Condensed Matter Physics 2007, Vol. 10, No 2(50), pp. 151–163
Hydrodynamic equations for dense fluid mixtures with
multistep interaction between particles
M.V.Tokarchuk1,2, Y.A.Humenyuk1∗
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., Lviv 79011, Ukraine
2 State University “Lvivska Politekhnika”, 12 Bandera Str., Lviv 79013, Ukraine
Received January 20, 2007, in final form April 23, 2007
Generalization of the kinetic equation for a dense fluid with a multistep interaction potential to the case of
a dense mixture is presented. Derivation of hydrodynamic equations is performed starting with the kinetic
equation for the one-particle distribution function and the transport equation for the potential energy density.
Expressions for momentum and heat fluxes are obtained. The transport equation for the kinetic energy density
is shown to contain a new term of the “source” type, which describes fast exchange processes between kinetic
and potential energies of the system.
Key words: kinetic theory, square-well potential, energy exchange, fluid mixture, multistep interaction
PACS: 05.20.Dd, 05.10.+y, 05.60.-k
1. Introduction
Construction of kinetic equations for dense fluids remains one of the main problems in the
kinetic theory. Besides the difficulties characteristic of moderate and high densities and connected
with the increasing role of interparticle correlations, that go beyond the pair collision approxima-
tion and take higher-order collisions into account, there appears another circumstance. When the
average interparticle distance decreases, the interaction energy contribution to the total energy
becomes appreciable. Apart from the short-range repulsion, the processes occurring between par-
ticles at distances of molecular attraction become important. Thus, a problem arises concerning
the construction of the kinetic equations, which, through the corresponding collision integral could
explicitly include microscopic processes due to the long-range attractive part of the interaction
potential.
Neither the standard Enskog theory (SET) [1] nor its modifications (MET) [2,3] and (RET)
[4,5] solve the problem as far as they are valid for hard spheres. The Enskog-type kinetic theories
[6,7] suggested for a “hard spheres + smooth tail” potential treat the long-range interactions by
a non-dissipative mean-field term. But it indirectly effects the transport coefficients through the
pair distribution function and the contributions to thermodynamic quantities.
The square-well (SW) potential is the simplest one for which in the case of dense fluid the inter-
molecular attraction can be irreversibly included into the kinetic equation. For the first time this
was performed in the so-called DRS theory [8]. Though the transport coefficients were calculated,
still substantial shortcomings consisted in disregarding the energy conservation law as well as in
the absence of the H theorem. These deficiencies were overcome in the revised version RDRS [9].
The H theorem was proved and the new kinetic theory was shown to give the correct equilibrium
solution. The theory got its continuation in [10], where it was shown that the DRS and RDRS bulk
viscosity coefficients differ from each other. Later, the spectrum of the linearized kinetic equation
of RDRS was investigated and the new mode of the exchange type was discovered in [11].
∗E-mail: josyp@ph.icmp.lviv.ua
c© M.V.Tokarchuk, Y.A.Humenyuk 151
M.V.Tokarchuk, Y.A.Humenyuk
The multistep (MS) interaction potential is more general and can better reproduce the course
of the realistic interaction both at short and at long intermolecular distances. The corresponding
kinetic equation for the one-component fluid was obtained in [12] based on the Zubarev’s non-
equilibrium statistical operator method starting with the Liouville equation. The H theorem was
proved [12] and the transport coefficients for one-component fluid Ar along the gas–liquid saturation
curve were calculated together with temperature and density dependencies [13,14].
Estimation of the effect of the soft attractive part on the transport coefficients in the high
density region remains rather important. Two ways, i.e., indirect and direct, of dealing with the soft
interaction can be distinguished. For the first one, the kinetic variational theory III, the stochastic
theory, and the renormalized Kirkwood theory have been analyzed in a recent short review [15].
Their capability of describing the self-diffusion coefficient as well as the bulk and shear viscosities
for the one-component Lennard-Jones (LJ) fluid was investigated. In the semi-phenomenological
approach of Morioka [16] a simple method was formulated for the shear viscosity calculation for
fluids with a soft attraction, based on the combination of the Enskog hard sphere result and the
evaluation of the average cross-section of the momentum transfer for the continuous potential.
For the second case, considerations involve time-correlation functions and memory-kernel meth-
ods starting with the Green-Kubo formulas rather than with an appropriate kinetic equation. In
[17] the results for the self-diffusion coefficient of a dense LJ fluid are obtained in the binary col-
lision approximation for the friction memory kernel. An approximate treatment of pair dynamics
for the SW potential described by a Smoluchowski diffusion-like equation is used in evaluating the
self-diffusion and shear viscosity coefficients [18]. The deviation from the Stokes-Einstein law is
observed if higher-particle time correlations are taken into account.
The SW interaction is applied to the calculation of the high-temperature correction to transport
coefficients for the moderate density fluid with a hard core [19]. Finally, both the square-well and
square-barrier potentials have been used to mimic cohesive forces and soft material effect for
inelastic particles in granular matter systems. In order to investigate the interplay between these
forces of different nature, the 1D dense inelastic hard rod gas was studied by the kinetic theory
and computer simulations [20].
