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

TL;DR

The paper investigates fluid flow in a villi-patterned channel driven by a propagating pendular-wave along rigid villi walls. Using a 2D lattice Boltzmann framework, it reveals a mixing boundary layer (MBL) above the villi, carrying semi-vortical structures that travel with the wave, and an axial steady flow in the lumen that, counterintuitively, can oppose wave propagation due to non-reciprocal trajectories of fluid trapped between adjacent villi. The authors develop scaling laws for axial and radial fluxes, identify a Stokes-to-inertial transition governing the MBL height, and show how inertia dynamically confines flows toward villi tips, producing a non-monotonic axial pumping with a maximum near Wo ≈ 2.8. They also introduce an effective boundary condition at the villus tips that captures essential dynamics without resolving individual villi, enabling coarse-grained organ-scale simulations. Overall, the work links active microstructure, pendular-wave kinematics, and finite inertia to novel mixing and transport phenomena with potential biomimetic and microfluidic applications.

Abstract

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.

Figures (10)

• Figure 1: (A) Leaf-like and ridge-like villi of the small intestine of rat (duodenum and ileum). (B) The planar 2D simulation domain consists of $N$ oscillating villi with periodic boundaries in the $\pm z$ direction, a wall at $r=0$, and an axis of symmetry at the channel center $r=R$. Each villus oscillates around its mean position (orange outline) with an amplitude $a$ and a phase lag $\Delta \phi$ relative to its neighbour. The mean positions of adjacent villi are separated by a constant pitch distance $P$. The instantaneous positions (filled green) of the villi show that when one section of the villi-wall contracts, the adjacent section relaxes, thereby pumping and drawing fluid in and out of the intervillous spaces. (C) Illustration of the imposed travelling wave and time-periodic motion of the villi-wall. (D) The same seven villi viewed from the top ($z$, $t$ axes), showing the contractions (in red) and expansions (in hatched purple) of the inter-villus gaps propagating in the $-z$ direction.
• Figure 2: Flow induced by non-propagation oscillations of the villi over two neighbouring villi at $\Wo=0.5$. Instantaneous flow at $t/T=0.3$ for (A) in-phase ($\DPhi=0$) and (B) out-of-phase ($\DPhi=\pi$) villi motion. The colour fields map the magnitude of instantaneous velocity ($|\mathbf{u}|/U_0$). Time-averaged steady streaming flow-fields ($\bm{u}^{ss}$) for (C) in-phase ($\DPhi=0$) and (D) out-of-phase ($\DPhi=\pi$) villi motion. The colour fields map the magnitude of $|\mathbf{u}^{ss}|/U_0$. The width of the streamlines is proportional to the local velocity magnitude. See also figures S2-S4 in supplementary material.
• Figure 3: Snapshots of the instantaneous flow field for $\tilde{a}=0.2$ and $\Wo=0.16$, at $t/T=0.45$, for (A)$\DPhi = \pi/2$ and (B)$\DPhi = \pi/4$. The dashed (magenta) line marks the approximate separation between the mixing layer and the advected layer. Note the absence of the advected layer in (B). See supplementary movies 1 and 2.
• Figure 6: Steady streaming flow-field ($\bm{u}^{ss}$) streamlines for $\tilde{a}=0.2$, plotted around a pair of adjacent villi for three increasing Womersley numbers, (A)$\Wo=0.16$, (B)$\Wo=1.58$, and (C)$\Wo=5.0$ (row-wise). For each $\Wo$, panels for five decreasing $\DPhi$ values (column-wise) are shown. The colour-map plots the axial component of the steady streaming flow (SSF), ${u}^{ss}_z$. Note that the SSF pattern shows axial periodicity over the intervillous distance $P$, for all cases. The dashed (orange) lines in each panel show the approximate mixing layer height $\ell$ seen in the respective instantaneous flow-fields.
• Figure 7: Plot of the mixing layer height $\ell$ against the Womersley number $\Wo = W / \StLayer$, for various $\DPhi$, for (A)$\tilde{a}=0.2$ and (B)$\tilde{a}=0.1$. Note the plateauing of the curves when $Wo \le 1$, and their rapid decrease with increasing $\Wo$, when $\Wo \ge 1$, indicating a flow regime transition. The measured mixing layer heights $\ell$ from (A) and (B), collapses as a double-power law when appropriately rescaled, as plotted in (C), delineating the two regimes. Note that in (C) we have not show the data-points for $\DPhi = 0$ and $\pi$. In (C), large $x$-axis shows the viscosity dominant regime, while at small $x$-axis shows the inertial regime. Contrast the scaling for $\ell$ in the inertial regime $(\propto \StLayer^{1/2})$ with that for the theoretical boundary layer ($\propto \StLayer$) for an oscillating flat plate schlichting_BoundaryLayerTheory_1960. Transition between the two regimes appears to be smooth, and we visually identify (0.83,6.66) as the critical transition point in (C). From this critical point, we obtain the functional transition curve: $\ell^c = 6.66 e^{-3\tilde{a}/2} W / \Wo$, which is plotted as dash-dotted (grey) lines in (A) and (B). These lines indicate the viscous to inertial regime transition, for all $\DPhi$ and $\tilde{a}$.