40 CHAPTER 4. ACOUSTOFLUIDIC FORCES ON PARTICLES
where ext is the stress exerted by the external ow. The radiation force is usually dened
by the time-average of Eq. (4.3) for a periodic state, where all transients have diminished
16. In the limit where the particle radius a is much smaller than the acoustic wavelength
a , relevant for microparticle acoustophoresis, the time-averaged radiation force can be
expressed in terms of the incoming acoustic elds pin and vin and the scattering coecients
f0 and f1 15,
Frad
= ��a3
2s
3
Re
n
f 0 pinrpin
o
�� 0 Re
n
f 1 vin rvin
o
: (4.4)
The scattering coecients f0 and f1 depend on the thermodynamic and transport properties
of the uid and the particle. For a thorough study of this, including eects of both
viscous and thermal boundary layers, the reader is referred to Karlsen and Bruus Ref. 16.
In the limit where the particle radius is much larger than the thicknesses of the viscous
and thermal boundary layers, a s and a t, the scattering coecients become
f0 = 1 �� ~s; ~s =
ps
s
; (4.5a)
f1 =
2(~ �� 1)
2~ + 1
; ~0 =
p
0
0
; (4.5b)
where superscript p refers to the properties of the particle. For a one-dimensional
standing acoustic wave of pressure p1(r; t) = Re
n
�� pa sin(k0y)e��i!t
o
, velocity v1(r; t) =
Re
n
iva cos(k0y)e��i!tey
o
and acoustic energy density Eac = 1
4sp2a
= 1
40v2
a, the acoustic
radiation force
Frad
1D
becomes
Frad
1D
= ��4a3k0Eac sin(2k0y)ey; (4.6a)
=
1
3
Re
n
f0
o
+
1
2
Re
n
f1
o
; (4.6b)
where is referred to as the contrast factor. When is positive, the radiation force is
directed towards the pressure nodes, and the particles are referred to as acoustically hard,
which is the case for particles with low compressibility and high mass density. All particles
treated in this thesis are acoustically hard.
The Stokes drag force induced by the external velocity eld vext is given by
Fdrag =
I
@
(t)
vext �� u
ext n dA = 6a
: (4.7)
Assuming periodic acoustic elds and an initially quiescent state of the uid, the timeaveraged
drag force becomes
vext
Fdrag
= 6a
2
��
u
; (4.8)
where
vext
2
is the steady external streaming velocity and
u
is the velocity of the particle
averaged over one oscillation period of the acoustic wave.