiative
envelope have large binding energies ($\sim 10^{50}$ erg), and merge during CE
even under the assumption that all the internal and recombination energy of the
envelope, as well as the energy from an inspiral, is used for ejection. This is
independent of core helium ignition for the donor, a condition under which
various rapid-population synthesis calculations assume a successful ejection is
possible. Moreover, we find that the critical mass ratio for instability is
such that for periods between $\sim 1-1000$ days merging binary BHs can be
formed via stable MT. A large fraction of these systems overflow their L$_2$
equipotential, in which case we find stable MT produces merging binary BHs even
under extreme assumptions of mass and angular momentum outflows. Our
conclusions are limited to the study of one donor star, but suggest that
population synthesis calculations overestimate the formation rate of merging
binary BHs produced by CE evolution, and that stable MT could dominate the rate
from isolated binaries.
