PHYSICAL REVIEW E 90, 043016 (2014)
Numerical study of thermoviscous effects in ultrasound-induced acoustic
streaming in microchannels
Peter Barkholt Muller* and Henrik Bruus†
Department of Physics, Technical University of Denmark, DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark
(Received 21 August 2014; published 21 October 2014)
We present a numerical study of thermoviscous effects on the acoustic streaming flow generated by an
ultrasound standing-wave resonance in a long straight microfluidic channel containing a Newtonian fluid. These
effects enter primarily through the temperature and density dependence of the fluid viscosity. The resulting
magnitude of the streaming flow is calculated and characterized numerically, and we find that even for thin
acoustic boundary layers, the channel height affects the magnitude of the streaming flow. For the special case
of a sufficiently large channel height, we have successfully validated our numerics with analytical results from
2011 by Rednikov and Sadhal for a single planar wall. We analyzed the time-averaged energy transport in
the system and the time-averaged second-order temperature perturbation of the fluid. Finally, we have made
three main changes in our previously published numerical scheme to improve the numerical performance:
(i) The time-averaged products of first-order variables in the time-averaged second-order equations have
been recast as flux densities instead of as body forces. (ii) The order of the finite-element basis functions
has been increased in an optimal manner. (iii) Based on the International Association for the Properties
of Water and Steam (IAPWS 1995, 2008, and 2011), we provide accurate polynomial fits in temperature
for all relevant thermodynamic and transport parameters of water in the temperature range from 10 to
50 ◦C.
DOI: 10.1103/PhysRevE.90.043016 PACS number(s): 47.15.−x, 43.25.Nm, 43.25.+y, 43.35.Ud
I. INTRODUCTION
Ultrasound acoustophoresis has been used to handle particles
of a few micrometers to tens of micrometers in microfluidic
channels 1, with applications in, e.g., up-concentration
of rare samples 2, cell synchronization 3, cell trapping 4,
cell patterning 5, cell detachment 6, cell separation 7, and
particle rotation 8. Control and processing of submicrometer
bioparticles have many applications in biomedicine and in
environmental and food analysis, however acoustophoretic
focusing of submicrometer particles by the primary
radiation force is hindered by the drag force from the acoustic
streaming flow of the suspending liquid. Consequently, there
is a need for understanding the acoustic streaming and for
developing tools for engineering acoustic streaming patterns
that allow for acoustic handling of submicrometer particles.
The theory of acoustic streaming, driven by the timeaveraged
shear stress near rigid walls in the acoustic boundary
layers of a standing wave, was originally described
by Lord Rayleigh 9. It was later extended, among others, by
Schlicting 10, Nyborg 11, Hamilton 12,13, and Muller
et al. 14. Recently, Rednikov and Sadhal 15 have included
the temperature dependence of the dynamic viscosity and
shown that this can lead to a significant increase in the
magnitude of the streaming velocity. In the present work, we
present a numerical study of this and related thermoviscous
effects.
A major challenge in numerical modeling of acoustic
streaming is the disparate length scales characterizing the bulk
of the fluid and the acoustic boundary layer, the latter often
*peter.b.muller@fysik.dtu.dk
†bruus@fysik.dtu.dk
being several orders of magnitude smaller than the former
in relevant experiments. One way to handle this problem is
to determine the first-order oscillatory acoustic field without
resolving the acoustic boundary layers, and from this calculate
an approximate expression for the time-averaged streaming
velocity at the boundary, acting as a boundary condition
for the steady bulk streaming 16,17. This method has the
advantage of being computationally less demanding. For
example, Lei et al. 18,19 used it to model streaming flow
in microfluidic channels in three dimensions, and they were
able to qualitatively explain several experimental observations
of streaming flow in microchannels and flat microfluidics
chambers. Another method is the direct numerical solution
of the full thermoviscous acoustic equations both in the
bulk and in the thin boundary layers, demanding a fine
spatial resolution close to rigid surfaces as developed by, e.g.,
Muller et al. 20. They obtained a quantitative description
of the physics of the thermoviscous boundary layers and the
acoustic resonance. The same model was later employed in
a quantitative comparison between numerics, analytics, and
experiments of microparticle acoustophoresis, demonstrating
good agreement 14. In a more recent study, the numerical
scheme was further used to demonstrate how simultaneous
actuation of the two overlapping half-wavelength resonances
of a nearly square channel can generate a single vortex
streaming flow that allows for focusing of submicrometer
particles, an effect demonstrated experimentally by focusing
0.5-μm − diam particles and E. coli bacteria 21.
In this paper, we extend our numerical model for a
rectangular microchannel 20 to include the thermoviscous
effects, which were treated analytically in the special case
of a single planar infinite rigid wall by Rednikov and Sadhal
15. The extension is done by including the dependence
on the oscillatory first-order temperature and density fields
1539-3755/2014/90(4)/043016(12) 043016-1 ©2014 American Physical Society