  Predicting the rheology of dense suspensions under inhomogeneous flow is
crucial in many industrial and geophysical applications, yet the conventional
`$\mu(J)$' framework is limited to homogeneous conditions in which the shear
rate and solids fraction are spatially invariant. To address this shortcoming,
we use particle-based simulations of frictionless dense suspensions to derive
new constitutive laws that unify the rheological response under both
homogeneous and inhomogeneous conditions. By defining a new dimensionless
number associated with particle velocity fluctuations and combining it with the
viscous number, the macroscopic friction and the solids fraction, we obtain
scaling relations that collapse data from homogeneous and inhomogeneous
simulations. The relations allow prediction of the steady state velocity,
stress and volume fraction fields using only knowledge of the applied driving
force.
