Acoustic Streaming is ideal for microfluidic systems because it arises from viscous forces which are the dominant forces in low Reynolds flows and which usually hamper microfluidic systems. Also, streaming force scales favorably as the size of the channel, conveying a fluid through which an acoustic wave propagates, decreases. Because of acoustic attenuation via viscous losses, a gradient in the Reynolds stresses is manifest as a body force that drives acoustic streaming as well as streaming from Lagrangian components of the flow. For more information on the basic theory of acoustic streaming please see Engineering Acoustics/Acoustic streaming. When applied to microchannels, the principles of acoustic streaming must include bulk viscous effects (dominant far from the boundary layer, though driven by boundary layer streaming), investigated in the classic solution developed extensively by Nyborg in 1953 as well as streaming inside the boundary layer. In a micromachined channel, the dimensions of the channels are on the order of boundary layer thickness, so both the inner and outer boundary layer streaming must be evaluated to have a precise prediction for flow rates in acoustic streaming micropumps. The derivation that follows is for a circular channel of constant cross section assuming that the incident acoustic wave is planar and bound within the channel filled with a viscous fluid. The acoustic wave has a known amplitude and fills the entire cross-section and there are no reflections of the acoustic wave. The walls of the channel are also assumed to be rigid. This is important, because rigid boundary interaction results in boundary layer streaming that dominates the flow profile for channels on the order of or smaller than the boundary layer associated with viscous flow in a pipe. This derivation follows from the streaming equations developed by Nyborg who starts with the compressible continuity equation for a Newtonian fluid and the Navier-Stokes and dynamic equations to get an expression for the net force per unit volume. Eckart uses the method of successive approximations with the pressure, velocity, and density expressed as the sum of first and second order terms. Since the first order terms account for the oscillating portion of the variables, the time average is zero. The second order terms arise from streaming and are time independent contributions to velocity, density, and pressure. These non-linear effects due to viscous attenuation of the acoustic radiation in the fluid are responsible for a constant streaming velocity[1]. Then, the expansion (through the method of approximations) of the variables are substituted into the standard force balance equations describing a fluid resulting in two equations[5] where: ( 1 )   − F = − ∇ p 2 + ( β μ + 4 3 μ ) ∇ ∇ u 2 − μ ∇ × ∇ × u 2 {\displaystyle (1)\ -F=-\nabla p_{2}+\left(\beta _{\mu }+{\frac {4}{3}}\mu \right)\nabla \nabla u_{2}-\mu \nabla \times \nabla \times u_{2}} ( 2 )   − F ≡ ρ 0 | ( u 1 ⋅ ∇ u 1 ) + u 1 ( ∇ ⋅ u 1 ) | {\displaystyle (2)\ -F\equiv \rho _{0}|\left(u_{1}\cdot \nabla u_{1}\right)+u_{1}\left(\nabla \cdot u_{1}\right)|} where the signifier | e x p r e s s i o n | {\displaystyle |expression|} denotes time average, F {\displaystyle F} is the body force density, β μ {\displaystyle \beta _{\mu }} is the bulk viscosity, p 2 {\displaystyle p_{2}} is the second order pressure, μ {\displaystyle \mu } is the dynamic viscosity, ρ 0 {\displaystyle \rho _{0}} is the density, u 2 {\displaystyle u_{2}} is the streaming velocity, and u 1 {\displaystyle u_{1}} is the acoustic velocity. The acoustic velocity, represented two dimensionally in axial and radial directions respectively, is described by: ( 3 )   u 1 x = V a e − ( α + i k ) x ( 1 − e − ( 1 + i ) ζ z ) e i ω t {\displaystyle (3)\ u_{1x}=V_{a}e^{-(\alpha +ik)x}\left(1-e^{-(1+i)\zeta z}\right)e^{i\omega t}} ( 4 )   u 1 z = − V a e − ( α + i k ) x ( α + i k ) ( 1 − e − ( 1 + i ) ζ z ) e i ω t ( 1 + i ) ζ {\displaystyle (4)\ u_{1z}={\frac {-V_{a}e^{-(\alpha +ik)x}\left(\alpha +ik\right)\left(1-e^{-(1+i)\zeta z}\right)e^{i\omega t}}{\left(1+i\right)\zeta }}} where ζ = ω ρ 0 2 μ {\displaystyle \zeta ={\frac {\omega \rho _{0}}{2\mu }}} where V a {\displaystyle V_{a}} is the acoustic velocity at the source, k = ω c 0 {\displaystyle k={\frac {\omega }{c_{0}}}} is the wave number, c 0 {\displaystyle c_{0}} is the velocity of sound in the fluid, and α {\displaystyle \alpha } is the acoustic absorption coefficient. The ζ {\displaystyle \zeta } term describes the viscous penetration depth, or how large the boundary layer is. The components of the acoustic velocity given in Equation (3) and (4) can be substituted into Equation (2) to solve for the first-order body force. This gives the one-dimensional body force per unit volume in axial and radial components, respectively: ( 5 )   F x v = ρ 0 V a 2 e − 2 α x {\displaystyle (5)\ F_{xv}=\rho _{0}V_{a}^{2}e^{-2\alpha x}} ( 6 )   F x b = 1 2 ρ 0 V a 2 e − 2 α x [ k e − ζ z ( c o s ( ζ z ) + s i n ( ζ z ) − e − ζ z ) + α e − ζ z ( e − ζ z − 3 ( c o s ( ζ z ) + s i n ( ζ z ) ) ] . {\displaystyle (6)\ F_{xb}={\frac {1}{2}}\rho _{0}V_{a}^{2}e^{-2\alpha x}\left[ke^{-\zeta z}\left(cos(\zeta z)+sin(\zeta z)-e^{-\zeta z}\right)+\alpha e^{-\zeta z}\left(e^{-\zeta z}-3(cos(\zeta z)+sin(\zeta z)\right)\right].} F x v {\displaystyle F_{xv}} and F x b {\displaystyle F_{xb}} are expressions for the body force due to viscous losses and due to the acoustic wave touching the rigid boundary[5]. With no-slip boundary conditions imposed on Equation (1), with Equations (5) and (6) inserted, the streaming velocity u 2 {\displaystyle u_{2}} can be found. The differential pressure is assumed to be zero and static head can be derived by evaluating Equation (1) with a boundary condition of zero net flow through any fluid element. The solution of Equation(1) for the streaming velocity profile in two terms relating to the viscous effects (outer boundary layer streaming) and the boundary layer effects (inner boundary layer streaming), respectively, results in: ( 7 )   u 2 v = α ρ 0 V a 2 h 2 2 μ ( z h ) ( 1 − z h ) {\displaystyle (7)\ u_{2v}={\frac {\alpha \rho _{0}V_{a}^{2}h^{2}}{2\mu }}\left({\frac {z}{h}}\right)\left(1-{\frac {z}{h}}\right)} ( 8 )   u 2 b = V a 2 4 c 0 [ 1 + 2 e − ζ z ( s i n ( ζ z ) − c o s ( ζ z ) ) + e − 2 ζ z ] {\displaystyle (8)\ u_{2b}={\frac {V_{a}^{2}}{4c_{0}}}\left[1+2e^{-\zeta z}\left(sin(\zeta z)-cos(\zeta z)\right)+e^{-2\zeta z}\right]} These two expressions are summed when calculating the velocity profile across the diameter of the pipe. With no-slip conditions, the outer boundary layer streaming contribution to the acoustic streaming velocity decreases as the diameter decreases with a with a profile similar to Hele-Shaw flow in infinitely wide rectangular channels [7]. Figure 1 shows this diameter scaling effect in water with an acoustic velocity A = .1 m / s {\displaystyle A=.1m/s} and a driving frequency of 2 MHz. Many groups such as Rife et al. [7], are underestimating the possibilities that acoustic streaming has to offer in channels less than 10 μ m {\displaystyle 10\mu m} because the inner boundary layer streaming velocity is ignored. The boundary layer effects are present regardless of diameter. In water, the acoustic boundary layer is about 1 micron, therefore, for pipes with diameters on the order ten microns or less, there is a marked increase in the streaming velocity. From the velocity profile of the inner boundary layer streaming in Figure 2, the contribution of the boundary layer factors in favorably as the diameter of the channel decreases. Note that the magnitude of the inner boundary layer streaming is not affected by the diameter and that the percentage of the channel experiencing the boundary streaming decreases as channel diameter increases. Then the total flow velocity profile, with both the viscous and boundary layer effects in Figure 3, takes on a flow profile that becomes more plug-like