Enskog–Vlasov equation—a nonlinear partial-integrodifferential equation, provides a robust framework for analyzing liquid dynamics and phase transition. The Vlasov force expanded using Taylor series yields Korteweg stress with two constant coefficients. The first coefficient yields the van der Waals like contribution to equilibrium pressure, while the second coefficient determines the interfacial stresses/surface tension. In this article, we express the second Korteweg constant coefficient as a function of temperature. The developed model effectively captures the liquid–vapor interfacial region in equilibrium, reproducing the sharp interfacial structure predicted by the full Enskog–Vlasov equation. Additionally, we compare our results with those obtained through a particle-based approach, as studied by Frezzotti et al. (2009) [“Direct simulation Monte Carlo applications to liquid–vapor flows,” Phys. Fluids 31, 062103 (2019)]. The proposed model balances simplicity with computational efficiency, comprehensively examined within the paper.
REFERENCES
1.
A.
Frezzotti
,
P.
Barbante
, and
L.
Gibelli
, “
Direct simulation Monte Carlo applications to liquid-vapor flows
,” Phys. Fluids
31
, 062103
(2019
).2.
S.
Corsetti
,
R. E.
Miles
,
C.
McDonald
,
Y.
Belotti
,
J. P.
Reid
,
J.
Kiefer
, and
D.
McGloin
, “
Probing the evaporation dynamics of ethanol/gasoline biofuel blends using single droplet manipulation techniques
,” J. Phys. Chem. A
119
, 12797
–12804
(2015
).3.
D.-H.
Kim
,
S.
Lim
,
J.
Shim
,
J. E.
Song
,
J. S.
Chang
,
K. S.
Jin
, and
E. C.
Cho
, “
A simple evaporation method for large-scale production of liquid crystalline lipid nanoparticles with various internal structures
,” ACS Appl. Mater. Interfaces
7
, 20438
–20446
(2015
).4.
O.
Denmead
,
F.
Dunin
,
R.
Leuning
, and
M.
Raupach
, “
Measuring and modelling soil evaporation in wheat crops
,” Phys. Chem. Earth
21
, 97
–100
(1996
).5.
G.
Hasanain
,
T.
Khallaf
, and
K.
Mahmood
, “
Water evaporation from freshly placed concrete surfaces in hot weather
,” Cem. Concr. Res.
19
, 465
–475
(1989
).6.
F. M.
Richter
, “
Timescales determining the degree of kinetic isotope fractionation by evaporation and condensation
,” Geochim. Cosmochim. Acta
68
, 4971
–4992
(2004
).7.
S. S.
Sazhin
, “
Modelling of fuel droplet heating and evaporation: Recent results and unsolved problems
,” Fuel
196
, 69
–101
(2017
).8.
M. P.
Brenner
,
S.
Hilgenfeldt
, and
D.
Lohse
, “
Single-bubble sonoluminescence
,” Rev. Mod. Phys.
74
, 425
(2002
).9.
J.
Lutišan
and
J.
Cvengroš
, “
Mean free path of molecules on molecular distillation
,” Chem. Eng. J. Biochem. Eng. J.
56
, 39
–50
(1995
).10.
C. J.
Knight
, “
Theoretical modeling of rapid surface vaporization with back pressure
,” AIAA J.
17
, 519
–523
(1979
).11.
I.
Eames
,
N.
Marr
, and
H.
Sabir
, “
The evaporation coefficient of water: A review
,” Int. J. Heat Mass Transfer
40
, 2963
–2973
(1997
).12.
G.
Fang
and
C. A.
Ward
, “
Temperature measured close to the interface of an evaporating liquid
,” Phys. Rev. E
59
, 417
(1999
).13.
D.
Enskog
, “
The numerical calculation of phenomena in fairly dense gases
,” Arkiv Mat. Astr. Fys.
16
, 1
–60
(1921
).14.
L. d
Sobrino
, “
On the kinetic theory of a van der Waals gas
,” Can. J. Phys.
45
, 363
–385
(1967
).15.
16.
A.
Frezzotti
, “
A particle scheme for the numerical solution of the Enskog equation
,” Phys. Fluids
9
, 1329
–1335
(1997
).17.
A.
Frezzotti
and
L.
Gibelli
, “
A kinetic model for equilibrium and non-equilibrium structure of the vapor-liquid interface
,” in AIP Conference Proceedings
(
American Institute of Physics
, 2003
), Vol.
663
, pp. 980
–987
.18.
A.
Frezzotti
,
L.
Gibelli
, and
S.
Lorenzani
, “
Mean field kinetic theory description of evaporation of a fluid into vacuum
,” Phys. Fluids
17
, 012102
(2005
).19.
S.
Busuioc
,
L.
