  Stellar-mass black holes (sBH) embedded in gaseous disks of active galactic
nuclei (AGN) can be important sources of detectable gravitational radiation for
LIGO/Virgo when they form binaries and coalesce due to orbital decay. In this
paper, we study the effect of gas dynamical friction (DF) on the formation of
BH binaries in AGN disks using $N$-body simulations. We employ two simplified
models of DF, with the force on the BH depending on $\Delta {\bf v}$, the
velocity of the sBH relative to the background Keplerian gas. We integrate the
motion of two sBH initially on circular orbits around the central supermassive
black hole (SMBH), and evaluate the probability of binary formation under
various conditions. We find that both models of DF (with different dependence
of the frictional coefficient on $|\Delta{\bf v}|$) can foster the formation of
binaries when the effective friction timescale $\tau$ satisfies $\Omega_{\rm
K}\tau\lesssim 20-30$ (where $\Omega_{\rm K}$ is the Keplerian frequency around
the SMBH): prograde binaries are formed when the DF is stronger (smaller
$\tau$), while retrograde binaries dominate when the DF is weaker (larger
$\tau$). We determine the distribution of both prograde and retrograde binaries
as a function of initial orbital separation and the DF strength. Using our
models of DF, we show that for a given sBH number density in the AGN disk, the
formation rate of sBH binaries increases with decreasing $\tau$ and can reach a
moderate value with a sufficiently strong DF.
