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A numerically efficient damping model for acoustic resonances in
microfluidic cavities
P. Hahn and J. Dual
Citation: Physics of Fluids 27, 062005 (2015); doi: 10.1063/1.4922986
View online: http://dx.doi.org/10.1063/1.4922986
View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/27/6?ver=pdfcov
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PHYSICS OF FLUIDS 27, 062005 (2015)
A numerically efficient damping model for acoustic
resonances in microfluidic cavities
P. Hahna) and J. Dual
Institute of Mechanical Systems (IMES), Department of Mechanical and Process
Engineering, ETH Zurich, Tannenstrasse 3, CH-8092 Zurich, Switzerland
(Received 15 January 2015; accepted 13 June 2015; published online 30 June 2015)
Bulk acoustic wave devices are typically operated in a resonant state to achieve
enhanced acoustic amplitudes and high acoustofluidic forces for the manipulation
of microparticles. Among other loss mechanisms related to the structural parts of
acoustofluidic devices, damping in the fluidic cavity is a crucial factor that limits
the attainable acoustic amplitudes. In the analytical part of this study, we quantify
all relevant loss mechanisms related to the fluid inside acoustofluidic micro-devices.
Subsequently, a numerical analysis of the time-harmonic visco-acoustic and thermo-
visco-acoustic equations is carried out to verify the analytical results for 2D and
3D examples. The damping results are fitted into the framework of classical linear
acoustics to set up a numerically efficient device model. For this purpose, all damping
effects are combined into an acoustofluidic loss factor. Since some components of the
acoustofluidic loss factor depend on the acoustic mode shape in the fluid cavity, we
propose a two-step simulation procedure. In the first step, the loss factors are deduced
from the simulated mode shape. Subsequently, a second simulation is invoked,
taking all losses into account. Owing to its computational efficiency, the presented
numerical device model is of great relevance for the simulation of acoustofluidic
particle manipulation by means of acoustic radiation forces or acoustic streaming.
For the first time, accurate 3D simulations of realistic micro-devices for the quanti-
tative prediction of pressure amplitudes and the related acoustofluidic forces become
feasible. C2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4922986]
I. INTRODUCTION
Ultrasonic standing waves within acoustofluidic devices can be used to exert forces on sus-
pended microparticles. These acoustofluidic forces can be traced back to the acoustic radiation
forces and drag forces due to acoustic streaming. Bulk acoustic devices are typically operated in the
low MHz regime and have a characteristic size ranging from millimeters to a few centimeters. They
provide a means to handle cells, bacteria, or other particles in a contactless fashion which is attrac-
tive for a wide range of lab-on-a-chip applications. A comprehensive review of theoretical work,
experimental setups as well as recent developments can be found in the acoustofluidics tutorial
series.1
With the growing performance of computing hardware and readily available simulation soft-
ware, numerical acoustofluidic device models are increasingly used to design new experimental
particle manipulation setups. Nevertheless, it is still prohibitively expensive to simulate acoustoflu-
idic micro-devices in their full complexity since the relevant physics exhibits a multiscale character.
First, there can be several orders of magnitude scale-difference between the time-harmonic fields
and the time-averaged effects of interest, leading to problems related to the limited numerical
accuracy. Second, the time scale-difference between the ultrasonic cycle and the transient or asymp-
totic behavior makes a solution in the time-domain very expensive. Third, the geometric scale-
difference between the size of the device and thin viscous or thermal boundary layers requires a fine
a)Electronic mail: hahnp@ethz.ch
1070-6631/2015/27(6)/062005/27/$30.00 27, 062005-1 ©2015 AIP Publishing LLC
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062005-2 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
discretization, resulting in extremely fine meshes and an enormous numerical effort. However, due
to the amplitude scale-difference between the time-harmonic and the time-averaged fields, the prob-
lem can be studied in good approximation by a perturbation expansion for all field variables in
the governing equations as detailed by Muller et al.2and Bruus3,4with and without consideration
of the thermal field, respectively. This perturbation approach resolves the multiscale issues related
to the time discretization and the magnitude differences between the first-order and second-order
fields. Nonetheless, the geometric scale-differences between the extremely thin viscous or thermal
boundary layers and the size of the device remain. Fortunately, the problem can be further simplified
if the validity of the model is limited to the bulk of the fluid and the structure, whereas deviations
from the exact solution need to be tolerated in the boundary layers. In order to do so, the thermal
field and the viscous effects in the fluid need to be analyzed.
The thermal field, created by thermoacoustic and thermoelastic coupling, has a boundary layer
structure due to the thermal coupling at interfaces, whereas the viscous boundary layer arises from
the no-slip boundary condition at the fluid cavity walls.5,6In the bulk, the first-order time-harmonic
motion is affected by the thermal field and the fluid viscosity only through their dissipative ef-
fect which limits the amplitudes of resonances. Many experimental setups work with low particle
concentrations where boundary-driven acoustic streaming inside the fluid cavity is dominated by
the boundary layers at the cavity walls rather than by the boundary layers surrounding the particles.
Under this assumption, the particles do not have to be considered in the calculation of the first-order
time-harmonic fluid motion which represents the first step in the simulation of radiation forces and
acoustic streaming. The radiation forces on small linear elastic spherical particles in inviscid7as
well as slightly viscous8fluids can be calculated analytically, whereas geometrically or mechani-
cally more complex particles need to be treated numerically.9,10 Particles that are in close proximity
to each other experience inter-particle or Bjerknes forces which again can be treated analytically for
spherical beads.11,12 Further, it has been shown that the boundary-driven acoustic streaming field in
the bulk of the fluid can be determined semi-analytically with good accuracy as long as the curva-
ture radii of the cavity walls are small in comparison to the viscous boundary layer thickness.13,14 In
essence, this means that the thermal field and the viscosity in the first-order fluiddynamic equations
can be dropped entirely in the device model as long as the dissipation, associated with both quan-
tities, is retained. Such a model can be used to accurately predict the acoustic radiation forces on
particles as well as the drag forces due to acoustic streaming anywhere in the bulk of the fluid cavity.
In this article, we analyze the damping of the first-order time-harmonic motion with a focus on the
losses in the fluid. For the first time, accurate 3D device simulations become numerically feasible
since the boundary layers do not have to be resolved in the device model. Our work represents the
missing link that allows to make not only qualitative but also quantitative predictions for both the
radiation forces and the acoustic streaming in realistic experimental devices.
In Sec. II, we give an introduction to the modeling of damped device resonances and we
demonstrate why the damping effects associated with the fluid are often the most important ones.
Subsequently in Sec. III, the different damping mechanisms are analyzed and quantified. After a
numerical comparison between our damping model and the full governing equations in Sec. IV,
the numerical device model is set up in Sec. V. It leverages the semi-analytical damping terms of
Sec. III to achieve high numerical efficiency.
II. DAMPED DEVICE RESONANCES
Device resonances are a first-order effect of constructive interferences between the ultrasonic
excitation and the repeated wave reflection at interfaces and boundaries. At the resonance frequen-
cies, the transducer provides high mechanical power which accumulates in the device, leading to
high time-harmonic amplitudes.15 Associated with the time-harmonic fields, there are a number of
damping mechanisms that, during each oscillation cycle, extract a certain fraction of the stored
kinetic and potential energies. These losses determine how many oscillation cycles it takes to
reach a stationary energy level; they limit the attainable amplitudes and they define the frequency
bandwidth of the resonance.9
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062005-3 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
A. Basic device model
Disregarding all damping mechanisms and boundary layer effects for now, the first-order
time-harmonic device motion can be described by a short set of equations. At this level of simplifi-
cation, the model does not contain the necessary damping information anymore. However, it can be
reintroduced as outlined in Sec. III. The motion of a compressible Newtonian fluid is governed by
the Navier-Stokes equations,16
ρf(˙vi+vjvi,j)=−p,i+(4/3η+ηB)vj,ji +η(vi,j j −vj,ji ),(1)
where body forces are omitted. The indicial notation refers all quantities to an orthonormal basis
with coordinates xi,i=1,2,3, whereas the derivative with respect to xiis denoted as, i, the dot
indicates a time derivative, and the Einstein summation convention for repeated indices is invoked.
The pressure and velocity fields are pand vi, respectively, the fluid density is denoted as ρf, the
dynamic viscosity is η, and the bulk viscosity is ηB. A perturbation of Eq. (1) is used to obtain
greatly simplified equations for the time-harmonic first-order fluid motion. It results in the lossy
Helmholtz equation,3
p,ii =−k2p,(2)
for the pressure field, whereas the complex wavenumber k=2π
λ=ω
cis a function of either the
complex acoustic wavelength λor the angular frequency ωand the complex speed of sound c.
Disregarding all boundary effects, the velocity field is then calculated according to
vi=i1
ρ0ωp,i,(3)
with the quiescent fluid density ρ0.
The time-harmonic solid dynamics can be defined by three equations.17 First, the dynamic
equilibrium,
σi j,j+ρω2ui=0,(4)
with the displacement field ui, the stress field σi j, and the material density ρ. Second, the kinematic
relations,
γi j =1
2ui,j+uj,i,(5)
with the strain field γi j and third, the constitutive law of the material which translates the strain field
into a stress field. Linearly, elastic materials show the functional relation σλ=σλγµwith greek
indices interpreted as double indices.17 Piezoelectric materials show an additional dependence on
the electric field Ekaccording to σλ=σλγµ,Ek.18,19 The solid and the fluid domains are coupled
through fluid-structure interaction at the fluid cavity walls.17,19
Mathematically, the damping can be modeled conveniently by an approach that implies a
complex representation of time-harmonic fields,
A(xi,t)=Re A(xi)eiωt,(6)
with time tand the complex-valued magnitude A(xi)as a function of the position vector xi. It is
understood that the differential equations above describe the magnitude of field variables corre-
sponding to A(xi), whereas the explicit position dependence will be omitted from here on. Dissipa-
tion can then be modeled by a complex speed of sound for fluids and a complex constitutive law for
solids.16,20 The corresponding complex speed of sound and wavenumber in the fluid,
c=c01+iϕ
2and k=k01−iϕ
2,(7)
are expressed in terms of the dimensionless loss factor ϕwhich traditionally only captures the
dissipation due to ultrasonic wave attenuation in the bulk of the fluid.3,16 The real-valued speed of
sound is denoted by c0and a real-valued wavenumber by k0=ω
c0. As demonstrated in Sec. III, it is
possible to use the same formulation to account for additional damping effects in the fluid cavity.
