We simulate the spatial and temporal evolution of inhomogeneous
flow fields in viscometric devices such as cylindrical Couette cells.
The computations focus on a class of two species elastic network models
which are prototypes for a model which can capture, in a
self-consistent manner, the creation and destruction of
elastically-active network segments as well as diffusive coupling
between the microstructural conformations and the local state of stress
in regions with large spatial gradients of local deformation. For each
of these models, the ‘flow curve’ of stress and apparent
shear rate resulting from an assumption of homogeneous deformation is
non-monotonic and linear stability analysis shows that the region of
non-monotonic response is unstable. Steady-state calculations
of the full inhomogeneous flow field lead to localized shear bands that grow linearly in extent across the gap as the apparent shear rate is incremented. Time-dependent calculations in step strain experiments and in start up of steady shear flow show that the velocity profile in the gap and the total stress measured at the bounding surfaces are coupled and evolve in a complex non-monotonic manner as the shear bands develop. These spatio-temporal dynamics are consistent with time-resolved particle imaging velocimetry measurements in both concentrated solutions of monodisperse entangled polymers and in wormlike micellar solutions. The computational results have a number of implications for experimental observations of ‘apparent’ or ‘gap-averaged’ quantities in nearly-viscometric devices, and lead to plateaus or ‘yield-like’ transitions in the steady flow curve and deviations from the Lodge-Meissner relation in non-ideal step shearing deformations.