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The cavitation tensile strength of a liquid for simple materials by the example of argon has been studied using molecular dynamics methods. Results on the negative tensile pressure have been obtained within the temperature range from 85 to 135 K. The tensile strength of liquid argon organization has been studied theoretically using the Redlich-Kwong equation of state. These approaches are in good agreement. Comparison with the earlier results of other authors has been performed. The test of the determination of the tensile pressure by molecular dynamics methods for homogeneous systems will make it possible to analyze qualitatively the cavitation strength in multicomponent systems as well as during consideration of heterogeneous nucleation, where the theoretical studies are extremely troublesome.
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ISSN 0018151X, High Temperature, 2015, Vol. 53, No. 3, pp. 406–412. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © V.L. Malyshev, D.F. Marin, E.F. Moiseeva, N.A. Gumerov, I.Sh. Akhatov, 2015, published in Teplofizika Vysokikh Temperatur, 2015, Vol. 53, No. 3,
pp. 423–429.
406
INTRODUCTION
According to the theory of the origin of cavitation,
it is considered that cavities (bubbles) are formed
when the local pressure in a liquid decreases to the
pressure of the saturated vapor. When a cavity in a homo
geneous liquid is formed, the continuity of the liquid
should be broken; therefore, the necessary pressure is
determined not by the pressure of the saturated vapor but
by the tensile strength of the liquid at the given tempera
ture [1]. The absolute value of the maximum negative
pressure that can be applied to the liquid is taken as the
upper boundary of the strength of the liquid.
There exist different theories describing the deter
mination of the tensile strength of the liquid. One of
the first is the theory of the internal pressure, which is
based on estimating the force of the intermolecular
interactions inside the system. This theory gives a
pressure value of –1700 atm for liquid argon. The sec
ond approach proposed by Temperly in 1947 is based
on consideration of the vanderWaals equation of
state [2]. This equation describes well the behavior of
the gas; however, it gives essential deviations from the
experimental data when describing the liquid state. On
the basis of this method, in 1975 Trevena calculated
the strength of the liquid for simple materials, such as
argon, oxygen, and nitrogen [3]. The calculated pres
sure for argon was –130 atm. The third theory elabo
rated by Fisher in 1948 [4] is based on the classical
nucleation theory [5]. The value calculated according
to this method was –190 atm for argon.
The tensile strength of the liquid was also studied
by experimental methods. Many researchers used the
method elaborated by Berthelet [6] in 1850. Later
Briggs [7] proposed to use
U
shaped capillaries, which
were drawn directly before the experiment for produc
ing the “pure” surface of the capillary. However, there is
a difficulty in determining the cavitation strength. For
example, the tensile strength of water in a glass Berthe
let tube is –50 atm, and in a steel tube it is13 atm.
According to the results of Beams’s experiments [8],
the cavitation strength for liquid argon was –12 atm.
Thus, the tensile strength of the liquid depends strongly
on the wall material, the quality of the surface purifica
tion, the presence of the gas and impurities in the liquid,
the purity of the experiment, and other factors.
Among recent works in this field are [9] and [10]. In
[9] the studies of explosive cavitation in superheated
liquid argon are compared with the homogeneous
nucleation theory. In [10] different theories on the ori
gin of cavitation, experimental results, and computer
simulation methods for liquid argon are compared.
Kuksin and Norman considered the phase transitions
of the first kind and mechanism of the destruction of
liquids [11, 12]. In their works the homogeneous
nucleation of bubbles in the liquid was formulated and
the growth of the formed cavities was studied using the
molecular dynamics methods.