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062005-4 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
For this purpose, the bulk attenuation loss factor ϕhas to be exchanged by the acoustofluidic loss
factor ¯ϕ(f)which is derived in Sec. III to account for further dissipation effects such as viscous
and thermal boundary layer losses at cavity walls and suspended particles. To express its strong
dependence on the resonance mode, the acoustofluidic loss factor is written in function of the
frequency f. The theory is appropriate only for small damping (ϕ≪1, ¯ϕ(f)≪1), which is why
linearizations are always applied implicitly in the following. The general constitutive law for a lossy
linear elastic material can be written as
σi j =ci j k lγk l,(8)
where the fourth-order stiffness tensor ci j kl is split up according to
ci jk l =c′
i jk l +ic′′
i jk l (9)
into the storage and the loss tensors c′
i jk l and c′′
i jk l , respectively.21 The number of storage and loss
parameters can be reduced considerably due to the intrinsic symmetry properties of the stiffness
tensor, whereas for isotropic materials, it can be reduced even further.19,22
B. Quantification of losses in a device
Typical acoustofluidic micro-devices consist of parts made of piezoelectric ceramics, silicon,
glass, metal, and often thin glue layers. Depending on the solid material’s constitutive behavior,
viscoelasticity and thermoelasticity are the most relevant sources of dissipation in the bulk ultra-
sonic field.16,20,23 In piezoelectric elements, there exists additional dielectric and piezoelectric dissi-
pation, leading to a complicated dependence on the driving conditions and the vibration mode.24,25
It is worth mentioning that material behavior, including the damping terms, can change consider-
ably with temperature and frequency. Additional to the dissipation in the volume, energy is also
lost over the surface of the device.15 Air damping refers to the energy loss into the surrounding air
and consists of viscous losses and radiation losses. Typically, air damping can be neglected as long
as squeezed film effects are avoided and the device dimensions are not extremely small. Anchor
losses comprise the energy flow into the device support or, e.g., connection tubes. If there are no
tubes and the device is placed on a low-impedance base made of foam or tissue paper, the anchor
losses become small compared to the other losses in the device. Often, however, they need to be
incorporated into a device simulation, which can become numerically expensive.
Both viscoelastic and thermoelastic dissipations in the bulk of the material induce a phase-shift
between the stress and the strain, which is caused by the relaxation times of the material.22,26 This
effect is accurately reproduced by the complex approach above. In order to calculate the rate at
which energy is dissipated inside the material, the definition of the differential strain energy density,
dE=σi jdγi j =σi j ˙γij dt,(10)
is used. Integration over the material volume Vand time-averaging as denoted by ⟨. . .⟩(Ref. 4)
gives the power dissipation Ψsinside the solid material,
Ψs=V
⟨σi j ˙γij ⟩dV=1
2V
Re[σi j ˙γ∗
i j]dV,(11)
where an asterisk denotes complex conjugation. Expressing the stress tensor through the constitu-
tive relation (Eqs. (8) and (9)), it becomes apparent that the energy dissipation rate depends on the
strain tensor distribution corresponding to the mode shape. Based on the weighting of storage and
loss terms for the specific vibration mode, a scalar loss-factor can be defined.22 It is often referred to
as the loss tangent tan ϵof the resonance mode which, for weak damping, is approximately equal to
the loss angle ϵ. The dissipated energy per cycle can then be calculated as
Ws
loss =2πϵ Ws
st,(12)
where the stored energy Ws
st is equal to twice the time-averaged strain energy in the solid. Based on
the complex model, it is also possible to calculate dissipation in the bulk of the fluid Ψfaccording to
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062005-5 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
Ψf=1
2V
Re[−pv∗
i,i]dV,(13)
which is equivalent to Eq. (11) with the acoustic diagonal stress tensor σij =−pδi j, where δi j is
the Kronecker delta. Here, the integration is performed over the whole fluid space V. Differentiating
Eq. (3) and replacing the Laplace of pwith Eq. (2), the dissipation rate is written in terms of
the acoustic pressure only. The definition of the wavenumber (Eq. (7)) and the magnitude of the
pressure field |p|=√pp∗lead to an expression that involves the bulk attenuation loss factor,
Ψf=1
2ρ0ωV
Re[ik2pp∗]dV=ωϕ
2ρ0c2
0V
|p|2dV.(14)
Correspondingly, the bulk dissipation per cycle reads
Wf
loss =πϕ
ρ0c2
0V
|p|2dV.(15)
With the stored energy in the fluid Wf
st being calculated by integration of twice the time-averaged
potential energy density,16
Wf
st =1
2ρ0c2
0V
|p|2dV,(16)
the bulk dissipation per cycle can be expressed as
Wf
loss =2πϕWf
st.(17)
It is important to define the stored energy Wf
st based on the potential energy since potential and
kinetic energies are not always the same. Even though the kinetic and the potential energy contents
of the system as a whole is equal for any time-harmonic oscillator at resonance, this does not
necessarily hold for individual parts of the oscillator. Also, driving the resonator off-resonance leads
to differences. It is apparent that Eq. (17) is of the same form as Eq. (12) for solids and it can be
shown that surface losses are also related to the energy density on the device surface.15 In summary,
the ratio between the total stored and the total lost energy in a device depends not only on the
individual loss terms but also on the energy distribution within the device. This explains why each
resonance mode can have a different quality factor (Q-factor), as defined by
Q=2πstored energy in the system
losses per cycle .(18)
Further, it also explains why dissipation in the fluid is of essential importance for the acoustic
amplitudes at the fluid resonances which are leveraged for the manipulation of particles. Here,
the energy is focused inside the resonating fluid cavity, which is a result of the relatively large
acoustic impedance gap between the fluid and the other device materials. As an example, the energy
distribution inside a typical acoustofluidic microdevice for the positioning of hollow glass particles
in the middle of the 1.2 mm ×1.2 mm ×0.2 mm fluid cavity is shown in Fig. 1.19 In the Finite
Element (FE) simulation model, there are no surface losses since it is assumed that the device is
placed on foam and has no tubing attached. Dissipation inside the silicon can be neglected due to
its minute loss factor,27 whereas the losses in the 20 µm thick epoxy glue layer between the silicon
and the piezoelectric element need to be considered. The energy fraction and total loss in each part
of the device are provided in Table I. The full Comsol FE model for the calculation of the energy
distribution is contained in the supplementary material.28 In descending order, the most relevant
losses appear in the water, in the glue layer, and in the transducer. Summation over the whole device
gives the total stored energy and the total loss per cycle, corresponding to a Q-factor of Q=166 for
this resonance mode (see Eq. (18)). A quick feasibility check is to compare the total power loss with
the electric power consumption of the transducer, which matches for the presented model. Further,
the aspect ratio of the resonance peak can be used to estimate the Q-factor according to Q=f0
∆f,
leading to a value of Q=163 for the given example (compare Fig. 2).29
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062005-6 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
FIG. 1. Energy distribution in an acoustofluidic micro-device excitation at 0.847 MHz and 5 Vpp. The color code shows the
time-averaged acoustic potential energy density on the surface of the fluid domain as well as the time-averaged strain energy
density on three cut planes through the device structure. The energy is densely focused in the resonating fluid cavity.
III. DAMPING MECHANISMS IN THE FLUID CAVITY
Losses due to the different damping mechanisms in the fluid cavity add up, so the total acoustoflu-
idic loss factor can be written as
¯ϕ(f)=1
ωWf
st(f)
n
Ψf
n(f)=
n
¯ϕn(f),(19)
with the individual losses defined by Ψf
n(f)or ϕn(f)as derived in the following paragraphs. The
superscript f for fluid and the explicit frequency dependence . . . (f)are omitted from here on. It
is understood that the acoustofluidic loss factor ¯ϕhas to be interpreted as a mode parameter rather
than a fluid property due to its dependence on the device geometry and the frequency. Losses can
be categorized into bulk-related, surface-related, viscous, or thermal effects. Further, there can be
radiation losses into adjacent fluid spaces which are often difficult to incorporate into a numerical
simulation. However, sometimes this case can be handled similar to the structural anchor losses.
Finally, the time-harmonic resonances are also subject to losses due to the nonlinear terms in the
Navier-Stokes equations or a nonlinear fluid response. All effects are analyzed in the following
paragraphs and evaluated for their relevance in the context of acoustofluidic micro-devices.
Measured values for the attenuation of ultrasound in water and other liquids are available in the
literature.30–32 Often, the spatial attenuation coefficient is given as αsin Np m−1or as asin dB m−1
which can be converted according to16
as≈8.7αsand αs=k0
2ϕ. (20)
It is important to note that these attenuation measurements only include the dissipation in the bulk
which, for micro-devices, is often much smaller than the boundary-related damping (see Sec. Vand
Fig. 18). In most cases, acoustofluidic devices are filled with water which we assume from here on.
The water properties at 25 ◦C are summarized in Table II. Regarding the temperature dependence
TABLE I. Stored energy and losses in an acoustofluidic micro-device. Values are provided for all individual device
components as well as for the summation over the whole device. The power dissipation in the piezoelectric transducer
contains mechanical, dielectric, and piezoelectric losses.
