P. B. MULLER et al. PHYSICAL REVIEW E 88, 023006 (2013)
generalizes to
v2y ( ˜ y,˜z
) = vstr
∞
am sin(mπ˜y
m=1
) A
(mα,˜z
)
⊥(mα
+bmA
−1,˜y
) cos(mπ˜z
), (25a)
v2z( ˜ y,˜z
) = vstr
∞
am cos(mπ˜y
m=1
) A
⊥(mα,˜z
)
(mα
+bmA
−1,˜y
) sin(mπ˜z
). (25b)
The two perpendicular-to-the-wall velocity conditions
(24b) and (24c) are automatically fulfilled as they by
construction are inherited from the original conditions (18b)
and (18c). The unknown coefficients am and bm are thus to
be determined by the parallel-to-the-wall conditions (24a)
and (24d).
Using v2y in the form of Eq. (25a), boundary condition
(24a) becomes
sin(π˜y
) =
∞
am sin(mπ˜y
m=1
⊥(mα
) + (−1)mbmA
−1,˜y
). (26)
The discrete Fourier transform of this equation, i.e., multiplying
by sin(jπ˜y
), where j is an arbitrary integer, and integrating
over ˜y
from −1 to 1, becomes
δj,1 =
∞
δj,m am + A⊥
m=1
j,m(α
−1) bm, j = 1,2,3, . . . , (27)
where the (j,m)th element A⊥
j,m of the α-dependent matrix A⊥
is given by
A⊥
j,m(α
−1) = (−1)m
1
−1
d ˜y A
⊥(mα
−1,˜y
) sin(jπ˜y
). (28)
Introducing the coefficient vectors a and b and the first unit
vector e1 with mth components am, bm, and δ1,m, respectively,
we can write Eq. (27) as the matrix equation
e1 = a + A⊥(α
−1) · b. (29)
Likewise, using Eq. (25b) and multiplying it by sin(jπ˜z
),
where j is an arbitrary integer, and integrating over ˜z
from −1
to 1, the zero-parallel-component boundary condition (24d)
can be written as the matrix equation
0 = A⊥(α) · a + b. (30)
Solving the equation systems (29) and (30), the coefficient
vectors a and b become
a = I − A⊥(α
−1)A⊥(α)−1· e1, (31a)
b = −A⊥(α) · a. (31b)
A comparison between results for the classical parallel-plate
geometry and the new results for the rectangular geometry
is shown in Fig. 3. It is seen how the velocity profile of the
rectangular channel solution Eq. (25) is suppressed close to
the wall in comparison to the parallel-plate channel solution
Eq. (19). Note that for the nth resonance kn = nπ/w, the unit
vector e1 in Eq. (31a) is replaced by (−1)n−1 en, with the sign
originating from the n-dependent phase shift in the streaming
boundary condition (11).
FIG. 3. (Color online) Analytical results comparing the streaming
velocity field v2 in the parallel plate and the rectangular channel.
(a) Color plot from 0 (black) to vstr (white) of the analytical expression
for v2 Eqs. (19) and (20) in the classical parallel-plate geometry
with a half-wave resonance λ/2 = w (n = 1). Due to symmetry, only
the left half (−1 < ˜y < 0) of the vertical channel cross section is
shown. (b) As in (a) but for v2 in the rectangular channel Eqs. (25)
and (31), including the first 20 terms of the Fourier series. (c) Line
plots of v2y ( ˜ y,0) in units of vstr along the left half of the center
line for the parallel-plate channel (dashed lines) and the rectangular
channel (full lines) for aspect ratios α = 0.1, 0.4, and 0.8 and the
half-wave resonance λ/2 = w. (d) As in (c) but for the full-wave
resonance λ = w (n = 2).
G. Acoustophoretic particle velocity
The forces of acoustic origin acting on a single microparticle
of radius a, density ρp, and compressibility κp undergoing
acoustophoresis with velocity up in a liquid of density ρ0,
compressibility κs , and viscosity η0 are the Stokes drag force
Fdrag = 6πη0av2 − up from the acoustic streaming v2
and the acoustic radiation force Frad. Given an observed maximum
acoustophoretic velocity of up 1mm/s for the largest
particles of diameter 2a = 5.0 μm, the Reynolds number for
the flow around the particle becomes ρ02aup/η 6 × 10−3,
and the time scale for acceleration of the particle becomes
τacc = (4/3)πa3ρp/6πηa ≈ 2 μs. Since the acceleration
time is much smaller than the time scale for the translation
of the particles τtrans = w/(2up) 0.1 s, the inertia of the
particle can be neglected, and the quasi-steady-state equation
of motion Fdrag = −Frad for a spherical particle of velocity
up then becomes
up = Frad
6πη0a
+ v2 = urad + v2, (32)
where urad is the contribution to the particle velocity from the
acoustic radiation force. The streaming velocity v2 is given
in the previous sections, while an analytical expression for
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