Ltp1 ~
1
ck
½aLtT1 {+:v1, (5b)
r0htv1 = 2 +p1 + g+2v1 + bg+(+?v1). (5c)
Here, Dth is the thermal diffusivity, g is the dynamic viscosity,
and b is the viscosity ratio, which has the value 1/3 for simple
liquids.23 A further simplification can be obtained when
assuming that all first-order fields have a harmonic time
dependence e2ivt inherited from the imposed ultrasound field
(eqn (4c)), because then p1 is eliminated by inserting eqn (5b)
with htp1 = 2ivp1 into eqn (5a) and (c). After using the
thermodynamic identity26 T0a2/(r0Cpk) = c 2 1, we arrive at
ivT1 z cDth +2T1 ~
c{1
a
+:v1, (6a)
ivv1 z n+2v1 zn b z i
1
cr0knv
+ð+:v1Þ~
a
cr0k
+T1, (6b)
where n = g/r0. From eqn (6) arise the thermal and the viscous
penetration depth dth and d, respectively (values for ultrasound
waves at 2 MHz in water at 25 uC),
r
dth~
ffiffiffiffiffiffiffiffiffiffi
2Dth
v
~0:15 mm, and d~
ffiffiffiffiffi
2n
v
r
~0:38 mm: (7)
These are the length scales over which the thermoacoustic fields
change from their bulk values to the boundary conditions of the
rigid walls stated in eqn (4).
B Second-order, time-averaged equations
For water and most other liquids, the thermal effects in the
above first-order equations are minute because of the smallness
of the pre-factor c 2 1 # 1022 and dth/d # 0.3. To simplify the
following treatment, we therefore neglect the coupling in the
second-order equations between the temperature field T2 and
the mechanical variables v2 and p2. Furthermore, the values of g
and b are kept fixed at the ones given at T = T0. The secondorder
continuity equation and Navier–Stokes equation are
htr2 = 2r0+?v2 2 +?(r1v1), (8a)
r0htv2 = 2+p2 + g+2v2 + bg+(+?v2)
2 r1htv1 2 r0(v1?+)v1, (8b)
and consequently, thermal effects enter solely through the
temperature-dependent first-order fields r1 and v1.
In a typical experiment on microparticle acoustophoresis, the
microsecond timescale 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 S…T, of the second-order
continuity equation and Navier–Stokes equation becomes27
r0+?Sv2T = 2+?Sr1v1T, (9a)
g+2Sv2T + bg+(+?Sv2T) 2 S+p2T
= Sr1htv1T + r0S(v1?+)v1T. (9b)
It is seen that products of first-order fields act as source terms (at
the right-hand sides) for the second-order fields (at the left-hand
sides). We note that for complex-valued fields A(t) and B(t) with
harmonic time-dependence e2ivt, the time average is given by the
real-part rule SA(t)B(t)T~
1
2
Re½A(0) B(0), where the asterisk
represents complex conjugation.
The second-order problem was solved in the case of the
infinite parallel-plate channel by Rayleigh,17,20 see Fig. 1.
Assuming a first-order bulk velocity field with only the
horizontal y-component v1y being non-zero and of the form v1y
= U1cos(2py/l)e2ivt, the resulting y-component Svbnd
2y T of Sv2T
just outside the boundary layers (in our notation at z # ¡h/2 2
3d), becomes
Svbnd
2y T~
3
8
U2
1
c0
sin 4p
y
l
, (10)
Relative to the first-order bulk velocity v1y, the second-order
field Svbnd
2y T is phase shifted by p/2, period doubled in space, and
smaller by a factor of U1/c0, where c0 is the speed of sound of the
liquid.
C Time-averaged acoustic forces on a single suspended
microparticle
Once the first- and second-order acoustic fields have been
calculated, it is possible to determine the time-averaged acoustic
forces on a single suspended particle. These are the acoustic
radiation force F rad due to the scattering of acoustic waves on
the particle and the Stokes drag force F drag from the acoustic
streaming.
The time-average acoustic radiation force F rad on a single
small spherical particle of radius a, density rp, and compressibility
kp in a viscous fluid is given by11
F rad ~{pa3 2k0
3
Re f1
p1
½ :+v1
½ +p1 { r0Re f2
v1
, (11)
where k0 = 1/(r0c0
2) is the compressibility of the fluid, and where
the pre-factors f1 and f2 are given by
f1ðk~Þ ~ 1{k~, with k~ ~
kp
k0
, (12a)
f2 ~r,~d
~
h i
2 1{C ~d
ðr~{1Þ
, with ~r ~
2~rz1{3C ~d
rp
r0
, (12b)
C ~d
~{
3
2
h i~ d, with ~d
1zi 1z~d
~
d
a
, (12c)
For the special case of the horizontal pressure half-wave
resonance, p1 = pasin(qy), with channel width w and wavenumber
q = p/w, the acoustic energy density is Eac~
1
4
k0p2
a~
1
4
r0U2
1. The
expression for the radiation force then simplifies to
This journal is The Royal Society of Chemistry 2012 Lab Chip, 2012, 12, 4617–4627 | 4619
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Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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