Gibelli
,
D. A.
Lockerby
, and
J. E.
Sprittles
, “
Velocity distribution function of spontaneously evaporating atoms
,” Phys. Rev. Fluids
5
, 103401
(2020
).20.
A.
Frezzotti
and
C.
Sgarra
, “
Numerical analysis of a shock-wave solution of the Enskog equation obtained via a Monte Carlo method
,” J. Stat. Phys.
73
, 193
–207
(1993
).21.
F. J.
Alexander
,
A. L.
Garcia
, and
B. J.
Alder
, “
A consistent Boltzmann algorithm
,” Phys. Rev. Lett.
74
, 5212
(1995
).22.
J. M.
Montanero
and
A.
Santos
, “
Monte Carlo simulation method for the Enskog equation
,” Phys. Rev. E
54
, 438
(1996
).23.
A.
Frezzotti
, “
Molecular dynamics and Enskog theory calculation of one dimensional problems in the dynamics of dense gases
,” Phys. A
240
, 202
–211
(1997
).24.
L.
Wu
,
Y.
Zhang
, and
J. M.
Reese
, “
Fast spectral solution of the generalized Enskog equation for dense gases
,” J. Comput. Phys.
303
, 66
–79
(2015
).25.
M.
Kon
,
K.
Kobayashi
, and
M.
Watanabe
, “
Method of determining kinetic boundary conditions in net evaporation/condensation
,” Phys. Fluids
26
, 072003
(2014
).26.
A.
Frezzotti
,
L.
Gibelli
,
D. A.
Lockerby
, and
J. E.
Sprittles
, “
Mean-field kinetic theory approach to evaporation of a binary liquid into vacuum
,” Phys. Rev. Fluids
3
, 054001
(2018
).27.
A. A.
Vlasov
, Many-Particle Theory and Its Application to Plasma
(
Gordon and Breach
, 1961
).28.
K.
Piechór
, “
Discrete velocity models of the Enskog-Vlasov equation
,” Transp. Theory Stat. Phys.
23
, 39
–74
(1994
).29.
X.
He
and
G. D.
Doolen
, “
Thermodynamic foundations of kinetic theory and lattice Boltzmann models for multiphase flows
,” J. Stat. Phys.
107
, 309
–328
(2002
).30.
M.
Sadr
and
M. H.
Gorji
, “
A continuous stochastic model for non-equilibrium dense gases
,” Phys. Fluids
29
, 122007
(2017
).31.
H.
Struchtrup
and
A.
Frezzotti
, “
Twenty-six moment equations for the Enskog–Vlasov equation
,” J. Fluid Mech.
940
, A40
(2022
).32.
D. M.
Anderson
,
G. B.
McFadden
, and
A. A.
Wheeler
, “
Diffuse-interface methods in fluid mechanics
,” Annu. Rev. Fluid Mech.
30
, 139
–165
(1998
).33.
H.
Struchtrup
,
H.
Jahandideh
,
A.
Couteau
, and
A.
Frezzotti
, “
Heat transfer and evaporation processes from the Enskog-Vlasov equation and its moment equations
,” Int. J. Heat Mass Transfer
223
, 125238
(2024
).34.
M.
Grmela
, “
Kinetic equation approach to phase transitions
,” J. Stat. Phys.
3
, 347
–364
(1971
).35.
N. F.
Carnahan
and
K. E.
Starling
, “
Equation of state for nonattracting rigid spheres
,” J. Chem. Phys.
51
, 635
–636
(1969
).36.
W.
Sutherland
, “
LII. The viscosity of gases and molecular force
,” London, Edinburgh, Dublin Philos. Mag. J. Sci.
36
, 507
–531
(1893
).37.
J. D.
Van der Waals
, Over de Continuiteit Van Den Gas-en Vloeistoftoestand
(
Sijthoff
, 1873
), Vol.
1
.38.
H.
Struchtrup
, Macroscopic Transport Equations for Rarefied Gas Flows
(
Springer
, 2005
).39.
M.
Torrilhon
, “
Modeling nonequilibrium gas flow based on moment equations
,” Annu. Rev. Fluid Mech.
48
, 429
–458
(2016
).40.
G. M.
Kremer
, An Introduction to the Boltzmann Equation and Transport Processes in Gases
(
Springer Science & Business Media
, 2010
).41.
V.
Giovangigli
, “
Kinetic derivation of diffuse-interface fluid models
,” Phys. Rev. E
102
, 012110
(2020
).42.
H.
Struchtrup
, Thermodynamics and Energy Conversion
(
Springer
, 2014
).43.
J.
Karkheck
and
G.
Stell
, “
Kinetic mean-field theories
,” J. Chem. Phys.
75
, 1475
–1487
(1981
).44.
F. C.
Dias
and
F.
Sharipov
, “
The structure of shock waves propagating through heavy noble gases: Temperature dependence
,” Shock Waves
31
, 609
–617
(2021
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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