ULTRASOUND-INDUCED ACOUSTOPHORETIC MOTION OF . . . PHYSICAL REVIEW E 88, 023006 (2013)
1
2h
z
0
−1
2h
−1
δ
δ
λ/2
2w 0 y 1
2w
FIG. 1. (Color online) A cross-sectional sketch in the yz plane of
the classical Rayleigh-Schlichting streaming pattern in the liquidfilled
gap of height h between two infinite, parallel rigid walls
(black) in the xy plane. The bulk liquid (light shade) supports a
horizontal standing sinusoidal pressure half-wave p1 (dashed lines)
of wavelength λ in the horizontal direction parallel to the walls. In
the viscous boundary layers (dark shade) of submicrometer thickness
δ, large shear stresses appear, which generate the boundary-layer
(Schlichting) streaming rolls (light thin lines). These result in
an effective boundary condition v
bnd
2y
(thick light arrows) with
periodicity λ/2 driving the bulk (Rayleigh) streaming rolls (black
thin lines). Only the top and bottom walls are subject to this effective
slip boundary condition.
the velocity gradients are large because the velocity field
changes from its bulk value to zero at thewalls across this layer
36–38. Inwater at ω/(2π) = 2 MHzit becomes δ ≈ 0.4 μm.
We focus on the transverse standing-wave resonance
sketched in Fig. 1, which is established by tuning of ω in
the time-harmonic boundary condition (4c) to achieve one of
the resonance conditions nλn/2 = w, n = 1,2,3, . . ., where
λn = 2πc0/ωn is the acoustic wavelength of the nth horizontal
resonance. The associated first-order fields v1, p1, and T1 in
the bulk of the channel take the form
v1 = va sin(kny + nπ/2) e
−iωnt ey , (7a)
p1 = pa cos(kny + nπ/2) e
−iωnt , (7b)
T1 = Ta cos(kny + nπ/2) e
−iωnt , (7c)
where kn = 2π/λn = nπ/w is the wave number of the nth
horizontal resonance, and the oscillation amplitudes of the
first-order fields, indicated by subscript “a”, are related through
|va/c0| ∼ |pa/p0| ∼ |Ta/T0| 1, with c0 being the isentropic
speed of sound in water. The spatial form of the standing-wave
resonance is determined entirely by the resonance frequency
and the geometry of the resonator, while its amplitude (here
va ≈ 104 vbc 22) is governed by the specific form of vbc and
of the Q factor of the resonance cavity. The acoustic energy
density Eac is constant throughout the cavity and given by
Eac = 14
ρ0v2
a
= 14
κ0p2
a . (8)
B. Second-order governing equations for v2
In a typical experiment on microparticle acoustophoresis,
the microsecond time scale of the ultrasound oscillations is not
resolved. It therefore suffices to treat only the time-averaged
equations. The time average over a full oscillation period,
denoted by the angled brackets . . ., of the second-order
continuity equation and Navier-Stokes equation becomes
ρ0∇ · v2 = −∇ · ρ1v1, (9a)
η0∇2v2 + βη0∇(∇ · v2) − ∇p2
= ρ1∂tv1 + ρ0(v1 · ∇)v1
−η1∇2v1 − βη1∇(∇ · v1)
−∇η1 · ∇v1 + (∇v1)T
−(β − 1)(∇ · v1)∇η1. (9b)
Here, η1 is the perturbation of the dynamic viscosity due
to temperature, η = η0 + η1 = η(T0) + ∂T η(T0) T1. From
Eq. (9) we notice that second-order temperature effects
enter only through products of first-order fields. Dimensional
analysis leads to a natural velocity scale u0 for second-order
phenomena given by
u0 = 4Eac
ρ0c0
= v2
a
c0
. (10)
C. Boundary condition for bulk streaming flow
The second-order problem (9) was solved analytically by
Lord Rayleigh 9,39 in the isothermal case (T = T0) for
the infinite parallel-plate channel in the yz plane with the
imposed first-order bulk velocity v1 Eq. (7a). The resulting
y component v
bnd
2y
of v2 just outside the boundary layers at
the top and bottom walls becomes
v
bnd
2y
= −vstr sin
nπ
2y
w
+ 1
, (11)
as sketched in 1 for the half-wave k1 = π/w. In Rayleigh’s
isothermal derivation, the amplitude vstr of the streaming
velocity boundary condition v
bnd
2y
becomes
0
str = 3
v
8
v2
a
c0
= 3
8
u0, (12)
where the superscript “0” refers to isothermal conditions.
Recently, Rednikov and Sadhal 18 extended this analysis
by including the oscillating thermal field T1 as well as
the temperature dependence η1(T ) of the viscosity. They
found that the amplitude of the streaming velocity boundary
condition vT
str then becomes
vT
str = 8
3
KT v0
str
= KT u0, (13a)
KT = 3
8
+ γ − 1
4
1 − (∂T η)p
η0αp
√
ν/Dth
1 + ν/Dth
, (13b)
where the superscript “T ” refers to inclusion of thermoviscous
effects leading to a temperature-dependent pre-factor multiplying
the temperature-independent result. For water at 25 ◦C
str = 1.26 v
we find vT
0
str using the material parameter values of
Table I, and in all calculations belowwe use this thermoviscous
value for vstr.
D. Second-order governing equations for bulk v2
In the bulk of the fluid the oscillating velocity and density
fields v1 and ρ1 are out of phase by π/2. Consequently
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