J. Chem. Phys. 152, 094503 (2020); https://doi.org/10.1063/1.5142364 152, 094503
© 2020 Author(s).
Bulk viscosity of liquid noble gases
Cite as: J. Chem. Phys. 152, 094503 (2020); https://doi.org/10.1063/1.5142364
Submitted: 12 December 2019 . Accepted: 13 February 2020 . Published Online: 03 March 2020
René Spencer Chatwell , and Jadran Vrabec
ARTICLES YOU MAY BE INTERESTED IN
The relation between molecular dynamics and configurational entropy in room
temperature ionic liquids: Test of Adam–Gibbs model
The Journal of Chemical Physics 152, 091101 (2020); https://doi.org/10.1063/1.5140569
The hard sphere diameter of nanocrystals (nanoparticles)
The Journal of Chemical Physics 152, 094502 (2020); https://doi.org/10.1063/1.5132747
The EXP pair-potential system. IV. Isotherms, isochores, and isomorphs in the two
crystalline phases
The Journal of Chemical Physics 152, 094505 (2020); https://doi.org/10.1063/1.5144871
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
Bulk viscosity of liquid noble gases
Cite as: J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364
Submitted: 12 December 2019 •Accepted: 13 February 2020 •
Published Online: 3 March 2020
René Spencer Chatwell and Jadran Vrabeca)
AFFILIATIONS
Thermodynamics and Process Engineering, Technische Universität Berlin, 10587 Berlin, Germany
a)Author to whom correspondence should be addressed: [email protected]
ABSTRACT
An equation of state for the bulk viscosity of liquid noble gases is proposed. On the basis of dedicated equilibrium molecular dynamics
simulations, a multi-mode relaxation ansatz is used to obtain precise bulk viscosity data over a wide range of liquid states. From this dataset,
the equation of state emerges as a two-parametric power function with both parameters showing a conspicuous saturation behavior over
temperature. After passing a temperature threshold, the bulk viscosity is found to vary significantly over density, a behavior that resembles
the frequency response of a one pole low-pass filter. The proposed equation of state is in good agreement with available experimental sound
attenuation data.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5142364
., s
I. INTRODUCTION
In contrast to prevailing opinion, bulk viscous effects have
widely been explored in fluid mechanics and even share a somewhat
controversial history. In 1845, Stokes1had argued geometrically that
Cauchy’s stress tensor field,2
Πij =(μb−2
3μs)∂nvnδij +μs(∂jvi+∂ivj), (1)
entails not only the well-known shear viscosity μsbut also the bulk
viscosity μb, which he concurrently postulated to be zero,
μb
!
=0. (2)
Stokes, whose proof can hardly be considered rigorous, had
limited his argument to incompressible fluids and remained skep-
tical that his finding (2) would emerge as universally valid. For
more than a century, this hypothesis was afflicted with miscon-
ceptions until the Royal Society hosted “a discussion on the first
and second viscosities of fluids,”3–18 reinvigorating investigations
of bulk viscous effects on a wide variety of fluid mechanical
problems.
Riemann’s solution19 of the Euler equation20 was among the
first. Retaining the bulk viscosity in the stress tensor field21 conclu-
sively solved the problem of insensibly small wave thicknesses that
were predicted22,23 in descriptions limited to shear viscosity.24 The
conjectured increase in wave thickness was qualitatively confirmed
in successive atomistic simulations of non-ideal liquids25,26 and in
numerical investigations of rarefied gases,27–29 given that the shock
structure is symmetric.30
Bulk viscous effects have long been considered mere higher-
order contributions.31 However, including the bulk viscosity in
the acoustic approximation32,33 straightforwardly explained the
observed second-order fields associated with ultrasonic waves. The
vorticity that is generated across a propagating wave’s free sur-
face induces a counter-oriented circulatory flow34–39 that is also
known as acoustic streaming or quartz-wind. In hypersonic bound-
ary layer approximations, bulk viscous effects are promoted to
second-order in the pressure distribution40 and even first-order
in the outer and inner flow velocities of high Reynolds number
flows.41
In fluids confined to capillaries, the bulk viscosity contributes
to first-order in radial pressure and to second-order in the axial
velocity.42–47
Bulk viscous behavior was also investigated for more complex
scenarios. An increased shock wave thickness affects the outer flow’s
adverse pressure gradient and consequently suppresses the shock
induced boundary layer separation,48 while additionally the shock
wave’s location and strength are much more accurately predicted
when bulk viscous effects are included.49 Likewise, bulk viscous
damping has been observed in compressible turbulent flows.50 A
non-zero bulk viscosity enhances kinetic energy dissipation while
additionally inhibiting the energy transfer between translational and
configurational energy, thus rendering the flow effectively incom-
pressible.51
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-1
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
Each of the aforementioned investigations, however, has suf-
fered from missing, incomplete, or unreliable bulk viscosity data and
was consequently restricted to predominantly qualitative results.
II. SOUND ATTENUATION
Since its introduction, the bulk viscosity was closely associ-
ated with linear acoustics. Stokes had originally proposed that sound
attenuation measurements could confirm his hypothesis (2) experi-
mentally. A propagating wave’s linear momentum is dissipated by a
surrounding atmosphere, which leads to the exponential decay of its
amplitude A(z) over traveled distance Δz=z2−z1and is measured
by an attenuation coefficient α,
A(z2)
A(z1)=exp(−αλ
Δz
λ), (3)
given here per wavelength λ, i.e., αλ=αλ. In a first estimate of αλ
for one-dimensional motion, limited to shear viscosity μs, Stokes
found that the attenuation factorizes into frequency ω= 2πfand a
transport function ξ,
αλ=ωξStokes =ωπ
K
4
3μs, (4)
where K=ρc2is the fluid’s low-frequency modulus of compres-
sion, ρits density, and cits thermodynamic speed of sound. Kirch-
hoff52 advanced the discussion by including heat conduction and
established that both effects superimpose linearly in the transport
function,
αclassical
λ=ω(ξStokes +ξKirchhoff)
=ωπ
K(4
3μs+(κ−1)γ
cp), (5)
where κ=cp/cvis the ratio of specific heats and γis the thermal
conductivity. The attenuation predicted by this classical theory cor-
responds ostensibly well with low frequency acoustic measurements.
