The Geometry of Contraction-Induced Flows

TL;DR

This work develops a geometric, Lagrangian framework to describe contraction-induced flows in elastic tubes with coupled transverse and longitudinal wall motions. By extending solid-boundary coordinates into the fluid via a time-dependent metric, the authors derive 1D and 3D formulations that remain valid at large deformations and enable analytic expressions for flow q and mean pumping ⟨q⟩ under radius- and length-imposed constraints as well as peristaltic waves. Key findings show that transverse and longitudinal contractions generate instantaneous flows of the same order in wall strain but in opposite directions, with the Poisson ratio ν governing the coupling and flow magnitude: incompressible walls suppress flow while auxetic walls enhance it; length-imposed waves can produce backflow, whereas radius-imposed waves can cause reflux and trapping. The framework extends naturally to networks of contracting vessels and is validated against COMSOL simulations, offering a comprehensive, adaptable description of contraction-driven pumping with potential applications to biological systems (e.g., esophagus, ureter) and engineered soft-fluidic devices.

Abstract

Peristalsis is the driving mechanism behind a broad array of biological and engineered flows. In peristaltic pumping, a wave-like contraction of the tube wall produces local changes in volume which induce flow. Net flow arises due to geometric nonlinearities in the momentum equation, which must be properly captured to compute the flow accurately. While most previous models focus on radius-imposed peristalsis, they often neglect longitudinal length changes - a natural consequence of radial contraction in elastic materials. In this paper, to capture a more accurate picture of peristaltic pumping, we calculate the flow in an elastic vessel undergoing contractions in the transverse and longitudinal directions simultaneously, keeping the geometric nonlinearities arising in the strain. A careful analysis requires us to study our fluid using the Lagrangian coordinates of the elastic tube. We perform analytic calculations of the flow characteristics by studying the fluid inside a fixed boundary with time-dependent metric. This mathematical manipulation works even for large-amplitude contractions, as we confirm by comparing our analytical results to COMSOL simulations. We demonstrate that transverse and longitudinal contractions induce instantaneous flows at the same order in wall strain, but in opposite directions. We investigate the influence of the wall's Poisson ratio on the flow profile. Incompressible walls suppress flow by minimizing local volume changes, whereas auxetic walls enhance flow. For radius-imposed peristaltic waves, wall incompressibility reduces both reflux and particle trapping. In contrast, length-imposed waves typically generate backflow, although trapping can still occur at large amplitudes for some Poisson ratios. These results yield a more complete description of peristalsis in elastic media and offer a framework for studying contraction-induced flows more broadly.

Figures (13)

