A Actuation
The ultrasound actuation is modeled using the velocity boundary
condition in eqn (4c) at the frequency 1.97 MHz to excite the
horizontal half-wave resonance in our model system Fig. 2(b).
Using the following three actuation modes for the velocity
boundary condition vbc in eqn (4c),
vA
bc(¡w/2,z) = ¡vA
bc;0, (17a)
vB
bcðy,{h=2Þ ~ vB
bc,0 sin 2p
y
w
, (17b)
vC
bc ðy,{h=2Þ ~ vC
bc,0
1
1
2
{
y
w
2
2
z
y
w
, (17c)
we show the expected result that this resonance is indeed excited
regardless of the detailed spatial dependence of vbc as long as the
oscillation frequency equals the resonance frequency.
For all three actuation modes, the amplitude of the velocity
boundary conditions is chosen in such a way that the line integral
of the absolute value |vbc| of the velocity along the perimeter hV
of the domain V is given in terms of the angular frequency v and
a characteristic value d0 of the displacement of an actuated
boundary,
þ
LV
jvX
bcj d‘~2hvd0, X~A,B,C: (18)
In particular, the normalization constant 2h is chosen so that
vA
bc;0= vd0, with d0 = 0.1 nm, a typical value of displacements,38
which results in resonance amplitudes in the range of those
measured in typical experiments.7,39–41
The first-order pressure fields resulting from the three different
actuation modes are shown in Fig. 4. It is seen that all of the
actuation modes excite the horizontal half-wave 1.97-MHz
resonance. Although the velocity boundary conditions have
been normalized, the amplitude of the resonance is different for
each of the three actuation modes, i.e. each actuation mode
couples to the resonance mode with its own strength. In the
studies presented in the rest of this paper, we have used the
velocity boundary condition eqn (17a), shown in Fig. 4(a), due to
its simplicity and strong coupling to the resonance mode.
B First-order fields
We now turn to a study of the first-order fields resulting from the
velocity boundary condition eqn (17a) and Fig. 4(a). In Fig. 5,
color plots of the pressure p1, temperature T1, horizontal velocity
Fig. 4 Three different actuation modes vbc (magenta arrows) of the
water-filled cavity. Color plot of the first-order pressure field p1 resulting
from the actuation, eqn (17). In all cases the actuation frequency is
1.97 MHz, corresponding to the lowest resonance frequency of the cavity,
and it is seen that all three actuation modes excite the horizontal half-wave
resonance. The pressure amplitude of the resonance mode is (a) 0.24 MPa
with side-wall actuation, (b) 0.16 MPa with anti-symmetric bottom-wall
actuation, and (c) 0.06 MPa with non-symmetric bottom-wall actuation.
Fig. 5 Color plots of the amplitudes of the oscillating first-order fields
in the water-filled channel at the horizontal standing half-wave 1.97-MHz
resonance excited by velocity boundary condition eqn (17a): (a)
pressure p1, identical to panel (a) in Fig. 4, (b) temperature T1, (c)
horizontal velocity v1y, and (d) vertical velocity v1z. The horizontal
velocity is much larger than the vertical velocity, arising because of
the interaction of the acoustic resonance with the bottom and top
walls. The sub-micrometer thin viscous boundary cannot be seen on
the 100-mm scale of the plot. The dashed white lines indicate the
domain for the line plots in Fig. 6.
4622 | Lab Chip, 2012, 12, 4617–4627 This journal is The Royal Society of Chemistry 2012
Downloaded by DTU Library on 27 February 2013
Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
View Article Online