Subsequently performed ultrasonic measurements,53–57 how-
ever, disclosed large deviations from the classical description (5)
and motivated Herzfeld58 and Kneser59,60 to introduce an additional
mechanism,
αλ=αclassical
λ+αexcess
λ. (6)
Both connected classical hydrodynamics to relaxation theory and
attributed the excess attenuation αexcess
λto a time lag that occurs dur-
ing the transfer of energy between the molecules’ translational and
internal degrees of freedom. In contrast, Tisza61 refrained from any
physical interpretation and incorporated the Herzfeld mechanism
into the transport function ξby force fitting the complex mechanism
of relaxation into a scalar valued bulk viscosity,62
αλ=ω(ξStokes +ξKirchhoff +ξTisza)
=ωπ
K(4
3μs+(κ−1)γ
cp
+μb). (7)
This extended theory has been validated experimentally up to mod-
erate ultrasonic frequencies, i.e., f≤280 MHz,63 over a wide range
of thermodynamic states.
In the extended vicinity of the critical point, however, an
anomalously high attenuation was observed64–66 that was conclu-
sively attributed to long-range density correlations,67–73 which are
alterations of structural relaxation effects.74–77 The total attenuation
partitions into the already established background part (7) and a
critical contribution,78,79
αtotal
λ=αλ+αcritical
λ. (8)
Acoustic dispersion, i.e., the frequency dependent speed of sound
c(ω), confines this critical attenuation to an extended critical region
in the thermodynamic state space, cf. supplementary material,
αcritical
λ=πI(˜
ω)
J(˜
ω)([c(ω)
c]2−1), (9)
where I(˜
ω),J(˜
ω)are improper integrals of a characteristic
frequency ˜
ω.
III. EXPERIMENTAL DATA
At present, the bulk viscosity is determined experimen-
tally either by non-resonant Rayleigh–Brillouin scattering80–90 or
ultrasonic attenuation measurements.91–121 Both techniques have
successfully been applied to a variety of substances, yet each mea-
surement series was restricted to selective thermodynamic states.
A substantially larger dataset, however, can be obtained for
liquid noble gases by utilizing their self-similar behavior. While all
available sound attenuation measurements for neon, argon, kryp-
ton, and xenon were evaluated, data subject to critical attenuation
FIG. 1. Overview of thermodynamic states at which experimental sound atten-
uation data are available for neon106 (circles), argon98–101,110,112,113,115,116
(squares), krypton115–117,119 (triangles), and xenon79,115,116,118,119,122 (diamonds).
Highlighted: Seven isotherms were selected T/Tc= 0.759, 0.76, 0.83, 0.86,
0.863, 0.91, and 0.93, while T/Tc= 0.759 and 0.863 were omitted in the plot
for visibility reasons, along which atomistic simulations were performed to com-
plement and extend the dataset to higher pressures p/pc≤21. All thermodynamic
states were reduced with the respective fluid’s critical pressure pcand density ρc.
The extended critical region was constructed on the basis of sound dispersion
measurements from the literature and is delimited by the dashed line.
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-2
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
TABLE I. The following parameters for the full, i.e., untruncated Lennard-Jones poten-
tial σ,ϵ, and atomic mass mwere used in the canonical transformation130 to reduce
the bulk viscosity and time t⋆=t√ϵ/(mσ2), where kBis Boltzmann’s constant.
ϵ/kB(K) σ(Å) m(u)
Ne 33.92 2.801 20.180
Ar 116.79 3.395 39.948
Kr 162.58 3.627 83.798
Xe 226.14 3.949 131.293
were identified and discarded, cf. Fig. 1. The bulk viscosity was cal-
culated on the basis of the extended classical theory (7) and in con-
trast to previous works, more accurate thermodynamic data were
available by resorting to most recent equations of state.123–127 Sub-
sequently, μbwas reduced by a canonical transformation resting on
the Lennard-Jones potential, i.e., μ⋆
b=μbσ2/√mϵ, with the required
set of parameters specified in Table I.
Each contribution to bulk viscosity is afflicted by a different
uncertainty. While the uncertainty of most recent thermodynamic
data are well established to range from Δρ/ρ= 0.005 to Δμs/μs
= 0.3 depending on substance and state, the attenuation coefficient
αλhas been determined by single measurements at the respective
state point, and thus, only its absolute maximum errors Δαλhave
been estimated. Moreover, it is not assured that systematic measure-
ment errors in the literature data, specifically diffraction effects,128
were properly accounted for. Consequently, the reduced bulk vis-
cosity’s uncertainty Δμbwas determined to be linearly affected129 by
its various contributions Δαλ,Δc,Δμs,Δγ,Δcv,Δcp,
Δμb=∣K
ωπ ∣Δαλ+⋯+∣(κ−1)γ
c2
p∣Δcp. (10)
FIG. 2. Overview of bulk viscosity data calculated from the extended classical the-
ory on the basis of experimental sound attenuation data, taken from the literature at
the state points indicated in Fig. 1.Highlighted: Experimental data at the selected
isotherms T/Tc= 0.76, 0.83, 0.86, 0.91, 0.96 with their respective uncertainties,
calculated from Eq. (10), indicating a monotonic decline of μbas a function of
density.
After selecting and evaluating the experimental data, a decline of
the bulk viscosity along each isotherm, i.e., from saturation line
toward higher density, can qualitatively be inferred. The effect tends
to increase with temperature, however, due to large errors neither
the function’s gradient nor the curvature can accurately be specified,
cf. Fig. 2.
IV. MOLECULAR DYNAMICS SIMULATION
To interpret and extend the experimental dataset to higher
densities, additional bulk viscosity data were sampled by equi-
librium molecular dynamics (EMD) simulations along seven
selected isotherms T/Tc= 0.759, 0.76, 0.83, 0.86, 0.863, 0.91,
0.93, cf. Fig. 1. The bulk viscosity was determined microscopi-
cally by time-autocorrelation functions of local small-scale, tran-
sient pressure fluctuations131,132 that are intrinsic in any fluid under
equilibrium.133 The Lennard-Jones interaction potential was used,
which has demonstrated to resolve such small-scale dynamics ade-
quately and hence successfully describes macroscopic transport in
liquid noble gases.134–138
In the present work, the bulk viscosity’s autocorrelation func-
tion BAwas sampled in the microcanonical (NVE) ensemble, uti-
lizing the fully open source program ms2.139 The necessary average
energies ⟨E⟩were determined by preceding canonical (NVT) ensem-
ble simulations for the given pair of temperature and density. Finite
size effects were minimized by placing N= 4096 particles in cubic
volumes with periodic boundary conditions and choosing a suffi-
ciently large cutoff radius rc≥5.5σ.140–142 The employed particle
number is well chosen, as simulations containing N= 12000 par-
ticles yielded virtually identical results, cf. supplementary material.
