  We study a model of a cyclic, spatially homogeneous, anisotropic 'mixmaster'
universe of Bianchi type IX, containing a radiation field with non-comoving
('tilted' with respect to the tetrad frame of reference) velocities and
vorticity. We employ a combination of numerical and approximate analytic
methods to investigate the consequences of the second law of thermodynamics on
the evolution. We model a smooth cycle-to-cycle evolution of the mixmaster
universe, bouncing at a finite minimum, by the device of adding a comoving
'ghost' field with negative energy density. In the absence of a cosmological
constant, an increase in entropy, injected at the start of each cycle, causes
an increase in the volume maxima, increasing approach to flatness, falling
velocities and vorticities, and growing anisotropy at the expansion maxima of
successive cycles. We find that the velocities oscillate rapidly as they evolve
and change logarithmically in time relative to the expansion volume. When the
conservation of momentum and angular momentum constraints are imposed, the
spatial components of these velocities fall to smaller values when the entropy
density increases, and vice versa. Isotropisation is found to occur when a
positive cosmological constant is added because the sequence of oscillations
ends and the dynamics expand forever, evolving towards a quasi de Sitter
asymptote with constant velocity amplitudes. The case of a single cycle of
evolution with a negative cosmological constant added is also studied.