The results of the measurement of the tensile
strength of the liquid show that quite high tensile stress
can exist in it. However, the measurement results have
a large scatter in works of different authors and in
those of the same group of experimenters. The scatter
of measurement results for the same liquid makes it
possible to suppose that regions of lowered and vari
Study of the Tensile Strength of a Liquid
by Molecular Dynamics Methods
V. L. Malyshev
a
,
b
, D. F. Marin
a
,
b
, E. F. Moiseeva
a
, N. A. Gumerov
a
,
c
, and I. Sh. Akhatov
a
,
d
a
Center for Micro and Nanoscale Dynamics of Disperse Systems, Bashkir State University, Ufa, Russia
b
Mavlyutov Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, Ufa, Russia
c
Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA
d
North Dakota State University, Fargo, ND, USA
email: victor.l.malyshev@gmail.com
Received February 14, 2014
Abstract
—The cavitation tensile strength of a liquid for simple materials by the example of argon has been
studied using molecular dynamics methods. Results on the negative tensile pressure have been obtained
within the temperature range from 85 to 135 K. The tensile strength of liquid argon organization has been
studied theoretically using the Redlich–Kwong equation of state. These approaches are in good agreement.
Comparison with the earlier results of other authors has been performed. The test of the determination of the
tensile pressure by molecular dynamics methods for homogeneous systems will make it possible to analyze
qualitatively the cavitation strength in multicomponent systems as well as during consideration of heteroge
neous nucleation, where the theoretical studies are extremely troublesome.
DOI:
10.1134/S0018151X15020145
HEAT AND MASS TRANSFER
AND PHYSICAL GASDYNAMICS
HIGH TEMPERATURE Vol. 53 No. 3 2015
STUDY OF THE TENSILE STRENGTH OF A LIQUID 407
able strength are formed in it, in which the breakup
occurs. These can be places of weaker adhesion of the
liquid to the container walls or “weak spots” in the liq
uid itself. The experimental studies make it possible to
assume that the appearance of the “weak spots” is due
to the presence of the impurities and the smallest gas
bubbles in the liquid. Thus, qualitative experimental
study of homogeneous nucleation requires expensive
equipment and highquality purification of the liquid.
The theoretical descriptions are based on approximate
models introducing their own errors in the results.
Therefore, study of this process on the molecular level
using computer simulation is the most available for
this type of problems.
This work presents the results of the calculation of
homogeneous nucleation in liquid argon in the
absence of any impurities using the molecular dynam
ics method. The theoretical calculation of the tensile
strength of the liquid was implemented according to
the Temperly method for the Redlich–Kwong equa
tion of state. Two approaches and the earlier results of
other authors were compared. The simulation results
show that molecular dynamics methods are an effi
cient means for solving similar problems. The experi
mental database is very limited, and the continuum
models mainly describe only the properties of simple
systems. Molecular dynamics methods make it possi
ble to simulate a wide range of problems associated
with homogeneous and heterogeneous nucleation.
Thus, the results obtained in this work are the basis for
solving more difficult problems associated with cavita
tion effects in a complex system.
The Redlich–Kwong Equation of State
Many works associated with the theoretical study
of the tensile strength of liquids are based on the van
derWaals equation of state [2]. However, it does not
describe very well the liquid state of the material [13].
In this work the twoparameter Redlich–Kwong equa
tion of state is considered [14]
where
P
is pressure,
R
is the universal gas constant,
T
is
temperature,
V
is the molar volume, and
a
and
b
are
gas parameters, which are calculated according to the
formula
The critical temperature and pressure parameters
for argon are
T
c
= 150.86 K,
P
c
= 48.6 atm, so the ther
modynamic constants of the equation of state for
0.5
,
()
RT a
PVb
TVV b
=−
+
22.5
0.4275 0.08664
,.
cc
cc
ab
PP
==
argon are the following:
a
= 1.65037 (J
2
K
0.5
)/(mol
2
Pa),
b
= 21.7231
×
10
–6
m
3
/mol.
The characteristic form of the Redlich–Kwong
equation is given in Fig. 1. The theoretical value of the
strength of the liquid is determined as the value of the
pressure at the point of the minimum for the corre
sponding isotherm. It is noted in the figure that at the
temperature
T
=
T
m
the minimum pressure value is
zero. Let us determine this temperature. Starting from
the equation of state, the temperature
T
m
has the value
For argon
T
m
= 135 K. Thus, the possible range of
the negative pressures is within temperatures from 83.8 K
(the melting temperature of argon) to 135 K. Table 1
shows the results of the tensile strength of the liquid for
argon obtained by different methods for the tempera
ture
T
= 85 K and the results calculated using the
Redlich–Kwong equation of state.