In order to adequately resolve both the existing small-scale dynam-
ics and also the slowly decaying pressure fluctuations,143 a reduced
integrator time step Δt⋆=5⋅10−4was specified and each autocor-
relation function BAwas sampled over a reduced time period of at
least t⋆≥14.6.
A. Relaxation ansatz
The fluid’s intrinsic small-scale pressure fluctuations have con-
clusively been established to relax in different modes.144–146 Each
mode decays exponentially over time following a Kohlrausch–
Williams–Watts function.147,148 For all of the investigated state
points, three superimposing relaxation modes were found to be
present, leading to the relaxation model’s analytical form,
BR(t)=Cfexp(−(t
δf)βf)+Cmexp(−(t
δm)βm)
+Csexp(−(t
δs)βs). (11)
The first term describes the fast, and the subsequent terms describe
the intermediate and slow modes, respectively. The weighting fac-
tors are constraint Cf+Cm+Cs= 1, and the Kohlrausch param-
eters δi,βiare a measure of relaxation time scale and distortion
from the exponential function, respectively. The eight independent
parameters of Eq. (11) were determined by fitting the relaxation
model BRto the sampled autocorrelation function BAat each state
point independently.
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-3
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
Each mode’s average relaxation time τiis properly defined
as integral mean value of its respective contribution BR,ito the
relaxation model,149
τi=1
Ci
lim
t→∞∫t
0dt(BR,i(t)). (12)
As originally proposed by Maxwell, the bulk viscosity μbis propor-
tional to the cumulative averaged relaxation time,
μb=Kr∑
i
Ciτi, (13)
with the proportionality constant Krbeing the fluid’s relaxation
modulus.150
V. RESULTS
A. Relaxation times
The present results are exemplary discussed for states along
the isotherm T/Tc= 0.86 yet are qualitatively similar for all other
investigated state points as disclosed in the supplementary material.
The sampled autocorrelation function BApartitions into three
segments, with each segment being dominated by a different mode,
cf. Fig. 3. The two dents in the sampled signal readily indicate the
fast and intermediate mode’s decay. Accompanying the latter, incip-
iently small scale oscillations are amplified, giving rise to substantial
noise contributions in the slow mode. After reconstructing its sig-
nal by post-processing, the good agreement between the sampled
autocorrelation function BAand the proposed relaxation model BR
becomes apparent.
More importantly, in contrast to the sampled autocorrelation
function that is plagued by noise, the employed relaxation model’s
time integral properly converges to a definite value, thus allowing to
determine the bulk viscosity unambiguously at each state point, cf.
Fig. 4.
FIG. 3. Comparison of sampled autocorrelation function and relaxation model
including all three relaxation modes. The gray line constitutes BAsampled at T/Tc
= 0.86 and ρ/ρc= 2.06, while the solid black line represents the relaxation model
(11), and its fast, intermediate, and slow modes are depicted by dashed, short-
dashed, and dashed–dotted lines, respectively. The post-processed simulation
signal (dotted line) is in good agreement with the relaxation model.
FIG. 4. Comparison of the integrated autocorrelation function BAwith the relaxation
model’s (11) integral at T/Tc= 0.86 and ρ/ρc= 2.06. Due to noise contribu-
tions to BA, the bulk viscosity μbis difficult to determine precisely by molecular
dynamics simulation (gray line). In contrast, the proposed relaxation model (solid
black line) converges toward an unambiguous value at finite times. The dashed,
short-dashed, and dotted–dashed lines represent the fast, intermediate, and slow
modes, respectively.
The average reduced relaxation times τiwere found to decline
exponentially with density for each mode and to differ roughly
by one order of magnitude among the modes up to ρ/ρc≤2.2.
While all relaxation modes are increasingly damped with rising den-
sity, facilitating shorter relaxation times, the slow mode is damped
disproportionately, cf. Fig. 5.
FIG. 5. Distribution of average reduced relaxation times τifor all three modes
along the isotherm T/Tc= 0.86. While each mode is identified to relax
exponentially, the slow mode is predominantly affected by increasing den-
sity. The symbols represent the relaxation times of the fast τs(white circles),
the intermediate τm(black triangles), and the slow mode τs(gray squares),
respectively.
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-4
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
B. Equation of state
The most recognized equation of state, of the very few that
have actually been discussed in the literature,151,152 relates the bulk
viscosity to a power function of density,
μb∝ρα, (14)
for which special case solutions with fixed exponents
2α=−1, 0, 1, 3 exist.153–156 This proportionality, which was origi-
nally proposed in the context of viscous cosmological fluids, is much
more universal and also applies to liquid noble gases if the observed
temperature dependence is included.
After consolidating all relaxation results, a threshold tempera-
ture Tt/Tc∼0.74 emerges below which the reduced bulk viscosity
indicates virtually no variation with density.
In contrast, at higher temperatures, μbincreases progressively
toward saturated liquid states, an effect that intensifies with rising
temperature, cf. Fig. 6.
TABLE II. Best fit parameters ai,bi,ciof Eq. (16).
aibici
α1−0.93 5.91 8.67
α2−0.53 −1.49 7.86
While any physically sound equation of state must necessar-
ily establish a unique one-to-one relation between μband each state
point, i.e., being bijective, entropy constraints additionally restrict
this equation to be non-negative157,158 and the present results fur-
ther specify it to be convex with monotonically decreasing gradient
and curvature. Satisfying all conditions, a reduced two-parametric
power function is proposed
μ⋆
b=(ρ
ρc−1)α1
+α2, (15)
FIG. 6. Outline of bulk viscosity variations from the saturation line toward high density along four isotherms (a) T/Tc= 0.759, (b) T/Tc= 0.83, (c) T/Tc= 0.86, and (d) T/Tc
= 0.863. The gray shaded areas represent the experimentally determined bulk viscosity, according to Eq. (7), including its absolute maximum error, i.e., μb±Δμb, according
to Eq. (10). The symbols indicate bulk viscosity data obtained by the employed relaxation ansatz (11) on the basis of the present EMD simulations. The solid line constitutes
the present equation of state (15).
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-5
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
FIG. 7. Variation of parameters α1(circles) and α2(squares) over temperature.