Parameter Symbol Transducer Silicon Glass Glue Water Device Unit
Stored energy Wst 14.0 20.4 11.1 3.7 102.8 152.0 nJ
Loss coefficient ¯ϕ(f)or ϵ0.01 0 0.0004 0.1 0.004 —
Power loss Ψ0.73 0 0.02 1.94 2.32 5.01 mW
Loss per cycle Wloss 0.86 0 0.03 2.30 2.58 5.77 nJ
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062005-7 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
FIG. 2. Resonance peak for the calculation of the Q-factor. The pressure magnitude at one corner of the fluid chamber
is plotted over frequency. The center frequency is f0=847 kHz and the −3 dB bandwidth is ∆f=5.2 kHz, leading to
Q=f0
∆f=163.
of the water properties, we refer to Muller et al.33 In each numerical simulation, the water prop-
erties should be adapted to the device temperature in order to obtain accurate results. It is worth
mentioning that the speed of sound is also slightly affected by the damping16,36 and the presence of
suspended particles.37 However, these two effects are very small for moderate damping and particle
concentrations, meaning that they are not of great practical relevance for numerical simulations
where other uncertainties dominate.
A. Viscous damping in the bulk
By the first-order perturbation of the Navier-Stokes equations (Eq. (1)), the viscous loss factor
in the bulk of the fluid can be derived as16
¯ϕ1=ω
ρ0c2
04
3η+ηB.(21)
The loss can be traced back to the shear relaxation time corresponding to the dynamic viscosity
ηand to the structural relaxation time of the fluid substance, associated with the bulk viscosity
ηB. The bulk viscosity can be neglected for a few fluids including monatomic gases for which
the Stokes hypothesis (ηB=0) holds. However, measurements demonstrate that the bulk viscosity
plays an important role for ultrasonic wave propagation in most liquids since the damping asso-
ciated with the bulk viscosity is often larger than that associated with the dynamic viscosity.38
For water at 25 ◦C, it has been shown experimentally by Holmes et al.32 that the bulk viscosity is
ηrt
B=2.485 mPa s and stays fairly constant in a range from 10 MHz to 100 MHz as expected for
a Newtonian fluid. These measurements agree with the ones of Dukhin and Goetz31 and are in line
with an earlier study of Liebermann30 at a frequency of 5 MHz where a value of ηrt
B=2.8 mPa s
has been found. Furthermore, Holmes et al.32 extend the work of Dukhin and Goetz31 regarding the
TABLE II. Water properties at a room temperature of 25 ◦C. The parameters are adopted from Muller et al.33 who deduced
their data from the IAPWS Formulation 2008 and 2011,34,35 whereas the bulk viscosity is taken from Holmes et al.32
Parameter Symbol Value Unit
Density ρrt
0997 kg/m3
Speed of sound crt
01497 m s−1
Dynamic viscosity ηrt 0.890 mPa s
Bulk viscosity ηrt
B2.485 mPa s
Dynamic viscosity temperature coefficient ηT−0.020 mPa s K−1
Bulk viscosity temperature coefficient ηT
B−0.064 mPa s K−1
Thermal conductivity κrt
th 0.607 W m−1K−1
Specific heat capacity crt
p4181 J kg−1K−1
Thermal expansion coefficient α2.573 ×10−4K−1
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062005-8 P. Hahn and J. Dual Phys. Fluids 27, 062005 (2015)
temperature dependence of the attenuation in the range of 7 ◦C–50 ◦C, whereas the ambient pressure
is kept constant. These measurements are used to calculate the temperature dependent bulk viscos-
ity. A linearization of the dynamic and the bulk viscosities around a temperature of T0=298.15 K or
25 ◦C according to
η=ηrt +ηT(T−298.15 K)and (22)
ηB=ηrt
B+ηT
B(T−298.15 K)(23)
is handy to use in numerical simulations and sufficiently accurate to model experimental setups
that are operated around room temperature (20 ◦C–30 ◦C). Herein, the temperature Tis entered in
K and the temperature coefficients ηTand ηT
Bfor the dynamic and the bulk viscosities, respectively,
are determined at constant ambient pressure. Values are provided in Table II. Since the viscosity
does not show significant frequency dependence for Newtonian fluids, it can be assumed that ηT
and ηT
Bare independent of frequency as well. In summary, Eqs. (22) and (23) can be combined
with Eq. (21) to approximate the viscous loss factor in the bulk of the fluid cavity around room
temperature for an ultrasonic frequency up to at least 100 MHz.
B. Thermal damping in the bulk
Any acoustic field also induces a fluctuating thermal field due to thermoacoustic coupling.2,33
Thermal conduction in the bulk of the fluid leads to a second loss factor,16
¯ϕ2=ω
ρ0c2
0(γ−1)κth
cp,(24)
which causes the so-called thermoelastic damping or TED. The expression depends on the ratio of
specific heats γ, the heat conductivity κth, and the specific heat at constant pressure cp. The ratio of
specific heats is deduced from the parameters given in Table II according to39
γ=T0α2c2
0
cp
+1,(25)
whereas the unperturbed fluid temperature is T0=298.15 K or 25◦C. The ratio between the thermal
and the viscous loss factors in the bulk takes the form
¯ϕ2
¯ϕ1
=(γ−1)κth
cp4
3η+ηB,(26)
which is independent of the frequency and equal to ¯ϕ2
¯ϕ1=4.2×10−4for water at 25◦C. This means
that thermal losses in the bulk of the fluid cavity can be neglected. There exists a second thermal
damping mechanism due to molecular thermal relaxation which, however, is even smaller for polar
liquids such as water and at typical experimental frequencies.16
C. Viscous damping at the cavity walls
The viscous boundary layer at the cavity walls (also called Stokes layer or acoustic boundary
layer) is a major source of dissipation in the fluid cavity. Due to the no-slip boundary condition at
the fluid structure interface, there exists a transition layer in which the tangential component of the
fluid velocity adapts to the velocity of the solid boundary. During each cycle, the viscous shear in
this layer dissipates an acoustic energy fraction which strongly depends on both the resonance mode
and the shape of the cavity. The layer thickness is characterized by the viscous penetration depth
(also called the characteristic viscous boundary layer thickness),
δ=2η
ρ0ω,(27)
which gives a value of δ=0.53 µm for water at 25◦C at a frequency of 1 MHz. This means that
the viscous boundary layer has a planar structure if the curvature radius of the cavity walls is much
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FIG. 3. Schematic illustration of the viscous boundary layer with the tangential fluid velocity component ξin the vicinity
of the wall. At x=0, the fluid follows the tangential wall motion ξb(y)and for δ≪x≪λ, it approaches ξa(y)which is the
tangential part of the inviscid acoustic velocity at the boundary.
larger than δ. For this special case, the problem can be reduced to two dimensions without loss of
generality and it is assumed that gradients in boundary-normal direction are much larger than those
in boundary-tangential direction, which is justified by δ≪λ. The transition of the tangential fluid
velocity from ξa(y)just outside of the boundary layer to the tangential velocity of the boundary
ξb(y)is illustrated in Fig. 3. The dissipation can be calculated most conveniently in the moving
coordinate system (x′, y′)which follows the fluid motion just outside of the boundary layer. From
this perspective, the fluid just outside the boundary layer is quiescent and the tangential wall vibra-
tion velocity assumes the tangential fluid-wall relative velocity. It is given by the velocity difference
ξdiff=ξa(y′)−ξb(y′)=ξa(y)−ξb(y)which is identical in both coordinate systems since the acous-
tic amplitudes are small compared to the wavelength. The dissipation in the viscous boundary layer
is now solely driven by the tangential wall vibration and it can be calculated as the product of the
viscous force on the vibrating wall and ξdiff. Therefore, the dissipation per unit surface is equal to
the product of the stress on the interface and ξdiff. In (x, y)coordinates, the first-order tangential
fluid velocity component in the boundary layer reads16
ξ=ξdiffe−(1+i)x
δ,(28)
the stress on the boundary is
τ=−η∂ξ
∂xx=0
=ηξdiff
1+i
δ,(29)
and the power dissipation becomes
Ψ3=S
⟨τ(xi,t)ξdiff(xi,t)⟩dS
=η
2δS
Re[(1+i)ξdiffξ∗
diff]dS
=ρ0ωδ
4S
|ξdiff|2dS,
(30)
where time-averaging is applied and the integration is performed over the whole fluid-structure
interface S. In a device simulation with acoustic fluid-structure interaction, there is always a guaran-
tee that the solid and the fluid velocities are equal in interface-normal direction. For this reason, it
is straight forward to determine the 3D generalization of the tangential velocity difference based on
the fluid and solid velocity fields,
ξdiffi=vi−˙uiand |ξdiffi|2=(vi−˙ui)(vi−˙ui)∗.(31)
Finally, the loss factor due to dissipation in the viscous boundary layer becomes
¯ϕ3=ρ0δ
4Wst S
|ξdiffi|2dS,(32)
whereas the integral over the fluid-structure interface is easily determined in a numerical simulation.
For the interaction with a non-moving wall (i.e., ˙ui=0), the expression coincides with the one that
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is usually given in the literature.5It is understood that Eq. (30) is not valid at sharp corners or edges
but the total error is negligible if the simulation does not include too many of those locations.
A very rough but simple estimation of the loss factor due to the viscous boundary layer at walls
can be derived if the tangential velocity difference ξdiffat the wall is dominated by the fluid velocity
vi. This assumption (ξdiffi≈vi) is usually justified if the cavity walls are acoustically stiff(high
acoustic impedance and large wall thickness). Further, the stored energy in the fluid is approximated
from the kinetic energy (Wst ≈1
2ρ0V|vi|2dV), Eq. (32), and it is assumed that the acoustic velocity
magnitudes at the cavity walls and in the bulk of the cavity are similar. This leads to
¯ϕ3≈
δS|vi|2dS
2V|vi|2dV≈δSδ
2V≤δS
2V,(33)
where Sδis the fraction of the cavity surface Son which the formation of a viscous boundary
layer is expected from the observation of a given acoustic mode shape. An upper limit of the
approximated loss factor is found by choosing Sδ=S. If the device contains acoustically passive
fluid spaces (i.e., fluid in connection tubes or unactuated fluid reservoirs), only the fluid volume V
that contains a strong acoustic field has to be considered.
