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.
REFERENCES
1.
G. 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
).2.
A. 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
).3.
L.
Rosenhead
, “Introduction. The second coefficient of viscosity: A brief review of fundamentals
,” Proc. R. Soc. London, Ser. A
226
, 1
(1954
).4.
P. E.
Doak
, “Vorticity generated by sound
,” Proc. R. Soc. London, Ser. A
226
, 7
(1954
).5.
E. G.
Richardson
, “Acoustic experiments relating to the coefficient of viscosity of various liquids
,” Proc. R. Soc. London, Ser. A
226
, 16
(1954
).6.
R. O.
Davies
, “Kinetic and thermodynamic aspects of the second coefficient of viscosity
,” Proc. R. Soc. London, Ser. A
226
, 24
(1954
).7.
G. I.
Taylor
, “The two coefficients of viscosity for an incompressible fluid containing air bubbles
,” Proc. R. Soc. London, Ser. A
226
, 34
(1954
).8.
G. I.
Taylor
, “Notes on the volume viscosity of water containing bubbles
,” Proc. R. Soc. London, Ser. A
226
, 38
(1954
).9.
R. O.
Davies
, “A note on Sir Geoffrey Taylor’s paper
,” Proc. R. Soc. London, Ser. A
226
, 39
(1954
).10.
H. O.
Kneser
, “Transport and relaxation phenomena
,” Proc. R. Soc. London, Ser. A
226
, 40
(1954
).11.
J. E.
Piercy
and J.
Lamb
, “Acoustic streaming in liquids
,” Proc. R. Soc. London, Ser. A
226
, 43
(1954
).12.
J. H.
Andreae
and J.
Lamb
, “Ultrasonic measurements and the second viscosity of carbon disulphide
,” Proc. R. Soc. London, Ser. A
226
, 51
(1954
).13.
J.
Meixner
, “On the thermodynamic theory of the second viscosity
,” Proc. R. Soc. London, Ser. A
226
, 51
(1954
).14.
S. M.
Karim
, “Experimental determination of the second viscosity
,” Proc. R. Soc. London, Ser. A
226
, 56
(1954
).15.
J. G.
Oldroyd
, “Note on the hydrodynamic and thermodynamic pressures
,” Proc. R. Soc. London, Ser. A
226
, 57
(1954
).16.
C. A.
Truesdell
, “The present status of the controversy regarding the bulk viscosity of fluids
,” Proc. R. Soc. London, Ser. A
226
, 59
(1954
).17.
E. N. da C.
Andrade
, “Review of discussion
,” Proc. R. Soc. London, Ser. A
226
, 65
(1954
).18.
R. O.
Davies
and L.
Rosenhead
, “The two viscosities of fluids
,” Nature
173
, 1209
(1954
).19.
B.
Riemann
, “Über die Fortpflanzung ebenener Luftwellen von endlicher Schwingungsbreite
,” in Collected Works of Bernhard Riemann
, edited by H.
Weber
(Dover
, 1953
), p. 157
.20.
L.
Euler
, “Principes généraux du mouvement des fluides
,” Mém. Acad. Sci. Berlin
11
, 274
(1755
).21.
D.
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/24900350.22.
R.
Becker
, “Stoßwelle und Detonation
,” Z. Phys.
8
, 321
(1921
).23.
H.
Grad
, “The profile of a steady shock wave
,” Commun. Pure Appl. Math.
5
, 257
(1952
).24.
W.
Tollmien
, H.
Schlichting
, H.
Görtler
, and F. W.
Riegels
, “Zur Theorie des Verdichtungsstoßes
,” in , edited by F. W.
Riegels
(Springer
, Berlin
, 1961
).25.
W. G.
Hoover
, “Structure of a shock-wave front in a liquid
,” Phys. Rev. Lett.
42
, 1531
(1979
).26.
B. L.
Holian
, W. G.
Hoover
, B.
Moran
, and G. K.
Straub
, “Shock-wave structure via nonequilibrium molecular dynamics and Navier-Stokes continuum mechanics
,” Phys. Rev. A
22
, 2798
(1980
).27.