Above the threshold temperature T/Tc∼0.74 indicated by the vertical line, the
bulk viscosity shows a substantial density dependence.
with parameters α1,α2depending exclusively on temperature,
αi=ai+bi⋅tanh(ci(T
Tc−1)). (16)
Both parameters, as specified in Table II, offer a salient saturation
behavior that closely resembles the frequency response of a one
TABLE III. Comparison of the bulk viscosity from the present equation of state
(15) with literature data that have either been obtained by molecular dynamics
simulation140,159 or by theoretical calculations.160
μbσ2/√mϵ μbσ2/√mϵ
T/Tcp/pcρ/ρcEq. (15) literature
0.795 0.545 2.196 1.17 1.20 ±0.37140
... 3.616 2.353 0.99 0.93 ±0.05
... 8.727 2.510 0.92 0.87 ±0.05
... 16.558 2.667 0.88 0.89 ±0.05
0.875 1.118 2.039 1.42 1.10 ±0.36
... 3.274 2.196 0.98 1.03 ±0.05
... 6.966 2.353 0.79 0.79 ±0.04
0.674 0.473 2.448 1.00 0.97 ±0.10159
0.679 0.707 ... 1.00 0.98 ±0.10
0.826 7.875 ... 0.89 0.75 ±0.08
0.699 17.065 2.773 0.94 1.10 ±0.11
0.763 21.558 ... 0.91 1.00 ±0.11
0.802 24.239 ... 0.87 0.96 ±0.10
0.605 3.117 2.640 0.96 1.05160
0.652 6.149 ... 0.96 0.93
0.667 7.126 ... 0.96 1.01
0.678 7.800 ... 0.96 1.00
0.747 12.086 ... 0.93 0.86
0.792 14.845 ... 0.89 0.86
FIG. 8. Illustration of the present equation of state (15). The solid lines constitute
the bulk viscosity’s variation over density at constant temperature, and the dashed
lines constitute its variation at constant pressure. The present results straightfor-
wardly explain the experimental results in Fig. 2, which already suggested the
bulk viscosity to be a monotonically declining function of density with increasing
gradient for higher temperatures.
pole low-pass filter, cf. Fig. 7, causing the fluid to become increas-
ingly more bulk viscous after passing the threshold temperature.
This observed temperature dependence is caused by a transition
from short-range order, that is present at low temperatures,150 to the
long-range density correlations that are intrinsic within the extended
critical region. The present equation of state is in good agreement
not only with experimental sound attenuation data but also with
concurrent MD simulations, cf. Table III.
VI. CONCLUSION
An equation of state for the bulk viscosity of liquid noble
gases is proposed. The bulk viscosity originates microscopically
from relaxations of small-scale pressure fluctuations which were
found to decay in three different modes following stretched expo-
nential functions. The slow mode was observed to be dispropor-
tionately affected by high density and the average relaxation times
τibetween the modes to differ roughly by one order of magni-
tude. Each mode was determined on the basis of an autocorrelation
function that was straightforwardly sampled by EMD simulations
at the respective state point. In order to adequately resolve the slow
mode, considerably long autocorrelation functions were necessary.
The equation of state emerges as a two-parametric power func-
tion with parameters depending exclusively on temperature, cf.
Fig. 8. This temperature dependence is attributed to a transition
from short-range order that is present at high densities to long-range
density correlations that arise when approaching the extended criti-
cal region. After a threshold T/Tc∼0.74 is passed, the fluid becomes
increasingly more bulk viscous. This effect causes sound attenua-
tion to rise progressively with temperature, closely resembling the
frequency response of a one pole low-pass filter. In addition, both
bulk viscosity coefficients were observed to exhibit opposing behav-
ior, i.e., an increase in bulk viscosity corresponds to a decline of
shear viscosity and vice versa, causing the viscosities’ ratio to peak
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-6
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
μb/μs∼5 at the highest investigated temperature close to the satura-
tion line.
SUPPLEMENTARY MATERIAL
See the supplementary material for the construction of the crit-
ical region, as well as for a full disclosure of all data associated with
this manuscript.
ACKNOWLEDGMENTS
This work was carried out under the auspices of the
Boltzmann–Zuse Society (BZS). All simulations were performed
either on the HPC clusters OCuLUS and Noctua at the Paderborn
Center for Parallel Computing (PC2) or on the Cray CX40 system
Hazel Hen at the High Performance Computing Centre Stuttgart
(HLRS) with resources allocated according to Grant No. MMHBF2.
REFERENCES
1G. G. Stokes, “On the theories of the internal friction of fluids in motion, and
of the equilibrium and motion of elastic solids,” Cambridge Philos. Soc. 8, 287
(1845).
2A. L. Cauchy, “Recherches sur l’equilibre et le mouvement intérieur des corps
solides ou fluides. Élastiques ou non élastiques,” Bull. Soc. Philomath. 2, 9 (1823).
3L. Rosenhead, “Introduction. The second coefficient of viscosity: A brief review
of fundamentals,” Proc. R. Soc. London, Ser. A 226, 1 (1954).
4P. E. Doak, “Vorticity generated by sound,” Proc. R. Soc. London, Ser. A 226, 7
(1954).
5E. G. Richardson, “Acoustic experiments relating to the coefficient of viscosity of
various liquids,” Proc. R. Soc. London, Ser. A 226, 16 (1954).
6R. O. Davies, “Kinetic and thermodynamic aspects of the second coefficient of
viscosity,” Proc. R. Soc. London, Ser. A 226, 24 (1954).
7G. I. Taylor, “The two coefficients of viscosity for an incompressible fluid
containing air bubbles,” Proc. R. Soc. London, Ser. A 226, 34 (1954).
8G. I. Taylor, “Notes on the volume viscosity of water containing bubbles,” Proc.
R. Soc. London, Ser. A 226, 38 (1954).
9R. O. Davies, “A note on Sir Geoffrey Taylor’s paper,” Proc. R. Soc. London,
Ser. A 226, 39 (1954).
10H. O. Kneser, “Transport and relaxation phenomena,” Proc. R. Soc. London,
Ser. A 226, 40 (1954).
11J. E. Piercy and J. Lamb, “Acoustic streaming in liquids,” Proc. R. Soc. London,
Ser. A 226, 43 (1954).
12J. H. Andreae and J. Lamb, “Ultrasonic measurements and the second viscosity
of carbon disulphide,” Proc. R. Soc. London, Ser. A 226, 51 (1954).
13J. Meixner, “On the thermodynamic theory of the second viscosity,” Proc. R.
Soc. London, Ser. A 226, 51 (1954).
14S. M. Karim, “Experimental determination of the second viscosity,” Proc. R. Soc.
London, Ser. A 226, 56 (1954).
15J. G. Oldroyd, “Note on the hydrodynamic and thermodynamic pressures,”
Proc. R. Soc. London, Ser. A 226, 57 (1954).
