2
relative to the radiation force and thus allowing the latter
also to dominate manipulation of sub-micrometer particles.
This might provide an alternative method to the
one proposed by Antfolk et al. 19, which used an almost
square channel with overlapping resonances to create a
streaming flow that did not counteract the focusing of
sub-micrometer particles.
To theoretically study the effects of a pulsed ultrasound
actuation, we need to solve the temporal evolution
of the acoustic resonance and streaming, which is
the topic of the present work. Numerical solutions of the
time-domain acoustic equations were used by Wang and
Dual 20 to calculate the time-averaged radiation force
on a cylinder and the steady streaming around a cylinder,
both in a steady oscillating acoustic field. However,
they did not present an analysis of the unsteady build-up
of the acoustic resonance and the streaming flow.
In this paper, we derive the second-order perturbation
expansion of the time-dependent governing equations for
the acoustic fields and streaming velocity, and solve them
numerically for a long straight channel with acoustically
hard walls and a rectangular cross section. The analysis
and results are divided into two sections: (1) A
study of the transient build-up of the acoustic resonance
and streaming from a initially quiescent state towards
a steady oscillating acoustic field and a steady streaming
flow. (2) An analysis of the response of the acoustic
field and the streaming flow to pulsed actuation, and
quantifying whether this can lead to better focusability
of sub-micrometer particles.
In previous studies, such as 2, 11, 17, only the periodic
state of the acoustic resonance and the steady timeaveraged
streaming velocity are solved. When solving
the time-dependent equations, we obtain a transient solution,
which may also be averaged over one oscillation
period to obtain an unsteady time-averaged solution.
II. BASIC ADIABATIC ACOUSTIC THEORY
In this section we derive the governing equations for
the first- and second-order perturbations to unsteady
acoustic fields in a compressible Newtonian fluid. We
only consider acoustic perturbation in fluids, and treat
the surrounding solid material as ideal rigid walls. Our
treatment is based on textbook adiabatic acoustics 21
and our previous study Ref. 11 of the purely periodic
state.
A. Adiabatic thermodynamics
We employ the adiabatic approximation, which assumes
that the entropy is conserved for any small fluid
volume 22. Consequently, the thermodynamic state of
the fluid is described by only one independent thermodynamic
variable, which we choose to be the pressure p.
See Table I for parameter values. The changes d in the
TABLE I. IAPWS parameter values for pure water at ambient
temperature 25 ◦C and pressure 0.1013 MPa. For references
see Sec. II-B in Ref. 11.
Parameter Symbol Value Unit
Acoustic properties:
Mass density 0 9.971 × 102 kg m−3
Speed of sound cs 1.497 × 103 m s−1
Compressibility s 4.477 × 10−10 Pa−1
Transport properties:
Shear viscosity 8.900 × 10−4 Pa s
Bulk viscosity b 2.485 × 10−3 Pa s
density from the equilibrium state is given by
d = s dp, (1)
where the isentropic compressibility s is defined as
s =
1
@
@p
s
=
1
c2s
. (2)
B. Governing equations
Mass conservation implies that the rate of change @t
of the density in a test volume with surface normal vector
n is given by the influx (direction −n) of the mass current
density v. In differential form by Gauss’s theorem it is
@t = ∇·
− v
. (3a)
Substituting @t and ∇ using Eq. (1), and dividing by
, the continuity equation (3a) becomes
s@tp = −∇· v − sv ·∇p. (3b)
Similarly, momentum conservation implies that the rate
of change @t(v) of the momentum density in the same
test volume is given by the stress forces acting on the
surface (with normal n), and the influx (direction −n) of
the momentum current density vv. In differential form,
neglecting body forces, this becomes
@t(v) = ∇·
− p 1 − vv
, (4a)
where the viscous stress tensor is defined as
=
∇v + (∇v)T
+
b −
2
3
(∇· v) 1. (4b)
Here 1 is the unit tensor and the superscript ”T” indicates
tensor transposition. Using the continuity equation
(3a), the momentum equation (4a) is rewritten into the
well-known Navier–Stokes form,
@tv = ∇·
− p 1
− (v ·∇)v, (4c)
which is useful when solving problems in the time domain.
The equations (3b) and (4c) constitutes the nonlinear
governing equations which we will study by applying
the usual perturbation approach of small acoustic
amplitudes.