PETER BARKHOLT MULLER AND HENRIK BRUUS PHYSICAL REVIEW E 90, 043016 (2014)
is generated in boundary layers near the top and bottom walls
and mainly conducted out of these, while some of the heat
is conducted into the bulk and out through the left and right
walls. The bulk viscosity ηb is often neglected whenworking at
frequencies around 2 MHz because of its small contribution to
the total dissipation, and the subsequent negligible influence on
the resonance curve and the streaming velocity field. However,
Fig. 7 clearly shows that the bulk absorption is important
for the spatial structure of the time-averaged temperature
field.
In Sec. IIB we stated that the changes in the dynamic
viscosity due to its temperature and density dependence
are 0.33% and 0.37%, respectively, for the amplitudes of
the acoustic oscillation used in this paper. It might seem
surprising that, first, such a small perturbation of the viscosity
can increase the magnitude of the streaming by 19% at
25 ◦C as stated in Sec. IV B (39% at 50 ◦C), and secondly,
the numerical results are in very good agreement with the
analytical expression from Ref. 15, which does not include
the density dependence of the dynamic viscosity of similar
magnitude as the temperature dependence. The explanation
lies within the spatial structure of the fields. From the timeaveraged
momentum Eq. (14), we see that the divergence of
the stress tensor leads to a term containing the gradient of
the viscosity perturbation ∇η1
· ∇v1
+ (∇v1)T. Here, η1
is proportional to T1 and ρ1, and since T1 changes on the
small length scale of the boundary layer, whereas ρ1 only
changes on the long length scales of the channel width,
we get ∇= ∇(T )
+ ∇(ρ)
∼ (T )
η1
η
η
η
/δt + (ρ)
()
1
1
1 η
1 /w ≈ η
T 1 /δt,
where the superscripts refer to the contribution from either the
temperature or the density perturbation. Consequently, with
respect to the acoustic streaming, the temperature dependence
of the dynamic viscosity is much more important than the
density dependence.
The significant increase of the acoustic streaming magnitude,
due to the temperature-induced viscosity perturbation,
influences the interplay between radiation forces and drag
forces on suspended particles 20,33. The steady temperature
rise of less than 1 mK, on the other hand, has a negligible
influence on acoustic handling of biological samples, however
other applications of acoustofluidics, such as thermoacoustic
engines, rely on the steady energy currents for pumping heat
from a low-temperature source to a high-temperature sink,
or inversely, for generating acoustic power from the heat flow
between a high-temperature source and a low-temperature sink
34,35.
VI. CONCLUSION
In this work, we have presented a full numerical study
of the acoustic streaming in the cross section of a long
straight microchannel including the temperature and density
dependence of the fluid viscosity and thermal conductivity.
The temperature dependence of the streaming amplitude
in the case of a deep microchannel agreed well with the
analytical single-wall result from 2011 by Rednikov and
Sadhal 15, whereas significant deviations were found for
shallow channels. This strong dependence of the streaming
amplitude on the channel height was explained qualitatively
with a simple one-dimensional backflow model.
Furthermore, we showed that a meaningful comparison of
solutions at different temperatures and off-resonance frequencies
could be performed by normalizing the secondorder
velocity field to the square of the first-order velocity
amplitude.
We have also solved the time-averaged second-order energy
conservation equation numerically and calculated the steady
temperature rise in the channel, and we analyzed the energy
transport in the system. For acoustophoretic devices, the
temperature rise is less than 1mKand has no consequences for
either operation conditions or biological samples. However, in
other applications such as thermoacoustic engines, the energy
transport is important.
Finally, we have provided polynomial fits in the temperature
range from 10 to 50 ◦C of the thermodynamic properties
and transport properties of water at ambient atmospheric
pressure based on data from IAPWS, which cover a much
wider range of temperatures and pressures. This allows for
easy implementation of the official parameter values for the
properties of water in other models working under the same
temperature and pressure conditions.
With the inclusion of the local perturbation in viscosity and
thermal conductivity, due to their temperature and density dependence,
we have solved the complete time-averaged secondorder
acoustic equations for a Newtonian fluid enclosed by
vibrating walls, with the one exception of the unknown density
dependence of the bulk viscosity. To further progress the
numerical analysis of microchannel acoustic streaming, one
should improve the modeling of the vibration of the walls,
preferably including the elastic waves in the surrounding solid
material. In the present model, the acoustic streaming velocity
field depends strongly on the choice of actuation conditions
on the walls.
ACKNOWLEDGMENTS
We thank Wolfgang Wagner, Ruhr-Universit¨at Bochum,
for providing us with the software FLUIDCAL, Version Water
(IAPWS-95), for calculating the thermodynamic properties
of water. This work was supported by the Danish Council
for Independent Research (DFF): Technology and Production
Sciences (Grant No. 11-107021).
APPENDIX A: IAPWS FORMULATION
To ease the use of the official IAPWS values for the
thermodynamic and transport properties of water in our
numerical analysis, we fit polynomials in temperature to the
data. The precise fitting procedure and its validation are
described in the following.
The data for the thermodynamic properties are obtained
from an Excel implementation 36 of the IAPWS Formulation
1995 24, in which the equation of state for water is fitted
using a function with 56 parameters covering the range Tmelt
T 1273.15 K and p 1000 MPa. The shear viscosity is
taken from the IAPWS Formulation 2008 25, and the thermal
conductivity is taken from the IAPWS Formulation 2011 26,
for which we have implemented the expressions stated in the
papers to extract data values in the temperature and density
range of interest to us. The data for the density derivatives of
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