16C. A. Truesdell, “The present status of the controversy regarding the bulk
viscosity of fluids,” Proc. R. Soc. London, Ser. A 226, 59 (1954).
17E. N. da C. Andrade, “Review of discussion,” Proc. R. Soc. London, Ser. A 226,
65 (1954).
18R. O. Davies and L. Rosenhead, “The two viscosities of fluids,” Nature 173, 1209
(1954).
19B. Riemann, “Über die Fortpflanzung ebenener Luftwellen von endlicher
Schwingungsbreite,” in Collected Works of Bernhard Riemann, edited by H. Weber
(Dover, 1953), p. 157.
20L. Euler, “Principes généraux du mouvement des fluides,” Mém. Acad. Sci.
Berlin 11, 274 (1755).
21D. Gilbarg and D. Paolucci, “The structure of shock waves in the
continuum theory of fluids,” J. Ration. Mech. Anal. 2, 617 (1953), see
https://www.jstor.org/stable/24900350.
22R. Becker, “Stoßwelle und Detonation,” Z. Phys. 8, 321 (1921).
23H. Grad, “The profile of a steady shock wave,” Commun. Pure Appl. Math. 5,
257 (1952).
24W. Tollmien, H. Schlichting, H. Görtler, and F. W. Riegels, “Zur Theorie des
Verdichtungsstoßes,” in Ludwig Prandtl Gesammelte Abhandlungen, edited by F.
W. Riegels (Springer, Berlin, 1961).
25W. G. Hoover, “Structure of a shock-wave front in a liquid,” Phys. Rev. Lett. 42,
1531 (1979).
26B. L. Holian, W. G. Hoover, B. Moran, and G. K. Straub, “Shock-wave structure
via nonequilibrium molecular dynamics and Navier-Stokes continuum mechan-
ics,” Phys. Rev. A 22, 2798 (1980).
27G. Emanuel and B. M. Argrow, “Linear dependence of the shock wave thickness
on bulk viscosity,” Phys. Fluids 6, 3203 (1994).
28T. G. Elizarova, A. A. Khokhlov, and S. Montero, “Numerical simulation of
shock wave structure in nitrogen,” Phys. Fluids 19, 068102 (2007).
29A. V. Chikitkin, B. V. Rogov, G. A. Tirsky, and S. V. Utyuzhnikov, “Effect of bulk
viscosity in supersonic flow past spacecraft,” Appl. Numer. Math. 93, 47 (2015).
30S. Taniguchi, T. Arima, T. Ruggeri, and M. Sugiyama, “Overshoot of the non-
equilibrium temperature in the shock wave structure of a rarefied polyatomic gas
subject to the dynamic pressure,” Int. J. Nonlinear Mech. 79, 66 (2016).
31M. van Dyke, “Second-order compressible boundary layer theory with applica-
tion to blunt bodies in hypersonic flow,” in Hypersonic Flow Research, edited by
F. R. Riddell (Academic Press, 1962), p. 37.
32P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Press,
1986).
33M. Trusler, Physical Acoustics and Metrology of Fluids (CRC Press, 1991).
34C. Eckart, “Vortices and streams caused by sound waves,” Phys. Rev. 73, 68
(1948).
35L. N. Liebermann, “Second viscosity of liquids,” Phys. Rev. 75, 1415 (1949).
36J. J. Markham, “Second-order acoustic fields: Streaming with viscosity and
relaxation,” Phys. Rev. 86, 497 (1952).
37P. J. Westervelt, “The theory of steady rotational flow generated by a sound
field,” J. Acoust. Soc. Am. 25, 60 (1953).
38W. L. Nyborg, “Acoustic streaming due to attenuated plane waves,” J. Acoust.
Soc. Am. 25, 68 (1953).
39J. Lighthill, “Acoustic streaming,” J. Sound Vib. 61, 391 (1978).
40G. Emanuel, “Effect of bulk viscosity on a hypersonic boundary layer,” Phys.
Fluids 4, 491 (1992).
41M. S. Cramer and F. Bahmani, “Effect of large bulk viscosity on large-Reynolds-
number flows,” J. Fluid Mech. 751, 142 (2014).
42H. R. van den Berg, C. A. ten Seldam, and P. S. van der Gulik, “Compressible
laminar flow in a capillary,” J. Fluid Mech. 246, 1 (1993).
43J. C. Harley, Y. Huang, H. H. Bau, and J. N. Zemel, “Gas flow in micro-
channels,” J. Fluid Mech. 284, 257 (1995).
44Y. Zohar, S. Y. K. Lee, W. Y. Lee, L. Jiang, and P. Tong, “Subsonic gas flow in a
straight and uniform microchannel,” J. Fluid Mech. 472, 125 (2002).
45D. C. Venerus, “Laminar capillary flow of compressible viscous fluids,” J. Fluid
Mech. 555, 59 (2006).
46E. G. Taliadorou, M. Neophytou, and G. C. Georgiou, “Perturbation solutions
of Poiseuille flows of weakly compressible Newtonian liquids,” J. Non-Newtonian
Fluid Mech. 163, 25 (2009).
47D. C. Venerus and D. J. Bugajsky, “Compressible laminar flow in a channel,”
Phys. Fluids 22, 046101 (2010).
48F. Bahmani and M. Cramer, “Suppression of large shock-induced separation in
fluids having large bulk viscosities,” J. Fluid Mech. 756, R2 (2014).
49S. Bhola and T. K. Sengupta, “Roles of bulk viscosity on transonic shock-
wave/boundary layer interactions,” Phys. Fluids 31, 096101 (2019).
50S. Chen, X. Wang, J. Wang, M. Wan, H. Li, and S. Chen, “Effects of bulk viscosity
on compressible homogeneous turbulence,” Phys. Fluids 31, 085115 (2019).
51S. Pan and E. Johnsen, “The role of bulk viscosity on the decay of compressible,
homogeneous, isotropic turbulence,” J. Fluid Mech. 833, 717 (2017).
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-7
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
52G. Kirchhoff, “Über den Einfluß der Wärmeleitung in einem Gas auf die
Schallbewegung,” Ann. Phys. 210, 177 (1868).
53N. Neklepajev, “Über die Absorption kurzer akustischer Wellen in der Luft,”
Ann. Phys. 340, 175 (1911).
54T. P. Abello, “Absorption of ultrasonic waves by various gases,” Phys. Rev. 31,
1083 (1928).
55W. H. Pielemeier, “The Pierce acoustic interferometer as an instrument for the
determination of velocity and absorption,” Phys. Rev. 34, 1184 (1929).
