## 2 Forces Induced by Acoustic Wave and Streaming Flow
### Force Acting on the Target in Streaming Flow  
Manipulate particles must be manipulated with the vibration-induced streaming flow. Although it is impossible to accurately calculate the mathematical model of the fluid, three equations--continuity, Navier-Stokes, and energy conservation--can be applied on the macro-scale to solve many fluid dynamics problems based on the continuum assumption [46]. Generally, the performance of microfluidics is not the same as that of macrofluids. However, the scale of micromanipulation is still much larger than the molecular size of the medium, other numerical analysis methods of micro-fluids are too complicated, and the continuous medium assumption is still reliable with certain accuracy. Therefore, particles can still be dynamically analyzed by the method described above. Hydrodynamics is highly nonlinear, and studies related to micromanipulation are numerous and complex [33; 47; 48]. Here, we summarize the simple mechanical principles involved in streaming flow rather than precise numerical solutions.  
For streaming flow with a low Reynolds number on the micro-scale, the influence of the fluid's convection acceleration can be ignored. So, the drag force \(\overset{\rightarrow}{F}_{d}\) of the spherical particles in streaming flow can be solved by using the Navier-Stokes equation (NSE):  
\[\overset{\rightarrow}{F}_{d}=-6\pi\mu R_{p}\overset{\rightarrow}{v}_{p} \tag{4}\]  
where \(\mu\) is fluid viscosity. The drag force is positively related to the particle velocity, and the direction is precisely the opposite. This result means that the fluid hinders the movement of the particles, but due to the existence of relative motion, the moving fluid can also cause the movement of the particles. For non-spherical particles, the way in which drag forces work is also different, for which readers can refer to related microfluidics dynamic works.  
In addition to the drag of the fluid, the particles are also affected by gravity and buoyancy, which is a fundamental mechanical equation:  
\[G_{p}+F_{B}=V\big{(}\rho_{p}-\rho_{f}\big{)}\mathbf{g}. \tag{5}\]  
Equation (5) calculates the sum of the gravity \(G_{p}\) and the buoyancy force \(F_{B}\) of particles, where \(\mathbf{g}\) is the acceleration due to gravity. Usually, particles with a density close to the fluid remain suspended in the medium. For particles with a density much smaller than the medium, especially microbubbles, the surrounding liquid moves during its accelerated motion. In other words, this is equivalent to adding additional mass to the particles, which is the virtual mass force [49]:  
\[\overset{\rightarrow}{F}_{v}=-V\rho_{f}K_{m}\overset{\rightarrow}{a}_{p} \tag{6}\]  
where \(\overset{\rightarrow}{a}_{p}\) is the acceleration of particles and \(K_{m}\) is the empirical coefficient of the virtual mass force, which is set to 0.5 in the calculation. The effect of this force can be ignored due to the large motion inertia for high-density particles.  
Considering only the above conditions, the force of the particles in the streaming flow field is concise. Streaming flow is always use to drive particle motion, so the direction of motion is determined by the direction of fluid microelements integration, which is almost unpredictable at the micro level. Regular vibrations of some geometric structures can produce corresponding regular flows. Therefore, manipulation is feasible using microfluid without fully grasping the movement details of each position of the microfluid field.