# Design and Experimental Verification of a PZT Pump with Streamlined Flow Tubes
## 2 Structure and Working Principle
### Working Principle  
During fluid flowing in the forward direction, the initial pressure of the flow tube is \(P\), the initial velocity is \(u\), the radius of the flow tube at Section 2.2 is \(R\), then the velocity and pressure are expressed as  
\[\left\{\begin{array}{l}P=P(x)\\ u=u(x,y)\end{array}\right. \tag{5}\]  
where \(x\) is the length of the flow tube, and \(y\) is the flow area radius.  
Therefore, according to the formulas (1)-(4), the relationship between the pressure change and the velocity are  
\[\left\{\begin{array}{l}\frac{dp}{dx}>0\;\frac{\partial^{2}u_{y}}{\partial y ^{2}}>0\;\frac{\partial u_{x}}{\partial x}<0\;(0<x<L_{1-2})\\ \frac{dp}{dx}<0\;\frac{\partial^{2}u_{y}}{\partial y^{2}}<0\;\frac{\partial u _{x}}{\partial x}>0\;(L_{1-2}<x<L_{2-3})\end{array}\right. \tag{6}\]  
where \(L_{1-2}\) is the length from Sections 1.1-2.2, \(L_{2-3}\) is the length from Sections 2.2-3.3.  
During backwards fluid flow, similarly, the coordinate system is established with the Section 3.3 as the origin, the relationship between the pressure change and the velocity are  
\[\left\{\begin{array}{l}\frac{dp}{dx}>0\;\frac{\partial^{2}u_{y}}{\partial y ^{2}}>0\;\frac{\partial u_{x}}{\partial x}<0\;(0<x<L_{2-3})\\ \frac{dp}{dx}<0\;\frac{\partial^{2}u_{y}}{\partial y^{2}}<0\;\frac{\partial u _{x}}{\partial x}>0\;(L_{2-3}<x<L_{1-2})\end{array}\right. \tag{7}\]  
According to Formulas (6) and (7), whether fluid flows forward or backward, the divergent or convergent trend is similar. For example, when fluid flows forward, the fluid begins with an adverse pressure gradient from Sections 1.1-2.2, where the internal pressure of the flow field increases while the velocity decreases. Then goes through a consequent pressure gradient from Sections 2.2-3.3, where the pressure decreases while velocity increases. Similarly, when it flows backwards, the adverse pressure gradient appears in Sections 3.3-2.2 and the consequent pressure gradient appears in Sections 2.2-1.1. The internal velocity and pressure change are opposite to the barotropic gradient. By changing the fluid flow trend, it is theoretically guaranteed that the velocity of the internal boundary layer will not appear to be zero, avoiding the occurrence of backflow and vortex, thereby improving the flow stability.  
When the piezoelectric vibrator is oscillating periodically, the volume of the pump chamber changes correspondingly. Scholars have divided this periodic change into two stages: one is the suction phase (suction stroke) when the internal volume of the pump chamber increases from minimum to maximum. The other one is the discharge phase (compression stroke) when the internal volume reduces from maximum to minimum. In both the suction and compression strokes, the flow trend changes in the streamlined flow tube of the SLFT PZT pump are always opposite to each other. Thus, at any stage, there are always obvious characteristics of unequal flow resistance, which can realize macroscopic fluid transportation.  
The three-dimensional model of the SLFT PZT pump, shown in Figure 2, was fabricated by Sterolithography Apparatus (iSLA660, ZRapid, Suzhou, China). The size parameters of the pump are shown in Table 1. The resonance impedance and resonance frequency of the piezoelectric vibrator (Dong Guan Cosson Electric Co, LTD, Dongguan, China) adopted in this paper are \(1.9\pm 0.5\) KHz and \(400\)\(\Omega\) respectively. The size parameters of the piezoelectric vibrator are shown in Table 2.