D. Thermal damping at the cavity walls
As mentioned above, thermoacoustic coupling leads to a fluctuating temperature field in the
fluid. In the bulk, this field is proportional to the acoustic pressure but at the cavity walls, thermal
boundary interaction creates a thermal boundary layer. The thickness of this boundary layer is
characterized by the thermal penetration depth,
δth =2Dth
ω,(34)
with the thermal diffusivity Dth =κth
ρ0cp, giving a value of δth =0.22 µm for water at 25 ◦C and a
frequency of 1 MHz. Regarding the loss in the thermal boundary layer, we refer to Swift40 who
derives the power dissipation at a planar interface,
Ψ4=ωδth
4ρ0c2
0
γ−1
1+ϵsS
|p|2dS,(35)
with the interface heat capacity ratio,
ϵs=ρ0cpδth tanh [(1+i)d/δth]
ρscs
pδs
th tanh (1+i)ds/δs
th,(36)
where the superscript “s” refers to the solid boundary. This ratio is derived for a solid plate of
thickness 2dsthat is surrounded by a layer of water (layer thickness d). The expression can be
simplified if we assume that both the solid structure and the fluid cavity are much thicker than the
respective thermal penetration depths (for values, see Table III). With d≫δth and ds≫δs
th, the heat
capacity ratio simplifies to
ϵs≈ρ0cpδth
ρscs
pδs
th
.(37)
Values for common wall materials are given in Table III. In summary, the loss factor due to the
thermal boundary layer reads
¯ϕ4=δth
4ρ0c2
0Wst
γ−1
1+ϵsS
|p|2dS,(38)
whereas in a numerical device simulation, the integral is conveniently determined. The ratio be-
tween the thermal and the viscous boundary layer losses cannot be determined explicitly because
the integrals in Eqs. (38) and (32) strongly depend on the mode shape and the geometry of the fluid
cavity. Nonetheless, if the tangential wall motion is neglected (assuming ˙ui=0), it is possible to
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TABLE III. Thermal parameters of common acoustofluidic device materials. Since they are technical products with varying
properties, the given values have to be interpreted as approximative values at room temperature. The heat capacity ratio for
the calculation of the thermal boundary layer losses are determined for an interface between water at 25 ◦C and the respective
wall material based on Eq. (37).
Parameter Symbol Silicon Glass Aluminum Steel Unit
Density ρ2336 2240 2700 7850 kg/m3
Thermal conductivity κth 150 1.1 237 43 W m−1K−1
Specific heat capacity cp703 750 897 490 J kg−1K−1
Thermal penetration depth at
1 MHz
δth 5.4 0.5 5.6 1.9 µm
Interface heat capacity ratio ϵs0.10 1.0 0.06 0.12 —
derive the general trend by assessing the maximum power loss per unit area. From Eqs. (35) and
(30), we get the maximum power loss per unit area corresponding to the thermal and the viscous
boundary layers,
ψ4=ωδth
4ρ0c2
0
γ−1
1+ϵs
|p|2(39)
and
ψ3=ρ0ωδ
4
|vi|2,(40)
respectively. The hatindicates the global maximum value of a field. Using the relation defined in
Eq. (3), the maximum magnitude of the fluid velocity field becomes
|vi|=1
ρ0ω
|p,i|and the maximum
magnitude of the pressure gradient field
|p,i|=k0
|p|leads to
ψ3=ωδ
4ρ0c2
0
|p|2.(41)
Thus, the average ratio between thermal and the viscous boundary layer loss factors is approxi-
mately,
¯ϕ4
¯ϕ3≈
ψ4
ψ3
=δth
δ
γ−1
1+ϵs
=γ−1
√Pr (1+ϵs),(42)
with the Prandtl number Pr =ηcp
κth , Pr =6.1 for water at 25 ◦C. This means, by tendency, the
thermal boundary layer losses are smaller than the viscous boundary layer losses by a factor of
approximately 250 (assuming silicon walls with ϵs=0.1), but there is a strong dependence on the
resonance mode and the fluid cavity. As an extreme example, a radial mode in a spherical cavity has
a vanishing viscous boundary layer loss but the thermal boundary layer loss might be large.
E. Viscous damping due to suspended particles
In the vicinity of suspended microparticles, the acoustic field shows a boundary layer structure
if there is a density difference between the particles and the fluid. Viscous shear in the boundary
layer around these suspended particles is another source of damping in microfluidic cavities. It is
understood that liquid droplets or gas bubbles can also cause viscous shear dissipation. The acoustic
interface conditions are significantly different from the ones for solid particles and, at least for
liquid droplets, the resulting viscous boundary layer expands into the suspended droplet. The related
damping would have to be derived from the existing work regarding the first-order velocity field for
these cases.41–43 In the present work, we focus on the damping related to solid particles. Classical
analytical derivations of the acoustic field and the resulting streaming field include the case of a
vibrating spherical particle44 as well as a spherical liquid drop at the velocity antinode of a 1D
field.45 These results have been derived under the assumption of a viscous boundary layer that is
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much smaller than the particle size which in turn is much smaller than the acoustic wavelength
(δ≪a≪λ). A recent work presented by Settnes and Bruus discusses the important case of a
small suspended particle at arbitrary locations in an arbitrary acoustic field.8An analytical result
of the velocity field is presented for the assumption of a slightly viscous fluid (δ, a≪λ). Based on
this velocity field, we derive the dissipation due to non-interacting suspended particles in the fluid
cavity. Spherical particles are best studied in polar coordinates due to the azimuthal symmetry (see
Fig. 4). For a<r≪λ, the velocity field components in the in polar coordinates read
vr=vier
i=
1−f2
a3
r3+2qa B h1
1(s)
ss=qr
cos θvin,(43a)
vθ=vieθ
i=
1+f2a3
2r3+qaB (sh1
1(s)),s
ss=qr
(−sin θ)vin,(43b)
where
q=1+i
δ,(43c)
B=−3f2
2aqh1
0(qa)−6h1
1(qa),(43d)
f2=21−χ(˜
δ)(˜ρ−1)
2 ˜ρ+1−3χ(˜
δ),with ˜ρ=ρp
ρ0
,and (43e)
χ(˜
δ)=−3
21+i(1+˜
δ)˜
δ, with ˜
δ=δ
a.(43f)
Herein, er
iand eθ
iare the unit vectors in radial and angular directions, ais the particle radius, ρpis
the particle density, vin =|vi|is the undisturbed fluid velocity magnitude at the particle location, and
h1
0and h1
1(s)are spherical Hankel functions of the first kind with order 0 and 1, respectively. Without
proof, we postulate that the viscous dissipation at suspended particles is primarily generated by the
tangential velocity gradients in the boundary layer. Similar as in Sec. III C, the power dissipation
per unit area is calculated in a moving coordinate system as the product of the tangential stress on
the particle surface and the tangential difference between the particle velocity and the velocity just
outside of the boundary layer. Given the velocity field and the dynamic viscosity of the fluid, the
tangential stress component τat the particle surface takes the form
τ=σθr|r=a=η[(1/r)vr,θ +vθ, r−(1/r)vθ]|r=a
=−6ia((1+i)a+iδ)(−1+˜ρ)
δ(9(1+i)aδ+9iδ2+a2(2+4 ˜ρ))ηvin sin θ.
(44)
FIG. 4. Illustration of the time-harmonic velocity field with the viscous boundary layer around a suspended particle. In the
shown example, the particle is located at the velocity antinode of a 1D ultrasonic standing wave but the analysis applies for
general acoustic fields and particle locations. It is performed in polar coordinates (r, θ) for a spherical particle (wavelength
λmuch smaller than the particle radius a) which is free to translate and to perform a breathing motion due to the first-order
acoustic forces.
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Since the particle is free to perform a translatory motion in the acoustic field, the tangential velocity
of the particle surface
ξsurf =vθ|r=a=−32a2+(3+3i)aδ+3iδ2
9(1+i)aδ+9iδ2+a2(2+4 ˜ρ)vin sin θ(45)
needs to be taken into account similar to the moving boundary in Sec. III C. The tangential velocity
ξout outside the boundary layer is not affected by the fluid viscosity. Therefore, it can either be ob-
tained by the inviscid linear acoustic theory or equivalently by neglecting the influence of viscosity
on the velocity field (Eqs. (43a)–(43f)) in the limit δ→0+. Choosing r=a+√δensures that the
velocity is evaluated outside of the boundary layer because √δconverges slower than δ. Still, the
maximal velocity is obtained which, for the inviscid limit, occurs on the surface at r=a. With
ξout =lim
δ→0vθ|r=a+√δ=−3 ˜ρ
1+2 ˜ρvin sin θ, (46)
the tangential velocity difference in the boundary layer becomes
ξdiff=ξout −ξsurf =−3(−1+˜ρ)3(1+i)aδ+3iδ2+a2(2+4 ˜ρ)
(1+2 ˜ρ)(9(1+i)aδ+9iδ2+a2(2+4 ˜ρ))·vin sin θ . (47)
Integration of the time-averaged area-specific power dissipation over the particle surface gives the
power dissipation corresponding to one suspended particle,
Ψp
5=S
⟨τ(θ, t)ξdiff(θ,t)⟩dS=π
θ=0
1
2Re [τξdiff]2πa2sin θdθ=A|vin|2,with
A=48a5(a+δ)ηπ(−1+˜ρ)2
δ(162a2δ2+162aδ3+81δ4+4a4(1+2 ˜ρ)2+36a3(δ+2δ˜ρ)) .
(48)
Even though the expression is lengthy, it is a simple product of |vin |2and the coefficient Awhich
only depends on four parameters (a,δ,η, and ˜ρ). The corresponding loss factor for a single particle
reads
¯ϕp
5=A
ωWst
|vi|2.(49)
Given that vin is the undisturbed acoustic velocity at the location of the particle, a 3D generalization
is obtained by setting |vin|2=|vi|2=viv∗
i, with the acoustic velocity vitaken from a device simula-
tion. Clearly, backscattering at boundaries or the scattering field of other particles is neglected here.
