  The inner part of a thin accretion disk around a Kerr black hole can serve as
an important tool to study the physics of the strong gravity regime. A tilt in
such a disk with respect to the black hole spin axis is particularly useful for
this purpose, as such a tilt can have a significant effect on the observed
X-ray spectral and timing features via Lense-Thirring precession. However, the
inner disk has been predicted to become aligned with the spin direction of the
black hole by the well-known Bardeen-Petterson effect. Here we calculate, both
analytically and numerically, the radial profile of the thin accretion disk
tilt angle in the viscous regime (i.e., $\alpha$ > H/R; $\alpha$ is the
Shakura-Sunyaev viscosity parameter, H is the disk thickness and R is the
radial distance). We show that the inner disk may not be aligned at all for
certain reasonable ranges of parameter values. This makes the inner accretion
disk particularly promising to probe the black hole parameters, and the
accretion process in the strong gravity region.