Such a scatter of values is possible due to the insuf
ficient highquality purification of the surface, the
presence of the gas and impurities in the liquid, the
purity of the experiment, and inaccuracy of the
applied models and equations of state.
THE MATHEMATICAL MODEL
In the molecular dynamics method, the positions
of molecules are determined from the solution of the
classical equations of motion:
where
r
i
is the radiusvector of the
i
th particle and
m
i
is the mass of the
i
th particle. With the exception of
32
(3 2 2)
.
ma
TbR
=
2
2
()
,() (),
N
ii i
ii
d
Ur
m
dt
==
rFr Fr r
P
A
V
T
>
T
m
T
<
T
m
T
=
T
m
Fig. 1.
Redlich–Kwong equation of state.
Table 1.
Comparison of the results of the tensile strength of liquid argon
Temperly theory Fisher theory Beams experiment Internal pressure Redlich–Kwong equation
–130 atm –190 atm –12 atm –1700 atm –370 atm
408
HIGH TEMPERATURE Vol. 53 No. 3 2015
MALYSHEV et al.
the simplest cases, this system of equations is solved
numerically according to the chosen algorithm (the
velocity Verlet method, leapfrog, etc.). However, first of
all it is necessary to calculate the force
F
(
r
i
)
acting on
the atom
i
, which is calculated in accordance with the
interaction potential
U
(
r
N
)
, where
r
N
= (
r
1
,
r
2
,
,
r
N
)
is
the set of distances from the
i
th particle to all other
particles.
The LennardJones potential is chosen as the
potential function, since the shortrange interaction
for simple monoatomic molecules is considered by the
example of argon:
The parameter values of the LennardJones poten
tial interaction are chosen as follows:
σ
= 3.4
Å
,
ε
=1.64
×
10
–21
J that corresponds to argon. The mass
of the molecule
m
= 66.4
×
10
–27
kg, and the integra
tion time step
Δ
t
= 2
×
10
–15
s. The cutoff radius is
chosen as
r
cutoff
= 8.
, since the small values of the
cutoff radius do not describe sufficiently well the prop
erties of the system [15].
12 6
() 4 .
j
jj
Ur rr
⎡⎤
⎛⎞ ⎛⎞
σσ
=ε −
⎢⎥
⎜⎟ ⎜⎟
⎢⎥
⎝⎠ ⎝⎠
⎣⎦
In this work the variables are nondimensionalized
according to [16]. The simulation region is a parallel
epiped, the sizes of which are determined from the
given density and the number of particles. The peri
odic boundary conditions are applied to the system in
all directions. The NVT ensemble (Number Volume
Temperature ensemble) is considered for the simula
tion of the statistical properties of the system in which
the kinetic energy of molecules is fixed. The mainte
nance of constant temperature is implemented using a
Berendsen thermostat [17].
CODE VERIFICATION
The free dynamics of the vapor–liquid argon
medium is considered for verification of the molecular
dynamics calculations.
The simulation region is a parallelepiped with sizes
L
x
=
L
y
= 24
σ
,
L
z
= 48
σ
. Liquid argon is placed in the
center with the dimensionless parameters
ρ
*
= 0.7 and
T
*
= 0. The scheme of the region is shown in Fig. 2.
The temperature of the system is kept constant by
the algorithm of the velocity correction (the Ber
endsen thermostat) during the first 50000 steps. Then
for 25000 steps the system is in the absence of thermo
statting for the establishment of the equilibrium state. For
the following 125000 steps (with an interval of 100 steps),
the macroscopic properties (e.g., density) are selected,
which then are averaged out over the whole set of the
particles.
To determine the density, the simulation region is
divided into layers parallel to the
L
x
L
y
plane. We
denote the number of such layers as
N
bin
; in this work
it is taken as 100. The density in each layer is deter
mined according to the formula
where
N
i
is the number of particles located in the
i
th
layer. The value
i
varies from 1 to
N
bin
.
Figure 3 shows the density profiles at different tem
peratures of the system. To make a symmetrical pic
ture, every 500 steps the particles were centered with
respect to the parallelepiped center with conservation
of the intermolecular distances.
