PETER BARKHOLT MULLER AND HENRIK BRUUS PHYSICAL REVIEW E 90, 043016 (2014)
in the dynamic shear viscosity, previously taken to be
constant. This has a significant influence on the shear stresses
in the thermoviscous boundary layers responsible for
generating the steady acoustic streaming. Furthermore, we
study the steady temperature rise and energy current densities
resulting from solving the time-averaged second-order energy
transport equation. Finally, we improve the convergence
properties of our previous numerical scheme 20 by
implementing the governing equations in a source-free flux
formulation and optimizing the order of the basis functions of
the finite-element scheme.
II. BASIC THEORY
In this section, we derive the governing equations for the
first- and second-order perturbations to the thermoviscous
acoustic fields in a compressible Newtonian fluid. We only
consider the acoustics in the fluid, and we treat the surrounding
walls as ideal hard walls. Our treatment is based on textbook
thermodynamics 22 and thermoviscous acoustics 23, but
in a source-free flux formulation suitable for our specific
numerical implementation. As water is our model fluid of
choice, we carefully implement the best available experimental
data for the thermodynamic and transport parameters provided
by the International Association for the Properties ofWater and
Steam (IAPWS).
A. Thermodynamics
The independent thermodynamic variables of the compressible
Newtonian fluid are taken to be the temperature T and the
pressure p 22. The dependent variables are the mass density
ρ, the internal energy ε per mass unit, and the entropy s per
mass unit. The first law of thermodynamics is usually stated
with s and ρ as the independent variables,
dε = T ds − pd
1
ρ
= T ds + p
ρ2 dρ. (1a)
By a standard Legendre transformation of ε to the Gibbs
free energy g per unit mass, g = ε − T s + p 1
ρ , we obtain the
first law with T and p as the independent variables,
dg = −s dT + 1
ρ
dp. (1b)
Due to their importance in thermoacoustics, we furthermore
introduce the following three thermodynamics coefficients: the
isobaric heat capacity cp per unit mass, the isobaric thermal
expansion coefficient αp, and the isothermal compressibility
κT , as
cp
= T
∂s
∂T
p
, (2a)
αp
= −1
ρ
∂ρ
∂T
p
, (2b)
κT
= 1
ρ
∂ρ
∂p
T
. (2c)
Moreover, as a standard step toward getting rid of explicit
references to the entropy, we derive from Eqs. (1b) and (2b)
the following Maxwell relation:
∂s
∂p
T
= − ∂ 2g
∂p∂T
= −
∂ ( 1
ρ )
∂T
p
= −1
ρ
αp. (2d)
Using Eqs. (2a)–(2d), we express ds and dρ in terms of dT
and dp,
T ds = cpdT −
αpT
ρ
dp, (3a)
1
ρ
dρ = κT dp − αpdT, (3b)
which combined with Eq. (1a) leads to dε in terms of dT and
dp,
ρ dε = (cpρ − αpp)dT + (κT p − αpT )dp. (3c)
Using Eqs. (3a)–(3c), small changes ds, dρ, and dε in the
dependent thermodynamic variables s, ρ, and ε away from
equilibrium can thus be expressed in terms of changes in
the independent thermodynamic variables T and p. In our
numerical analysis, the default unperturbed equilibrium state
is the one at ambient temperature T0
= 25.0 ◦C and pressure
p0
= 0.1013 MPa.
B. Physical properties of water according to IAPWS
The theoretical treatment of thermoviscous acoustics requires
detailed knowledge of the dependence on temperature
and density (or temperature and pressure) of the physical
properties of the fluid of choice. In the present paper, we use
the parameter values for water supplied by the IAPWS in its
thorough statistical treatment of large data sets provided by
numerous experimental groups 24–26.
The values of the thermodynamic properties are taken from
the IAPWS Formulation 1995 24, the shear viscosity is
taken from the IAPWS Formulation 2008 25, the thermal
conductivity is taken from the IAPWS Formulation 2011 26,
while the bulk viscosity is taken from Holmes, Parker, and
Povey 27, who extended the work by Dukhin and Goetz
28. The IAPWS data set spans a much wider range in
temperature and ambient pressures than needed in our work,
and it is somewhat complicated to handle. Consequently,
to ease the access to the IAPWS data in our numerical
implementation, we have carefully fitted the temperature
dependence of all properties at atmospheric pressure by
fifth-order polynomials in temperature in the range from
10 to 50 ◦C, as described in detail in Appendix A. In
the specified range, the differences between our fits and
the IAPWS data are negligible. In Table I, we have listed
the physical properties of water at ambient temperature and
pressure.
The thermodynamic coefficients of Eq. (3) are by definition
evaluated at the equilibrium state T = T0 and p = p0, leaving
all acoustics perturbations to enter only in the small deviations,
e.g., dT = T1
+ T2. On the other hand, the transport
coefficients of the fluid depend on the acoustic perturbation.
To avoid the ambiguity of the pressure p as either the ambient
pressure outside the fluid or the intrinsic pressure (cohesive
energy) of the fluid, we use Eq. (3b) to change the variable
from pressure p to density ρ in our treatment of the IAPWS
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