as the diameter of the channel decreases. Driving frequency does have an effect on the velocity profile for a channel of constant diameter that experiences a sizable contribution from the boundary layer. The frequency dependence on the inner boundary layer contribution is evident for a 10 μ m e t e r {\displaystyle 10\mu meter} channel with typical paramaters for water and an acoustic velocity A = .1 m / s {\displaystyle A=.1m/s} in Figure 4. Note that the viscous contribution to acoustic streaming is also shown, but does not exhibit a frequency dependence. For small channels (less than 10 microns), the inner boundary layer streaming affects a more sizable portion of the channel at lower frequencies. The total acoustic streaming flow profile is then given in Figure 5. From this plot, matching the driving frequency to channel geometry is important to achieve the maximum flow velocity for micro-nano fluidic devices. In microfluidic systems, a piezoelectric actuator can be used to impart an acoustic wave field in a liquid. This effect is even imparted through the walls of the device. The advantage is that the actuator does not need be in contact with the working fluid. Since the streaming effect occurs normal to the resonator, there may be difficulties in coupling an actuator with typical micromachining techniques which generally yield 2-D layouts of microfluidic networks. The solutions developed for acoustic streaming assume that the acoustic wave is planar with respect to the channel axis. Therefore a configuration that results in the most predictable flow is one in which the acoustic wave source (piezoelectric bulk acoustic resonator) is placed such that the channel is axially oriented to the normal of the actuator surface. Figure 6 shows such a configuration from a view looking down onto the device. The piezo actuators are in black. This cartoon of a micromachined device is based on one created by Rife et al. [7]. The dimensions of their device where on the order of 1.6 mm square (much greater than the size of the boundary layer), which makes their predictions using classical solutions by Nyborg that do not include inner boundary layer streaming valid, as can be seen in Figure 3 where channels much larger than the boundary layer size are relatively uninfluenced by that part of the acoustic streaming. However, employing this configuration in the context of microfabrication techniques is difficult for very small channels. Rife et al. [7], managed to place piezoelectric actuators oriented perpendicular to the opened ends of channels milled into a block of PMMA manually, albeit their channel dimensions are much larger than that at which the boundary layer effects dominate or contribute significantly. For smaller channels, the only option is to put the actuators on the underside or top of a micro machined fluidic circuit [8]. This configuration, shown in Figure 7 , results in acoustic wave reflections. Reflections or standing waves will complicate the streaming analysis. Another option for instigating acoustic streaming results from the attenuation of surface acoustic waves (SAW) in contact with a fluid medium [6]. In this case, the transverse component of a Rayleigh wave (or a Lamb wave) propagating along a surface in contact with a fluid is effectively transferred into a compression wave in the fluid. The energy in the SAW is dissipated by the fluid and little disturbance is felt in the substrate far from the SAW source (interdigital piezo actuators). Figure 8 is a cartoon of the principle. This is the case so long as the velocity of the SAW is greater than the acoustic velocity of the liquid. The compression wave radiating from the surface leaves at the Rayleigh angle given by: ( 9 )   φ R = a r c s i n ( V a V R ) {\displaystyle (9)\ \phi _{R}=arcsin({\frac {V_{a}}{V_{R}}})} where V R {\displaystyle V_{R}} is the velocity of the surface acoustic wave. Given that angle then, theoretically, two actuators producing SAWs could be placed