P. B. MULLER et al. PHYSICAL REVIEW E 88, 023006 (2013)
TABLE I. Model parameters for water and polystyrene given at
temperature T = 25 ◦C and taken from the literature as indicated or
derived based on these.
Parameter Symbol Value Unit
Water
Densitya ρ0 998 kg m−3
Speed of sounda c0 1495 m s−1
Viscositya η 0.893 mPa s
Specific heat capacitya cp 4183 J kg−1 K−1
Heat capacity ratio γ 1.014
Thermal conductivitya kth 0.603 Wm−1 K−1
Thermal diffusivity Dth 1.44×10−7 m2 s−1
Compressibility κs 448 TPa−1
Thermal expansion coeff. αp 2.97×10−4 K−1
Thermal viscosity coeff.a (∂T η)p
η0
−0.024 K−1
Polystyrene
Densityb ρps 1050 kg m−3
Speed of soundc cps 2350 m s−1
Poisson’s ratiod σps 0.35
Compressibilitye κps 249 TPa−1
aCOMSOL Multiphysics Material Library 40.
bReference 41.
cReference 42.
dReference 43.
eCalculated as κps = 3(1−σps)
1+σps
1
ρpsc2
ps
(see Ref. 44).
ρ1v1 = 0, and the source term in the second-order continuity
equation (9a) vanishes. As a result, the time-averaged secondorder
velocity field v2 is divergence free or incompressible in
the bulk. Hence, the continuity equation and the Navier-Stokes
equation for the bulk streaming velocity field reduce to
∇ · v2 = 0, (14a)
η0∇2v2 − ∇p2 = ρ1∂tv1 + ρ0(v1 · ∇)v1
−η1∇2v1 − βη1∇(∇ · v1)
−∇η1 · ∇v1 + (∇v1)T
−(β − 1)(∇ · v1)∇η1. (14b)
Only the y component of the source terms on the right-hand
side of Eq. (14b) is nonzero in the bulk, and it depends only on
y and not on z. Consequently, their rotation is zero, and they
can be reformulated as a gradient term absorbed together with
∇p2 into an effective pressure gradient ∇χ given by
∇χ = ∇p2 + ρ1∂tv1 + ρ0(v1 · ∇)v1
−η1∇2v1 − βη1∇(∇ · v1)
−∇η1 · ∇v1 + (∇v1)T
−(β − 1)(∇ · v1)∇η1. (15)
Using this, the system of bulk equations reduces to the standard
equation of incompressible creeping flow
∇ · v2 = 0, (16a)
η0∇2v2 = ∇χ. (16b)
These equations together with appropriate boundary conditions,
to be discussed below, govern the steady bulk streaming
velocity field v2 in the microchannel.
E. Streaming in a parallel-plate channel
Based on Rayleigh’s analysis, we first study the analytical
solution for v2 in the special case of a standing half-wave
(n = 1) in the parallel-plate channel shown in 1. We choose
the symmetric coordinate system such that −w/2 < y <
w/2 and−h/2 < z < h/2, and introduce nondimensionalized
coordinates ˜y
and ˜z
by
˜y
= 2y
w
with − 1 < ˜y < 1, (17a)
˜z
= 2z
h
with − 1 < ˜z < 1, (17b)
α = h
w
the aspect ratio. (17c)
˜z
In this case, using Eq. (11), the boundary conditions for
v2( y,˜ ) are
v2y = vstr sin(π˜y
) for ˜z
= ±1, (18a)
v2z =0 for ˜z
= ±1, (18b)
v2y =0 for ˜y
= ±1, (18c)
∂y v2z =0 for ˜y
= ±1, (18d)
˜y
where Eqs. (18c) and (18d) express the symmetry condition at
the wall-less vertical planes at = ±1. Rayleigh focused his
analysis of the parallel plate geometry on shallow channels for
which α 1. Here, α = 0.4, derived from the aspect ratio of
the microchannel described in Sec. III and in Refs. 26,28,45,
and consequently we need to solve the case of arbitrary α. We
find
v2y ( ˜ y,˜z) = vstr sin(π˜y
) A
(α,˜z
), (19a)
v2z( ˜ y,˜z
) = vstr cos(π˜y
) A
⊥(α,˜z
), (19b)
and A
where the α- and z-dependent amplitude functions A
⊥
for the velocity component parallel and perpendicular to the
first-order wave, respectively, are given by
A
(α,˜z
) = B(α){1 − πα coth(πα) cosh(πα˜z
)
+πα˜z
sinh(πα˜z
)}, (20a)
A
⊥(α,˜z
) = παB(α){coth(πα) sinh(πα˜z
)
−˜z
cosh(πα˜z
)}, (20b)
B(α) = sinh(πα)
sinh(πα) cosh(πα) − πα
, (20c)
(α,±1) = 1 and A
⊥(α,±1) = 0. In Rayleigh’s wellcited
shallow-channel limit α 1, the amplitude functions
with A
reduce to
(α,˜z
) ≈ 3
A
2
˜z
2 − 1
2
for α 1, (21a)
A
⊥(α,˜z
) ≈ πα
2
(˜z
− ˜z
3) for α 1. (21b)
The analytical solution of v2 for λ/2 = w is illustrated in
Figs. 2(a) and 2(b) for channel aspect ratios α = 1.2 and 0.2.
We note that the maximum streaming velocity is near the
top and bottom walls. For the shallow channel Fig. 2(b),
there is furthermore a significant streaming velocity along the
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