If particles of uniform size and density are distributed over the fluid cavity, the loss factor due to the
viscous boundary layer surrounding these particles can then be defined as
¯ϕ5=A
VpωWst V
C|vi|2dV,(50)
with the spherical particle volume Vp=4
3πa3and the volumetric particle concentration Cwhich
may vary over the location within the fluid cavity. A further simplification is possible if the particles
are uniformly distributed over the fluid cavity. With the constant volumetric concentration C=Cc,
the loss factor is calculated as
¯ϕCc
5=CcAV|vi|2dV
VpωWst ≈3CcA
2πa3ω ρ0
.(51)
Herein, the stored energy is approximated by the kinetic energy according to Wst ≈ρ0
2|vi|2dV
which may lead to differences in some cases (see Sec. II B). Analyzing acoustic wave propagation
in suspensions, Atkinson and Kytömaa37 solve the linearly perturbed coupled system of equations
with momentum exchange between the fluid and the evenly distributed particles. Even though their
derivation is quite different, the loss factor that can be deduced from the wavenumber provided
in Eq. (42) of Ref. 37 is identical to our result in Eq. (51). Nevertheless, our solution is more
versatile since it does not require the approximation Wst =1
2ρ0c2
0V|p|2dV≈ρ0
2V|vi|2dVwhich
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always holds for the wave propagation described by Atkinson and Kytömaa37 but not necessarily for
resonators that involve several components. Furthermore, Eq. (50) also allows for spatially varying
particle concentrations. The presence of particles also affects the speed of sound in the suspension.37
However, since the derivation above is limited to low particle concentration where particles do not
interact significantly, the change in the speed of sound is neglected here.
F. Radiation losses
Some acoustofluidic devices are designed as flow-through chips where long fluid-filled tubes or
channels are directly in contact with the fluid cavity. Other acoustofluidic devices are only actuated
in a small region, whereas large parts of the device are passive and only contribute by the fact that
waves can leak into them. In these situations, a fraction of the ultrasonic energy is lost through
radiation into the passive parts of the fluid domain. However, it is advantageous to exclude these
passive parts from the numerical model for computational efficiency. Based on the 1D concept of
acoustic impedance, an impedance boundary condition can be formulated that models planar wave
radiation from the water into a neighboring fluid of density ρmand speed of sound cm.3It can be
written as
nip,i=iωρ 0
ρmcm
p,(52)
with the interface normal vector nipointing into the neighboring medium. If the adjacent medium is
water, the expression simplifies to
nip,i=ikp .(53)
When modeling realistic devices, it is not straight forward to decide if the use of the impedance
boundary condition is justified. Two conditions have to hold. First, the radiated waves have to be
planar to the interface. Second, once the wave has entered the adjacent fluid space, reflection back
into the device must not occur or has to be negligible at least. This condition is satisfied if the waves
have enough space to decay sufficiently before they are scattered. The distance is characterized by
the attenuation length L=1
2αsover which the acoustic intensity decreases to 1
eof its initial level.
At least within twice this distance or further, the waves should not be scattered at any obstacle or
boundary. Thus, the impedance boundary condition is well suited to model the radiation loss into
the fluid contained in long and narrow (width ≪λ) tubes or channels that have sound-hard walls.
For more complex situations, perfectly matched layers (PMLs) are often the better choice since they
are more versatile. For example, they allow the approximative modeling of backscattering due to
leaky side walls and, in contrast to the impedance boundary condition, they are not limited to planar
wave radiation.46,47 Still, it remains difficult to generate accurate simulation results of complex
device designs where large parts need to be excluded from the model for numerical efficiency. The
losses due to radiation can be very significant since it is a full first-order effect. Compared to the
other loss mechanisms in the fluid cavity which essentially only affect the amplitudes at resonances,
the radiation losses can directly influence the acoustic field at all frequencies. This means if radia-
tion losses are modeled in a device simulation, no additional loss factor has to be considered since
the loss is already covered by the impedance condition or the PML. Nevertheless, if one is interested
in the radiated power, it can be calculated according to
Ψ6=S
⟨pvini⟩dS=S
Re[pv∗
i]nidS,(54)
where the acoustic intensity pvigives the area-specific energy flux vector and time averaging has
been applied. The integration has to be performed over the lossy boundary or the interface to the
PML region.
G. Other losses
It can be expected that there is damping due to the thermal boundary layer around suspended
particles. However, the arguments made in Sec. III D suggest that the thermal boundary layer loss
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is typically smaller than the viscous boundary layer loss. Therefore, we do not analyze this effect in
detail.
Nonlinearities in the fluiddynamic equations extract energy from the linear time-harmonic
motion. If the amplitudes are small, the problem can be solved with the perturbation approach
mentioned in Sec. I. In this case, the losses due to nonlinearities are identical to the power that
drives the acoustic streaming and the particle motion due to the radiation forces. Assuming the
system has reached a quasi-stationary state48 and the particles move at their terminal velocity vt
i, the
power loss due to the radiation forces Frad
ion a single particle can be expressed as
Ψp
7=Frad
ivt
i.(55)
Further, assuming small spherical particles of radius a, the radiation force can be deduced from
the acoustic field using the theories mentioned in Sec. Iand the terminal velocity is determined by
balancing the Stokes drag forces with the radiation force,
Frad
i=6πη avt
i.(56)
Given the position-dependent volumetric particle concentration Cand the particle volume Vp=
4
3πa3, the loss due to the radiation forces in a fluid-particle suspension becomes
Ψ7=1
8π2ηa4V
C(Frad
iFrad
i)dV,(57)
which is an expression of higher order magnitude and hence negligible.
Similarly, the losses due to acoustic streaming can be written as
Ψ8=V
FR
ivs
idV,(58)
where FR
i=ρ0⟨vivi,j+vivj,j⟩is a volume force due to the spatial variation of the Reynolds stress
and vs
iis the streaming velocity field.49 Since both terms are second order in magnitude, the loss
due to streaming can be neglected. However, if the acoustic amplitudes are high or the fluid exhibits
a nonlinear elastic response, the situation becomes much more complicated and the losses due to
nonlinearities can become relevant.
IV. VALIDATION
In this section, we provide a validation for the damping mechanisms related to the viscous layer
at walls (Sec. III C) and suspended particles (Sec. III E) as well as the thermal boundary layer
damping (Sec. III D). We focus on these effects since they rely on assumptions which we validate
with analytical and numerical solutions of the viscoacoustic and thermoacoustic equations. Derived
by a first-order perturbation of the Navier-Stokes equations, the time-harmonic viscoacoustic equa-
tions take the form3,50
(c2
1−c2
2)vj,ji +c2
2vi,j j +ω2vi=0,(59)
with
c1=c01+i ¯ϕ1and c2=(1+i)ωη
2ρ0
.(60)
The time-harmonic pressure field follows from the linear equation of state and the continuity
equation. It can be written in terms of the velocity field according to
p=ic2
0ρ0
ωvi,i.(61)
This set of equations (59)–(61) is suited to model viscous boundary layer loss as well as viscous
dissipation in the bulk.
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Regarding the thermoacoustic equations, we refer to Blackstock,6Morse and Ingard,5as well as
Muller et al.,2but we apply a different notation for consistency. Here, the time-harmonic tempera-
ture and velocity field are governed by the coupled equations,
iωT+γDthT,ii =γ−1
αvj,j(62)
and
c2
01
γ+i ¯ϕ1−c2
2vj,ji +c2
2vi,j j +ω2vi=−iωαc2
0
γT,i.(63)
Once the thermal field and the velocity field is known, the pressure field can be deduced according
to
p=c2
0ρ0
γαT+i1
ωvi,i.(64)
This set of equations (62)–(64) is suited to model viscous and thermal boundary layer losses as well
as viscous and thermal dissipations in the bulk. In the numerical examples, the boundary layers are
resolved in computationally expensive FE simulations. The computed time-harmonic amplitudes at
resonance are compared to regular acoustic simulations where the boundary layer loss factors, as
derived in Sec. III, emulate the dissipation. The computations are performed on a regular computer
(quad-core Intel i7-2600K CPU, 32 GB RAM (DDR-3 1600 MHz), Windows 7 (64-bits)) using the
FE software Comsol Multiphysics version 4.3a.
A. Validation for viscous damping at cavity walls
The viscous boundary layer loss for a monochromatic plane standing wave along the axis of a
long rigid pipe with a circular cross section and an inner radius of Rpis derived by Kinsler et al.16
based on an impedance calculation (see Fig. 5). This derivation involves the analytic solution of the
boundary layer under the assumption Rp≫δand it leads to a Q-factor of Q=Rp
δor equivalently a
loss factor of ¯ϕ3=δ
Rp. For the presented situation, all terms in Eq. (32) can be determined analyt-
ically based on the acoustic velocity field (vx=sin(k0x), vy=0, v z=0). From the time-averaged
kinetic acoustic energy density follows the stored energy Wst =1
2ρ0πR2
p|vx|2dxand the surface
integral can be written as S|ξdiffi|2dS=2πRp|vx|2dx. With this, we obtain
¯ϕ3=ρ0δ
4Wst S
|ξdiffi|2dS=δ
Rp
,(65)
which is equal to the result reported by Kinsler et al.16 In this example, the system consists of only
one resonating part (the water domain). This justifies the use of the kinetic energy since both kinetic
and potential energies contribute equally to the stored energy.
The second validation example covers resonances between the side walls of a long rigid
fluid channel of rectangular cross section (see Fig. 6). This leads to a 2D problem for which
an analytic solution to Eqs. (59)–(61) has been derived by Karlsen et al.36 This solution pre-
dicts a Q-factor of Q=h
δor a loss factor of ¯ϕ3=δ
h, where his the height of the fluid channel.
FIG. 5. Rigid pipe with a standing wave along its axis. The zero-slip boundary condition at the rigid wall creates a viscous
boundary layer in which the velocity in x-direction decays from the bulk velocity vxto zero.