Here, we consider a generalization of the kinetic equation for a system of particles with the
multistep interaction to the case of dense fluid mixture and are concerned with the issue of deriving
the hydrodynamic equations. In order to get an adequate kinetic description, the transport equation
for the interaction energy density needs also to be considered as it was done in [9–11]. However, in
these works the proper attention was not paid to derivation of the kinetic energy density transport
equation starting with the corresponding kinetic equation. The many-component version is of
great interest since it permits to investigate diffusion and thermal diffusion processes as well as
to elucidate the effect of the long-range interaction on the related transport coefficients, which is
lacking in both SW and MS potentials.
The paper is organized as follows. In section 2 the kinetic equation is presented. In section 3
transport equations for the partial mass densities, average velocity, and kinetic energy density are
derived. Section 4 is devoted to the potential energy density. In the last section we give some
conclusions.
2. Kinetic equation
We will consider a many-component system of classical particles with M components. The
particles interact by pair central forces as it holds for simple fluids. A realistic potential such as
the Lennard-Jones type is approximated by a multistep function. It consists of the hard core part
and a set of repulsive and attractive walls of finite heights (figure 1). The following parameters
fully define the geometry of the MS potential: σij0, σrijl, σaijl determine allocations of the hard-core,
repulsive, and attractive walls, Krij is the number of the repulsive walls (not including the hard
core), Kaij is the number of the attractive walls. Parameters rijl, aijl denote the values of the MSP
between walls, while
∆rijl = rijl − rij,l+1, ∆aijl = aijl − aij,l−1 (1)
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Hydrodynamic equations for fluid mixtures with multistep interaction
rij
φij(rij)
σrij0
σrij1 σrij2
σaij1σ
a
ij2 σ
a
ij3 σ
a
ij4 σ
a
ij5
εrij1
εrij2
εrij3 ε
a
ij0
εaij1
εaij2
εaij3
εaij4
εaij5=0
∆εrij1
∆εrij2 ∆εaij1
∆εaij2
∆εaij3
LJ potential
MS potential
Figure 1. The model multistep potential, Krij = 2, Kaij = 5.
characterize the heights of the walls and are introduced in such a way that ∆rijl, ∆aijl > 0. The
plateaux are numbered in the direction of rij increasing: from the hard core for the repulsive part
and from the first attractive wall for the attractive one. For the bottom of the well we use two
designations rij,Krij+1 and 
a
ij0. Besides, σij0 ≡ σrij0, rij0 = ∞, aijKaij = 0.
The distinctive feature of the interactions between two particles at the walls is that they are
instantaneous: the interaction time tends to zero τ int → 0+. Due to this, ternary and higher-order
processes at the walls are much less probable and we can restrict ourselves in the kinetic equation
to the pair collision approximation. The kinetic equation for the one-component case was obtained
in [12] based on the theoretical scheme for constructing the kinetic theory of dense fluid within
the framework of the non-equilibrium statistical operator method by Zubarev. We generalize the
theory to the mixture and write the corresponding kinetic equation for the one-particle distribution
function of species i in the form:
[∂t + vi · ∇]fi(r,vi, t) = IE+MSPi [f2], (2)
where ∂t ≡ ∂/∂t, ∇ ≡ ∂/∂r.
The collision integral IE+MSPi consists of two contributions corresponding to the structure of
the MS potential:
IE+MSPi [f2] =
M
∑
j=1
{
IEij [f ij2 ] + IMSPij [f ij2 ]
}
, (3)
where f ij2 are the two-particle distribution functions. The Enskog type collision integral IEij describes
pair interactions at the hard core, while IMSPij contributes due to the processes at repulsive and
attractive walls of finite heights.
IEij [f ij2 ] = σ2ij0
∫
dvjdσˆ vjiσ θ(vjiσ)[f ij2 (r,v′i, r+ σij0,v′j)+ − f ij2 (r,vi, r− σij0,vj)+], (4)
where vi, vj and v′i, v′j are pre- and postcollision velocities of particles i, j at the hard core contact,
σij0 = σij0σˆ, vjiσ = (vj − vi)· σˆ, the unit vector σˆ is used to determine the mutual arrangement
of the two particles and in the direct collision (vi,vj) → (v′i,v′j) it is directed from the center of
particle j to the center of particle i,
v′i = vi + 2Mjivji · σˆσˆ, v′j = vj − 2Mijvji · σˆσˆ, (5)
153
M.V.Tokarchuk, Y.A.Humenyuk
Mji = mj/(mi +mj), Mij = mi/(mi +mj).
The contribution IMSPij in equation (3) reads:
IMSPij [f ij2 ] =
∑
q=r,a
Kqij
∑
l=1
∑
p=⊕,	,⊗
Iqpijl[f
ij
2 ], (6)
where the term under summation describes the pair (ij)-processes at the wall of type q (“r” or
“a”) with number l. There are three types of such processes depending on the interplay between
the value of the kinetic energy of relative motion and the height of the potential wall {q, l} as well
as on the character of the relative motion (approaching or mutual removal):
• p = ⊕ descending and acceleration at the wall (entering): each of two particles receives an
additional momentum along the line of centers; the two momenta are directed one against
another (for a wall of “a”-type) and one from another (for an “r”-type wall); the interaction
energy decreases;
• p = 	 ascending and deceleration at the wall (escape): each particle loses a lot of momentum
along the line of centers; the interaction energy increases;
• p = ⊗ reflection from the wall (bound state): the kinetic energy of relative motion along the
line of centers is not enough to overcome the potential barrier; both the kinetic and potential
energies do not change.
The most inner sum ∑p in equation (6) incorporates the described three types of processes.
Together with the hard core collision, thereafter denoted as (E), these pair processes can be divided
into two subsets correspondingly to either the two-particle both kinetic and interaction energies
conserve separately or the exchange between them occurs: nonexchange (E, ⊗) and exchange
processes (⊕, 	).
For convenience we will ascribe formal numerical values to the introduced symbolic values of
parameters q and p:
q =
(
r
a
)
=
(
−1
+1
)
, p =