G.
Emanuel
and B. M.
Argrow
, “Linear dependence of the shock wave thickness on bulk viscosity
,” Phys. Fluids
6
, 3203
(1994
).28.
T. G.
Elizarova
, A. A.
Khokhlov
, and S.
Montero
, “Numerical simulation of shock wave structure in nitrogen
,” Phys. Fluids
19
, 068102
(2007
).29.
A. 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
).30.
S.
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
).31.
M.
van Dyke
, “Second-order compressible boundary layer theory with application to blunt bodies in hypersonic flow
,” in Hypersonic Flow Research
, edited by F. R.
Riddell
(Academic Press
, 1962
), p. 37
.32.
P. M.
Morse
and K. U.
Ingard
, Theoretical Acoustics
(Princeton University Press
, 1986
).33.
34.
C.
Eckart
, “Vortices and streams caused by sound waves
,” Phys. Rev.
73
, 68
(1948
).35.
L. N.
Liebermann
, “Second viscosity of liquids
,” Phys. Rev.
75
, 1415
(1949
).36.
J. J.
Markham
, “Second-order acoustic fields: Streaming with viscosity and relaxation
,” Phys. Rev.
86
, 497
(1952
).37.
P. J.
Westervelt
, “The theory of steady rotational flow generated by a sound field
,” J. Acoust. Soc. Am.
25
, 60
(1953
).38.
W. L.
Nyborg
, “Acoustic streaming due to attenuated plane waves
,” J. Acoust. Soc. Am.
25
, 68
(1953
).39.
J.
Lighthill
, “Acoustic streaming
,” J. Sound Vib.
61
, 391
(1978
).40.
G.
Emanuel
, “Effect of bulk viscosity on a hypersonic boundary layer
,” Phys. Fluids
4
, 491
(1992
).41.
M. S.
Cramer
and F.
Bahmani
, “Effect of large bulk viscosity on large-Reynolds-number flows
,” J. Fluid Mech.
751
, 142
(2014
).42.
H. 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
).43.
J. C.
Harley
, Y.
Huang
, H. H.
Bau
, and J. N.
Zemel
, “Gas flow in micro-channels
,” J. Fluid Mech.
284
, 257
(1995
).44.
Y.
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
).45.
D. C.
Venerus
, “Laminar capillary flow of compressible viscous fluids
,” J. Fluid Mech.
555
, 59
(2006
).46.
E. 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
).47.
D. C.
Venerus
and D. J.
Bugajsky
, “Compressible laminar flow in a channel
,” Phys. Fluids
22
, 046101
(2010
).48.
F.
Bahmani
and M.
Cramer
, “Suppression of large shock-induced separation in fluids having large bulk viscosities
,” J. Fluid Mech.
756
, R2
(2014
).49.
S.
Bhola
and T. K.
Sengupta
, “Roles of bulk viscosity on transonic shock-wave/boundary layer interactions
,” Phys. Fluids
31
, 096101
(2019
).50.
S.
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
).51.
S.
Pan
and E.
Johnsen
, “The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence
,” J. Fluid Mech.
833
, 717
(2017
).52.
G.
Kirchhoff
, “Über den Einfluß der Wärmeleitung in einem Gas auf die Schallbewegung
,” Ann. Phys.
210
, 177
(1868
).53.
N.
Neklepajev
, “Über die Absorption kurzer akustischer Wellen in der Luft
,” Ann. Phys.
340
, 175
(1911
).54.
T. P.
Abello
, “Absorption of ultrasonic waves by various gases
,” Phys. Rev.
31
, 1083
(1928
).55.
W. H.
Pielemeier
, “The Pierce acoustic interferometer as an instrument for the determination of velocity and absorption
,” Phys. Rev.
34
, 1184
(1929
).56.
E.
Großmann
, “Schallabsorptionsmessung in Gasen bei hohen Frequenzen
,” Ann. Phys.
405
, 681
(1932
).57.