opposite each other to produce a standing wave field in the fluid across the channel and a traveling planar wave parallel to the channel axis. Figure 9 shows how this could be done. Finally, a very interesting pump that uses acoustic standing waves and a diffuser nozzle is shown in Figure 10, which has been developed by Nabavi and Mongeau. While this pump does not use the same acoustic streaming principles, it is included because it uses acoustic waves to generate a flow. The standing wave, induced by relatively small movements of the piston, has a maximum pressure at the anti-node and a min at the node. Positioning the inlet and the outlet at these two locations allows fluid to enter the chamber immediately after the part of the cycle where the pressure at the anti-node has overcome the discharge pressure at the diffuser nozzle. Most importantly, the diffuser nozzle outlet has an asymmetric resistance. After the fluid is ejected and the pressure in the chamber is temporarily lower than the ambient pressure the fluid does not flow right back in the outlet but enters at the pressure node, where pressure is less. This clever design allows for a valveless pumping apparatus. The forward and back flow resistance of the diffuser nozzle are not the same, so a net mass flow out of the resonance chamber is observed. Another interesting pump for precise positioning of fluid droplets in microchannels that uses similar acoustic standing waves in a resonance chamber has been developed by Langelier et al.. Instead of a piezoelectric membrane generating the acoustic standing waves in the resonance chamber, the resonance chamber is filled with air and connected to a larger container that has a speaker at one end. Multiple quarter wavelength resonance chambers are tuned to specific frequencies, each with a different length and width. Different pipes connected to each resonance chamber can then be activated with one source, each one independently depending on which frequencies the speaker is emitting. Just like the acoustic standing wave pump of Nabavi and Mongeau, an outlet is located at the point of peak pressure amplitude, which in this case is at the end of the resonance chamber. With a rectification structure, an oscillating flux of fluid out of the resonance chamber is converted into a pulsed flow in the microfluidic channel. K. D. Frampton, et al., "The scaling of acoustic streaming for application in micro-fluidic devices," Applied Acoustics, vol. 64, pp. 681-692, 2003. J. Lighthill, "ACOUSTIC STREAMING," Journal of Sound and Vibration, vol. 61, pp. 391-418, 1978. W. L. Nyborg, "ACOUSTIC STREAMING DUE TO ATTENUATED PLANE WAVES," Journal of the Acoustical Society of America, vol. 25, pp. 68-75, 1953. K. D. Frampton, et al., "Acoustic streaming in micro-scale cylindrical channels," Applied Acoustics, vol. 65, pp. 1121-1129, Nov 2004. C. Eckart, "VORTICES AND STREAMS CAUSED BY SOUND WAVES," Physical Review, vol. 73, pp. 68-76, 1948. G. Lindner, "Sensors and actuators based on surface acoustic waves propagating along solid-liquid interfaces," Journal of Physics D-Applied Physics, vol. 41, 2008. J. C. Rife, et al., "Miniature valveless ultrasonic pumps and mixers," Sensors and Actuators a-Physical, vol. 86, pp. 135-140, Oct 2000. K. Hashimoto, et al., "Micro-actuators employing acoustic streaming caused by high-frequency ultrasonic waves," Transducers 97 - 1997 International Conference on Solid-State Sensors and Actuators, Digest of Technical Papers, Vols 1 and 2, pp. 805-808, 1997. Nabavi, M. and L. Mongeau (2009). "Numerical analysis of high frequency pulsating flows through a diffuser-nozzle element in valveless acoustic micropumps." Microfluidics and Nanofluidics 7(5): 669-681. S. M. Langelier, et al., "Acoustically driven programmable liquid motion using resonance cavities," Proceedings of the National Academy of Sciences of the United States of America, vol. 106, pp. 12617-12622, 2009.
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