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FIG. 6. Rigid channel of rectangular cross section with its coordinate system. The standing wave is excited between the two
side walls, whereas the velocity in the bulk (vy) decays to zero in the viscous boundary layers at the top and bottom.
Again, all terms in Eq. (32) can be determined analytically based on the acoustic velocity field
(vx=0, vy=cos(π y /w), v z=0). The stored energy is calculated from the kinetic acoustic energy
density according to Wst =1
2ρ0h|vy|2dy. The surface integral S|ξdiffi|2dSis split in two parts.
First, the integration along the top and bottom of the channel giving 2 |vy|2dy. Second, the inte-
gration over the side walls leading to 2 |vx|2dxwhich is equal to zero because the fluid velocity
component vxis zero. Finally, we obtain the expected result,
¯ϕ3=ρ0δ
4Wst S
|ξdiffi|2dS=δ
h.(66)
Experimental measurements51 of the Q-factor for the same resonance mode and channel dimensions
w=380 µm and h=160 µm lead to values between Q=200 and Q=600 or an acoustofluidic
loss factor of 1.7×10−3≤¯ϕ≤5.0×10−3. The analytical model by Karlsen et al.36 and a numerical
model by Muller et al.2predict loss factors of ¯ϕ3=2.38 ×10−3and ¯ϕ3=2.39 ×10−3, respectively.
Assuming that damping in this setup is predominantly caused by the viscous boundary layer, the
results indicate that the models are accurate.
To investigate the effect of sharp edges and curved boundaries, we turn to a more complex
chamber geometry for which numerical treatment is required. As outlined above, the derivation
of the boundary layer loss factor is based on the assumption that the boundary layer has a planar
structure. At corners and edges, this requirement is not satisfied. Furthermore, if the boundary layer
thickness becomes comparable with the curvature radius of the boundary, the assumption does
not hold any more either. Both situations lead to an error in the predicted loss factor, which is
studied using the fluid channel shown in Fig. 7. On the one hand, the reference result is determined
by solving time-harmonic viscoacoustic equations (Eqs. (59)–(61)). On the other hand, acoustics
with acoustofluidic loss factor damping (also called regular acoustics here) is implemented by the
Helmholtz equation (Eqs. (2) and (3)) using the complex speed of sound (Eq. (7)) with the viscous
boundary layer loss factor (Eq. (32)). For clarity, all other loss mechanisms are not implemented
here, meaning ¯ϕ1=0 in Eq. (60) and ¯ϕ=¯ϕ3in Eq. (7). The fluid (water with the properties defined
in Table II) is excited by the side walls that both move with vy=ωd0where d0=0.1 nm and
vz=0. All other boundaries are static, whereas the boundary conditions are implemented as a
zero-slip boundary condition for the viscoacoustic equation and velocity continuity in boundary
normal direction for the regular acoustic equations. The mesh for the viscoacoustic FE simulation
is very fine at the walls and uses 39 012 elements of cubic order to ensure an accurate resolution of
FIG. 7. Fluid channel with a sharp interior edge and a curved surface. A resonance of the fluid channel in the y-z-plane is
excited by the moving side walls and studied in 2D numerical simulations. Viscous boundary layers form at all surfaces since
the resonance mode has a complex shape.
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FIG. 8. This graph shows the pressure magnitude at point P for the viscoacoustic reference simulation and the regular
acoustic simulation with acoustofluidic loss factor damping. Pressure peaks appear at the fluid resonance. In contrast, the loss
factor according to Eq. (32) is amplitude independent and does not change significantly at the resonance.
the boundary layer. Each simulation takes 37 s of computing time on a regular personal computer
(PC). The mesh for the regular acoustic simulation is much coarser and only refined around the
sharp edge at the origin. It uses 2774 elements of quadratic order. Each acoustic simulation consists
of two steps. In the first step, the fields are simulated using an estimated fluid loss factor (1 ×10−2
is chosen here). Based on these results, the correct acoustofluidic loss factor is calculated from
Eq. (32) which is not affected by the estimated loss factor. In the second step, the field is solved
using the acoustofluidic loss factor. The whole procedure takes 0.4 s of computing time on the PC.
The simulations are performed for the frequency range between 1.5 MHz and 1.65 MHz in steps
of 500 Hz. The frequency-dependent pressure magnitude at point P is plotted in Fig. 8. Character-
izing the fluid resonance, the peaks appear at almost the same frequency (viscoacoustic resonance
frequency frv =1.5690 MHz and acoustic resonance frequency fra =1.5730 MHz corresponding to
0.3% error) and they have a very similar height (viscoacoustic resonance pressure prv =0.355 MPa
and acoustic resonance pressure pra =0.361 MPa corresponding to 1.6% error). Considering that a
major fraction of the dissipation occurs in the boundary layer close to the origin where the boundary
layer is not planar, this level of accuracy is remarkable. The viscoacoustic and the regular acoustic
fields at the resonance peaks are shown in Fig. 9. A comparison shows that the mode shapes are
identical outside the very thin boundary layer. In order to further investigate the limits of validity
of the approximate solution, the viscosity is increased successively. For both the reference and the
approximate simulations, the pressure magnitude at resonance (prv and pra evaluated at P) and the
resonance frequencies ( frv and fra) are plotted over the viscosity (see Fig. 10). The discrepancy
between the two models grows with increasing viscosity. At a hundredfold increased viscosity, the
error (2.7% error in the resonance frequency and 6.6 % error in the amplitude) becomes critical
for accurate device simulations. The graph stops at a hundredfold viscous fluid because the losses
becomes so high ( ¯ϕ3=0.051 equivalent to Q=19.7) that the channel would not be suited for
resonant acoustic manipulation any more. Figure 11 shows that the mode shape is still accurately
modeled under these conditions with only slight differences due to the increased boundary layer
FIG. 9. 2D time-harmonic field in a rigid water-filled channel obtained from viscoacoustics (left) and regular acoustics with
acoustofluidic loss factor damping (right) for the resonance at frv =1.5690 MHz and fra =1.5730 MHz, respectively. The
magnitude of the velocity field is shown in (a) with a magnified view on the sharp edge. In contrast to the regular acoustic
simulation, the viscoacoustic simulation includes the very thin (here, δ=0.43 µm) viscous boundary layer at the walls. The
absolute pressure field which does not have a boundary layer structure is shown in (b) for both models.
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FIG. 10. Simulation results for increasing fluid viscosity starting at the value of water at 25 ◦C up to a hundredfold. The
pressure and the resonance frequency are given for both viscoacoustics (prv,frv) and regular acoustics with acoustofluidic
loss factor damping (pra,fra). Also, the acoustofluidic loss factor of the regular acoustic simulation is provided. Viscosity,
pressure, and loss factor are plotted in logarithmic scale, the resonance frequency is given in linear scale, and dots mark the
simulated values.
thickness. It is emphasized that this example is already a worst-case scenario, combining a relatively
small cavity volume with a large boundary layer volume and an edge at the velocity maximum.
B. Validation for thermal damping at cavity walls
For the validation of the thermal boundary layer loss factor ¯ϕ4according to Eq. (38), a radial
resonance in a rigid and isothermal circular 2D cavity is studied. The 3D equivalent is a radial mode
in a rigid capillary of circular cross section. The reference simulation implements Eqs. (62)–(64),
whereas in the regular acoustic simulation, Eqs. (2) and (3) are solved with damping by the com-
plex speed of sound (Eq. (7)). In order to focus on the thermal boundary layer dissipation, all
other damping mechanisms that are inherently incorporated in the reference simulation (viscous
boundary layer loss as well as viscous and thermal dissipations in the bulk) are set to zero if
possible. The chosen example has the advantage of a vanishing viscous boundary layer since the
tangential velocity at the wall is zero. Viscous dissipation in the bulk is avoided by setting ¯ϕ1in
Eq. (63) to zero. However, thermal damping in the bulk cannot be avoided easily and is retained
in both reference and the regular acoustic models. For the latter, this means that dissipation is
implemented by ¯ϕ=¯ϕ2+¯ϕ4. Fluid parameters are adopted from Table II with exception of the
thermal conductivity which is increased by a factor of 10 to κth =6.07 W m−1K−1with the intention
to reduce the computational effort related to the resolution of the thermal boundary layer. The
diameter of the circular cross section is 380 µm, the interface heat capacity ratio is set to ϵs=0 for
the isothermal boundary condition (T=T0), and the characteristic thermal boundary layer thickness
at the first radial resonance at around 4.8 MHz is δth ≈0.3µm. Excitation is provided by the
boundary which moves with the velocity vi=niωd0, where niis the outward pointing boundary
normal vector and d0=0.1 nm. The pressure magnitude in the center of the circle is plotted over
the frequency range between 4.8040 MHz and 4.8055 MHz as shown in Fig. 12. Another set of
FIG. 11. Magnitude of the resonant velocity field at high viscosity (100 times the viscosity of water at 25 ◦C) for both
viscoacoustic and the regular acoustic simulation models. Here, the field differences due to the viscous boundary layer
(δ=4.3µm) in close proximity to the channel walls start to become visible but the predicted field magnitudes are still
similar.