⊕ descending
	 ascending
⊗ reflection

 =


+1
−1
0

 . (7)
This permits to present the collision integrals Iqpijl for all p and q in the following compact form.
The reflection processes are described with
Iq⊗ijl [f
ij
2 ] = (σqijl)2
∫
dvjdσˆ vjiσ θ(vjiσ) θ(vqijl − vjiσ)
× [f ij2 (r,v′i, r− qσqijl,v′j)−q − f
ij
2 (r,vi, r+ qσqijl,vj)−q], (8)
and
Iqpijl[f
ij
2 ] = (σqijl)2
∫
dvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)
× [f ij2 (r,vqpil , r+ qpσ
q
ijl,v
qp
jl )−qp − f
ij
2 (r,vi, r− qpσqijl,vj)+qp] (9)
corresponds to the descending or ascending processes if the appropriate value of p is chosen. In
these expressions numerical values of p and q are used to determine the position of particle j for
the function f ij2 , the argument of the θ function in equation (9), and to fix the type of the limiting
value (right or left) for the function f ij2 (..)−q in expression (8), and f ij2 (..)∓qp in expression (9). In
all other cases the parameters q and p are symbols used for designation. Owing to the discontinuity
of the pair potential each f ij2 is also discontinuous in the coordinate space. Its right (+) and left
(−) limiting values in equations (4), (8), (9) mean:
f ij2 (r, ., r+ σij0, .)± = f ij2 (r, ., r+ σ±ij0σˆ, .) = limδ→0+ f
ij
2 (r, ., r+ (σij0 ± δ)σˆ, .). (10)
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Hydrodynamic equations for fluid mixtures with multistep interaction
The other symbols introduced in the above expressions mean: σqijl= σ
q
ijlσˆ, v′i, v′j are velocities
of particles i and j just after the reflection from the q-type wall with number l which are to be
determined from the same relations (5) as for the hard core collisions; vqpil , v
qp
jl are velocities just
after the process of type p at wall {q, l} and can be determined from the pair collision rules for
processes of the exchange type (p = ⊕,	):
vqpil = vi +Mji[vji · σˆ −
√
(vji · σˆ)2 + p(vqijl)2] σˆ,
vqpjl = vj −Mij [vji · σˆ −
√
(vji · σˆ)2 + p(vqijl)2] σˆ, (11)
where vqijl = (2∆
q
ijl/µij)1/2 is the wall height in velocity units, µij = mimj/(mi + mj) is the
reduced mass.
To a certain degree the instantaneous processes at the walls correspond to noncompleted scat-
tering trajectories in the region of smooth behaviour of the intermolecular potential of the real
fluid. To a greater extent this is associated with the sloping attractive part. Instead of dealing with
these complicated contributions from the noncompleted trajectories we effectively replace them
with the instantaneous processes at the walls. From this viewpoint the MS potential dominates
over the square-well potential since it treats the long-range attraction effects more accurately.
As the collision integrals contain such pair processes, when during the typical time interval
of changing of the one-particle distribution function, the two particles i and j are allowed to
interact only at one wall, while successive interactions at two or more neighbouring walls which
are described with more complicated terms [21,22] are out of consideration, the following condition
must be satisfied [13,14]:
lfree  ∆σ, (12)
where lfree is the mean free path, ∆σ is the minimum separation between two walls ∆σ =
min |σqij,l+1 − σ
q
ijl|, i, j = 1 ÷ M, q = r, a, l = 1 ÷ K
q
ij . The density restriction appears from
the estimation for lfree given, for example, for the one-component dense fluid in [13,14]
∆σ
σ0
 1
4
√
2pi nσ30 g2(σ+0 )
, (13)
and results in the possibility of applying the introduced kinetic equation to high densities only.
The estimation (13) remains valid for the case of mixture as well.
If the number of walls increases and at the same time separations between them decrease,
then the MSP approximates the realistic potential better and better. Eventually, condition (12) is
violated and we cannot proceed further but are forced to seek an optimal solution to the choice of
the MS potential parameters. Nevertheless, if we formally consider such a limiting behaviour of the
theory for φMSPij → φtailij , where φtailij is, for example, the “hard spheres + smooth tail” interaction,
then as shown in [14], the irreversible MSP collision integral is reduced to the well-known term of
the mean-field type [6]. Works [21,22] present the analysis of successive processes of the “attractive
wall – hard core – attractive wall” type and some others for the SW case as well as clears up their
effects on transport coefficients of moderately dense gas.
3. Hydrodynamic level of description
Local densities of the conserved quantities, mass, momentum, and total energy form the basis
of the hydrodynamic level. In turn, these are equivalent to the partial mass densities, ρi(r, t), the
average mass velocity, V(r, t), and the internal energy density, e(r, t). The latter is equal to the
total energy e(r, t) minus the convective part and has two contributions:
e(r, t) = ek(r, t) + ep(r, t), (14)
where ek is the kinetic energy density in the local reference system, ep is the density of the potential
energy of interaction.
155
M.V.Tokarchuk, Y.A.Humenyuk
Define total densities ρ(r, t), ρ(r, t)V(r, t), ek(r, t) through the corresponding partial densities,
which are first moments of fi(r,vi, t):


