## 2 Forces Induced by Acoustic Wave and Streaming Flow
### Acoustic Radiation Force on the Targets  
Acoustic radiation force based on first-order acoustic pressure is the simplest acoustic field force model. Generally, both the static and dynamic field devices mentioned above use the Gor'kov potential in the acoustic field. This was first described mathematically in [PERSON]'s research [37; 38], but its basic phenomena had been known for some years before as acoustic standing wave in the Kundt tube, which was used to test the speed of sound separation in various mediums. After a period of development, [PERSON] proposed an elegant model to explain and describe the potential energy and force of standing waves and traveling wave fields [39]. When the diameter of particles is much smaller than the incident wavelength of sound (\(kR_{p}\ll 1\), where \(k=\frac{2\pi}{\lambda}\) and \(R_{p}\) is the particle radius), the acoustic field force, \(F(\mathbf{r})\), resulting from a potential field (note that it is a scalar), \(U(\mathbf{r})\), can be found as [40; 41]:  
\[ \begin{split} U(\mathbf{r})=V\Bigg{(}\!\!\!\left(1-\frac{\kappa_{p}}{ \kappa_{f}}\right)\!\!\!E_{\text{pot}}(\mathbf{r})-\frac{3\left(\rho_{p}-\rho_{f} \right)}{\left(2\rho_{p}+\rho_{f}\right)}E_{kin}(\mathbf{r})\!\!\!\Bigg{)}\\ \text{E}_{\text{pot}}(\mathbf{r})\propto\langle\left|p_{1}\right|^{2} \rangle\text{ and }\text{E}_{kin}(\mathbf{r})\propto\langle\left|\overset{\rightarrow}{ \left|p_{p}\right|^{2}}\rangle\end{split}  \tag{1}\]where \(V\) is the volume of the particles being manipulated (usually a sphere, \(V=4\pi R_{p}{}^{3}/3\)); \(\kappa_{(\cdot)}\) and \(\rho_{(\cdot)}\) are the compressibility and density, respectively; the subscripts \(p\) and \(f\) denote the particles and fluid, respectively. The time-averaged radiation potential energy \(U\) of the particle of volume \(V\) located at \(\mathbf{r}\) is related to the time-averaged kinetic energy (\(E_{kin}\), which is proportional to the mean squared particle velocity, \(\langle\overline{|p_{\rm p}|}^{2}\rangle\)) and potential energy densities (\(\mathrm{E_{pot}}\), which is proportional to the mean squared pressure, \(\langle\left|p_{1}\right|^{2}\rangle\)), ignoring the flow of the fluid in the sound field and the force between the particles. Note that for a fluid, \(\kappa_{f}=1/\rho\mathrm{c}^{2}\), where \(\mathrm{c}\) is the wave speed. The premise of this formula is that the particles must be spherical and small enough, and the flow in the field and the interaction between multiple particles are negligible. The acoustic streaming more significantly influences the small particles than the ARF, according to [PERSON] et al. [42]. For some large or non-spherical particles, many methods are available for analyzing the ARF in typical conditions via numerical analysis [43; 44]. In methods operating with dynamic acoustic fields, the flow of fluid is generally considered an interference term. However, the microfluid is an alternative to micromanipulation, which we discuss later. Acoustic streaming induced by second-order sound pressure and Reynolds pressure is a popular method of fluid generation, which involves the precise application of the nonlinear part of the acoustic radiation force. The influence of acoustic streaming on many systems is vital because second-order acoustic pressure is ubiquitous in various systems based on ARF. The reason why the quadratic term is generally neglected is that the influence of the quadratic term on the operation accuracy of the particles that are small enough is limited [42]. However, to cope with exceptional cases such as large particle size (\(kk_{p}\geq 1\)) for which the influence of the second-order term cannot be ignored, many studies have numerically analyzed second-order acoustic pressure. The most common approach has been to integrate the second-order acoustic pressures around a boundary that encloses the particle.