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FIG. 12. This graph shows the pressure magnitude at the center of the circular fluid space for the thermoacoustic reference
simulation and the regular acoustic simulation with acoustofluidic loss factor damping. Pressure peaks appear at the fluid
resonance. Dissipation occurs due to thermal effects in the bulk and in the thermal boundary layer. The amplitudes are
unrealistically high since other damping mechanisms are avoided as intended.
simulations (not presented here) was conducted to ensure that differences between the two simula-
tion models can be attributed to the error in the thermal boundary layer loss factor. Except for an
adiabatic boundary condition that avoids the formation of the thermal boundary layer, the simula-
tion conditions are identical. Here, the simulated field magnitudes and the resonance frequencies
show an almost perfect match. It indicates that under the given simulation conditions, the loss
factor ¯ϕ2due to thermal damping in the bulk does not contribute significantly to discrepancies
between the two simulation methods. Furthermore, it was checked that the numerical damping of
the FE simulation is negligible. For these reasons, the simulation results shown in Fig. 12 allow
a first-order validation of the loss factor ¯ϕ4. The resonance frequencies are almost identical (ther-
moacoustic resonance frequency frt =4.804 815 MHz and regular acoustic resonance frequency
fra =4.804 865 MHz corresponding to 0.01herror) and the pressure magnitudes at resonance
match very well (thermoacoustic resonance pressure prt =333.76 MPa and regular acoustic reso-
nance pressure pra =333.42 MPa corresponding to 1.0herror). At resonance, the acoustofluidic
loss factor becomes ¯ϕ=¯ϕ2+¯ϕ4=1.75 ×10−5with the bulk loss factor ¯ϕ2=2.08 ×10−7two or-
ders of magnitude smaller than the boundary layer loss factor ¯ϕ4=1.73 ×10−5. The simulated
fields at resonance are provided for the thermoacoustic reference and the regular acoustic simulation
as shown in Fig. 13. In summary, excellent quantitative and qualitative agreement is achieved for
both the pressure and the velocity field. Furthermore, a number of numerical experiments have been
conducted with other chamber geometries and changed thermal fluid parameters, indicating that
the damping factor ¯ϕ4follows the correct trends and it is still accurate if both thermal and viscous
boundary layers are present.
C. Validation for viscous damping due to suspended particles
Here, we provide a numerical validation of the loss factor ¯ϕp
5(see Eq. (49)) due to the viscous
boundary layer surrounding a single suspended particle. The geometry of the rectangular fluid
cavity which contains one microparticle is shown in Fig. 14. The fluid cavity is chosen much larger
than the particle to minimize undesired scattering effects. If the particle is large, its scattering field
can lead to considerable viscous boundary layer dissipation at the cavity side walls. In the reference
simulation, the fluid is modeled by the time-harmonic viscoacoustic equations (Eqs. (59)–(61)),
whereas the particle is modeled as a linear elastic solid with coupling at the interface (velocity
continuity and stress continuity). Excitation is provided by the left and the right walls where the
fluid velocity vx=0, vy=ωd0, and vz=0 with d0=0.1 nm is enforced. At all other walls, slip
boundary conditions are applied with only the velocity component in boundary normal direction
constrained to zero. To avoid viscous damping in the bulk, ¯ϕ1in Eq. (60) is set to zero. In this way,
the dissipation in the viscous boundary layer at the particle is essentially the only source of damping
in the simulation. The same setup is also studied using a regular acoustic simulation (Eq. (2) and
(3)) with the complex speed of sound (Eq. (7)) and the loss factor ¯ϕp
5(Eq. (49)) which emulates
viscous boundary layer damping due to the suspended particle. Here, the particle does not need
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FIG. 13. 2D acoustic field at the first radial resonance in a fluid filled channel of circular cross section (380 µm in diameter)
with rigid and isothermal walls. The velocity and the pressure magnitude are shown in (a) and (b), respectively, for both
thermoacoustics (top) and regular acoustics with acoustofluidic loss factor damping (bottom). There is no viscous boundary
layer. However, the magnified view on the temperature field of the thermoacoustic simulation in (c) shows a very thin (here,
δth =0.3µm) thermal boundary layer.
to be modeled explicitly since ¯ϕp
5requires only knowledge of the acoustic velocity at the particle
location without the presence of the particle. The fluid properties are taken from Table II with the
dynamic viscosity increased by a factor of 100 to η=890 mPa s to keep the computational effort
of these 3D numerical simulations at an acceptable level. Still, each viscoacoustic simulation takes
200 s of computing time, while the two-step regular acoustic simulations finishes in less than 2 s. In
all simulations, the half-wave resonance between the two side walls at roughly 0.75 MHz is excited
and the particle’s elastic properties are kept constant (Pyrex with a Young’s modulus E=62 GPa
and a Poisson ratio of ν=0.24) since they do not affect the damping up to first order. In the
first set of simulations, the particle radius is a=10 µm, the density ratio is ˜ρ=10 (the particle
is ten times denser than the fluid), and it is positioned in the center of the cavity at the velocity
antinode. The pressure magnitude at point P in the corner of the cavity (compare Fig. 14) is plotted
over frequency for both numerical models (compare Fig. 15). Similar as in Figs. 8and 12, the
viscoacoustic reference simulation has its resonance at a slightly lower frequency with a relative
FIG. 14. Geometry (in mm) and coordinate system of the fluid cavity for the study of dissipation in the viscous boundary
layer around a suspended spherical particle. The excitation is provided through the vibration of the cavity walls to both sides.
If not explicitly stated differently, the particle is located in the center of the cavity ( yp=0.5 mm).
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FIG. 15. This graph shows the pressure magnitude at point P (see Fig. 14) of the cavity for the viscoacoustic reference
simulation and the regular acoustic simulation with acoustofluidic loss factor damping. The pressure peaks of almost identical
height indicate the fluid resonance in both numerical models. In this simulation setup, dissipation in the viscous boundary
layer around the suspended particle is the dominating source of damping.
frequency difference so small (0.04h) that it is irrelevant in practical simulations. The pressure
magnitude at resonance is accurately modeled with an error of 0.3%. The velocity fields obtained
with both models are illustrated in Fig. 16(a). In the following simulation setups, we study if the
loss factor ¯ϕp
5reflects the trends accurately. For this reason, three important simulation parameters
are modified separately while keeping all other parameters fixed. Figure 16 shows the simulated
velocity field for both models at their individual resonance frequency. In Fig. 16(b), the position of
the particle is changed to yp=0.25 mm; in (c), the particle radius is a=20 µm; in (d), the density
ratio is ˜ρ=1.1; and in (e), the density ratio is ˜ρ=0.1. In all cases, the velocity field magnitudes
in the bulk match with an error of around 1h, whereas the maximum magnitudes range over more
than three decades between the individual simulation setups. The same applies for the pressure
fields which are not shown here. It shall be noted that, similar to the simulations of Sec. IV B,
the acoustic amplitudes are unrealistically high due to the absence of other damping effects. As
predicted by Eq. (49), the damping tends to zero as the density ratio ˜ρbetween the particle and the
fluid approaches unity. This effect can be traced back to the vanishing boundary layer as the particle
and the surrounding fluid are moving with the same translational velocity. This result is especially
relevant for practical applications because some typical particles (e.g., biological cells, bacteria, or
FIG. 16. 3D simulation results of the half-wave resonance field in a rectangular fluid cavity, containing one microparticle
of varying density, position, and size. For each case, the plots allow a comparison between the viscoacoustic reference
simulation and the regular acoustic simulation with acoustofluidic loss factor damping. For the latter, particles are only
shown for reference and are not included in the simulation. The velocity magnitude is visualized on three horizontal cut
planes, showing that differences between the two simulation models are only considerable in the thin viscous boundary layer
at the particle, which is shown in a magnified view. (a) Particle of radius a=10 µm and density ratio ˜ρ=10 in the center
of the cavity at yp=0.5 mm. (b) Particle at different positions (a=10 µm, ˜ρ=10, and yp=0.25 mm). (c) Larger particle
radius (a=20 µm, ˜ρ=10, and yp=0.5 mm). (d) Density of the particle is similar to the fluid density (a=10 µm, ˜ρ=1.1,
and yp=0.5 mm). (e) Particle that is significantly lighter than the fluid (a=10 µm, ˜ρ=0.1, and yp=0.5 mm).
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copolymer beads) have densities that are very close to water density. In this case, the damping due
to the viscous boundary layer around those particles is minor.
V. IMPLEMENTATION OF AN EFFICIENT NUMERICAL DEVICE MODEL WITH ACCURATE
DAMPING
The analytical and semi-analytical components of the acoustofluidic loss factor derived in
Sec. III can be fitted into numerical device models as outlined in Sec. II A. Since there is no need to
resolve the boundary layers, the numerical models stay computationally efficient. Nevertheless, all
dissipation mechanisms in the fluid cavity are emulated accurately and amplitudes can be predicted
realistically if the acoustofluidic loss factor is considered in the device model. The only difficulty is
that the boundary layer loss factors implicitly depend on the acoustic mode shape inside the fluid
domain, which is not known a priori. This represents a loop-dependence of the numerical model
on its own solution, whereas the standard way to solve such a problem is to employ a nonlinear
solver that finds the simulation result in an iterative fashion. The major drawback of nonlinear
solvers is a drastically increased computing time due to heavier memory usage and an increased
number of numerical unit operations. However, given that the components of the acoustofluidic loss
factor only depend on the mode shape and not on the acoustic amplitudes, we suggest a simple and
numerically more efficient solving procedure that was already applied in Sec. IV. In essence, the
device simulation is performed twice, using a linear solver: once with a roughly estimated fluid loss
factor to find the correct acoustic mode shape at a potentially wrong amplitude and a second time
with the calculated acoustofluidic loss factor to obtain the final solution which is both qualitatively
and quantitatively accurate. In order to give an example, a device for the positioning of hollow glass
particles in the center of a square fluid cavity is chosen. This device already served as an example in
Fig. 2where the energy distribution was analyzed. The exact device geometry is shown in Fig. 17.
It is assumed that the fluid cavity as well and the two inlet channels and reservoirs is filled with a
water-particle suspension that contains 0.1 volume-percent of hollow glass particles ( ˜ρ=0.6) that
FIG. 17. Geometry of the 3D acoustofluidic micro-device for the positioning of hollow particles in a square fluidic chamber.
Between the piezoelectric transducer and the silicon, there is a 20 µm thin layer of epoxy glue that is modeled using the
thin layer approximation. In contrast, the glass lid and the silicon layer are directly bonded to each other. At all fluid-solid
interfaces, fluid structure interaction is implemented.