R. W.
Curtis
, “An experimental determination of ultrasonic absorption an reflexion coefficients in air and in carbon dioxide
,” Phys. Rev.
46
, 811
(1934
).58.
K. F.
Herzfeld
and F. O.
Rice
, “Dispersion and absorption of high frequency sound waves
,” Phys. Rev.
31
, 691
(1928
).59.
H. O.
Kneser
, “Schallabsorption in mehratomigen Gasen
,” Ann. Phys.
408
, 337
(1933
).60.
H. O.
Kneser
, “Schallabsorption und -dispersion in Flüssigkeiten
,” Ann. Phys.
424
, 277
(1938
).61.
L.
Tisza
, “Supersonic absorption and Stokes’ viscosity relation
,” Phys. Rev.
61
, 531
(1942
).62.
W. E.
Meador
, G. A.
Miner
, and L. W.
Townsend
, “Bulk viscosity as a relaxation parameter: Fact or fiction?
,” Phys. Fluids
8
, 258
(1996
).63.
R. A.
Rapuano
, “Ultrasonic absorption from 75 to 280 Mc/sec
,” Phys. Rev.
72
, 78
(1947
).64.
W. G.
Schneider
, “Sound velocity and sound absorption in the critical region
,” Can. J. Chem.
29
, 243
(1951
).65.
A. G.
Chynoweth
and W. G.
Schneider
, “Ultrasonic propagation in xenon in the region of its critical temperature
,” J. Chem. Phys.
20
, 1777
(1952
).66.
H. D.
Parbrook
and E. G.
Richardson
, “Propagation of ultrasonic waves in vapours near the critical point
,” Proc. Phys. Soc., Sect. B
65
, 437
(1952
).67.
M.
Fixman
, “Ultasonic attenuation in the critical region
,” J. Chem. Phys.
33
, 1363
(1960
).68.
W.
Botch
and M.
Fixman
, “Sound absorption in gases in the critical region
,” J. Chem. Phys.
42
, 199
(1965
).69.
M.
Fixman
, “Transport coefficients in the gas critical region
,” J. Chem. Phys.
47
, 2808
(1967
).70.
K.
Kawasaki
, “Correlation-function approach to the transport coefficients near the critical point. I
,” Phys. Rev.
150
, 291
(1966
).71.
K.
Kawasaki
, “Sound attenuation and dispersion near the liquid-gas critical point
,” Phys. Rev. A
1
, 1750
(1970
).72.
L. 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
).73.
L. P.
Kadanoff
and J.
Swift
, “Transport coefficients near the liquid-gas critical point
,” Phys. Rev.
166
, 89
(1968
).74.
L.
Hall
, “The origin of excess ultrasonic absorption in water
,” Phys. Rev.
71
, 318
(1947
).75.
L.
Hall
, “The origin of ultrasonic absorption in water
,” Phys. Rev.
73
, 775
(1948
).76.
D.
Sette
, “Ultrasonic absorption in liquid mixtures and structural effects
,” J. Chem. Phys.
21
, 558
(1953
).77.
K. F.
Herzfeld
and T. A.
Litovitz
, Absorption and Dispersion of Ultrasonic Waves
(Academic Press
, 1959
).78.
C. W.
Garland
, D.
Eden
, and L.
Mistura
, “Critical sound absorption in xenon
,” Phys. Rev. Lett.
25
, 1161
(1970
).79.
P. 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
).80.
P. A.
Fleury
and J.-P.
Boon
, “Brillouin scattering in simple liquids: Argon and neon
,” Phys. Rev.
186
, 244
(1969
).81.
A. S.
Pine
, “Velocity and attenuation of hypersonic waves in liquid nitrogen
,” J. Chem. Phys.
51
, 5171
(1969
).82.
B. 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
).83.
X.
Pan
, M. N.
Shneider
, and R. B.
Miles
, “Coherent Rayleigh-Brillouin scattering
,” Phys. Rev. Lett.
89
, 183001
(2002
).84.
J.
