Operating Principles of Peristaltic Pumping through a Dense Array of Valves

TL;DR

This paper models peristaltic pumping with dense arrays of ideal valves, showing how valve density affects flow, pressure, and optimal valve number, with implications for biological and engineered flow systems.

Contribution

It introduces a continuum approximation for dense valve arrays in peristaltic pumping, enabling analytical treatment of complex nonlinear flow systems.

Findings

Flow is linear in peristaltic amplitude for small forces

Flow is independent of pumping direction with sinusoidal peristalsis

Optimal valve density balances flow enhancement and suppression

Abstract

Immersed nonlinear elements are prevalent in biological systems that require a preferential flow direction, such as the venous and the lymphatic system. We investigate here a certain class of models where the fluid is driven by peristaltic pumping and the nonlinear elements are ideal valves that completely suppress backflow. This highly nonlinear system produces discontinuous solutions that are difficult to study. We show that as the density of valves increases, the pressure and flow are well-approximated by a continuum of valves which can be analytically treated, and we demonstrate through numeric simulation that the approximation works well even for intermediate valve densities. We find that the induced flow is linear in the peristaltic amplitude for small peristaltic forces and, in the case of sinusoidal peristalsis, is independent of pumping direction. Despite the continuum approximation used, the physical valve density is accounted for by modifying the resistance of the fluid appropriately. The suppression of backflow causes a net benefit in adding valves when the valve density is low, but once the density is high enough, valves predominately suppress forward flow, suggesting there is an optimum number of valves per wavelength. The continuum model for peristaltic pumping through an array of valves presented in this work can eventually provide insights about the design and operating principles of complex flow networks with a broad class of nonlinear elements.