  We provide a clear evidence that a two species mesoscopic Lattice Boltzmann
(LB) model with competing short-range attractive and mid-range repulsive
interactions supports emergent Herschel-Bulkley (HB) rheology, i.e. a power-law
dependence of the shear-stress as a function of the strain rate, beyond a given
yield-stress threshold. This kinetic formulation supports a seamless transition
from flowing to non-flowing behaviour, through a smooth tuning of the
parameters governing the mesoscopic interactions between the two species. The
present model may become a valuable computational tool for the investigation of
the rheology of soft-glassy materials on scales of experimental interest.