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
).85.
X.
Pan
, M. N.
Shneider
, and R. B.
Miles
, “Power spectrum of coherent Rayleigh-Brillouin scattering in carbon dioxide
,” Phys. Rev. A
71
, 045801
(2005
).86.
A. 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
).87.
X.
He
, H.
Wei
, J.
Shi
, J.
Liu
, S.
Li
, W.
Chen
, and X.
Mo
, “Experimental measurement of bulk viscosity of water based on stimulated Brillouin scattering
,” Opt. Commun.
285
, 4120
(2012
).88.
Z.
Gu
and W.
Ubachs
, “A systematic study of Rayleigh-Brillouin scattering in air, N2, and O2 gases
,” J. Chem. Phys.
141
, 104320
(2014
).89.
Z.
Gu
, W.
Ubachs
, and W.
van de Water
, “Rayleigh-Brillouin scattering of carbon dioxide
,” Opt. Lett.
39
, 3301
(2014
).90.
Y.
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
).91.
G. W.
Pierce
, “Piezoelectric crystal oscillators applied to the precision measurement of the velocity of sound in air and CO2 at high frequency
,” Proc. Am. Acad. Arts Sci.
60
, 271
(1925
).92.
M.
Kohler
, “Schallabsorption in Mischungen einatomiger Gase
,” Ann. Phys.
431
, 209
(1941
).93.
J. M. M.
Pinkerton
, “The absorption of ultrasonic waves in liquids and its relation to molecular constitution
,” Proc. Phys. Soc., Sect. B
62
, 129
(1949
).94.
M.
Pancholy
, “Temperature variation of velocity and absorption coefficient of ultrasonic waves in heavy water
,” J. Acoust. Soc. Am.
25
, 1003
(1953
).95.
T. A.
Litovitz
and E. H.
Carnevale
, “Effect of pressure on sound propagation in water
,” J. Appl. Phys.
26
, 816
(1955
).96.
W.
Tempest
and H. D.
Parbrook
, “The absorption of sound in argon, nitrogen and oxygen
,” Acustica
7
, 354
(1957
).97.
D. G.
Naugle
and C. F.
Squire
, “Ultrasonic attenuation in liquid argon
,” J. Chem. Phys.
42
, 3725
(1965
).98.
D. G.
Naugle
, “Excess ultrasonic attenuation and intrinsic volume viscosity in liquid argon
,” J. Chem. Phys.
44
, 741
(1966
).99.
D. G.
Naugle
, J. H.
Lunsford
, and J. R.
Singer
, “Volume viscosity in liquid argon at high pressure
,” J. Chem. Phys.
45
, 4669
(1966
).100.
D. S.
Swyt
, J. F.
Havlice
, and E. F.
Carome
, “Ultrasonic absorption in liquid argon
,” J. Chem. Phys.
47
, 1199
(1967
).101.
W. M.
Madigosky
, “Density dependence of bulk viscosity in argon
,” J. Chem. Phys.
46
, 4441
(1967
).102.
J. R.
Singer
and J. H.
Lunsford
, “Ultrasonic attenuation and volume viscosity in liquid nitrogen
,” J. Chem. Phys.
47
, 811
(1967
).103.
J. R.
Singer
, “Excess ultrasonic attenuation and volume viscosity in liquid methane
,” J. Chem. Phys.
51
, 4729
(1969
).104.
A. E.
Victor
and R. T.
Beyer
, “Ultrasonic absorption in liquid oxygen and nitrogen
,” J. Chem. Phys.
52
, 1573
(1970
).105.
J.
Allegra
, S.
Hawley
, and G.
Holton
, “Pressure dependence of the ultrasonic absorption in toluene and hexane
,” J. Acoust. Soc. Am.
47
, 144
(1970
).106.
E. V.
Larson
, D. G.
Naugle
, and T. W.
Aldair
, “Ultrasonic velocity and attenuation in liquid neon
,” J. Chem. Phys.
54
, 2429
(1971
).107.
S. K.
Kor
, O. N.