56E. Großmann, “Schallabsorptionsmessung in Gasen bei hohen Frequenzen,”
Ann. Phys. 405, 681 (1932).
57R. W. Curtis, “An experimental determination of ultrasonic absorption an
reflexion coefficients in air and in carbon dioxide,” Phys. Rev. 46, 811 (1934).
58K. F. Herzfeld and F. O. Rice, “Dispersion and absorption of high frequency
sound waves,” Phys. Rev. 31, 691 (1928).
59H. O. Kneser, “Schallabsorption in mehratomigen Gasen,” Ann. Phys. 408, 337
(1933).
60H. O. Kneser, “Schallabsorption und -dispersion in Flüssigkeiten,” Ann. Phys.
424, 277 (1938).
61L. Tisza, “Supersonic absorption and Stokes’ viscosity relation,” Phys. Rev. 61,
531 (1942).
62W. E. Meador, G. A. Miner, and L. W. Townsend, “Bulk viscosity as a relaxation
parameter: Fact or fiction?,” Phys. Fluids 8, 258 (1996).
63R. A. Rapuano, “Ultrasonic absorption from 75 to 280 Mc/sec,” Phys. Rev. 72,
78 (1947).
64W. G. Schneider, “Sound velocity and sound absorption in the critical region,”
Can. J. Chem. 29, 243 (1951).
65A. G. Chynoweth and W. G. Schneider, “Ultrasonic propagation in xenon in the
region of its critical temperature,” J. Chem. Phys. 20, 1777 (1952).
66H. D. Parbrook and E. G. Richardson, “Propagation of ultrasonic waves in
vapours near the critical point,” Proc. Phys. Soc., Sect. B 65, 437 (1952).
67M. Fixman, “Ultasonic attenuation in the critical region,” J. Chem. Phys. 33,
1363 (1960).
68W. Botch and M. Fixman, “Sound absorption in gases in the critical region,”
J. Chem. Phys. 42, 199 (1965).
69M. Fixman, “Transport coefficients in the gas critical region,” J. Chem. Phys. 47,
2808 (1967).
70K. Kawasaki, “Correlation-function approach to the transport coefficients near
the critical point. I,” Phys. Rev. 150, 291 (1966).
71K. Kawasaki, “Sound attenuation and dispersion near the liquid-gas critical
point,” Phys. Rev. A 1, 1750 (1970).
72L. P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V.
Palciauskas, M. Rayl, and J. Swift, “Static phenomena near critical points: Theory
and experiment,” Rev. Mod. Phys. 39, 395 (1967).
73L. P. Kadanoff and J. Swift, “Transport coefficients near the liquid-gas critical
point,” Phys. Rev. 166, 89 (1968).
74L. Hall, “The origin of excess ultrasonic absorption in water,” Phys. Rev. 71, 318
(1947).
75L. Hall, “The origin of ultrasonic absorption in water,” Phys. Rev. 73, 775
(1948).
76D. Sette, “Ultrasonic absorption in liquid mixtures and structural effects,”
J. Chem. Phys. 21, 558 (1953).
77K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves
(Academic Press, 1959).
78C. W. Garland, D. Eden, and L. Mistura, “Critical sound absorption in xenon,”
Phys. Rev. Lett. 25, 1161 (1970).
79P. E. Mueller, D. Eden, C. W. Garland, and R. C. Williamson, “Ultrasonic
attenuation and dispersion in xenon near its critical point,” Phys. Rev. A 6, 2272
(1972).
80P. A. Fleury and J.-P. Boon, “Brillouin scattering in simple liquids: Argon and
neon,” Phys. Rev. 186, 244 (1969).
81A. S. Pine, “Velocity and attenuation of hypersonic waves in liquid nitrogen,”
J. Chem. Phys. 51, 5171 (1969).
82B. Y. Baharudin, D. A. Jackson, P. E. Schoen, and J. Rouch, “Bulk viscosity of
liquid argon, krypton and xenon,” Phys. Lett. A 51, 409 (1975).
83X. Pan, M. N. Shneider, and R. B. Miles, “Coherent Rayleigh-Brillouin scatter-
ing,” Phys. Rev. Lett. 89, 183001 (2002).
84J. Xu, X. Ren, W. Gong, R. Dai, and D. Liu, “Measurement of the bulk viscosity
of liquid by Brillouin scattering,” Appl. Opt. 42, 6704 (2003).
85X. Pan, M. N. Shneider, and R. B. Miles, “Power spectrum of coherent Rayleigh-
Brillouin scattering in carbon dioxide,” Phys. Rev. A 71, 045801 (2005).
86A. S. Meijer, A. S. de Wijn, M. F. E. Peters, N. J. Dam, and W. van de Water,
“Coherent Rayleigh-Brillouin scattering measurements of bulk viscosity of polar
and nonpolar gases, and kinetic theory,” J. Chem. Phys. 133, 164315 (2010).
87X. He, H. Wei, J. Shi, J. Liu, S. Li, W. Chen, and X. Mo, “Experimental mea-
surement of bulk viscosity of water based on stimulated Brillouin scattering,”
Opt. Commun. 285, 4120 (2012).
88Z. Gu and W. Ubachs, “A systematic study of Rayleigh-Brillouin scattering in
air, N2, and O2gases,” J. Chem. Phys. 141, 104320 (2014).
89Z. Gu, W. Ubachs, and W. van de Water, “Rayleigh-Brillouin scattering of
carbon dioxide,” Opt. Lett. 39, 3301 (2014).
90Y. Ma, H. Li, Z. Gu, W. Ubachs, Y. Yu, J. Huang, B. Zhou, Y. Wand, and
K. Liang, “Analysis of Rayleigh-Brillouin spectral profiles and Brillouin shifts in
nitrogen gas and air,” Opt. Express 22, 2092 (2014).
91G. W. Pierce, “Piezoelectric crystal oscillators applied to the precision measure-
ment of the velocity of sound in air and CO2at high frequency,” Proc. Am. Acad.
Arts Sci. 60, 271 (1925).
92M. Kohler, “Schallabsorption in Mischungen einatomiger Gase,” Ann. Phys.
431, 209 (1941).
93J. M. M. Pinkerton, “The absorption of ultrasonic waves in liquids and its
relation to molecular constitution,” Proc. Phys. Soc., Sect. B 62, 129 (1949).
94M. Pancholy, “Temperature variation of velocity and absorption coefficient of
ultrasonic waves in heavy water,” J. Acoust. Soc. Am. 25, 1003 (1953).