ρ
ρV
ek

 =
M
∑
i=1


ρi
ρiVi
eki

 =
M
∑
i=1
∫
dvi fi(r,vi, t)


mi
mivi
1
2miv2i

 . (15)
The transport equations for the introduced variables can be obtained in a general form by
differentiating definitions (15) with t and using equation (2):
∂t〈ψi〉i +∇·〈viψi〉i − [〈∂tψi〉i + 〈vi ·∇ψi〉i] =
∫
dvi IE+MSPi [f
ij
2 ]ψi, (16)
where
ψi = {mi, mivi, 12miv
2
i } and 〈A〉i =
∫
dvi fi(r,vi, t)A (17)
is a notation of averaging. Each term in the left-hand side of equation (16) can be immediately
averaged for each ψi and we receive for total quantities:
∂tρ+∇· [ρV],
∂t(ρV) +∇· [ρVV + Pk],
∂t(ek + 12ρV
2) +∇·
[
(ek + 12ρV
2)V + qk + Pk ·V
]
, (18)
where Pk = ∑i〈micici〉i, qk =
∑
i〈 12mic2i ci〉i, ci = vi −V is the heat velocity.
After summing over all species the right-hand side (r.h.s.) of equation (16) reads:
M
∑
i,j=1
∫
dvi
{
IEij [f ij2 ] + IMSPij [f ij2 ]
}
ψi. (19)
Further transformations of this expression having, for example, for IE the form
M
∑
i,j=1
σ2ij0
∫
dvidvjdσˆ . . .
[
f ij′2 − f ij2
]
ψi, (20)
where f ij′2 denotes f ij2 (r,v′i, r + σij0,v′j)+ (see equation (4)), consist in reducing to a divergence
of some flux:
∇·
M
∑
i,j=1
σ2ij0
∫
dvidvjdσˆ . . . σˆ f ij2 [ψ′i − ψi] . (21)
Here ψ′i denotes the corresponding postcollision value of ψi. The same should be done for all the
processes at walls in equation (19). Quantities under the divergence are related to the fluxes of
momentum and kinetic energy. The described transformations are carried out due to the well-known
symmetrization procedure with the use of the momentum and energy conservation laws.
For the unexchange processes all the details of derivation are similar to the case of the Enskog
kinetic equation, e.g. see [6], and are presented elsewhere [23]. The final results for the r.h.s. of
equation (19) for the hard-core and reflection processes are:
RE =


0
−∇· PE
−∇· (qE + PE ·V)

 , R⊗ =


0
−∇· P⊗
−∇· (q⊗ + P⊗ ·V)

 . (22)
For the exchange processes, the symmetrization scheme has some peculiarities related to the ap-
pearance of the source term, Appendix A. The final result reads:
Rex =


