1. Introduction
It is well established that when granular matter is externally excited it can be modeled as a gas of
inelastic hard spheres (IHS). In the simplest version of the model, the spheres are assumed to be completely
smooth (i.e., with no rotational degrees of freedom) so that the inelasticity of collisions is characterized by a (positive) constant coefficient of normal restitution
. The case
corresponds to elastic collisions (molecular gases). In the low-density regime, the time evolution of the one-particle velocity distribution function is given by the Boltzmann kinetic equation properly adapted to account for the inelastic nature of collisions [
1]. Needless to say, the knowledge of the distribution function provides all the relevant information on the state of the gas at both microscopic and macroscopic levels.
However, the fact that the collision rate for hard spheres is proportional to the relative velocity of the two colliding particles hinders the search for solutions to the Boltzmann equation. In particular, this difficulty (which is also present for molecular gases) prevents the possibility of expressing the associated collisional moments of the Boltzmnn operator in terms of a finite number of velocity moments. This precludes for instance the derivation of
exact analytical results for the transport properties of the gas. For this reason, most of the analytical results derived for IHS are based on the truncation of a series expansion of the distribution function in the powers of Laguerre (or Sonine) polynomials. In the case of elastic collisions, the above problem for the collisional kernel of hard spheres can be overcome by assuming that the particles interact via a repulsive potential inversely proportional to the fourth power of the distance (Maxwell molecules) [
2]. For this interaction potential, the collision rate is independent of the relative velocity and so, any collisional moment of degree
k can be expressed in terms of velocity moments of a degree smaller than or equal to
k [
2]. Thanks to this property, nonlinear transport properties can be exactly obtained [
2,
3] and, when properly reduced, they exhibit a good agreement with results derived for other interaction potentials.
In the case of granular gases (namely, when the collisions are inelastic), one can still introduce the so-called inelastic Maxwell models (IMM) (see for instance Refs. [
4,
5,
6] for some of the first papers where IMM were introduced). These models share with elastic Maxwell molecules the property that the collision rate is independent of the relative velocity, but, on the other hand, their collision rules are the same as for IHS. Thus, although IMM cannot be represented by any interaction potential, its use allows one to obtain
exact analytical results of the
inelastic Boltzmann equation. In fact, IMM qualitatively keep the correct structure and properties of the nonlinear macroscopic equations and obey Haff’s law [
7]. In any case, as Ref. [
8] claims, one can introduce Maxwell models in the framework of the Boltzmann equation at the level of the cross section without any reference to a specific interaction potential. Recently [
9], an inelastic
rough Maxwell model has been also introduced in the granular literature.
The simplifications introduced by IMM in the kernel of the Boltzmann collision operator has allowed, in some cases, the determination of the dynamic properties of granular gases without employing uncontrolled approximations. For this reason, the Boltzmann equation for IMM has received a great attention of physicists and mathematicians in the last few years, especially in the study of overpopulated high energy tails in homogeneous states [
10,
11,
12] and in the evaluation of the transport coefficients [
13,
14]. The existence of high energy tails of the Boltzmann equation is common for IHS and IMM; however, the quantitative predictions of IMM differ from those obtained from IHS. A one-dimensional IMM has also been employed to study the two-particle velocity correlations [
15]. It is important to remark that most of the problems analyzed in the context of IMM have been focused on simple (monocomponent) granular gases. Much less is known in the case of inelastic Maxwell mixtures. For this sort of system, Marconi and Puglisi have studied the high velocity moments in the free cooling [
16] and driven [
17] states for the one-dimensional case (
). For arbitrary dimensions and in the tracer limit, Ben-Naim and Krapivsky [
18] have analyzed the velocity statistics of an impurity in a uniform granular gas, while the fourth cumulant of the velocity distribution in the homogenous cooling state (HCS) has been also obtained [
14].
Beyond the second degree velocity moments (which are directly related with the transport properties), Garzó and Santos [
19] have computed all the third and fourth degree velocity moments of the Boltzmann collision operator for a monocomponent granular gas of IMM. In addition, the collisional rates associated with the isotropic velocity moments
and the anisotropic moments
and
have been independently evaluated in Refs. [
10,
20,
21]. Here,
, where
is an arbitrary function of the velocity
and
is the one-particle velocity distribution function. All the above calculations have been performed for an arbitrary number of dimensions,
d. To the best of our knowledge, the above papers are the only works where the computation of the high-degree collisional moments of IMM has been carried out.
