  We consider laminar flow in channels constrained geometrically to remain
between two parallel planes; this geometry is typical of microchannels obtained
with a single step by current microfabrication techniques. For pressure-driven
Stokes flow in this geometry and assuming that the channel dimensions change
slowly in the streamwise direction, we show that the velocity component
perpendicular to the constraint plane cannot be zero unless the channel has
both constant curvature and constant cross-sectional width. This result implies
that it is, in principle, possible to design "planar mixers", i.e. passive
mixers for channels that are constrained to lie in a flat layer using only
streamwise variations of their in-plane dimensions. Numerical results are
presented for the case of a channel with sinusoidally varying width.