• Figure 1: A fluid-filled cylindrical tube undergoing an arbitrary axisymmetric deformation. The material configuration $\Omega_0$ is described using the coordinates $\vec{X}(R,\Phi,X) = X\vec{e}_X + R\vec{e}_R(\Phi)$. When restricted to the solid boundary, these are the Lagrangian coordinates of the solid membrane $S_0$. The current configuration $\Omega(t)$ is described using the Eulerian coordinates $\vec{x}(r,\phi,x) = x \vec{e}_x + r \vec{e}_r(\phi)$. The surface $S_0$ is mapped to $S(t)$ via the map $\vec{x}_s$. This map is extended into the tube's interior via the map $\vec{x}_\Sigma$ which specifies how a surface in the fluid region deforms from $\Sigma_0(X)$ to $\Sigma(X,t)$. In this example, the left side of the tube is contracted radially and elongated longitudinally as illustrated by the yellow slice.
• Figure 2: A tube is contracting peristaltically with period $T$. The velocity field is displayed in three different coordinate systems at five different time points: $t=0$, $t=T/4$, $t=T/2$, $t=3T/4$, and $t=T$. The trajectories of points on the wall are shown in red. Particles begin their trajectories at $t=0$ at the locations marked by hollow circles, and the current locations are marked by filled-in circles. In the left column, the Eulerian velocity field $\vec{V}(\vec{x},t)$ is plotted and points on the boundary are trace out loops. This represents the trajectories an observer would actually see. In the center column, the material velocity field $\vec{v}(\vec{X},t)$ is plotted, and the boundary points are stationary. The grid of lines of constant $X$ and $R$ in the material configuration deform to form the distorted grid in the current configuration. In the right column, the material velocity in the co-moving frame $\vec{\tilde{v}}(\vec{\tilde{X}})$ is plotted. The co-moving frame travels at velocity $c \vec{e}_X$, so particles on the boundary travel at velocity $-c \vec{e}_X$. The velocity field is independent of time in the co-moving frame. One could also consider Eulerian coordinates in the co-moving frame $\tilde{\Omega}(t)$, but this is not used in the paper. The wave satisfies equations \ref{['eq:dxsdX_longitudinal']} and \ref{['eq:ur_longitudinal']} with $\epsilon = 0.45$, $\nu=0.5$, and $\Delta \bar{p}_\lambda = 0$.
• Figure 3: Uniform contractions in a finite tube of rest radius $R_0$ and rest length $L_0$. Two types of boundary conditions are considered: $(a)-(d)$ radius-imposed contraction and $(e)-(h)$ length-imposed contraction. $(a)$ During radius-imposed contraction, circumferential stresses (represented by red lines around the circumference) impose a radius of $r_s = R_0 + u_r$ on the tube, while the condition that $\sigma_{XX}=0$ determines the length of the tube according to \ref{['eq:dudx_radius_imposed']}. The results for the dimensionless $(b)$ length $l_s/L_0$, $(c)$ volume $\text{Vol}/\text{Vol}_0$, and $(d)$ conductance $\kappa/\kappa_0$ are plotted as a function of the applied radius. Different colors show different values of the Poisson ratio $\nu$. The solid curves show the analytical results \ref{['eq:ls_radial']}, \ref{['eq:vol_radial']}, and \ref{['eq:kappa_radial']}, while points show results from COMSOL. $(e)$ During length-imposed peristalsis, longitudinal stresses (represented by red lines along the length) impose a length $l_s = L_0 + \frac{\partial u_x}{\partial X}L_0$, while the condition that $\sigma_{\Phi \Phi} = 0$ determines the radius fo the tube according to \ref{['eq:ur_length_imposed']}. The results for the dimensionless $(f)$ radius $r_s/R_0$, $(g)$ volume $\text{Vol}/\text{Vol}_0$, and $(h)$ conductance $\kappa/\kappa_0$ are plotted as a function of the applied length.
• Figure 4: Instantaneous Eulerian velocity field $\vec{V}(\vec{x},0)$ in the lab frame $\Omega(t=0)$ and example particle trajectories $\vec{x}_p(t)$ for a tube driven by a radius-imposed peristaltic wave traveling to the right. Solutions for a purely radial wall motion $\nu=0$ and an incompressible tube with $\sigma_{XX}=0$ are compared. Each row corresponds to different choices of the characteristic strain $\epsilon$ and adverse pressure $\Delta \bar{p}_\lambda$. Gridlines which are equally spaced for a tube at rest help to visualize longitudinal displacements. The length of the arrows indicates the magnitude and direction of $\vec{V}$. Particles begin their trajectories at $t=0$ at the locations marked by hollow circles and end their trajectories at $t=3T$ at the locations marked by filled-in circles.
• Figure 5: Streamlines in the co-moving wave-frame $\tilde{\Omega}_0$ and the corresponding particle wave speed distribution are plotted for a tube driven by a radius-imposed peristaltic wave using the same parameters as in figure \ref{['fig:radialVelocityField']}. At $\tilde{R}=0$, $\tilde{\psi}=0$ by convention. At $\tilde{R}=R_0$, $\tilde{\psi} = \tilde{q}$; its value is noted. Positive values of $\tilde{\psi}$ (negative values of $\tilde{\psi}/\tilde{q}$) indicate trapping while negative values of $s_p$ indicate reflux. For visualization, regions where particles undergo trapping are colored dark blue, regions where particles undergo reflux are colored red, and regions where particles travel forward at a speed less than $c$ are colored light blue.