# Engineering Acoustics/Acoustic Micro Pumps

## Application to Micro Scale Pipes

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.[^1] 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.[^2] 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[^3] 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.[^4] 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[^5] 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:

$(1)\  -F=-\nabla p_2 + \left(\beta_\mu + \frac{4}{3}\mu \right)\nabla \nabla u_2 - \mu \nabla\times\nabla\times u_2$

$(2)\  -F\equiv \rho_0 |\left(u_1\cdot\nabla u_1\right) + u_1\left(\nabla\cdot u_1\right)|$

where the signifier $|expression|$ denotes time average, $F$ is the body
force density, $\beta_\mu$ is the bulk viscosity, $p_2$ is the second
order pressure, $\mu$ is the dynamic viscosity, $\rho_0$ is the density,
$u_2$ is the streaming velocity, and $u_1$ is the acoustic velocity. The
acoustic velocity, represented two dimensionally in axial and radial
directions respectively, is described by:

$(3)\  u_{1x}=V_ae^{-(\alpha+ik)x} \left(1-e^{-(1+i)\zeta z}\right) e^{i\omega t}$

$(4)\  u_{1z}=\frac{-V_ae^{-(\alpha+ik)x} \left(\alpha+ik\right) \left(1-e^{-(1+i)\zeta z}\right) e^{i\omega t}}{\left(1+i\right)\zeta}$

where

$\zeta=\frac{\omega\rho_0}{2\mu}$

where $V_a$ is the acoustic velocity at the source,
$k=\frac{\omega}{c_0}$is the wave number, $c_0$ is the velocity of sound
in the fluid, and $\alpha$ is the acoustic absorption coefficient. The
$\zeta$ 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:

$(5)\  F_{xv}=\rho_0 V_a^2 e^{-2\alpha x}$

$(6)\  F_{xb}=\frac{1}{2}\rho_0 V_a^2 e^{-2\alpha x}\left[ke^{-\zeta z}\left(cos(\zeta z)+sin(\zeta z)-e^{-\zeta z}\right)+\alpha e^{-\zeta z}\left(e^{-\zeta z}-3(cos(\zeta z)+sin(\zeta z)\right)\right].$

$F_{xv}$ and $F_{xb}$ 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 $u_2$ 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:

$(7)\  u_{2v}=\frac{\alpha\rho_0 V_a^2 h^2}{2\mu}\left(\frac{z}{h}\right)\left(1-\frac{z}{h}\right)$

$(8)\  u_{2b}=\frac{V_a^2}{4c_0}\left[1+2e^{-\zeta z} \left(sin(\zeta z)-cos(\zeta z)\right)+e^{-2\zeta z} \right]$

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
$A=.1m/s$ and a driving frequency of 2 MHz. !Figure 1: Scaling of the
velocity profile with different channel dimensions in
microns.{width="380"}
Many groups such as Rife *et al.* \[7\], are underestimating the
possibilities that acoustic streaming has to offer in channels less than
$10\mu m$ 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.[^6] 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. !Figure 2: Percentage of channel diameter feeling influence
of inner boundary layer streaming for different channel
dimensions.{width="380"}
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. !Figure 3:
Normalized total acoustic streaming velocity profile showing effect of
channel
diameter.{width="380"}
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 $10\mu meter$ channel with typical
paramaters for water and an acoustic velocity $A=.1m/s$ 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. !Figure 4: The
inner boundary layer streaming is shown to take up a larger percentage
of the channel dimensions as the driving frequency decreases, while the
viscous contribution is
unnaffected.{width="380"}
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. !Figure 5: Total acoustic streaming velocity profile with
effect of driving frequency. The match of 2Mhz and 10 micron channel
yields the greatest volume
flow.{width="380"}

## Actuation in Microfluidics

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.[^7] 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.[^8] Figure 6
shows such a configuration from a view looking down onto the device.
![Figure 6: Micro pump utilizing acoustic streaming based on one
developed by Rife *et al.* \[7\]. The BAW piezo actuators are in black
and the arrows indicate the direction of
flow.](Acoustic_micro_pump1.JPG "Figure 6: Micro pump utilizing acoustic streaming based on one developed by Rife et al. [7]. The BAW piezo actuators are in black and the arrows indicate the direction of flow."){width="380"}
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.
![Figure 7: Planar micromachined micro pump based off one created by
Hashimoto *et al.* \[8\]. Every other BAW is actuated to drive flow in
one direction. The black rectangles represent activated BAW piezo
devices. By shutting them off and actuating the brown ones, the flow can
be
reversed.](Acoustic_micro_pump_2.jpg "Figure 7: Planar micromachined micro pump based off one created by Hashimoto et al. [8]. Every other BAW is actuated to drive flow in one direction. The black rectangles represent activated BAW piezo devices. By shutting them off and actuating the brown ones, the flow can be reversed."){width="380"}

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. ![Figure 8: Conversion of SAW,
in a solid substrate in contact with a liquid, to planar acoustic
pressure waves
\[6\].](SAW_interaction_with_liquid.jpg "Figure 8: Conversion of SAW, in a solid substrate in contact with a liquid, to planar acoustic pressure waves [6]."){width="380"}
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:

$(9)\ \phi_R = arcsin(\frac{V_a}{V_R})$

where $V_R$ 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. !Figure 9: Parallel SAW acoustic
streaming
micro-pump{width="380"}
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.[^9] !Figure 10: Valveless acoustic standing wave
micro
pump.{width="380"}
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.*.[^10] 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.

## References


[^1]: K. D. Frampton, et al., \"The scaling of acoustic streaming for
    application in micro-fluidic devices,\" Applied Acoustics, vol. 64,
    pp. 681-692, 2003.

[^2]: J. Lighthill, \"ACOUSTIC STREAMING,\" Journal of Sound and
    Vibration, vol. 61, pp. 391-418, 1978.

[^3]: W. L. Nyborg, \"ACOUSTIC STREAMING DUE TO ATTENUATED PLANE
    WAVES,\" Journal of the Acoustical Society of America, vol. 25, pp.
    68-75, 1953.

[^4]: K. D. Frampton, et al., \"Acoustic streaming in micro-scale
    cylindrical channels,\" Applied Acoustics, vol. 65, pp. 1121-1129,
    Nov 2004.

[^5]: C. Eckart, \"VORTICES AND STREAMS CAUSED BY SOUND WAVES,\"
    Physical Review, vol. 73, pp. 68-76, 1948.

[^6]: G. Lindner, \"Sensors and actuators based on surface acoustic
    waves propagating along solid-liquid interfaces,\" Journal of
    Physics D-Applied Physics, vol. 41, 2008.

[^7]: J. C. Rife, et al., \"Miniature valveless ultrasonic pumps and
    mixers,\" Sensors and Actuators a-Physical, vol. 86, pp. 135-140,
    Oct 2000.

[^8]: 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.

[^9]: 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.

[^10]: 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.