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FIG. 18. Acoustofluidic loss factor and its individual components as calculated from device simulations in the frequency
range from 0.1 MHz to 2.0 MHz. For the simulated device and the given operating conditions, the viscous boundary layer
loss factor is the dominating term which can be seen best in (a) where linear axes are used (the curve of ¯ϕ3is almost
identical to the one of the ¯ϕ). The frequency response of all acoustofluidic loss factor components is analyzed best using a
semi-logarithmic plot as seen in (b).
are distributed homogeneously. All material parameters and the driving conditions are provided in
the supplementary material including the complete FE simulation model.28 Here, the commercial
FE software package Comsol Multiphysics is used for both the modeling and system solving.
The FE simulation mesh consists of 60 847 tetrahedral elements of quadratic order which en-
sures a good resolution of all fields inside the solid and fluid device components. Each simulation
takes 42 s and occupies 5.5 GB of memory. Even though the evaluation of the different acoustoflu-
idic loss factor components involves surface and volume integrals, the computing time associated
with these operations is negligible in comparison with the system solving. In order to identify trends
in the different fluid loss factors, a frequency sweep between 0.1 MHz and 2.0 MHz is performed
in steps of 5 kHz. For each simulated frequency, the loss factors ¯ϕ1to ¯ϕ5(according to Eqs. (21),
(26), (32), (38), and (50)) are summarized (see Eq. (19)) to obtain the total acoustofluidic loss factor
¯ϕin the fluid cavity. The values are plotted on both a linear and a logarithmic scale in Fig. 18. In
the energy distribution analysis of the same device in Sec. II B, we employ an acoustofluidic fluid
loss factor of 0.004 for the resonance at 0.84 MHz. This is a result of the acoustofluidic loss factor
simulation shown in Fig. 18. Based on the calculated results, there is a number of observations that
can be made. As expected from the equations, the loss factors ¯ϕ1and ¯ϕ2associated with the bulk
losses show no mode shape dependence and they scale linearly with the frequency f. In contrast,
the loss factors ¯ϕ3, ¯ϕ4, and ¯ϕ5associated with boundary layer losses vary drastically with changing
acoustic mode shapes. Nevertheless, a general trend can clearly be seen in the frequency response
of both ¯ϕ3and ¯ϕ4. Looking at Eqs. (32) and (38), it is easily observed that these loss factors scale
with 1/f. This is consistent with the simplified approximation ¯ϕ3≈δSδ
2Vwhere the characteristic
viscous boundary layer thickness δscales with 1/f. Even though the loss factor ¯ϕ5due to the
viscous boundary layer around suspended particles scales in the same manner if the particle size
ais much larger than δ, it is noted that this loss factor displays a different scaling if aand δ
become comparable (see ¯ϕ5in Fig. 18(b) at low versus high frequency). This scaling effect can be
traced back to the complicated frequency dependence of the parameter Aaccording to Eq. (48).
Furthermore, it can be seen that the thermal boundary layer loss factor ¯ϕ4is smaller than the viscous
boundary layer loss factor ¯ϕ3by a factor of approximately 250 as predicted in Sec. III D. This
leads to another important observation, which is seen best in Fig. 18(a) where the linear axis is
used. For the simulation device, the total acoustofluidic loss factor ¯ϕis mainly affected by the loss
factor ¯ϕ3due to the viscous boundary layer at walls. Of course, this can change if a higher particle
concentration or another particle type with a different density ratio ˜ρis used. Furthermore, the
different frequency scalings of the individual loss factors imply that the bulk losses become more
important or even dominant at higher frequencies as often used in surface acoustic wave (SAW)
devices. Finally, the geometric scaling can also be deduced from the equations of Sec. III. While
the wall-related losses (due to ¯ϕ3and ¯ϕ4) scale with the fluid-structure interface area, the volume
related losses (due to ¯ϕ1, ¯ϕ2, and ¯ϕ5) scale with the fluid volume. This means that for a decreased
fluid cavity size, the wall-related losses become more prominent, whereas for larger fluid cavities,
the importance of volume related losses increases.
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Also shown in Fig. 18 is the value of the rule-of-thumb estimation (denoted as ˜ϕ3=δSδ
2V
according to Eq. (33)) of the viscous boundary layer loss factor. Since the main focus lies on
resonances in the square fluid cavity, the surface area and the volume term in Eq. (33) are calculated
according to only this part of the fluid cavity. During the frequency sweep, the acoustic mode shapes
and the location of viscous boundary layers change. Therefore, the upper limit of the approximation
with Sδ=S=3.76 mm2is chosen. The cavity volume is V=0.288 mm3and the characteristic
boundary layer thickness is calculated according to Eq. (27) with the water properties given in
Table II. At frequencies above the first cavity resonance at around 0.55 MHz, the simplified formula
provides a reasonable estimation for both the trend and the average value of ¯ϕ3. Below the first
cavity mode, there are structural device resonances where the vibration velocity of the fluid cavity
walls is not negligible with respect to the time-harmonic fluid velocity. Due to the violation of this
major assumption in the derivation of the estimation formula, the large discrepancy between ˜ϕ3and
¯ϕ3in the frequency range below 0.55 MHz is explained. Even though ˜ϕ3can never exactly predict
the loss factor ¯ϕ3since the mode-shape dependence is completely neglected, it still is a useful
measure that allows to decide if a fluid cavity is generally suited for resonant acoustic manipulation.
VI. CONCLUSION
Numerical simulation has become a valuable tool for the design, the analysis, and the under-
standing of acoustofluidic micro-devices. Despite the drastically increased performance of numer-
ical hard- and software, modeling the full physical complexity of 3D acoustofluidic devices is
still far beyond current computing capabilities. Therefore, researchers resort to perturbation ap-
proaches which can resolve the numerical difficulties associated with the multi-scale nature of the
relevant physical phenomena. If, further, the viscous and thermal boundary layers are neglected,
the governing equations can be simplified to a basic device model which is numerically suited for
full 3D simulations. However, in the course of this model simplification, the damping information
associated with a number of viscoacoustic, thermoacoustic, and acoustofluidic effects in the fluid
is lost. Nonetheless, damping is an important part of the model since it determines the attainable
acoustic amplitudes as well as the acoustofluidic force magnitudes at each device resonance. For
this reason, it is important to re-incorporate the damping information into the numerical model
by the acoustofluidic loss factor which needs to be set for each device component. Solid device
components have loss factors that, at given frequency and temperature, are predominantly material
parameters and they can be found in the literature or deduced from wave propagation experiments.
In contrast, the acoustofluidic loss factor of fluid cavities does not only depend on the fluid prop-
erties as it is crucially affected by the cavity geometry and the acoustic mode shape. By the example
of a typical silicon micro-device and a detailed energy distribution analysis, we have demonstrated
that dissipation in the fluid cavity represents a significant part of the total device loss. For this
reason, the basic device model cannot be used for a quantitative prediction of acoustofluidic forces
unless the acoustofluidic loss factor is determined in a physically correct fashion.
An extensive literature review has been carried out to identify all relevant loss effects inside
acoustofluidic cavities and to find relevant studies which are scattered in the literature. In the
present work, we have extended previous results towards the specific needs of the acoustofluidic
community. Analytic and semi-analytic formulations have been provided for a number of effects
in a unified notation. Further, we have carried out a thorough analytical and numerical validation
for the acoustofluidic loss factor components related to the viscous boundary layer at walls and
suspended particles as well as for the thermal boundary layer at walls. Besides an error estimation
for the acoustic amplitudes and the resonance frequencies, they also allow to assess the limits of
validity for this simplified damping model. In summary, the acoustofluidic loss factor damping
model can be applied in a very wide range of acoustofluidic device simulations, whereas acoustic
amplitudes and acoustofluidic forces can be predicted at high accuracy. With the simulation of a
typical acoustofluidic micro-device, we have demonstrated that, for the first time, we can implement
and efficiently solve a complex 3D device model that incorporates fluid damping mechanisms in
a physically accurate way. Furthermore, we have identified trends in the individual acoustofluidic
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loss factor components that allow to make predictions on which damping mechanisms are the most
relevant ones for any given acoustofluidic setup.
As part of our study, we have derived the damping due to suspended solid microparticles. To
complete the picture, future work may be directed toward deriving the damping related to suspended
liquid droplets and gas bubbles. As a further improvement of the presented damping model, the
temperature distribution inside the fluid cavity could be taken into account since it affects the
thermo-physical properties of the fluid. However, this would require a computationally expensive
thermo-fluidic device simulation, taking into account the localized dissipation due to the different
acoustofluidic damping mechanisms. Having developed a thorough understanding about the damp-
ing mechanisms in the fluid cavity, two other difficulties in the context device modeling remain to
be addressed. First, the damping properties of some solid device components, e.g., the glue layers
are often neither to be found in the literature nor given by the manufacturer. Second, the numerical
modeling of anchor losses from the acoustofluidic device into supporting structures or attached
tubing is challenging to handle and typically associated with a high numerical effort.
An interesting experimental implication of our findings is that the dependence of the viscous
boundary layer loss on the acoustic mode and cavity geometry can be exploited in new device
designs. Acoustofluidic cavities can be shaped to favor one specific mode in a sense that this mode
is less damped than others. It could be useful for multi-wavelength cavities where, e.g., the lateral
resonances often need to be suppressed to obtain accurate 1D line patterns. Given that the viscous
boundary layer damping is often by far the most prominent source of dissipation in the fluid cavity,
efforts can also be made to design fluid cavities of very high Q-factor for enhanced acoustofluidic
forces.
As the main result of this work, we provide the complete 3D FE model of the acoustoflu-
idic device including the implementation of the acoustofluidic loss factor in the supplementary
material.28 It is computationally efficient and allows the quantitative prediction of the acoustic
amplitudes inside realistic acoustofluidic devices. With augmentations, the model can accurately
predict the acoustic radiation forces on particles as well as the acoustic streaming velocities inside
acoustofluidic cavities.
ACKNOWLEDGMENTS
We would like to acknowledge financial support by the Swiss National Science Foundation,
SNF project 2000211-126986.
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