Instability of shear flows in spatially periodic domains

Force-driven parallel shear flow in a spatially periodic domain is shown
to be linearly unstable with respect to both the Reynolds number and the
domain aspect ratio. This finding is confirmed by computer simulations,
and a simple expression is derived to determine stable flow conditions.
Periodic extensions of Couette and Poiseuille flows are unstable at
Reynolds numbers two orders of magnitude smaller than their aperiodic
equivalents because the periodic boundaries impose fundamentally different
constraints. This instability has important implications for designing
computational models of nonlinear dynamic processes with periodicity.