One-dimensional model of valveless pumping in a closed loop and a numerical solution

The development of a theoretical model of valveless pumping and its numerical solution is presented in this work, applied for the case of a closed hydraulic loop, consisting of a soft and a rigid tube. A periodic compression and decompression of the soft tube causes a unidirectional flow, under certain conditions. The integration of the governing flow equations (continuity and momentum), over the tube cross-sectional area results in a quasi-one-dimensional unsteady model. A system of nonlinear partial differential equations of the hyperbolic type is solved numerically, employing three finite difference schemes: Lax-Wendroff, MacCormack, and Dispersion Relation Preserving, the last being the most accurate one. When the excitation takes place far from the midlength of the soft tube, a phase difference between the pressures at the two edges of each tube is developed, being in advance the one that is closer to the excitation area. Increasing the tube occlusion or the length of the excited part of the loop the mean flow rate increases and maximizes at the natural frequency of the loop. The direction of the maximum mean flow rate, for a given tube occlusion, is from the excitation area toward the edge of the stiff tube, which is located closer to the excitation area. Varying the excitation frequency both above and below the resonance frequency, local flow rate extremes appear, manifesting the complex character of the valveless pumping phenomenon.