  We investigate the effect of black hole spin on warped or misaligned
accretion discs - in particular i) whether or not the inner disc edge aligns
with the black hole spin and ii) whether the disc can maintain a smooth
transition between an aligned inner disc and a misaligned outer disc, known as
the Bardeen-Petterson effect. We employ high resolution 3D smoothed particle
hydrodynamics simulations of $\alpha$-discs subject to Lense-Thirring
precession, focussing on the bending wave regime where the disc viscosity is
smaller than the aspect ratio $\alpha \lesssim H/R$. We first address the
controversy in the literature regarding possible steady-state oscillations of
the tilt close to the black hole. We successfully recover such oscillations in
3D at both small and moderate inclinations ($\lesssim 15^{\circ}$), provided
both Lense-Thirring and Einstein precession are present, sufficient resolution
is employed, and provided the disc is not so thick so as to simply accrete
misaligned. Second, we find that discs inclined by more than a few degrees in
general steepen and break rather than maintain a smooth transition, again in
contrast to previous findings, but only once the disc scale height is
adequately resolved. Finally, we find that when the disc plane is misaligned to
the black hole spin by a large angle, the disc 'tears' into discrete rings
which precess effectively independently and cause rapid accretion, consistent
with previous findings in the diffusive regime ($\alpha \gtrsim H/R$). Thus
misalignment between the disc and the spin axis of the black hole provides a
robust mechanism for growing black holes quickly, regardless of whether the
disc is thick or thin.
