(ii) The time-averaged second-order flow Sv2T is calculated by
implementing eqn (9) in the Laminar Flow Physics Interface,
modified to include the addition of the time-averaged first-order
products from step (i) on the right-hand sides: the right-hand
side of eqn (9a) is included as a mass source term by adding a
so-called weak contribution to the governing equations,
{
1
2
ð
V
½LxRe(r1v
1x)zLyRe(r1v
1y)~p2 dV (p˜2 being the pressure test
function), while the right-hand side of eqn (9b) is added
straightforwardly as a body force term. Furthermore, the fourthorder
non-linear term r0(Sv2T?+)Sv2T is kept in the laminar flow
equations in COMSOL to enhance numerical stability.
(iii) The acoustic radiation forces are determined using eqn
(11) with the first-order fields of step (i).
(iv) Finally, the time-dependent motion of the particles is
determined using the COMSOL Particle Tracing Module only
taking into account the radiation force of step (iii) and the drag
force of eqn (14).
The solution strategy was carried out on a computational
mesh large enough for all dependent variables to reach
convergence, while taking special care to properly resolve the
acoustic boundary layer with an adequate computational mesh,
see Section III E. This fine mesh was used when determining the
first-order fields and the time-averaged second-order fields. In
the subsequent simulation of the time-dependent particle motion,
the flow field and radiation forces were interpolated to a coarser
mesh to speed up the transient solving procedure substantially.
D Computer hardware requirements
The computation was performed on a DELL Precision 7500
workstation running Windows 7 (64-bit) equipped with 48 GB
RAM (1333 MHz) and two hexa-core Intel Xeon X5650
processors of clock frequency 2.66 GHz and 12 MB cache.
When calculating the first-order acoustic fields in step (i), we
used the mesh found by the mesh-convergence analysis described
in the following subsection, and this resulted in about 3 6 106
degrees of freedom, a calculation time of 4.5 min, and a peak
RAM usage of 64% or 31 GB. The calculation of the secondorder
acoustic fields in step (ii) required around 5 6 105 degrees
of freedom and took 2 min, while having a peak RAM usage of
19% or 9 GB. The computation time for steps (iii) and (iv) was
less than 15 s for calculation of 144 particle trajectories of
100 time steps and solved on a coarser mesh resulting in about
9 6 104 degrees of freedom.
E Mesh convergence
The computational mesh is generated from a maximum element
size length dmesh at the domain boundaries hV and a maximum
element size in the bulk of the domain V given by 10dmesh. For
illustrative purposes, the computational mesh shown in Fig. 3(a)
is a coarse mesh with 1204 elements and dmesh = 20d, or d/dmesh =
0.05, where d is the boundary layer thickness defined in eqn (7).
In order to verify the correctness of the solution, a meshconvergence
analysis is required. The solutions are compared for
decreasing mesh element size dmesh to determine the point at
which the solution becomes independent of any further decrease
of dmesh. We define a relative convergence parameter C(g) for a
solution g with respect to a reference solution gref taken to be the
Fig. 3 (a) The computational mesh for a maximum element size of dmesh
= 20d at the boundaries, resulting in a coarse mesh with only 1204
triangular elements. (b) Semi-logarithmic plot of the relative convergence
parameter C, eqn (16), for the physical fields as the size of the mesh
elements is decreased. The dashed line indicates a threshold of C = 0.002,
chosen as a trade off between accuracy and computational time. For the
second-order velocity field to get below this convergence threshold, a
maximum element size of dmesh = 0.5d or d/dmesh = 2.0 is needed at the
boundaries of the domain (dash-dotted line).
solution for the smallest value of dmesh,
C(g)~
ffiÐffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (g{gref )2 dy dz
Ð (gref )2 dy dz
s
: (16)
For gref we have chosen dmesh = 0.3d or d/dmesh = 3.3, which
resulted in 2.6 6 105 triangular mesh elements.
The exponential convergence of both first- and second-order
fields for dmesh , d shows up as straight lines in the semilogarithmic
plots of Fig. 3(b). The time-averaged second-order
velocity field Sv2T converges considerably slower than the firstorder
fields, as it depends on the gradients of the first-order
fields, and thus demands better resolution. In order to obtain a
relative convergence of the second-order velocity field below
0.002 (dashed line), a maximum element size of dmesh = 0.5d or
d/dmesh = 2.0 is needed at the boundaries. This mesh size, which
results in 1.2 6 105 triangular elements, is used for the results
presented in this paper.
IV Results
The following results are aimed at showing the insensitivity of
the horizontal half-wave resonance to the specific form of the
ultrasound actuation, at characterizing the first- and secondorder
acoustic fields, and at investigating the dependence of the
acoustophoretic microparticle motion on system geometry and
material parameters.
This journal is The Royal Society of Chemistry 2012 Lab Chip, 2012, 12, 4617–4627 | 4621
Downloaded by DTU Library on 27 February 2013
Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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