Starting from the built distributions (Fig. 3), it is
possible to determine the density of liquid argon from
the central part of the plot and gaseous argon from the
extreme regions. By calculating in such way the density
at different temperatures, it is possible the build the sat
uration curve. The initial density of argon located in the
parallelepiped
ρ
* = 0.7 (
ρ
= 1182 kg/m
3
). The temper
ature
T
*
varies within the range from 0.7 to 1.25 (from
85 to 150 K, respectively, in dimensional quantities).
Figure 4 shows the obtained results in dimensional
quantities.
The saturation curve for argon was also determined
by other authors [18, 19]. However, it is complicated to
bin
,
i
i
xyz
NN
LLL
ρ=
L
y
L
x
L
z
Fig. 2.
Scheme of the simulation region.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
48444032241602012 2884 36
z
,
σ
0.9
ρ
*
1
2
3
4
5
Fig. 3.
Density profile at different temperatures: (
1
)
T
* =
0.7008, (
2
) 0.9053, (
3
) 1.0048, (
4
) 1.0988, (
5
) 1.2123.
HIGH TEMPERATURE Vol. 53 No. 3 2015
STUDY OF THE TENSILE STRENGTH OF A LIQUID 409
perform a comparison with their results, since quite
often only figures are presented without tables and
data. In this respect, Fig. 4 shows the calculation
results by molecular dynamics method and experi
mental data from [20]. It can be seen from the plot that
the numerical calculation results are in good agree
ment with the experiment.
The dynamics of the system consisting of many
particles has stochastic properties that are described in
detail in [21, 22]. We consider two trajectories calcu
lated from the same initial conditions but with differ
ent numerical integration steps
Δ
t
= 1 fs and
Δ
t
'
= 2 fs.
The first trajectory is denoted as (
r
i
(
t
),
v
i
(
t
)
, and the
second is denoted as ( . Averaged over the
trajectory, the differences of the coordinates and
velocities of the first and second trajectories in the
coinciding moments of time
t = k
Δ
t = k
'
Δ
t
'
are given in Fig. 5.
It is seen well from the plot that the “time of mem
ory” of the systems is on the order of 5 ps, which is in
good agreement with [22].
SIMULATION OF THE CAVITATION
STRENGTH OF THE LIQUID
We consider the molecular dynamics simulations of
the cavitation strength. The temperature and the den
sity corresponding to the liquid state of argon are cho
sen according to the saturation curve (Fig. 4). The ini
tial density is
ρ
= 1350 kg/m
3
, and the temperature of
the system is
T
= 85 K. The simulation region is a
cube, the sizes of which are determined from the given
density and the number of particles. In this case the
region has sizes
L
x
=
L
y
=
L
z
= 43.09
σ
. At the initial
moment of time, the particles are distributed uni
formly over the total simulation region. The tempera
ture in the system is kept by the Berendsen thermostat,
which is used at each time. Each 5000 steps the simu
lation region is increased in all directions by the value
of
0.02
σ
, and also the intermolecular distances are
increased by the value corresponding to the tension of
the region. In [12, 23] the spall strength of simple liq
uids was studied. The destruction model used in [23]
contributes changes to the value of the spalling
strength of hexane by 20%. The calculations show that
for different tension velocities (
0.01
σ
–0.04
σ
) the ten
sile pressure remains the same. Thus, it is possible to
note the weak dependence of the maximally reachable
pressure on the tension velocity that was also noted in
[12]. The calculations show that the value of the cavita
tion strength depends weakly on the tension velocity.
'
'
((), ())
ii
tt
rv
22
1
22
1
1'
() ( () ()),
1'
() ( () ())
N
ii
i
N
ii
i
rt rt rt
N
ttt
N
=
=
Δ= −
Δ= −
vvv
The pressure in the system was calculated using the
method of virial sums according to the formula [24]
where
P
is pressure,
V
is volume,
T
is temperature,
N
is the number of particles, and
r
ij
and
f
ij
are the radius
vector and force of the interaction between
i
and
j
particles,
...
is time averaging. The summation is
performed over all atoms located in the system, and
the interaction force is calculated according to the
LennardJones potential. The pressure is calculated
every 2000 steps. For higher accuracy the pressure is
calculated without using the cutoff radius, since it is
very sensitive to its value.