0
−∇· Pex
−∇· (qex + Pex ·V) + sk

 , (23)
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Hydrodynamic equations for fluid mixtures with multistep interaction
where the source term sk is of the form:
sk(r, t) =
M
∑
i,j=1
∑
q=a,r
Kqij
∑
l=1
∑
p=±1
1
2p∆
q
ijl(σ
q
ijl)2
×
∫
dvidvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)f
ij
2 (r,vi, r− qpσqijl,vj)qp. (24)
Expressions for all the contributions to P and q are given in Appendix B, equations (49), (50),
(51).
Collecting the contributions to the r.h.s., given by equations (22) and (23) we finally get the
system of transport equations for the one-particle macroscopic quantities:
∂tρi +∇· [ρiV] +∇· Jmdi = 0,
∂t[ρV] +∇· [ρVV] +∇· Pk+E+MSP = 0, (25)
∂t[ek + 12ρV
2] +∇· [(ek + 12ρV
2)V] +∇· [qk+E+MSP + Pk+E+MSP ·V] = sk,
where Jmdi = ρi(Vi −V), PMSP = P⊗ + Pex, qMSP = q⊗ + qex.
The appearance of the source term sk in the last transport equation was not stressed in any of
the previous articles dealing with both the SW interaction [8–11,24] and the MS potential [12–14].
It is related to accounting for the processes at the walls of finite height in an irreversible way and
describes the exchange rate between the kinetic and potential energy densities. Therefore, sk has
no analogue both in the case of the Enskog equation including its mean-field extensions [6,7], and
in the case of the Boltzmann kinetic equation. In the next section the relation of this term to the
transport equation for ep is considered.
4. Interaction energy density and the closure relation
The density of the potential energy of interaction ep(r, t) is expressed by means of the two-
particle distribution functions f ij2 :
ep(r, t) = 12
M
∑
i,j=1
∫
dxidxjf ij2 (xi, xj , t)φMSPij (rij) δ(r− ri). (26)
Its transport equation can be derived from the second equation of the BBGKY hierarchy with the
appropriate choice of the collision operators which correspond to the MS potential. It reads:
∂tep(r, t) +∇· [ep(r, t)V(r, t) + qp(r, t)] = sp(r, t), (27)
where
qp(r, t) = 12
M
∑
i,j=1
∫
dxidxjci f ij2 (xi, xj , t)φMSPij (rij) δ(r− ri), (28)
sp(r, t) = −
M
∑
i,j=1
∑
q=a,r
Kqij
∑
l=1
∑
p=±1
1
2p∆
q
ijl(σ
q
ijl)2
×
∫
dvidvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)f
ij
2 (r,vi, r− qpσqijl,vj)qp. (29)
qp(r, t) is the peculiarly molecular or “heat” flux of the potential energy density. The source
sp(r, t) in the r.h.s. like IE+MSPi [f2] describes the change of ep due to collisions. However, only the
exchange-type processes p = ⊕,	 contribute. sp(r, t) is identical (with the opposite sign) to its
kinetic counterpart (24):
sp(r, t) = −sk(r, t).
157
M.V.Tokarchuk, Y.A.Humenyuk
The result for sp is also obtainable using heuristic ideas about the number of collisions [23] and
is to be considered as a direct generalization to the MSP case of that used in [9–11] in which,
however, a connection to the transport equation for ek was not mentioned. Due to the observed
property the transport equation for the total energy density does not contain any source term as
it should be for a density of a conserved quantity.
In order to close the equations for fi and ep a closure relation should be supplied. This can be
done in the spirit of [9] (also see [10,11,24]) using the approximation which ignores pair correlations
in the velocity space:
f ij2 (xi, xj , t) = fi(xi, t)fj(xj , t) gij2 (ri, rj , t), (30)
where gij2 is a functional of the local number densities nk(r, t) and the reciprocal quasi-temperature
β(r, t) (also called the inverse potential energy temperature)
gij2 (ri, rj , t) = gij2 (ri, rj |{nk}, β)
so that gij2 has the same cluster expansion (n-vertex, f -bond) as the equilibrium counterpart. But
in nonequilibrium case nk(r, t) replaces each nk and βij(ri, rj , t) = [β(ri, t) + β(rj , t)]/2 replaces
1/kBT at each bond. β(r, t) is a Lagrange multiplier conjugated to the potential energy density
[9,11,24] and is treated in the theory using the transport equation for ep(r, t). The functional gij2
