Computational models of valveless pumping using the immersed boundary method

Mathematical models of valveless pumping can be represented by either a closed loop system or an open tube system. In this paper, we present a three-dimensional model of valveless pumping in a closed loop system. We also present a two-dimensional model using an open elastic cylinder contained in a rigid tank. In both models, we take the periodic compress-and-release action at the asymmetric location of the soft tube and observe the existence of a net flow and the important features of valveless pumping that have been reported in the previous models or experiments. The innovative idea of this work is that we explain the existence of a net flow by introducing the concept of the signed area of the flow-pressure loop over one cycle, which represents the power in the system. The direction and the magnitude of a net flow can also be explained by the sign and the amount of power, which is work done on the fluid by the fluid pressure and the elastic wall over one period, respectively.