The complexity of the calculation algorithm of the
pair interaction by the direct calculation method is
O
(
N
2
)
, where
N
is the number of particles. Therefore,
<
=+
1
,
3ij ij
ij
PV NT
rf
160
150
140
130
120
110
100
90
80 150012003000600 900
ρ
, kg/m
3
T
, K
Fig. 4.
Saturation curve: line – experiment, points – cal
culation by the molecular dynamics method.
10
2
10
0
10
–2
10
–4
10
–6
10
–8
1210840 62
Δ
r
2
(
t
)
Δv
2
(
t
)
t
, ps
Fig. 5.
Normalized averaged recession of coordinates and
velocities on two trajectories (the logarithmic scale on the
ordinate) calculated from the identical initial conditions with
steps
Δ
t
= 1 fs and
Δ
t
' = 2 fs (
N
= 64000,
ρ
= 1350 kg/m
3
,
T
=85 K).
410
HIGH TEMPERATURE Vol. 53 No. 3 2015
MALYSHEV et al.
significant computing resources are required for cal
culating large systems. In this work we used a special
data structure elaborated by the authors and a hetero
geneous workstation with two 6core CPU Intel Xeon
5660 2.8 GHz (a total of 12 physical cores and 12 vir
tual cores using the Hyper–Threading technology),
32 GB RAM, and four GPU NVIDIA Tesla C2075
with 6 GB RAM. The code on GPU was written using
the NVIDIA CUDA technology, and the parallel
working of several GPU was implemented using the
OpenMP technology. Calculations were performed
using numbers with a floating point of double accu
racy. The comparison with the package LAMMPS was
also performed concerning the performance. The sim
ulator was chosen in which a method is implemented
that makes it possible to reduce the computing com
plexity of the whole algorithm and the GPU. The
comparison was performed with the same physical
parameters and the uniform distribution of molecules
over the region. The package USER–CUDA was used
for calculating on the GPU in LAMMPS. The compar
ison was performed on identical computing stations
described above. The comparison showed that the per
formance of the program elaborated by the authors is up
to three times higher than the power of LAMMPS on
one GPU. A detailed description of the molecular
dynamic acceleration using GPU can be found in
[25]. The calculation results for different sizes of sys
tems showed that the calculation of the pressure has no
essential deviations for the sizes of systems from
N
=
8000 to
N
= 512000 particles. Therefore, in this work
the value of the number of particles
N
= 64000 was
chosen for the sake of convenience of the calculation
and larger visibility.
The plot of the pressure of the system as a function
of time is shown in Fig. 6. Under the tension of the
region, the pressure in the system drops to a certain
minimum value showing the maximum absolute neg
ative pressure that the liquid can endure. Since the
pressure has some fluctuations, a certain time domain
containing the minimum value is chosen, over which
the averaging is performed for determining the cavita
tion strength. In Fig. 6 the maximum absolute negative
pressure is
P
*
–0.88, which is reached at approxi
mately the 330000th step (
t
0.66
ns). At the subse
quent moments of time, the pressure has a sharp step
and goes into the stationary region. The cavitation
bubble is formed at the moment of a sharp increase in
the pressure in the system. Figure 7 shows the cut of
the region, in which the bubble is located, and its iso
–0.9
100000 200000 300000 400000 500 000
Step
P
*
–0.8
–0.7
–0.6
–0.5
–0.4
–0.3
–0.2
Fig. 6.
Time variation of the pressure (
T
= 85 K,
ρ
=
1350 kg/m
3
,
N
= 64000).
(a)
(b)
Fig. 7.
Formation of the cavitation bubble (
T
= 85 K,
ρ
=
1350 kg/m
3
,
N
= 64000,
t
= 0.94 ns, size of the region 15
×
15
×
15 nm
3
): (a) cut of the region, (b) isosurface.
HIGH TEMPERATURE Vol. 53 No. 3 2015
STUDY OF THE TENSILE STRENGTH OF A LIQUID 411
surface (the level surface built on the density value
ρ
=
642 kg/m
3
).