is discontinuous at each point of discontinuity of the MSP and obeys:
gij2 (r, r± σqijl, t)−qp = epβij∆ε
q
ijlgij2 (r, r± σqijl, t)qp.
Thus, the equations for fi and ep become closed. The closure problem in the kinetic theory of fluids
with attraction was analyzed in detail in [7] within the framework of the approach of maximization
of entropy subject to some constraints (see also [6,25]). Some useful analysis is given in [26,27].
In approximation (30) the expression for qp reduces to
qp(r, t) = 12
M
∑
i=1
Jndi (r, t)
M
∑
j=1
∫
drjnj(rj , t)gij2 (r, rj , t)φMSPij (|r− rj |), (31)
where Jndi = ni(Vi − V) is the number density flux. Under assumption (30) the heat flux qp is
of intrinsically diffusive nature and vanishes for a one-component fluid qp |M=1= 0. Perhaps, this
result can be used as another way of varifying the applicability of assumption (30) specifically for
the systems with the MS potential, when the comparison of theoretical and computer simulation
results is made.
If we subtract the convective term (ρV 2)/2 from equation (25) and combine the result with
equation (27), the equation for the internal energy density is received:
∂te+∇· [eV + qk+E+MSP+p] = − Pk+E+MSP :∇V† (32)
with the extra contribution qp contrary to the total stress tensor. The right-hand side term de-
scribes the rate of internal energy change due to the processes of the viscous molecular friction (†
designates the transpose of ∇V).
5. Summary
We have presented the kinetic equation for dense fluid mixtures with the multistep potential of
interaction. Together with the transport equation for the potential energy density it generalizes the
corresponding one-component case [12–14] as well as the kinetic equation for SW particles [8,9], if
only one attractive wall is retained. If all the wall heights are set equal to zero, we reproduce the
kinetic equation of RET for mixtures [4]. Each of the transport equations for ek and ep contains the
source-type term, sk and sp, which are identical with each other but differ in the sign and describe
the energy exchange phenomena in a non-equilibrium state but should vanish in equilibrium.
158
Hydrodynamic equations for fluid mixtures with multistep interaction
The situation for the transport equations substantially differs from that for the hard sphere
model and in our opinion is typical of the systems with the essential part of the interaction energy
where the energy exchange can occur. As it is known, when the dense system approaches equilib-
rium, the fast kinetic processes are superimposed with slower hydrodynamic processes related to
the local conservation laws – one says that kinetics and hydrodynamics are closely connected with
each other [28,29]. In our case this connection is realized by means of the potential energy density
which appears as a contribution to the slow variable, the total energy density, and simultaneously
it manifests the non-hydrodynamic (kinetic) behavior due to the presence of the source in its
transport equation caused by some subset of the microscopic pair processes, namely, the exchange
processes. As a consequence, the relaxation scenario complicates. Besides the one-particle distri-
bution function relaxation, another type of relaxation exists which is determined by the exchange
processes between the kinetic and potential energies. Exactly this gives rise to the appearance of
an additional exchange mode investigated in [11] by means of the linearized kinetic theory for the
one-component fluid of square-well particles. It is clear that such a behavior can be better seen in
the lower temperature region where the relative part of the interaction energy is more appreciable.
Appendix
A. Symmetrization of the exchange term
The exchange contribution to the r.h.s. of equation (19) can be decomposed onto the sum of
the inverse (*) and direct (0) terms:
Rex = R∗ +R0 =
∑
ij
∑
ql
∑
p=±1
[
Rqp∗ijl +R
qp0
ijl
]
, (33)
Rqp∗ijl +R
qp0
