Diagram of the experimental setup. An elastic tube is connected to a rigid tube as shown. R.S. indicates rigid sections. The tubes are filled with water. A cylinder depresses the elastic tube periodically. The movement of flow visualization particles in a transparent section of the rigid tubing is captured by a video camera. The illuminated portion of the transparent section is indicated by the dashed line. The arrows in the rigid sections indicate the direction of flow observed by Liebau, which we confirm to be the primary flow direction.
Diagram of the forcing mechanism. A motor drives the periodic motion of a lever, to which a cylinder is attached, so that the cylinder compresses the elastic tube. A spring keeps the lever in contact with the motor at all times. A potentiometer at the base of the lever measures its phase within a driving period (not shown).
Experimental results showing the velocity in the rigid section as a function of time. Left: Plot of vs time for the tube with thickness , forcing frequency , and forcing offset . Right: Plot of vs time for , , and . This plot shows a resonance.
Solid line: Plot of experimentally measured vs time for , , and . Dashed line: Scaled, shifted height of the forcing mechanism. Dotted line: Height at which the forcing mechanism contacts the elastic tube. At point A, the forcing mechanism contacts the elastic tube and acceleration begins. At point B, the elastic tube is nearly closed by the forcing mechanism and the flow decelerates rapidly. At point C, the elastic tube opens and the flow accelerates again. At point D, the forcing mechanism disengages from the elastic tube and a damped oscillation begins.
Experimental results showing the average velocity in the rigid section as a function of forcing frequency. Left: Plot of vs forcing frequency comparing results for different rigidities and forcing offsets. Right: Zoom-in of the lower frequency portion of the plot on the left.
Experimental results showing the average velocity in the rigid section as a function of forcing location. Left: Plot of vs forcing offset using the tube with at various lower frequencies. Right: Same as left but at higher frequencies.
Diagram showing the components and variables in our model.
Left: The elastic tube, represented by the horizontal lines, is impacted by the forcing mechanism, represented by the circle. At position , is the vertical distance from the center of the elastic tube to the forcing mechanism. Right: Cross section of the elastic tube at position . The unshaded area is that which intersects the forcing mechanism, and we let be equal to the total shaded area. The first term in Eq. (9) corresponds to the area of the lightly shaded region, and the second term corresponds to the area of the darkly shaded region.
Plots of vs time. The black lines show model results. The gray lines show the experimental results. Upper left: , , and . Upper right: , , and . Lower left: , , and . Lower right: , , and .
Plots of vs forcing frequency for the less rigid elastic tube. The solid lines show the model results. The dashed lines show the experimental results. Left: . Right: .
Plots of vs forcing offset for the more rigid elastic tube. The solid lines show the model results. The dashed lines show the experimental results. Upper left: . Upper right: . Lower left: (the resonance frequency). Lower right: .
Comparison of the average flux generated by the full model (shown by the solid lines) with that predicted by the asymptotic formula in Eq. (18) (shown by the dashed lines). Both plots are for the less rigid elastic tube. Left: , and we have chosen in the asymptotic formula to fit the results of the model. Right: , and we have chosen to fit the results of the model.
The solid line shows vs Reynolds number for the less rigid elastic tube and . The dashed line shows the flux of a traditional pump operating at the same frequency which displaces a volume in every period.
Plots of vs forcing frequency for the less rigid elastic tube with reduced from its experimentally measured value by a factor of 10. The solid lines show the model results. The dashed lines show the experimental results. Left: . Right: . Particularly in this second plot, resonancelike effects can be seen in the model.
Physical constants and parameters used throughout the paper.
Values of dimensionless constants for , , and .
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