Physiological transport of fluid at small scales is often achieved by microscopic active fingerlike structures. It is recognized that they have to move in a non-symmetric fashion in order to break the symmetry of creeping flow and to induce a net movement of the fluid. However, in the limit of low, but non-vanishing, Reynolds number, irreversible flow on long time scales could also be generated by symmetric oscillations of these microstructures. Inspired by small intestine villi, we reported three dimensional direct numerical simulations of the irreversible part of the flow, namely steady streaming flow (SSF), generated by an array of oscillating fingerlike structures. In order to capture these second order flow phenomena, the algorithm was based on a combination of lattice-Boltzmann methods with two relaxation times and the smoothed profile method. SSF was confined inside a steady viscous boundary above the villi. Two steady vortices at the tip of the villi characterized this flow which induced mass transfers between the bulk and the periphery. Strikingly, the spatial extension of these vortices was not solely governed by the Stokes boundary layer but also by the lateral confinement between the villi. Moreover, secondary vortices outside the steady boundary layer were also observed. These findings were rationalized in a state diagram showing three regimes of SSF. Finally, orders of magnitude showed that SSF should contribute to the transport of particles, such as bacteria or nano-particles, on a layer a few hundred micrometers above the villi and on a time scale of few minutes.
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