ijl = (σ
q
ijl)2
∫
dvidvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)
× [f ij2 (r,vqpil , r+ qpσ
q
ijlσˆ,v
qp
jl )−qp − f
ij
2 (r,vi, r− qpσqijlσˆ,vj)qp]ψi(r,vi). (34)
In the term for inverse collisions Rqp∗ijl we change the integration variables from Γ = (vi,vj , σˆ) to
Γ∗ = (vqpil ,v
qp
jl , σˆ). The inverse collision rule reads:
vi = vqpil +Mji
[
vqpjil · σˆ −
√
(vqpjil · σˆ)2 + p˜ν
q
ijl
]
σˆ,
vj = vqpjl −Mij
[
vqpjil · σˆ −
√
(vqpjil · σˆ)2 + p˜ν
q
ijl
]
σˆ, (35)
where vqpjil = v
qp
jl − v
qp
il is the relative velocity after the {qlp} process, p˜ = −p, ν
q
ijl = (v
q
ijl)2. This
rule can be deduced from relations (11) using the arguments of symmetry and equality of rights
for process p and its inverse p˜.
The Jacobian and the scalar product
JΓ→Γ∗ =
|vqpjil · σˆ|
√
(vqpjil · σˆ)2 − p ν
q
ijl
, vji · σˆ =
√
(vqpjil · σˆ)2 − p ν
q
ijl (36)
substituted into equation (34) give for the inverse collision contribution:
Rqp∗ijl = (σ
q
ijl)2
∫
dvqpil dv
qp
jl dσˆ (v
qp
jil · σˆ) θ
(√
(vqpjil · σˆ)2 − p ν
q
ijl + p−12 v
q
ijl
)
× f ij2 (r,vqpil , r+ qpσ
q
ijlσˆ,v
qp
jl )−qpψi(r,vi). (37)
The argument of the θ function in this formula can be reduced to a simpler form for the considered
cases p = ±1: vqpjil · σˆ + p˜−12 v
q
ijl.
159
M.V.Tokarchuk, Y.A.Humenyuk
Further, we denote variables vqpil ,v
qp
jl by va, vb. Then ψi(r,vi) which means “the value of the
molecular quantity before the process {qlp}” (e.g. before descending at the wall: p = +1), has, after
denoting in terms of the new variables va, vb, the opposite sense – “the value after the process
{qlp˜}” (respectively, after ascending at the wall: p˜ = −1). The designation must be changed as
follows: ψi(r,vi) → ψqp˜il (r,v
qp˜
al ). In terms of the new variables equation (37) becomes:
Rqp∗ijl = (σ
q
ijl)2
∫
dvadvbdσˆ (vba · σˆ) θ(vbaσ + p˜−12 v
q
ijl)
× f ij2 (r,va, r− qp˜σqijlσˆ,vb)qp˜ ψ
qp˜
il (r,v
qp˜
al ). (38)
The received expression substantially differs in arguments of the functions θ and f ij2 from Rqp0ijl
in equation (34) and should be combined with that from the process of p˜ type at the same wall,
that is to say with Rqp˜0ijl . This is the very circumstance that makes different the two symmetrization
procedures for the nonexchange and exchange processes, arising due to their different symmetry
properties. In an nonexchange process (E or ⊗) the inversion of the relative velocity normal com-
ponent occurs while for an exchange one this component keeps its direction. This yields:
Rqp∗ijl +R
qp˜0
ijl = (σ
q
ijl)2
∫
dvadvbdσˆ vbaσ θ(vbaσ + p˜−12 v
q
ijl)
× f ij2 (r,va, r− qp˜σqijlσˆ,vb)qp˜[ψ
qp˜
il (r,v
qp˜
al )− ψi(r,va)]. (39)
Having in mind this special feature we must rearrange the direct (0) and inverse (∗) contributions
from processes p = ±1 in accordance with the following rule:
∑
p=±1
[Rqp∗ijl +R
qp0
ijl ] =
∑
p=±1
[Rqp˜∗ijl +R
qp0
ijl ] =
∑
p=±1
Rq[p˜∗+p0]ijl . (40)
Now, carry out the symmetrization with respect to indices i, j for the expression
Rq[p˜∗+p0]ijl = (σ
q
ijl)2
∫
dvadvbdσˆ vbaσ θ(vbaσ + p−12 v
q
ijl)
× f ij2 (r,va, r− qpσqijlσˆ,vb)qp[ψ
qp
il (r,v
qp
al )− ψi(r,va)], (41)
which is obtained from equation (39) by the replacement of p˜ with p. To this end we perform the
following manipulations:
a) permute indices i→← j and corresponding arguments of f ji2 ;
b) change the designations of velocities va→←vb;
c) choose σˆ′ = −σˆ as a new variable and denote it again by σˆ.
Then, the term with permuted ji indices reads:
Rq[p˜∗+p0]jil = (σ
q
ijl)2
∫
dvadvbdσˆ vbaσ θ(vbaσ + p−12 v
q
ijl)
× f ij2 (r+ qpσqijl,va, r,vb)qp[ψ
qp
jl (r,v
qp
bl )− ψj(r,vb)]. (42)
Here f ij2 describes such a configuration in which position r is occupied by particle j with velocity
vb while particle i with velocity va lies in the position r + qpσqijl. The conservation laws with
corresponding spatial coordinates read:
ψj(r,vb) + ψi(r+ qpσqijl,va) = ψ
qp
jl (r,v
qp
bl ) + ψ
qp
il (r+ qpσ
q
ijl,v
qp
al ) + ∆ψ
qp
ijl, (43)
where
∆ψqpijl =


