13
up. We have also found that for a resonance with quality
factor Q, the amplitude of the oscillatory second-order
velocity component is a factor of Q larger than what is
expected from dimensional analysis, which results in a
more restrictive criterion for the validity of the perturbation
expansion, compared to the usual one based on the
first-order perturbation expansion.
Furthermore, contrary to a simple scaling analysis of
the time scales involved in the fast build-up of radiation
forces and slow build-up of drag-induced streaming
forces, we have found that pulsating oscillatory boundary
actuation does not reduce the time-integrated streaminginduced
drag force relative to the time-integrated radiation
force. As a result, pulsating actuation does not prevent
streaming flows perpendicular to the pressure nodal
plane from destroying the ability to focus small particles
by acoustophoresis.
ACKNOWLEDGMENTS
This work was supported by the Danish Council for
Independent Research, Technology, and Production Sciences
(grant no. 11-107021).
Appendix A: Amplitude of the second-order
oscillatory velocity field
Extending to second order the one-dimensional example
given in Ref. 30, we derive in this appendix the order
of magnitude of the second-order oscillatory component
v2!
2 , which was stated in Eq. (27).
Like
g2
denotes time-averaging over one oscillation
period, Eq. (30), and in the periodic state equals the
zero-order temporal Fourier component of the field, then
g2!
2 (r) denotes the complex amplitude of the oscillatory
second-order mode and is given by the second-order
Fourier component
g2!
2 (r) =
1
T
Z t+T/2
t−T/2
g2(r, t′)e−i2!t′ dt′. (A1)
By using the general formula for the real part
of any complex number Z, ReZ = 1
2 (Z + Z∗),
the product A(r, t)B(r, t) of two oscillating fields
A(r, t) = Re
Ae−i!t
and B(r, t) = Re
Be−i!t
can be
decomposed into a steady component and an oscillatory
component
A(t)B(t) = 1
2
Ae−i!t + A∗ei!t
1
2
Be−i!t + B∗ei!t
= 1
2 Re
h
A∗B
i
+ 1
2 Re
h
ABe−i2!t
i
, (A2)
from which we introduce the following notation
AB
≡ 1
2 Re
h
A∗B
i
, (A3)
��
AB
2!
≡ 1
2AB, (A4)
where A and B could be any first-order fields.
The governing equations for the oscillatory secondorder
component v2!
2 can be derived from Eqs. (7)
and (8) and in the one-dimensional problem treated in
Ref. 30, where the top and bottom walls are not taken
into account, they become
��
2!
−i2!sp2!
2 = −@yv2!
2 − s
v1@yp1
(A5a)
−i2!0v2!
2 = −@yp2!
2 +
�� 4
3 + b
@ 2
y v2!
2
−
��
1(−i!v1)
2!
− 0
��
v1@yv1
2!
. (A5b)
Applying the 2!-rule of Eq. (A4) and the mass continuity
Eq. (5), the two last terms of Eq. (A5b) cancel. Inserting
Eq. (A5a) into Eq. (A5b), the governing equation for v2!
2
becomes
4k2
0 v2!
2 + (1 − i4��)@ 2
y v2!
2 +
1
2
s@y(v1@yp1) = 0, (A6)
where �� is the non-dimensional bulk damping coefficient
given by �� = !
20c2s
4
3 + b
, and k0 = !
cs
is the
wavenumber. For the fundamental half-wave resonance,
the spatial dependence of the source term @y(v1@yp1) is
sin(2k0y), and the guess for the inhomogeneous solution
to Eq. (A6) thus becomes
v2!,inhom
2 = C sin(2k0y). (A7)
Inserting the inhomogeneous solution Eq. (A7) into the
governing equation (A6), we note that the first term cancels
with the “1” in the parentheses of the second term,
and the order of magnitude of the inhomogeneous solution
thus becomes
|v2!
2 | = C ∼
1
��
s|v1||p1| ∼
1
��3
v2b
c
cs ∼ Q3 v2b
c
cs
, (A8)
which is the result stated in Eq. (27).
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J. Nilsson, S. Radel, S. Sadhal, and M. Wiklund,
Lab Chip 11, 3579 (2011).
2 P. B. Muller, R. Barnkob, M. J. H. Jensen, and H. Bruus,
Lab Chip 12, 4617 (2012).
3 R. Barnkob, P. Augustsson, T. Laurell, and H. Bruus,
Phys Rev E 86, 056307 (2012).
4 L. Rayleigh, Philos Trans R Soc London 175, 1 (1884).
5 H. Schlichting, Phys Z 33, 327 (1932).
6 W. L. Nyborg, J Acoust Soc Am 30, 329 (1958).