Awasthi
, G.
Rai
, and S. C.
Deorani
, “Structural absorption of ultrasonic waves in methanol
,” Phys. Rev. A
3
, 390
(1971
).108.
S. K.
Kor
, S. C.
Deorani
, and B. K.
Singh
, “Origin of ultrasonic absorption in methanol
,” Phys. Rev. A
3
, 1780
(1971
).109.
S. 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
).110.
J. A.
Cowan
and R. N.
Ball
, “Temperature dependence of bulk viscosity in liquid argon
,” Can. J. Phys.
50
, 1881
(1972
).111.
G.
Rai
, B. K.
Singh
, and O. N.
Awasthi
, “Structural absorption of ultrasonic waves in associated liquids
,” Phys. Rev. A
5
, 918
(1972
).112.
J. 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
).113.
P. W.
Ward
, J. A.
Cowan
, and R. K.
Pathria
, “Critical attenuation of ultrasound in argon
,” Can. J. Phys.
53
, 29
(1975
).114.
G. J.
Prangsma
, A. J.
Alberga
, and J. J. M.
Beenakker
, “Ultrasonic determination of the volume viscosity of N2, CO, CH4 and CD4 between 77 and 300 K
,” Physica
64
, 278
(1973
).115.
S. A.
Mikhailenko
, B. G.
Dudar
, and V. A.
Shmidt
, “Volume viscosity and relaxation times in monoatomic classical fluids
,” Fiz. Nizk. Temp.
1
, 224
(1975
).116.
P.
Malbrunot
, A.
Boyer
, E.
Charles
, and H.
Abachi
, “Experimental bulk viscosities of argon, krypton, and xenon near their triple point
,” Phys. Rev. A
27
, 1523
(1983
).117.
J. A.
Cowan
and R. N.
Ball
, “Ultrasonic attenuation and bulk viscosity in liquid krypton
,” Can. J. Phys.
58
, 74
(1980
).118.
J. A.
Cowan
and J. W.
Leech
, “Ultrasonic attenuation and bulk viscosity of liquid xenon
,” Can. J. Phys.
59
, 1280
(1981
).119.
J. A.
Cowan
and J. W.
Leech
, “Critical region ultrasonic attenuation in the condensed inert gases
,” Can. J. Phys.
61
, 895
(1983
).120.
A. S.
Dukhin
and P. J.
Goetz
, “Bulk viscosity and compressibility measurement using acoustic spectroscopy
,” J. Chem. Phys.
130
, 124519
(2009
).121.
M. 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
).122.
J.
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
).123.
C.
Tegeler
, R.
Span
, and W.
Wagner
, “A new equation of state for argon covering 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
).124.
E. W.
Lemmon
and R. T.
Jacobsen
, “Viscosity and thermal conductivity equations for nitrogen, oxygen, argon and air
,” Int. J. Thermophys.
25
, 21
(2004
).125.
E. W.
Lemmon
and R.
Span
, “Short fundamental equations of state for 20 industrial fluids
,” J. Chem. Eng. Data
51
, 785
(2006
).126.
M. 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
.127.
R.
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 , edited by R. W.
Fast
(Springer
, 1986
), pp. 1189
–1197
.128.
L.
Claes
, L. M.
Hülskämper
, E.
Baumhögger
, N.
Feldmann
, R. S.
Chatwell
, J.
Vrabec
, and B.
Henning
, “Acoustic absoprtion measurement for the determination of the bulk viscosity of pure fluids
,” TM - Tech. Mess.
86
, 2
–6
(2019
).129.
J. R.
Taylor
, Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements
, 2nd ed. (University Science Books
, 1997
).130.
G.
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
).131.
M. S.
Green
, “Markoff random process and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids
,” J. Chem. Phys.
22
, 398
(1954
).132.
R.
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
).133.
R.
Kubo
, “The fluctuation-dissipation theorem
,” Rep. Prog. Phys.
29
, 255
(1966
).134.
L.
