  It was shown in [PRL 114, 138301 (2015)] that a remarkably simple dynamical
model exhibits many of the complex flow regimes and non-equilibrium phase
transitions characteristic of complex fluids. By removing extraneous detail,
this simplest microscopic model of non-Newtonian flow can reveal the universal
physics relevant to all complex fluids. Here we present more detailed results
and a full derivation of the model's compact mean-field constitutive relation,
with great potential scope for insights into universality and tractable
mathematics. By enforcing local conservation of angular momentum, the
one-dimensional (1D) XY-model (originally used for equilibrium magnetic
systems) can be driven into various flow regimes, including simple Newtonian
behaviour, shear banding, solid-liquid coexistence and slip-plane motion. The
model demonstrates that the phenomenon of shear banding does not rely on
details of tensorial stress fields, but can exist in 1D.
