Hydrodynamics in a villi-patterned channel due to pendular-wave activity

Inspired by small intestine motility, we investigate the flow induced by a propagating pendular-wave along the walls of a channel lined with rigid, villi-like microstructures. The villi undergo harmonic axial oscillations with a phase lag relative to their neighbours, generating travelling patterns of intervillous contraction. Using two-dimensional lattice Boltzmann simulations, we resolve the flow within the villi zone and the lumen, sampling small to moderate Womersley numbers. We uncover a mixing boundary layer (MBL) just above the villi, composed of semi-vortical structures that travel with the imposed wave. In the lumen, an axial steady flow emerges, surprisingly oriented opposite to the wave propagation direction, contrary to canonical peristaltic flows. We attribute this flow reversal to the non-reciprocal trajectories of fluid trapped between adjacent villi, and derive a geometric scaling law that captures its magnitude in the Stokes regime. The MBL thickness is found to depend solely on the wave kinematics given by intervillous phase lag in the low-inertia limit. Above a critical threshold, oscillatory inertia induces dynamic confinement, limiting the radial extent of the MBL and leading to non-monotonic behaviour of the axial steady flux. We further develop an effective boundary condition at the villus tips, incorporating both steady and oscillatory components across relevant spatial scales. This framework enables coarse-grained simulations of intestinal flows without resolving individual villi. Our results shed light on the interplay between active microstructure, pendular-wave and finite inertia in biological flows, and suggests new avenues for flow control in biomimetic and microfluidic systems.