0
0
−p∆qijl

 (44)
160
Hydrodynamic equations for fluid mixtures with multistep interaction
respectively for the mass, momentum, and energy. Substitute equation (43) into equation (42) and
take half of the sum of the obtained result and equation (41):
1
2 [R
q[p˜∗+p0]
ijl +R
q[p˜∗+p0]
jil ] = 12 (σ
q
ijl)2
∫
dvadvbdσˆ vbaσ θ(vbaσ + p−12 v
q
ijl)
×
{
[ψqpil (r,v
qp
al )− ψi(r,va)]f
ij
2 (r,va, r− qpσqijl,vb)qp
− [ψqpil (r+ qpσ
q
ijl,v
qp
al )− ψi(r+ qpσ
q
ijl,va)]f
ij
2 (r+ qpσqijl,va, r,vb)qp
}
+ 12 (−∆ψ
qp
ijl)(σ
q
ijl)2
∫
dvadvbdσˆ vbaσ θ(vbaσ + p−12 v
q
ijl)f
ij
2 (r+ qpσqijl,va, r,vb)qp. (45)
The first term in the r.h.s. can be written as a divergence, if we present the difference of the
products within the curly brackets as
∫
dx
{
δ(x− r)− δ(x− [r+ qpσqijl])
}
[ψqpil (x,v
qp
al )− ψi(x,va)] f
ij
2 (x,va,x− qpσqijl,vb)qp, (46)
and make use of the relation
δ(x− [r+ y])− δ(x− r) = ∇r · y
∫ 1
0
dλ δ(x− [r+ λy]). (47)
The resulting expression reads:
1
2R
q[p˜∗+p0]
[ij+ji]l = −∇· 12qp(σ
q
ijl)3
∫
dvidvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)σˆ
×
1
∫
0
dλ[ψqpil (r+ λqpσ
q
ijl,v
qp
il )− ψi(r+ λqpσ
q
ijl,vi)]
× f ij2 (r+ λqpσqijl,vi, r+ (λ− 1)qpσ
q
ijl,vj)qp
+ 12 (−∆ψ
qp
ijl)(σ
q
ijl)2
∫
dvidvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)f
ij
2 (r,vi, r− qpσqijl,vj)qp, (48)
where in the second term we have returned to the previous presentation of f ij2 , carrying out
the inverse manipulations at indices, velocities, and σˆ. Just this term forms the source sk. The
summation
∑
ij
∑
ql
∑
p=±1 gives the final results (23).
B. Contributions to fluxes
Here, expressions for the nonexchange (E, ⊗) and exchange (ex) contributions to the stress
tensor and the heat flux are given which emerge after the symmetrization and appear for the first
time in (22), (23).
(
P
q
)E
=
∑
ij
σ3ij0
2
∫
dvidvjdσˆ vjiσ θ(vjiσ)σˆ
1
∫
0
dλ
×
(
mi[c′i − ci]
mi
2 [c′2i − c2i ]
)
f ij2 (r+ λσij0σˆ,vi, r+ (λ− 1)σij0σˆ,vj)+, (49)
(
P
q
)⊗
=
∑
ij
∑
q=a,r
(−q)
Kqij
∑
l=1
(σqijl)
3
2
∫
dvidvjdσˆ vjiσ θ(vjiσ) θ(vqijl − vjiσ) σˆ
×
1
∫
0
dλ
(
mi[c′i − ci]
mi
2 [c′2i − c2i ]
)
f ij2 (r− λqσqijlσˆ,vi, r− (λ− 1)qσ
q
ijlσˆ,vj)−q, (50)
161
M.V.Tokarchuk, Y.A.Humenyuk
(
P
q
)ex
=
∑
ij
∑
q=a,r
Kqij
∑
l=1
∑
p=±1
qp (σ
q
ijl)
3
2
∫
dvidvjdσˆ vjiσ θ(vjiσ + p−12 v
q
ijl)σˆ
×
1
∫
0
dλ
(
mi[cqpil − ci]mi
2 [(c
qp
il )2 − c2i ]
)
f ij2 (r+ λqpσqijlσˆ,vi, r+ (λ− 1)qpσ
q
ijlσˆ,vj)qp. (51)
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Рiвняння гiдродинамiки для гуcтих cумiшей газiв зi
cходинковою взаємодiєю мiж чаcтинками
М.В.Токарчук1,2, Й.А.Гуменюк1
1 Iнcтитут фiзики конденcованих cиcтем Нацiональної академiї наук України
Львiв 79011, вул. Свєнцiцького 1, Україна
2 Нацiональний унiверcитет “Львiвcька полiтехнiка”, Львiв 79013, вул. Бандери 12, Україна
Отримано 20 сiчня 2007 р., в остаточному виглядi – 23 квiтня 2007 р.
Проведено узагальнення кiнетичного рiвняння для гуcтого газу з багатоcходинковим мiжчаcтинко-
вим потенцiалом на випадок гуcтої cумiшi. Здiйcнено виведення рiвнянь гiдродинамiки, виходячи з
кiнетичного рiвняння для одночаcтинкової функцiї розподiлу та рiвняння переноcу для гуcтини по-
тенцiальної енергiї. Отримано загальнi вирази для потокiв iмпульcу i тепла. Показано, що рiвняння
переноcу для гуcтини кiнетичної енергiї мiстить новий доданок типу “джерело”, який описує швидкi
процеcи обмiну мiж кiнетичною i потенцiальною енергiями системи.
Ключовi слова: кiнетична теорiя, потенцiал прямокутної ями, енергетичний обмiн, газова сумiш,
багатосходинкова взаємодiя
PACS: 05.20.Dd, 05.10.+y, 05.60.-k
163
164