Some physical experiments exhibit size-dependency for fluid flows at small scales. This in turn necessitates the introduction of couple-stresses in the corresponding continuum theory. The resulting size-dependent couple stress fluid mechanics can be used to explore a range of such non-Newtonian flow behavior at micro- and nano-scales, and also to bridge between atomistic and classical continuum theories. Here we concentrate on two-dimensional flow and examine the effects of couple-stresses by developing and then applying a stream function-vorticity computational fluid dynamics formulation. Details are provided both on the governing equations for size-dependent flow and on the corresponding numerical implementation. Afterwards, the formulation is applied to the lid-driven cavity problem to examine the behavior of the flow as a function of the length scale parameter I. The investigation covers a range of Reynolds numbers, and includes an evaluation of the critical value beyond which a stationary response is no longer possible. The additional boundary conditions associated with consistent couple stress theory are found to play an important role in determining the flow pattern and critical Reynolds number. (C) 2015 Elsevier B.V. All rights reserved.