Thus, it is possible to determine the cavitation
strength of the liquid by calculating the point of the
minimum on the pressure plot. The pressure values for
other temperatures were calculated analogously. Table 2
shows the results obtained using the molecular
dynamics method and from the Redlich–Kwong
equation of state (according to the Temperly method).
Figure 8 shows the results of determination of the
cavitation strength obtained by the molecular dynam
ics method and from the Redlich–Kwong equation of
state (Table 2). It is possible to note that the results of
molecular dynamics and the equation of state are in
good agreement. The obtained results show that it is
possible to use the Redlich–Kwong equation of state
in simple systems for determining the tensile strength
without using moleculardynamic simulation.
CONCLUSIONS
The study of the cavitation tensile strength of a
monoatomic liquid was described well by molecular
400
350
300
250
200
150
100
50
0120100 11090 130
T
, K
–P
, atm
Fig. 8.
Tensile strength of the liquid: curve is Redlich–
Kwong equation of state, points is molecular dynamics
method.
dynamics methods even for small systems (with the
number of molecules from 8000). In the given
approach the decrease in the pressure in the system
was reached due to the tension of the simulation
region and the tension rate does not play an essential
role. Good agreement between the molecular
dynamics results and the theoretical results on the
basis of the Redlich–Kwong equation of state was
obtained. The moleculardynamic approach will
make it possible to determine the tensile pressure in
different multicomponent and heterogeneous sys
tems, in which it is already impossible to use the
equation of state.
ACKNOWLEDGMENTS
This work was supported by the Ministry of Educa
tion and Science of the Russian Federation, project
no. 11.G34.31.0040.
REFERENCES
1. Knapp, R., Daily, J., and Hammitt, F.,
Cavitation
, New
Yor k : M cG ra w Hi ll , 1 97 0.
2. Temperly, H.N.V.,
Proc. Phys. Soc.,
1947, vol. 59,
p. 199.
3. Trevena, D.H.,
J. Phys. D: Appl. Phys.
, 1975, vol. 8,
p. L144.
4. Fisher, J.C.,
J. Appl. Phys.
, 1948, vol. 19, p. 1062.
5. Zel’dovich, Ya.B.,
Zh. Eksp. Teor. Fiz.
, 1942, vol. 12,
no. 11, p. 525.
6. Bertholet, M.,
Ann. Chim. Phys.
, 1850, vol. 30, p. 232.
7. Briggs, L.J.,
J. Appl. Phys.
, 1955, vol. 26, p. 1001.
8. Beams, J.W.,
Phys. Fluids
, 1959, vol. 2, no. 1, p. 1.
9. Vinogradov, V.E., Pavlov, P.A., and Baidakov, V.G.,
J. Chem. Phys.
, 2008, vol. 128, p. 234508.
10. Kalikmanov, V.I., Wolk, J., and Kraska, T.,
J. Chem.
Phys.
, 2008, vol. 128, p. 124506.
11. Kuksin, A.Yu., Norman, G.E., and Stegailov, V.V.,
High Temp.
, 2007, vol. 45, no. 1, p. 37.
12. Kuksin, A.Yu., Norman, G.E., Pisarev, V.V., Ste
gailov, V.V., and Yanilkin, A.V.,
High Temp.
, 2010, vol. 48,
no. 4, p. 511.
13. HoYoung, Kwak and Panton, R.L.,
J. Phys. D: Appl.
Phys
, 1984, vol. 18, p. 647.
14. Redlich, O. and Kwong, J.N.S.,
Chem. Rev.
, 1949,
vol. 44, no. 1, p. 233.
15. Wang, D., Zeng, D., and Cai, Z.,
J. Chongqing Univ.
(Engl. Ed.)
, 2002, vol. 1, no. 2, p. 60.
16. Allen, M.P. and Tildesley, D.J.,
Computer Simulation of
Liquids
, Oxford: Claredon, 1987.
17. Berendsen, H.J.C., Postma, J.P.M., van Gunsteren, W.F.,
DiNola, A., and Haak, J.R.,
J. Chem. Phys.
, 1984, vol. 81,
no. 8, p. 3684.
Table 2.
