  The stationary hydrodynamic equations for the transonic accretion disks and
flows around rotating black holes are presented by using the Kerr-Schild
coordinate where there is no coordinate singularity at the event horizon. We
use two types of the causal viscosity prescription, and the boundary conditions
for the transonic accretion flows are given at the sonic point. For one type of
the causal viscosity prescription we also add the boundary conditions at the
viscous point where the accreting radial velocity is nearly equal to the
viscous diffusion velocity. Based on the formalism for the transonic accretion
disks, after we present the calculation method of the transonic solutions, the
horizon-penetrating transonic solutions which smoothly pass the event horizon
are calculated for several types of the accretion flow models: the ideal
isothermal flows, the ideal and the viscous polytropic flows, the advection
dominated accretion flows (ADAFs) with the relativistic equation of state, the
adiabatic accretion disks, the standard accretion disks, the supercritical
accretion disks. These solutions are obtained for both non-rotating and
rotating black holes. The calculated accretion flows plunge into black hole
with finite three velocity smaller than the speed of light even at the event
horizon or inside the horizon, and the angular velocities of the accretion flow
at the horizon are generally different from the angular velocity of the
frame-dragging due to the black hole's rotation. These features contrast to the
results obtained by using the Boyer-Lindquist coordinate with the coordinate
singularity at the horizon.
