∂ t ρ 2 = − ρ 0 ∇ ⋅ v 2 − ∇ ⋅ ( ρ 1 v 1 ) _=-_\nabla \cdot _-\nabla \cdot (__)} , ρ 0 ∂ t v 2 = − ∇ p 2 + μ ∇ 2 v 2 + β μ ∇ ( ∇ ⋅ v 2 ) − ρ 1 ∂ t v 1 − ρ 0 ( v 1 ⋅ ∇ ) v 1 _\partial __=-\nabla _+\nabla ^_+\nabla (\nabla \cdot _)-_\partial __-_(_\cdot \nabla )_ . for the first-order equations, taking the time derivation of the navier-stokes equation and inserting the mass conservation, a combined equation can be found: 1 c f 2 ∂ t 2 p 1 = [ 1 − ( 1 − β ) μ ρ 0 c f 2 ∂ t ] ∇ 2 p 1 ^\partial _^= . this is an acoustic wave equation with viscous attenuation. physically, p 1 _ and v 1 _ can be interpreted as the acoustic pressure and the velocity of the acoustic particle. the second-order equations can be considered as governing equations used to describe the motion of fluid with mass source [ − ∇ ⋅ ( ρ 1 v 1 ) ) ] and force source [ − ρ 1 ∂ t v 1 − ρ 0 ( v 1 ⋅ ∇ ) v 1 ] . generally, the acoustic streaming is a steady mean flow, where the response time scale is much smaller than the one of the acoustic vibration. the time-average term v 2 ¯ _ is normally used to present the acoustic streaming. by using ∂ t ρ 2 ¯ = 0 _=0} , the time-average second-order equations can be obtained: ρ 0 ∇ ⋅ v 2 ¯ = − ∇ ⋅ ( ρ 1 v 1 ) ¯ _\nabla \cdot _=-\nabla \cdot __)}}} , μ ∇ 2 v 2 ¯ + β μ ∇ ( ∇ ⋅ v 2 ¯ ) − ∇ p 2 ¯ = ρ 1 ∂ t v 1 ¯ − ρ 0 ( v 1 ⋅ ∇ ) v 1 ¯ \nabla ^_+\nabla (\nabla \cdot _)-_=_\partial __-__\cdot \nabla )_ . fig. 3: cross-section of acoustic streaming around a solid cylindrical pillar it is important to note that the time-averaging of pure first-order terms lead to their cancellation, since they are by definition harmonic. this means that they are pure sine waves, and thus have a mean of 0, which leads to the cancellation of any term that contains them. second-order terms are, however, not harmonic, and do not get cancelled out by time-averaging. this is most important for understanding acoustic streaming: first-order terms, related to simple oscillatory motion, have much larger magnitudes than second-order terms, and thus are dominant in the oscillation time-scale. those first-order terms, however, being pure sines, in a quasi-steady state, repeat after each oscillation cycle, yielding no net fluid flow. second-order terms, instead, are not harmonic, and thus can have a cumulative effect which, despite being smaller, can add up over many oscillation cycles, leading to the development of the net steady-state flow we identify as acoustic streaming. in determining the acoustic streaming, the second-order equations are thus most important. since navier-stokes equations can only be analytically solved for simple cases, numerical methods are typically used, with the finite element method (fem) the most common technique. it can be employed to simulate the acoustic streaming phenomena. figure 3 is one example of acoustic streaming around a solid circular pillar, which is calculated by fem. as mentioned, acoustic streaming is driven by mass and force sources originating from the acoustic attenuation. however, these are not the only driven forces for acoustic streaming. the boundary vibration may also contribute, especially to "boundary driven streaming". for these cases, the boundary condition should also be processed by the perturbation approach and be imposed on the two order equations accordingly. particle motion the motion of a suspended particle whose gravity is balanced by the buoyancy force in an acoustic field is determined by two forces: the acoustic radiation force and stokes drag force. by applying newton's law, the motion can be described as: m d u d t = f rad + f drag =^+^ , f drag = 6 π a μ ( v − u ) ^=6\pi (-)} . where v is the fluid velocity, u is the velocity of particle. for applications in a static flow, the fluid velocity comes from the acoustic streaming. the magnitude of acoustic streaming depends on the power and frequency of the input and the properties of the fluid media. for typical acoustic-based microdevices, the operating frequency may be from the khz to the mhz range. the vibration amplitude is in a range of 0.1 nm to 1 μm. assuming the fluid used is water, the estimated magnitude of acoustic streaming is in the range of 1 μm/s to 1 mm/s. thus, the acoustic streaming should be smaller than the main flow for most continuous flow applications. the drag force is mainly induced by the main flow in those applications. applications cell separation cells with different densities and compression strengths can theoretically be separated with acoustic force. it has been suggested that acoustic tweezers could be used to separate lipid particles from red blood cells. this is a problem during cardiac surgery supported by a heart-lung machine, for which current technologies are insufficient. according to the proposal, acoustic force applied to blood plasma passing through a channel will cause red blood cells to gather in the pressure node in the center and the lipid particles to gather in antinodes at the sides (see figure 4). at the end of the channel, the separated cells and particles exit through separate outlets. the acoustic method might also be used to separate particles of different sizes. according to the equation of primary acoustic radiation force, larger particles experience larger forces than smaller particles. shi et al. reported using interdigital transducers (idts) to generate a standing surface acoustic wave (ssaw) field with pressure nodes in the middle of a microfluidic channel, separating microparticles with different diameters. when introducing a mixture of particles with different sizes from the edge of the channel, larger particles will migrate toward the middle more quickly and be collected at the center outlet. smaller particles will not be able to migrate to the center outlet before they are collected from the side outlets. this experimental setup has also been used to separate blood components, bacteria, and hydrogel particles. 