4
where Ppump is the total power delivered by the forced
vibration of the sidewalls, and Pdis is the total power
dissipated due to viscous stress. The quality factor Q of
a resonant cavity is given by
Q = 2
Energy stored
Energy dissipated per cycle
= !
Efd
ac
Pfd
dis
. (15)
G. Summary of theory
Throughout this paper we refer to two kinds of solutions
of the acoustic energy and velocity fields: unsteady
non-periodic solutions obtained from Eqs. (5)-(8)
and steady periodic solutions obtained from Eqs. (9)-
(12). When presenting the unsteady non-periodic solutions,
they are often normalized by the steady periodic
solution, to emphasize how close it has converged towards
this solution.
III. NUMERICAL MODEL
The numerical scheme solves the governing equations
for the acoustic field inside a water domain enclosed by
a two-dimensional rectangular microchannel cross section.
The vibrations in the surrounding chip material and
piezo transducer are not modeled. The water domain is
surrounded by immovable hard walls, and the acoustic
field is excited by oscillating velocity boundary conditions,
representing an oscillating nm-sized displacement
of the walls. A sketch of the numerical model is shown in
Fig. 1(a). We exploit the symmetry along the horizontal
center axis z = 0, reducing our computational domain
by a factor of two. The system is also symmetric about
the vertical center axis y = 0, however, our attempts to
use this symmetry introduced numerical errors, and consequently
it was not exploited in the numerical model.
The model used to calculate the steady streaming flow
in the time-periodic case is a simplification of the model
presented in Ref. 11, whereas the model used to solve
the time-dependent problem is new.
A. Governing equations
The governing equations are solved using the commercial
software Comsol Multiphysics 24 based on the finite
element method 25. To achieve greater flexibility
and control, the equations are implemented through
mathematics-weak-form-PDE modules and not through
the built-in modules for acoustics and fluid mechanics.
The governing equations are formulated to avoid evaluation
of second-order spatial derivatives and of timederivatives
of first-order fields in the second-order equations,
as time-derivatives carry larger numerical errors
compared to the spatial derivatives. To fix the numerical
solution of the second-order equations, a zero spatial
FIG. 1. (Color online) (a) Sketch of the rectangular computational
domain in the yz-plane representing the upper half
of a rectangular cross section of a long straight microchannel
of width w = 380 μm and height h = 160 μm as in 23.
The thick arrows indicate in-phase oscillating velocity actuation
at the left and right boundaries. (b) The three black
points indicate positions at which the velocity components
(gray arrows), defined in Eq. (29), are probed. (c) Sketch of
the spatial mesh used for the discretization of the physical
fields. (d) A zoom-in on the mesh in the upper left corner.
average of the second-order pressure is enforced by a Lagrange
multiplier. For the time-domain simulations we
use the generalized alpha solver 26, setting the alpha
parameter to 0.5 and using a fixed time step t. Furthermore,
to limit the amount of data stored in Comsol,
the simulations are run from Matlab 27 and long timemarching
schemes are solved in shorter sections by Comsol.
Comsol model files and Matlab scripts are provided
in the Supplemental Material 28.
B. Boundary conditions
The acoustic cavity is modeled with stationary hard
rigid walls, and the acoustic fields are exited on the side
walls by an oscillating velocity boundary condition with
oscillation period t0 and angular frequency !,
t0 =
2
!
. (16)
The symmetry of the bottom boundary is described by
zero orthogonal velocity component and zero orthogonal
gradient of the parallel velocity component. The explicit