  Microscopic flows are almost universally linear, laminar and stationary
because Reynolds number, $Re$, is usually very small. That impedes mixing in
micro-fluidic devices, which sometimes limits their performance. Here we show
that truly chaotic flow can be generated in a smooth micro-channel of a uniform
width at arbitrarily low $Re$, if a small amount of flexible polymers is added
to the working liquid. The chaotic flow regime is characterized by randomly
fluctuating three-dimensional velocity field and significant growth of the flow
resistance. Although the size of the polymer molecules extended in the flow may
become comparable with the micro-channel width, the flow behavior is fully
compatible with that in a table-top channel in the regime of elastic
turbulence. The chaotic flow leads to quite efficient mixing, which is almost
diffusion independent. For macromolecules, mixing time in this microscopic flow
can be three to four orders of magnitude shorter than due to molecular
diffusion.
