  Describing effects of small but finite inertia on suspended particles is a
fundamental fluid dynamical problem that has never been solved in full
generality. Modern microfluidics has turned this academic problem into a
practical challenge through the use of high-frequency oscillatory flows,
perhaps the most efficient way to take advantage of inertial effects at low
Reynolds numbers, to precisely manipulate particles, cells and vesicles without
the need for charges or chemistry. The theoretical understanding of flow forces
on particles has so far hinged on the pioneering work of Maxey and Riley (MR in
the following), almost 40 years ago. We demonstrate here theoretically and
computationally that oscillatory flows exert previously unexplained,
significant and persistent forces, that these emerge from a combination of
particle inertia and spatial flow variation, and that they can be
quantitatively predicted through a generalization of MR.