95T. A. Litovitz and E. H. Carnevale, “Effect of pressure on sound propagation in
water,” J. Appl. Phys. 26, 816 (1955).
96W. Tempest and H. D. Parbrook, “The absorption of sound in argon, nitrogen
and oxygen,” Acustica 7, 354 (1957).
97D. G. Naugle and C. F. Squire, “Ultrasonic attenuation in liquid argon,” J. Chem.
Phys. 42, 3725 (1965).
98D. G. Naugle, “Excess ultrasonic attenuation and intrinsic volume viscosity in
liquid argon,” J. Chem. Phys. 44, 741 (1966).
99D. G. Naugle, J. H. Lunsford, and J. R. Singer, “Volume viscosity in liquid argon
at high pressure,” J. Chem. Phys. 45, 4669 (1966).
100D. S. Swyt, J. F. Havlice, and E. F. Carome, “Ultrasonic absorption in liquid
argon,” J. Chem. Phys. 47, 1199 (1967).
101W. M. Madigosky, “Density dependence of bulk viscosity in argon,” J. Chem.
Phys. 46, 4441 (1967).
102J. R. Singer and J. H. Lunsford, “Ultrasonic attenuation and volume viscosity
in liquid nitrogen,” J. Chem. Phys. 47, 811 (1967).
103J. R. Singer, “Excess ultrasonic attenuation and volume viscosity in liquid
methane,” J. Chem. Phys. 51, 4729 (1969).
104A. E. Victor and R. T. Beyer, “Ultrasonic absorption in liquid oxygen and
nitrogen,” J. Chem. Phys. 52, 1573 (1970).
105J. Allegra, S. Hawley, and G. Holton, “Pressure dependence of the ultrasonic
absorption in toluene and hexane,” J. Acoust. Soc. Am. 47, 144 (1970).
106E. V. Larson, D. G. Naugle, and T. W. Aldair, “Ultrasonic velocity and
attenuation in liquid neon,” J. Chem. Phys. 54, 2429 (1971).
107S. K. Kor, O. N. Awasthi, G. Rai, and S. C. Deorani, “Structural absorption of
ultrasonic waves in methanol,” Phys. Rev. A 3, 390 (1971).
108S. K. Kor, S. C. Deorani, and B. K. Singh, “Origin of ultrasonic absorption in
methanol,” Phys. Rev. A 3, 1780 (1971).
109S. K. Kor, S. C. Deorani, B. K. Singh, R. Prasad, and U. Tandon, “Ultrasonic
study of structural relaxation in ethanol,” Phys. Rev. A 4, 1299 (1971).
110J. A. Cowan and R. N. Ball, “Temperature dependence of bulk viscosity in
liquid argon,” Can. J. Phys. 50, 1881 (1972).
111G. Rai, B. K. Singh, and O. N. Awasthi, “Structural absorption of ultrasonic
waves in associated liquids,” Phys. Rev. A 5, 918 (1972).
112J. A. Cowan and P. W. Ward, “Ultrasonic attenuation of the bulk viscosity of
liquid argon near the critical point,” Can. J. Phys. 51, 2219 (1973).
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-8
Published under license by AIP Publishing
The Journal
of Chemical Physics ARTICLE scitation.org/journal/jcp
113P. W. Ward, J. A. Cowan, and R. K. Pathria, “Critical attenuation of ultrasound
in argon,” Can. J. Phys. 53, 29 (1975).
114G. J. Prangsma, A. J. Alberga, and J. J. M. Beenakker, “Ultrasonic determination
of the volume viscosity of N2, CO, CH4and CD4between 77 and 300 K,” Physica
64, 278 (1973).
115S. A. Mikhailenko, B. G. Dudar, and V. A. Shmidt, “Volume viscosity and
relaxation times in monoatomic classical fluids,” Fiz. Nizk. Temp. 1, 224 (1975).
116P. Malbrunot, A. Boyer, E. Charles, and H. Abachi, “Experimental bulk viscosi-
ties of argon, krypton, and xenon near their triple point,” Phys. Rev. A 27, 1523
(1983).
117J. A. Cowan and R. N. Ball, “Ultrasonic attenuation and bulk viscosity in liquid
krypton,” Can. J. Phys. 58, 74 (1980).
118J. A. Cowan and J. W. Leech, “Ultrasonic attenuation and bulk viscosity of
liquid xenon,” Can. J. Phys. 59, 1280 (1981).
119J. A. Cowan and J. W. Leech, “Critical region ultrasonic attenuation in the
condensed inert gases,” Can. J. Phys. 61, 895 (1983).
120A. S. Dukhin and P. J. Goetz, “Bulk viscosity and compressibility measurement
using acoustic spectroscopy,” J. Chem. Phys. 130, 124519 (2009).
121M. J. Holmes, N. G. Paker, and M. J. W. Povey, “Temperature dependence
of bulk viscosity in water using acoustic spectroscopy,” J. Phys.: Conf. Ser. 269,
012011 (2011).
122J. Thoen and C. W. Garland, “Sound absorption and dispersion as a function
of density near the critical point of xenon,” Phys. Rev. A 10, 1311 (1974).
123C. Tegeler, R. Span, and W. Wagner, “A new equation of state for argon cov-
ering the fluid region for temperatures from the melting line to 700 K at pressures
up to 1000 MPa,” J. Phys. Chem. Ref. Data 28, 779 (1999).
124E. W. Lemmon and R. T. Jacobsen, “Viscosity and thermal conductivity
equations for nitrogen, oxygen, argon and air,” Int. J. Thermophys. 25, 21
(2004).
125E. W. Lemmon and R. Span, “Short fundamental equations of state for 20
industrial fluids,” J. Chem. Eng. Data 51, 785 (2006).
126M. L. Huber, “Models for viscosity, thermal conductivity, and surface tension
of selected pure fluids as implemented in REFPROP v10.0,” Technical Report No.
8209, National Institute for Standards and Technology, 2014.
127R. Katti, R. T. Jacobsen, R. B. Stewart, and M. Jahangiri, “Thermodynamic
properties of neon for temperatures from the triple point to 700 K at pressures to
700 MPa,” in Advances in Cryogenic Engineering, edited by R. W. Fast (Springer,
1986), pp. 1189–1197.
128L. Claes, L. M. Hülskämper, E. Baumhögger, N. Feldmann, R. S. Chatwell,
J. Vrabec, and B. Henning, “Acoustic absoprtion measurement for the determi-
nation of the bulk viscosity of pure fluids,” TM - Tech. Mess. 86, 2–6 (2019).
