Stochastic Hard-Sphere Dynamics for Hydrodynamics of Non-Ideal Fluids Page: 4 of 6
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Isotropic DSMC (I-DSMC). The cost is that is the compu-
tational efficiency is reduced by a factor of 2 - 3 due to the
need to perform neighbor searches.
Because the collisional momentum exchange mAvj in
traditional DSMC is not correlated with the displacement
Ar2; between the colliding particles, the virial (Avj - Aryj)
vanishes giving an ideal-gas pressure. In order to introduce
a non-trivial equation of state it is necessary to either give
an additional displacement to the particles that is parallel
to Av2i, or to bias the momentum exchange Av~i to be
(statistically) aligned to Arg. The former approach has
already been investigated in the Consistent Boltzmann Al-
gorithm (CBA) [11]; however, CBA is not thermodynami-
cally consistent since it modifies the compressibility with-
out affecting the density fluctuations (i.e., the structure of
the fluid is still that of a perfect gas). A fully consistent
approach is to require that the particles collide as if they
are elastic hard spheres of diameter equal to the distance
between them at the time of the collision. Such collisions
produce a positive virial only if the particles are approach-
ing each other, vn - vj - f-j > 0, therefore, we reject
collisions among particles that are moving apart. Further-
more, as for hard spheres, it is necessary to collide pairs
with probability that is linear in vn, which requires a fur-
ther increase of the rejection rate and thus decrease of the
efficiency. The EOS of a fluid with no internal energy must
be linear in temperature, which from the virial theorem
pc ~ (Avj - Arj), ~ F8c T, implies that the local col-
lisional frequency F8c must be proportional to the square
root of the local temperature Tc, Fs, ~ T,, as for hard
spheres. Without rejection based on vn or v,, tempera-
ture fluctuations would not be consistently coupled to the
local pressure pc because pc ~ T, instead of pc ~ Tc.
For DSMC the collisional rules can be manipulated arbi-
trarily to obtain the desired transport coefficients, however,
for non-ideal fluids thermodynamics eliminates some of the
freedom. Note that one can in fact mix SHSD collisions
with I-DSMC collisions, that either take into account or
ignore the vn requirement, to introduce more tunable pa-
rameters in SHSD. The efficiency is significantly enhanced
when the fraction of accepted collisions is increased, how-
ever, the compressibility is also increased at a comparable
collision rate.
For sufficiently small time steps, the SHSD fluid can be
considered as a simple modification of the standard hard-
sphere fluid. Particles move ballistically in-between colli-
sions. When two particles i and j are less than a diameter
apart, r. < D, there is a probability rate (3X/D)vnE(vn)
for them to collide as if they were elastic hard spheres with
a variable diameter Ds = rg. Here E is the Heaviside func-
tion, and X is a dimensionless parameter determining the
collision frequency. The prefactor 3/D has been chosen so
that for an ideal gas the average collisional rate would be
X times larger than that of a low-density hard-sphere gas
with density (volume fraction) 0 = 7rND3/(6V).
In order to understand properties of the SHSD fluid asa function of 0 and X, we consider the equilibrium pair
correlation function 92 at low densities, where correlations
higher than pairwise can be ignored. We consider the cloud
of point walkers ij representing the N(N - 1)/2 pairs of
particles, each at position r r2 - r and with veloc-
ity v = v,- vj. At equilibrium, the distribution of the
point walkers in phase space will be f(v, r) = f(vr, r) ~
g2(r) exp(-mvl/4kT). Inside the core r < D this distribu-
tion of pair walkers satisfies a kinetic equation
t+vn r vFf,
where Fo 3X/D is the collision frequency. At equilibrium,
Of/Ot 0 and vn cancels, consistent with choosing collision
probability linear in vWn. Thus d92/dx 3Xg20(1-x), with
solution 92(x) exp [3X(x - 1)] for x < 1 and 92(x) 1
for x > 1, where x = r/D. Indeed, numerical experiments
confirmed that at sufficiently low densities the equilibrium
92 for the SHSD fluid has this exponential form inside the
collision core.
This low density result is equivalent to g =
exp[-U(r)/kT], where U(r)/kT 3X(1 - x)0(1 - x) is an
effective linear core pair potential similar to the quadratic
core potential used in DPD. Remarkably, it was found nu-
merically that this weakly-repulsive potential can predict
exactly 92(x) at all liquid densities. Figure 1 shows a com-
parison between the pair correlation function of the SHSD
fluid on one hand, and a Monte Carlo calculation using the
linear core pair potential on the other, at several densities.
Also shown is a numerical solution to the hyper-netted chain
(HNC) integral equations for the linear core system. The
excellent agreement at all densities permits the use of the
HNC result in practical applications, notably the calcula-
tion of the transport coefficients.075
05
025025 05 075
I 125 IS 175
x-I/DFigure 1: (Color online) Equilibrium pair correlation function of
the SHSD fluid (solid symbols), compared to MC (open symbols)
and HNC calculations (solid lines) for the linear core system, at
various densities and x = 1.
An exact BBGKY-like hierarchy of Master equations for
the s-particle distribution functions of the SHSD fluid is
given in Ref. [12]. For the first equation of this BBGKY
hierarchy, valid at low densities, we can neglect correlations
other than pair ones and approximate f2(ri, vi, r2, v2)- SHSD--01 X=1
. SHSD 0-05 =1
SHSD9 01.0 X1
-HNC - a -01
o MC m ar
-- La dn
-
' ' ' ' ' ' . . . I . . . I .
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Donev, A; Alder, B J & Garcia, A L. Stochastic Hard-Sphere Dynamics for Hydrodynamics of Non-Ideal Fluids, article, February 26, 2008; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc900182/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.