to the detailed geometry and boundary conditions, but part of
the more conceptual difficulties with acoustic streaming in
acoustofluidics may be related to the lack of theoretical analysis
in the experimental relevant limit where the microchannel height
h is equal to one or a few times half the acoustic wavelength l,
i.e. h . l. The classical Rayleigh–Schlichting boundary-layer
theory for acoustic streaming,17–20 see Fig. 1, is valid in the limit
of thin boundary layers in medium-sized channels, d % h % l,
and a later extension13 is valid in the limit of thin boundary
layers in shallow channels, d . h % l. Moreover, in contrast to
rectangular channel cross sections of experimental relevance, the
classical analysis of the parallel-plate channel and recent
numerical studies of it21 do not include the effects of the vertical
side walls. One exception is the special case of gases in shallow,
low-aspect-ratio channels studied by Aktas and Farouk.22
The push within contemporary acoustofluidics for handling
smaller particles like bacteria, viruses, and large biomolecules,
and for doing so with better accuracy, emphasizes the urgency of
performing a numerical analysis of microparticle acoustophoresis
including acoustic radiation forces, streaming flows, and
boundary layers. Based directly on the governing equations, we
provide such an analysis in this paper for a simple, yet
experimentally relevant microsystem. In Section II we present
the governing thermoacoustic equations to first and second order
in the external ultrasound actuation. In Section III we describe
the model system, the numerical implementation of it, as well as
mesh-convergence analysis. In Section IV this is followed by the
results for first-order fields, time-averaged second-order fields,
and microparticle acoustophoresis as function of particle size
and material properties. We end with a concluding discussion in
Section V.
II Governing equations
The governing perturbation equations for the thermoacoustic
fields are well-known textbook material.23–25 The full acoustic
problem in a fluid, which before the presence of any acoustic
wave is quiescent with constant temperature T0, density r0, and
pressure p0, is described by the four scalar fields pressure p,
temperature T, density r, and entropy s per mass unit as well as
the velocity vector field v. Changes in r and s are given by the
two thermodynamic relations
dr = ck r dp 2 a r dT, (1a)
ds~
Cp
T
dT{
a
r
dp, (1b)
which, besides the specific heat capacity Cp at constant pressure,
also contain the specific heat capacity ratio c, the isentropic
compressibility k, and the isobaric thermal expansion coefficient
a given by
c ~
Cp
CV
, (2a)
k ~
1
r
Lr
Lp
s
, (2b)
a ~{
1
r
Lr
LT
p
: (2c)
Eqn (1) can be used to eliminate r and s, so that we only need to
deal with the acoustic perturbations in temperature T, pressure p,
and velocity v. Taking first and second order (subscript 1 and 2,
respectively) into account, we write the perturbation series
T = T0 + T1 + T2, (3a)
p = p0 + p1 + p2, (3b)
v = v1 + v2. (3c)
We model the external ultrasound actuation through boundary
conditions on the first-order velocity v1 while keeping the
temperature constant,
T = T0, on all walls, (4a)
v = 0, on all walls, (4b)
n?v1 = vbc(y,z)e2ivt, added to actuated walls. (4c)
Here n is the outward pointing surface normal vector, and
v is the angular frequency characterizing the harmonic time
dependence.
A First-order equations
To first order in the amplitude of the imposed ultrasound field,
the thermodynamic heat transfer equation for T1, the kinematic
continuity equation expressed in terms of p1, and the dynamic
Navier–Stokes equation for v1, become
LtT1 ~ Dth +2T1 z
aT0
r0Cp
Ltp1, (5a)
Fig. 1 A sketch of the classical Rayleigh-Schlichting streaming pattern
in a liquid-filled gap of height h between two infinite, parallel rigid walls
(black lines). The bulk liquid (light blue) supports a horizontal standing
sinusoidal pressure wave (magenta line) of wavelength l in the horizontal
direction parallel to the walls. In the viscous boundary layers (dark blue)
of sub-micrometer thickness d, large shear stresses appear, which
generate the boundary-layer (Schlichting) streaming rolls (yellow).
These then drive the bulk (Rayleigh) streaming rolls (red). The streaming
pattern is periodic in the horizontal direction with periodicity l/2, and
thus only the top and bottom walls are subject to the no-slip boundary
condition.
4618 | Lab Chip, 2012, 12, 4617–4627 This journal is The Royal Society of Chemistry 2012
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Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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