On the other hand, as said before, the results for granular mixtures modeled as IMM are more scarce. In particular, given that most of the works have been focused on the computation of the transport coefficients, only the first- and second-degree collisional moments have been considered [
14,
22,
23,
24,
25,
26]. Thus, it would be convenient (especially for simulators) to extend the results displayed in Ref. [
19] for the third- and fourth-degree collisional moments to the realistic case of granular binary mixtures. This is the main objective of the present paper. However, due to the long and complex algebra involved in the general problem, here we will consider situations where the mean flow velocity
(
) of each species is equal to the mean flow velocity
of the mixture. This means that no diffusion processes are present in the mixture (i.e.,
). Although this limitation restricts the applicability of the present results to general nonequilibrium situations, they are still useful for contributing to the advancement in the knowledge of exact properties of IMM in some specific situations. Among the different problems, we can mention the relaxation of the third and fourth degree moments towards the HCS (starting from arbitrary initial conditions) and the study of the combined effect of shearing and inelasticity on the high-degree moments in a binary mixture under uniform shear flow (USF).
Some previous results derived in the HCS for IMM in the monodisperse case [
10,
18] have demonstrated that for
, the (scaled) velocity distribution function
has a high-velocity tail of the form
(
c being the (scaled) velocity of the particle). The exponent
obeys a transcendental equation whose solution is always larger than four (
), except for the one-dimensional case (
) [
11]. Consequently, for any value of
and
, the corresponding (scaled) velocity moments of degree
k equal to or less than four tend towards well-defined values in the long-time limit (namely, they are always
convergent). An interesting issue is to explore whether or not the convergence of the moments of degree
for the single gas case is also present for inelastic binary mixtures and, if so, to what extent. An indirect way of answering this question is through the knowledge of the high degree velocity moments (beyond the second ones) of the velocity distribution function of each species. These moments play a relevant role, for instance, in the high velocity region. Surprisingly, our results for binary mixtures show that the (anisotropic) third and fourth degree moments could diverge in time for given values of the parameters of the mixture. Therefore, in contrast to the findings for the monocomponent
granular gases for
, only the (scaled) moments of degree equal to or smaller than 2 are always convergent in the HCS for arbitrary values of the parameters of the mixture. This is one of the main conclusions of the work.
Apart from the HCS, another interesting application of our results refers to the USF. For monocomponent granular gases, previous results [
27] have shown that, for a given value of the coefficient of restitution
, the (scaled) symmetric fourth-degree moments diverge in time for shear rates larger than a certain critical value
. The value of
decreases with decreasing
(increasing dissipation). Given that the analysis for general sheared binary mixtures is quite intricate, we consider here the limiting case where the concentration of one of the species is negligible (and so, it is present in tracer concentration). This limit allows one to express the moments of the tracer species in terms of the known moments of the excess gas. In particular, the knowledge of the second-degree moments provides the dependence of the temperature ratio on the parameters of the mixture. As occurs in the HCS, there is a breakdown of the energy equipartition; this behavior is produced here by the combined effect of both the shear rate and the inelasticity in collisions. In particular, in contrast to the HCS, we find a non-monotonic dependence of the temperature ratio on the (reduced) shear rate for given values of the coefficients of restitution. In addition, although the third-degree moments can be also divergent (as in the case of the HCS for mixtures and in contrast to the results reported for simple gases [
27]); surprisingly, they become convergent for shear rates larger than a certain critical value.
The plan of the paper is as follows. In
Section 2, the Boltzmann kinetic equation for inelastic Maxwell mixtures is presented. Next, the so-called Ikenberry polynomials [
2]
of degree
are introduced and their collisional moments
with
, and the four associated with the Boltzmann collision operators
are evaluated in
Section 3. Some technical details involved in the calculations are relegated to the
Appendix A. The time relaxation problem of the (scaled) moments towards their asymptotic values in the HCS is studied in
Section 4, while an study of the regions of the parameter space where the third- and fourth-degree moments can be divergent is presented in
Section 5.
Section 6 deals with the USF problem, where we pay special attention to the second- and third-degree moments of the tracer species. Its time evolution is studied in
Section 7. We close the paper in
Section 8, with a brief discussion of the results derived in this paper.
2. Boltzmann Kinetic Equation for Inelastic Maxwell Mixtures
We consider a granular binary mixture made of particles of diameters
and masses
(
). In the absence of external forces and assuming molecular chaos, the one-particle velocity distribution function
of species
r obeys the Boltzmann equation
where
is the Boltzmann collision operator for collisions between particles of species
r and
s. If the granular mixture is modeled as a gas of IHS, then, to determine any collisional moment of
, one needs to know
all the degree moments of the distributions
and
. This means that one has to resort to approximate forms of the distributions
and
to estimate the collisional moments of
. Usually, the lowest order in a Sonine polynomial expansion of these distributions is considered [
28]. This problem is also present in the conventional case of molecular binary mixtures (elastic collisions). However, if one assumes that the collision rate of the two colliding spheres is constant (IMM), the collisional moments of the operator
can be given in terms of velocity moments of the distributions
and
without knowing their explicit forms. This is the main advantage of using IMM instead of IHS.
