ULTRASOUND-INDUCED ACOUSTOPHORETIC MOTION OF . . . PHYSICAL REVIEW E 88, 023006 (2013)
FIG. 2. (Color online) Analytical results for the streaming velocity
v2 in parallel-plate channels. (a) Plot of the analytical
expressions (19) and (20) for v2 (arrows) and its magnitude color
plot from 0 (black) to vstr (white) in the vertical yz cross section
of a parallel-plate channel (Fig. 1) with λ/2 = w (n = 1) and aspect
ratio α = 1.2. (b) The same as (a), but for α = 0.2. (c) The same as
(b) but for a standing full wave λ = w (n = 2). (d) Line plot of the
amplitude v2y ( ˜ y,0) of the streaming velocity, in units of vstr, along
the first half of the center axis white dashed lines in (a) and (b) with
λ/2 = w for aspect ratios α = 0.2, 0.5, 0.8, and 1.2. (e) Line plot of
the maximum v2y ( ˜ y,0)max of the center-axis streaming velocity, in
units of vstr, as function of aspect ratio for the resonances nλ/2 = w,
with n = 1, 2, and 3, respectively.
horizontal center line ˜z= 0. However, the amplitude of this
velocity decreases for increasing aspect ratio α as shown in
Fig. 2(d).
This special case of the pure sinusoidal horizontal boundary
condition (18a) can readily be generalized to any horizontal
boundary condition by a Fourier expansion in wave number
km = 2π/λm = mπ/w, where m is a positive integer,
v2y = vstr f (˜y
) for ˜z
= ±1, (22a)
f (˜y
) =
∞
m=1
am sin(mπ˜y
). (22b)
As the governing equations (16) for the second-order bulk
flow are linear, we can make a straightforward generalization
of Eq. (19), and the two velocity components of the superposed
solution for v2 become
v2y ( ˜ y,˜z
) = vstr
∞
m=1
am sin(mπ˜y
) A
(mα,˜z
), (23a)
v2z( ˜ y,˜z
) = vstr
∞
m=1
am cos(mπ˜y
) A
⊥(mα,˜z
), (23b)
where the wave index m multiplies both the horizontal
coordinate ˜y
and the aspect ratio α. Note that A
(mα,±1) = 1
and A
⊥(mα,±1) = 0. The resulting steady effective pressure
χ is just the weighted sum
of the partial pressures χm of each
Fourier component =
∞
χ m=1 amχm.
In Fig. 2(c) is shown the streaming velocity field for
the higher harmonic boundary condition f (˜y
) = sin(nπ˜y
)
with n = 2. Furthermore, Fig. 2(e) shows how the maximum
v2y ( ˜ y,0)max of the center-axis streaming velocity decays as
function of aspect ratio α for n = 1, 2, and 3. Given sufficient
room, the flowrolls decay in the vertical direction on the length
scale of λn/4. Since n is the number of half wavelengths
of the first-order resonance pressure across the channel, we
conclude that the streaming amplitude in the center of the
channel decreases for higher harmonic resonances.
F. Streaming in a rectangular channel
Moving on to the rectangular channel cross section, we
note that the only change in the problem formulation is to
substitute the symmetry boundary conditions (18c) and (18d)
by no-slip boundary conditions, while keeping the top-bottom
slip boundary conditions (18a) and (18b) unaltered:
v2y = vstr sin(π˜y
) for ˜z
= ±1, (24a)
v2z =0 for ˜z
= ±1, (24b)
v2y =0 for ˜y
= ±1, (24c)
v2z =0 for ˜y
= ±1. (24d)
If we want to use the solution obtained for the parallel-plate
˜y
channel, we need to cancel the vertical velocity component
v2z on the vertical walls at = ±1. This leads us to consider
the problem rotated 90◦, where the first-order velocity field is
parallel to the vertical walls (interchanging y and z), and the
fundamental wavelength is λ/2 = h, and the aspect ratio is
w/h = α
−1. As the governing equations for the bulk flow (16)
are linear, we simply add this kind of solution to the former
solution and determine the Fourier expansion coefficients such
that the boundary conditions (24) are fulfilled. Given this, (23)
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