Value of the tensile strength of liquid argon at dif
ferent temperatures obtained by the molecular dynamics
method (MD) and from the Redlich–Kwong equation of
state (RK)
T
, K 85 93 100 107 115 123 130
P
, MD, atm 367 292 229 179 119 75 32
P
, RK, atm 370 282 216 159 104 56 21
412
HIGH TEMPERATURE Vol. 53 No. 3 2015
MALYSHEV et al.
18. Bo, Shi,
PhD Dissertation
, Los Angeles: University of
California, United States, 2006.
19. Cosden, I.A. and Lukes, J.R.,
J. Heat Transfer
, 2011,
vol. 133, no. 10, p. 101501.
20. Vargaftik, N.B.,
Spravochnik po teplofizicheskim svoist
vam gazov i zhidkostei
(A Reference Book on Thermal
and Physical Properties of Gases and Liquids), Mos
cow: Nauka, 1972.
21. Bazhirov, T.T., Norman, G.E., and Stegailov, V.V.,
J. Phys.: Condens. Matter
, 2008, vol. 20, no. 11, p. 114113.
22. Norman, G.E. and Stegailov, V.V.,
Mat. Model.
, 2012,
vol. 24, no. 6, p. 3.
23. Kuksin, A.Yu., Norman, G.E., Pisarev, V.V., Ste
gailov, V.V., and Yanilkin, A.V.,
Phys. Rev. B
, 2010,
vol. 82, p. 174101.
24. Rapaport, D.C.,
The Art of Molecular Dynamics Simu
lation
, Cambridge: Cambridge University Press, 2004.
25. Malyshev, V.L., Marin, D.F., Moiseeva, E.F., Gume
rov, N.A., and Akhatov, I.Sh.,
Vestnik NNGU
, 2014,
no. 3, p. 126.
Translated by L. Mosina
... The error in p stays below 5% for |ū ′ | 0.15, but exceeds 30% for u ′ 0.5. If Eqs. (30), (31) are deemed to be exact along a considered isentrope, they yield ...
... where, according to (31), ...
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A novel phase-flip model is proposed for thermodynamically consistent and computationally efficient description of spallation and cavitation in pure liquids within the framework of ideal hydrodynamics. Aiming at ultra-fast dynamic loads, the spall failure of a liquid under tension is approximated as an instantaneous decomposition of metastable states upon reaching the spinodal stability limit of an appropriate two-phase liquid-gas equation of state. The spall energy dissipation occurs as entropy jumps in two types of discontinuous solutions, namely, in hypersonic spall fronts and in pull-back compression shocks. Practical application of the proposed model is illustrated with numerical simulations and a detailed analysis of a particular problem of symmetric plate impact. The numerical results are found to be in good agreement with the previously published molecular-dynamics simulations. Also, new approximate nonlinear formulae are derived for evaluation of the strain rate, fractured mass, and spall strength in terms of the observed variation of the free-surface velocity. The new formula for the spall strength clarifies complex interplay of the three first-order nonlinear correction terms and establishes a universal value of the correction factor for attenuation of the spall pulse, which in the limit of weak initial loads is independent of the equation of state.
... The developed high-performance programming code for the MD simulations has been applied to several physical problems, such as cavitation in liquid argon ( fig. 2) [17], surface bubble dynamics in a liquid flow ( fig. 3) [18], particle movement by surface nanobubble (fig. 4) [19]. ...
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Some further experiments on the behaviour of water under tension are described, which appear to confirm the conclusions of the first two papers in this series. It appears that, under favourable conditions, water in a glass tube can support tensions as high as 60 atmospheres. A simple theoretical investigation shows that the commonly held view that the tensile strength of a liquid should be numerically equal to the "intrinsic pressure" is false. For water, a reasonable theoretical value would be 500 to 1000 atmospheres for the tensile strength, which is higher than anything that has been actually measured, though two possible explanations of this discrepancy can be suggested. In any case, it is clear that the discrepancy is much less serious than is usually supposed. The same theoretical considerations can be applied to the experiments of Kenrick, Gilbert and Wismer on the superheating of liquids, and their results can be accounted for without any extra assumptions.