# Simulation Analysis and Experiment of Piezoelectric Pump with Tapered Cross-Section Vibrator
## 5 Simulation Analysis of Pumping Performance Based on Fluid-Solid Coupling
### Flow-Field Analysis of the Piezoelectric Pump  
The second-order mode of the tapered cross-sectional vibrator is the closest to the swimming pattern of the fish, and the driving effect of the second-order bending vibration on the fluid is greater than other modes. Thus, to obtain the second-order vibration mode when simulating the fluid-solid two-way coupling, the amplitude of the driving AC voltage of the vibrator was set at 150 V, and the driving frequency was the second-order mode frequency of the vibrator, which means the boundary potential was \(V=150 sin(2\pi ft)\), and the transient calculation step was \(10^{-4}\) s, and the calculated total time was 0.015 s.  
Simulation results can be seen from the fluid streamline diagram (Figure 8), which indicated that two fluid vortices were generated in the flow field around the vibrator. One field appeared in the two-thirds of the vibrator away from the fixed end, and the other appeared in the tail region, in the one-third area of the vibrator from the swing end. Comparing with the fluid velocity field diagram as showed in Figure 9, it can be seen that the fluid near the swing end has much higher velocity than that around the fixed end of the vibrator. This is basically consistent with the characteristics of tuna swimming discussed in Section 2 above. It is the continuous driving force produced by the large swing of the vibrator tail that drives the fluid to form unidirectional flow. It can be further inferred that near the second-order mode shape, there is a node on the tail of the vibrator as showed in Figure 10. The second-order mode is divided into two parts  
Figure 7 shows that there is an obvious fluid disturbance region in the flow field of the vibrator tail. In order to further analyze the actual pattern of the flow field, the following analysis was focused on the vibrator vibration and the flow field in the tail region and fluid velocity distribution during a typical vibration period T. The fluid velocity distribution at each typical time point is shown in Figure 11.  
Figure 11: _Cont._  
Figure 10: Schematic diagram of the vibrator node. Figure 11: _Cont._  
According to the fluid disturbance area of the flow field at the tail region in Figure 9, when the tail of the vibrator (or point P) moves to the position at time T/2 (as shown in Figure 11(e-1)-(e-3)), the fluid driven by the vibrator tail fin reaches the maximum velocity of 1.12 m/s. The angle \(\theta\) between the NP and the equilibrium position is 0 at this time according to Figure 10, and the driving velocity of the NP on the surrounding fluid is the largest in the x direction, while the velocity component is 0 along the y direction; when the vibrator swings up and down from the equilibrium position, the angle \(\theta\) gradually increases, which makes the driving effect of the NP on the surrounding fluid gradually decrease along the x direction, and the fluid velocity reduces; when the point P moves to the position at the times T/4 and 3T/4, the fluid velocities are the smallest, 0.56 m/s and 0.64 m/s, respectively. According to the vibration position of the vibrator at time T/4 and time 3T/4, the point P reaches the maximum displacement after going upward and downward (as shown in Figure 11(c-1)-(c-3) and (g-1)-(g-3)). At these two typical moments, the vibrator tail (or point P) begins to go backward, causing an instantaneous motion suppression, which leads to a decrease in the speed of fluid flow. The formation of pumping flow is caused by the vibrator driving the fluid continuously. The superposition of all the continuous fluid movement in the fluid domain forms a unidirectional flow from the inlet to the outlet. Based on the above analysis, it can be seen that the driving of the vibrator tail plays a major role of forming the pumping flow. In the above analysis, there will be effects of gravity, but it is overall downward along the equilibrium position because of gravity influence, the amplitude of the vibrator is almost not affected, so the flow rate is almost not affected.