3d cell focusing fluorescence-activated cell sorters (facs) can sort cells by focusing a fluid stream containing the cells, detecting fluorescence from individual cells, and separating the cells of interest from other cells. they have high throughput but are expensive to purchase and maintain, and are bulky with a complex configuration. they also affect cell physiology with high shear pressure, impact forces and electromagnetic forces, which may result in cellular and genetic damage. acoustic forces are not dangerous to cells, and there has been progress integrating acoustic tweezers with optical/electrical modules for simultaneous cell analysis and sorting, in a smaller and less-expensive machine. acoustic tweezers have been developed to achieve 3d focusing of cells/particles in microfluidics. a pair of interdigital transducers (idts) are deposited on a piezoelectric substrate, and a microfluidic channel is bonded with the substrate and positioned between the two idts. microparticle solutions are infused into the microfluidic channel by a pressure-driven flow. once an rf signal is applied to both idts, two series of surface acoustic waves (saw) propagate in opposite directions toward the particle suspension solution inside the microchannel. the constructive interference of the two saws results in the formation of a ssaw. leakage waves in the longitudinal mode are generated inside the channel, causing pressure fluctuations that act laterally on the particles. as a result, the suspended particles inside the channel will be forced toward either the pressure nodes or antinodes, depending on the density and compressibility of the particles and the medium. when the channel width covers only one pressure node (or antinode), the particles will be focused in that node. in addition to focusing in a horizontal direction, cells/particles can also be focused in the vertical direction. after ssaw is on, the randomly distributed particles are focused into a single file stream (fig. 10c) in the vertical direction. by integrating a standing surface acoustic wave (ssaw)-based microdevice capable of 3d particle/cell focusing with laser-induced fluorescence (lif) detection system, acoustic tweezers are developed into a microflow cytometer for high-throughput single cell analysis. the tunability offered by chirped interdigital transducers renders it capable of precisely sorting cells into a number (e.g., five) of outlet channels in a single step. this is a major advantage over most existing sorting methods, which typically only sort cells into two outlet channels. noninvasive cell trapping and patterning a glass reflector with etched fluidic channels is clamped to the pcb holding the transducer. cells infused into the chip are trapped in the ultrasonic standing wave formed in the channel. the acoustic forces focus the cells into clusters in the center of the channel as illustrated in the inset. since the trapping occurs close to the transducer surface, the actual trapping sites are given by the near-field pressure distribution as shown in the 3d image. cells will be trapped in clusters around the local pressure minima creating different patterns depending on the number of cells trapped. the peaks in the graph correspond to the pressure minima. manipulation of single cell, particle, or organism fig. 4: traces of single cell manipulation manipulating single cells is important to many biological studies, such as in controlling the cellular microenvironment and isolating specific cells of interest. acoustic tweezers have been demonstrated to manipulate each individual cell with micrometer-level resolution. cells generally have a diameter of 10–20 μm. to meet the resolution requirements of manipulating single cells, short-wavelength acoustic waves should be employed. in this case, a surface acoustic wave (saw) is preferred to a bulk acoustic wave (baw), because it allows using shorter-wavelength acoustic waves (normally less than 200 μm). ding et al. reported a ssaw microdevice that is able to manipulate single cells with prescribed paths. figure 6 records a demonstration that the movement of single cells can be finely controlled with acoustic tweezers. the working principle of the device lies in the controlled movement of pressure nodes in an ssaw field. ding et al. employed chirped interdigital transducers (idts) that are able to generate ssaws with adjustable positions of pressure nodes by changing the input frequency. they also showed that the millimeter-sized microorganism c. elegan can be manipulated in the same manner. they also examined cell metabolism and proliferation after acoustic treatment, and found no significant differences compared to the control group, indicating the non-invasive nature of acoustic base manipulation. in addition to using chirped idts, phaseshift-based single particle/cell manipulation has also been reported. manipulation of single biomolecules sitters et al. have shown that acoustics can be used to manipulate single biomolecules such as dna and proteins. this method, which the inventors call acoustic force spectroscopy, allows measuring the force response of single molecules. this is achieved by attaching small microspheres to the molecules at one side and attaching them to a surface at the other. by pushing the microspheres away from the surface with a standing acoustic wave the molecules are effectively stretched out. manipulation of organic nano-materials polymer-dispersed liquid crystal (pdlc) displays can be switched from opaque to transparent using acoustic tweezers. a saw-driven pdlc light shutter has been demonstrated by integrating a cured pdlc film and a pair of interdigital transducers (idts) onto a piezoelectric substrate. manipulation of inorganic nano-materials acoustic tweezers provide a simple approach for tuneable nanowire patterning. in this approach, ssaws are generated by interdigital transducers, which induced a periodic alternating current (ac) electric field on the piezoelectric substrate and consequently patterned metallic nanowires in suspension. the patterns could be deposited onto the substrate after the liquid evaporated. by controlling the distribution of the ssaw field, metallic nanowires are assembled into different patterns including parallel and perpendicular arrays. the spacing of the nanowire arrays could be tuned by controlling the frequency of the surface acoustic waves. selective manipulation while most acoustic tweezers are able to manipulate a large number of objects collectively, a