P. B. MULLER et al. PHYSICAL REVIEW E 88, 023006 (2013)
the magnitude of radiation-dominated acoustophoretic particle
velocities parallel to the optical axis in similar simple
half-wave resonance systems. However, in more complex
configurations, or in the case of small particles dragged along
by acoustic streaming rolls, more advanced techniques are
necessary, which are able to resolve three-dimensional (3D)
particle positions and three-component (3C) motion. Among
these techniques, those based on μPIV have issues regarding
the depth of correlation between adjacent planes 30,31,
while classical 3D particle tracking techniques require either
stereomicroscopes with tedious calibration protocols, or fast
confocal microscopes with great losses in light intensity due
to the use of pinholes 32.
In this work, an analytical and experimental analysis is
presented with the aim to improve the understanding of the full
3D character of ultrasound-induced acoustophoretic motion of
microparticles. In particular, analytical expressions for this
motion are obtained by extending the classical results for
Rayleigh streaming in shallow parallel-plate channels to also
cover rectangular channels of experimental relevance. The
analytical results are compared with measurements of the 3D
motion of particles in an acoustofluidic microchip performed
by use of astigmatism particle tracking velocimetry (APTV)
33–35. APTV is a very precise single-camera tracking
method which allows a time-resolved, reconstruction in 3D of
the trajectories of microparticles in acoustophoretic motion.
The technique is applicable to general 3D acoustophoretic
motion of microparticles influenced by both the acoustic
radiation force and the Stokes drag from acoustic streaming.
The paper is organized as follows. In Sec. II we derive an
analytical expression of acoustic streaming in long, straight
channels with rectangular cross section, and we analyze the
implications of this streaming for acoustophoretic motion
of suspended microparticles. This is followed in Sec. III
by a description of the experimental methods, in particular
the astigmatism particle tracking velocimetry technique. In
Sec. IV we compare the theoretical and experimental results
for the acoustophoretic microparticle motion, and finally in
Secs.Vand VI we discuss the results and state our conclusions.
II. THEORY
The governing perturbation equations for the thermoacoustic
fields are standard textbook material 36–38. The full
acoustic problem in a fluid, which before the presence of
any acoustic wave is quiescent with constant temperature T0,
density ρ0, and pressure p0, is described by the four scalar
fields temperature T , density ρ, pressure p, and entropy s per
mass unit as well as the velocity vector field v. Changes in ρ
and s are given by the two thermodynamic relations
dρ = γκs ρ dp − αp ρ dT, (1a)
ds = cp
T
dT − αp
ρ
dp, (1b)
which besides the specific heat capacity cp at constant pressure
also contain the specific heat capacity ratio γ , the isentropic
compressibility κs , and the isobaric thermal expansion
coefficient αp given by
γ = cp
cv
= 1 +
α2
pT0
ρ0cpκs
, (2a)
κs = 1
ρ
∂ρ
∂p
s
, (2b)
αp = −1
ρ
∂ρ
∂T
p
. (2c)
The energy (heat), mass (continuity), and momentum (Navier-
Stokes) equations take the form
ρT ∂t + (v·∇)s = σ
:∇v + ∇·(kth∇T ), (3a)
∂tρ = −∇ · (ρv), (3b)
ρ∂t + v ·∇v = −∇p + ∇·η{∇v + (∇v)T}
+(β − 1)∇(η∇·v), (3c)
13
where η is the dynamic viscosity, β is the viscosity ratio,
which has the value for simple liquids 36, kth is the thermal
conductivity, and σ
is the viscous stress tensor. As in Ref. 22,
we model the external ultrasound actuation through boundary
conditions of amplitude vbc on the first-order velocity v1 while
keeping T constant,
T = T0 on all walls, (4a)
v = 0 on all walls, (4b)
n · v1 = vbc(y,z) e
−iωt added to actuated walls. (4c)
Here, n is the outward pointing surface normal vector, and
ω is the angular frequency characterizing the harmonic time
dependence written using complex notation.
A. First-order fields in the bulk
To first order in the amplitude vbc of the imposed ultrasound
field we can substitute the first-order fields ρ1 and s1 in
the governing equation (3) using Eq. (1). The heat transfer
equation for T1, the kinematic continuity equation expressed
in terms of p1, and the dynamic Navier-Stokes equation for v1
then become
∂tT1 = Dth∇2T1 + αpT0
ρ0cp
∂tp1, (5a)
∂tp1 = 1
γκs
αp∂tT1 − ∇·v1, (5b)
∂tv1 = − 1
ρ0
∇p1 + ν∇2v1 + βν ∇(∇·v1). (5c)
Here, Dth = kth/(ρ0cp) is the thermal diffusivity, and ν =
η0/ρ0 is the kinematic viscosity. A further simplification
can be obtained when assuming that all first-order fields
have a harmonic time dependence −e
iωt inherited from the
imposed ultrasound field (4c). Then, p1 can be eliminated by
inserting Eq. (5b), substituting ∂tp1 = −iωp1, into Eqs. (5a)
and (5c). Solutions of Eq. (5) describe the formation of thin
thermoviscous boundary layers at rigid walls. In the viscous
boundary layer of thickness
δ =
2ν
ω
, (6)
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