Verlet
, “Computer “experiments” on classical fluids. I. Properties of Lennard-Jones molecules
,” Phys. Rev.
159
, 98
(1967
).135.
D.
Levesque
and L.
Verlet
, “Computer “experiments” on classical fluids. III. Time-dependent self-correlation functions
,” Phys. Rev. A
2
, 2514
(1970
).136.
D.
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
).137.
S. V.
Lishchuk
, “Role of three-body interactions in formation of bulk viscosity in liquid argon
,” J. Chem. Phys.
136
, 164501
(2012
).138.
F.
Jaeger
, O. K.
Matar
, and E. A.
Müller
, “Bulk viscosity of molecular liquids
,” J. Chem. Phys.
148
, 174504
(2018
).139.
G.
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 simulation tool for thermodynamic properties, release 3.0
,” Comput. Phys. Commun.
221
, 343
(2017
).140.
K.
Meier
, A.
Laesecke
, and S.
Kabelac
, “Transport coefficients of the Lennard-Jones model fluid. III. Bulk viscosity
,” J. Chem. Phys.
122
, 014513
(2005
).141.
V. G.
Baidakov
and S. P.
Protsenko
, “Metastable Lennard-Jones fluids. III. Bulk viscosity
,” J. Chem. Phys.
141
, 114503
(2014
).142.
A.
Zaragoza
, M. A.
Gonzales
, L.
Joly
, I.
López-Montero
, M. A.
Canales
, A. L.
Benavides
, and C.
Valeriani
, “Molecular dynamics study of nanoconfined TIP4P/2005 water: How confinement and temperature affect diffusion and viscosity
,” Phys. Chem. Chem. Phys.
21
, 13653
(2019
).143.
G. S.
Fanourgakis
, J. S.
Medina
, and R.
Prosmiti
, “Determining the bulk viscosity of rigid water models
,” J. Phys. Chem. A
116
, 2564
(2012
).144.
G.-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
).145.
G.
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
.146.
J. 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
).147.
R.
Kohlrausch
, “Theorie des elektrischen Rückstandes in der Leidner Flasche
,” Ann. Phys.
167
, 179
(1854
).148.
G.
Williams
and D. C.
Watts
, “Non-symmetrical dielectric relaxation behaviour arising from a simple empirical decay function
,” Trans. Faraday Soc.
66
, 80
(1970
).149.
R. D.
Mountain
and R.
Zwanig
, “Shear relaxation times of simple fluids
,” J. Chem. Phys.
44
, 2777
(1966
).150.
T. 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
.151.
G. L.
Murphy
, “Big-bang model without singularities
,” Phys. Rev. D
8
, 4231
(1973
).152.
V. A.
Belinskii
and I. M.
Khalatnikov
, “Influence of viscosity on the character of cosmological evolution
,” Zh. Eksp. Theor. Fiz.
69
, 401
(1975
).153.
I.
Brevik
, E.
Elizalde
, S.
Nojiri
, and S. D.
Odintsov
, “Viscous little rip cosmology
,” Phys. Rev. D
84
, 103508
(2011
).154.
R.
Colistete
, J. C.
Fabris
, J.
Tossa
, and W.
Zimdahl
, “Bulk viscous cosmology
,” Phys. Rev. D
76
, 103516
(2007
).155.
M.
Cruz
, N.
Cruz
, and S.
Lepe
, “Accelerated and decelerated expansion in a causal dissipative cosmology
,” Phys. Rev. D
96
, 124020
(2017
).156.
B.
Li
and J. D.
Barrow
, “Does bulk viscosity create a viable unified dark matter model?
,” Phys. Rev. D
79
, 103521
(2009
).157.
158.
D.
Jou
, J.
Casas-Vázquez
, and G.
Lebon
, Extended Irreversible Thermodynamics
(Springer
, 2010
).159.
P.
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
).160.
K.
Tankeshwar
, “Bulk viscosity and the relation between transport coefficients
,” Phys. Chem. Liq.
24
, 91
(1991
).© 2020 Author(s).
2020
Author(s)
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