129J. R. Taylor, Introduction to Error Analysis: The Study of Uncertainties in
Physical Measurements, 2nd ed. (University Science Books, 1997).
130G. Rutkai, M. Thol, R. Span, and J. Vrabec, “How well does the Lennard-Jones
potential represent the thermodynamic properties of noble gases?,” Mol. Phys.
115, 1104 (2017).
131M. S. Green, “Markoff random process and the statistical mechanics of time-
dependent phenomena. II. Irreversible processes in fluids,” J. Chem. Phys. 22, 398
(1954).
132R. Kubo, “Statistical-mechanical theory of irreversible processes. I. General
theory of simple applications to magnetic and conductive problems,” J. Phys. Soc.
Jpn. 12, 570 (1957).
133R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255
(1966).
134L. Verlet, “Computer “experiments” on classical fluids. I. Properties of
Lennard-Jones molecules,” Phys. Rev. 159, 98 (1967).
135D. Levesque and L. Verlet, “Computer “experiments” on classical fluids. III.
Time-dependent self-correlation functions,” Phys. Rev. A 2, 2514 (1970).
136D. Levesque, L. Verlet, and J. Kürkijarvi, “Computer “experiments” on classical
fluids. IV. Transport properties and time-correlation functions of the Lennard-
Jones liquid near its triple point,” Phys. Rev. A 7, 1690 (1973).
137S. V. Lishchuk, “Role of three-body interactions in formation of bulk viscosity
in liquid argon,” J. Chem. Phys. 136, 164501 (2012).
138F. Jaeger, O. K. Matar, and E. A. Müller, “Bulk viscosity of molecular liquids,”
J. Chem. Phys. 148, 174504 (2018).
139G. Rutkai, A. Köster, G. Guevara-Carrion, T. Janzen, M. Schappals,
C. W. Glass, M. Bernreuther, A. Wafai, S. Stephan, M. Kohns, S. Reiser,
S. Deublein, M. T. Horsch, H. Hasse, and J. Vrabec, “ms2: A molecular simula-
tion tool for thermodynamic properties, release 3.0,” Comput. Phys. Commun.
221, 343 (2017).
140K. Meier, A. Laesecke, and S. Kabelac, “Transport coefficients of the Lennard-
Jones model fluid. III. Bulk viscosity,” J. Chem. Phys. 122, 014513 (2005).
141V. G. Baidakov and S. P. Protsenko, “Metastable Lennard-Jones fluids. III. Bulk
viscosity,” J. Chem. Phys. 141, 114503 (2014).
142A. Zaragoza, M. A. Gonzales, L. Joly, I. López-Montero, M. A. Canales,
A. L. Benavides, and C. Valeriani, “Molecular dynamics study of nanocon-
fined TIP4P/2005 water: How confinement and temperature affect diffusion and
viscosity,” Phys. Chem. Chem. Phys. 21, 13653 (2019).
143G. S. Fanourgakis, J. S. Medina, and R. Prosmiti, “Determining the bulk
viscosity of rigid water models,” J. Phys. Chem. A 116, 2564 (2012).
144G.-J. Guo, Y.-G. Zhang, K. Refson, and Y.-J. Zhao, “Viscosity and stress
autocorrelation function in supercooled water: A molecular dynamics study,”
Mol. Phys. 100, 2617 (2002).
145G. Delgado-Barrio, P. Villareal, G. Winter, J. S. Medina, B. González, J. V.
Alemán, J. L. Gomez, P. Sangrá, J. J. Santana, and M. E. Torres, “Viscosity of
liquid water via equilibrium molecular dynamics simulations,” in Frontiers in
Quantum Systems in Chemistry and Physics, edited by S. Wilson, P. J. Grout,
G. Delgado-Barrio, and P. Piecuch (Springer, 2008), pp. 351–361.
146J. S. Medina, R. Prosmiti, P. Villarreal, G. Delgado-Barrio, G. Winter,
B. González, J. V. Alemán, and C. Collado, “Molecular dynamics simulations of
rigid and flexible water models: Temperature dependence of viscosity,” Chem.
Phys. 388, 9 (2011).
147R. Kohlrausch, “Theorie des elektrischen Rückstandes in der Leidner Flasche,”
Ann. Phys. 167, 179 (1854).
148G. Williams and D. C. Watts, “Non-symmetrical dielectric relaxation
behaviour arising from a simple empirical decay function,” Trans. Faraday Soc.
66, 80 (1970).
149R. D. Mountain and R. Zwanig, “Shear relaxation times of simple fluids,”
J. Chem. Phys. 44, 2777 (1966).
150T. A. Litovitz and C. M. Davis, “Structural and shear relaxation in liquids,”
in Peroperties of Gases, Liquids and Solutions, Physical Acoustics, edited by
W. P. Mason (Academic Press, 1965), Vol. 2, pp. 281–349.
151G. L. Murphy, “Big-bang model without singularities,” Phys. Rev. D 8, 4231
(1973).
152V. A. Belinskii and I. M. Khalatnikov, “Influence of viscosity on the character
of cosmological evolution,” Zh. Eksp. Theor. Fiz. 69, 401 (1975).
153I. Brevik, E. Elizalde, S. Nojiri, and S. D. Odintsov, “Viscous little rip cosmol-
ogy,” Phys. Rev. D 84, 103508 (2011).
154R. Colistete, J. C. Fabris, J. Tossa, and W. Zimdahl, “Bulk viscous cosmology,”
Phys. Rev. D 76, 103516 (2007).
155M. Cruz, N. Cruz, and S. Lepe, “Accelerated and decelerated expansion in a
causal dissipative cosmology,” Phys. Rev. D 96, 124020 (2017).
156B. Li and J. D. Barrow, “Does bulk viscosity create a viable unified dark matter
model?,” Phys. Rev. D 79, 103521 (2009).
157S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, 1962).
158D. Jou, J. Casas-Vázquez, and G. Lebon, Extended Irreversible Thermodynamics
(Springer, 2010).
159P. Borgelt, C. Hoheisel, and G. Stell, “Exact molecular dynamics and kinetic
theory results for thermal transport coefficients of the Lennard-Jones argon fluid
in a wide range of states,” Phys. Rev. A 42, 789 (1990).
160K. Tankeshwar, “Bulk viscosity and the relation between transport coeffi-
cients,” Phys. Chem. Liq. 24, 91 (1991).
J. Chem. Phys. 152, 094503 (2020); doi: 10.1063/1.5142364 152, 094503-9
Published under license by AIP Publishing
