Peristaltic pumping with valves in low Reynolds numbers

Frequently in the vascular systems of living organisms, flow has to propagate against negative pressure gradients. For example, in humans, the pressure provided by the pumping heart alone is not enough to combat gravity and viscous forces and return the blood from the lower limbs back to the heart. Inspired by similar challenges in the lymphatic system, we employ a one-dimensional fluid model to study a pumping mechanism consisting of a viscous fluid in a contracting tube with valves. A simple model for valves is proposed to understand the nonlinear fluid-structure interaction. The valves open and close according to the sign of the pressure drop across the valve, in such a way that promotes unidirectional flow. We begin by demonstrating that oscillations in the pressure or cross-sectional area can lead to positive time-averaged flow even with an overall negative pressure gradient. Peristaltic pumping is modeled by imposing an oscillatory function that represents a contraction wave governing the cross-sectional area of the tube. It is demonstrated that, in some cases, it does not matter which direction the contraction wave propagates when calculating the average flow in the presence of valves. Furthermore, the effect of valve placement on the flow is investigated.