  A thin viscous accretion disc around a Kerr black hole, which is warped due
to the Lense-Thirring (LT) effect, was shown to cause the spin axis of the
black hole to precess and align with the outer disc. We calculate the total LT
torque acting on the black hole, and compute the alignment and precession
time-scales for both persistent and transient accretors. In our analysis, we
consider the contribution of the inner disc, as it can stay misaligned with the
black hole spin for a reasonable range of parameter values. We find that the
alignment time-scale increases with a decrease in the Kerr parameter below a
critical Kerr parameter value, contrary to earlier predictions. Besides, the
time-scales are generally longer for transience than the time-scales calculated
for persistent accretion. From our analysis of the transient case, we find that
the black hole in the low mass X-ray binary (LMXB) 4U 1543-47 could be
misaligned, whereas that in the LMXB XTE J1550-564 has aligned itself with the
outer disc. The age of the LMXB H 1743-322 is estimated assuming a misaligned
disc. We also find that the black hole in a typical Galactic LMXB can take a
significantly longer time to align than what was estimated in the past. This
may have an important implication on the measurement of black hole spin using
the continuum X-ray spectral fitting method.