The Boltzmann collision operator
for IMM is [
28]
Here,
is the number density of species
r,
is an effective collision frequency (it can be considered as a free parameter of the model),
is the total solid angle in
d dimensions, and
refers to the constant coefficient of the restitution for
r-
s collisions. In addition, the double primes on the velocities denote the initial values
that lead to
following a binary collision:
where
,
is the relative velocity of the colliding pair and
is a unit vector directed along the centers of the two colliding spheres.
Apart from the densities
, the granular temperature
T is defined as
where
is the concentration or mole fraction of species
r (
is the total number density) and
is the partial temperature of species
r. In Equation (
6), we have introduced the peculiar velocity
,
being the mean flow velocity, defined as
Here,
is the mass density of species
r and
is the total mass density. The second identity in Equation (
7) defines the partial mean flow velocities
. In addition, the mass flux of species
r is given by
. As said in
Section 1, for the sake of simplicity, we will assume in this paper that the mass fluxes vanish (i.e.,
).
To evaluate the collisional moments of the Boltzmann operator
, a useful identity for an arbitrary function
is
where
denotes the post-collisional velocity.
Ikenberry Polynomials
In the case of Maxwell models (both elastic and inelastic), it is convenient to introduce the Ikenberry polynomials [
2]
of degree
. The Ikenberry polynomials are defined as
. Here, as noted in Ref. [
19], the polynomial
is obtained by subtracting from
that homogeneous symmetric polynomial of degree
q in the components of
such as to annul the result of contracting the components of
on any pair of indices. The polynomials functions
of degree smaller than or equal to four are
Let us introduce here the notation
Equation (
15) gives the definition of the velocity moments of the distribution
, while Equation (
16) provides the definition of the collisional moments of the Boltzmann operator
.
Note that
,
(conservation of mass),
(since we have assumed that
) and
where
is the partial pressure of species
r. Moreover,
where
is the partial pressure tensor of species
r and
where
is the partial contribution to the total heat flux due to species
r.
The remaining third degree moments and the moments of degree are not directly related to hydrodynamic quantities. However, they provide indirect information on the velocity distribution function .
4. Relaxation to the HCS
The results derived in the preceding section can be applied to several interesting situations. In this paper, we will consider first the most basic problem in a granular mixture: the time evolution of the moments of a degree less than or equal to four (both isotropic and anisotropic) in the HCS. The HCS is a
homogeneous state where the granular temperature
monotonically decays in time. In this case, the set of Boltzmann kinetic equations (
1) for
and
becomes
In the HCS, the granular temperature
decreases in time due to collisional dissipation. A steady state can be achieved if some sort of thermostat (which injects energy into the system) is introduced in the system to compensate for the energy dissipated by collisions. Here, we will assume that the granular mixture is
undriven and hence,
T depends on time.
In the context of IMM, it has been proven for single granular gases [
29,
30] that, provided that
has a finite moment of some degree higher than two,
asymptotically tends toward a self-similar solution of the form
where
is an isotropic distribution of the scaled velocity
c. However, the exact form of the distribution
is not known to date.
At a hydrodynamic level, the only relevant balance equation is that of the temperature
. Its time evolution equation can be easily obtained from the moments
and
and it is given by
where we have taken into account that the time evolution of the partial temperatures
can be derived from the velocity moments
as
Here, we recall that
and
is given by Equation (
27). The time evolution of the temperature ratio
follows from Equation (
39) as
After a transient period, it is expected that the mixture achieves a hydrodynamic regime where all the time dependence of
only occurs through the granular temperature
. This necessarily implies that the three temperatures
,
and
are proportional to each other and their ratios are all constant. This does not necessarily mean that all three temperatures are equal (as in the case of elastic collisions) since the value of
must be obtained from Equation (
40). In fact, in the hydrodynamic regime,
and so the condition of equal partial cooling rates [
] provides the dependence of the temperature ratio on the parameters of the mixture [
31].
Figure 1 shows the dependence of the temperature ratio
on the (common) coefficient of restitution
for a three-dimensional binary mixture (
) with
,
, and three different values of the mass ratio. To compare with the results obtained from IHS, we chose
to obtain the same
[Equation (
27) for IMM] as that of IHS. In the case of IHS, the quantities
are evaluated by approximating the distributions
and
by Maxwellian distributions defined at temperatures
and
, respectively [
22]. With this choice, in the case
, one achieves the expression
where
. Theoretical results for IMM [with the choice (
41)] are compared against Monte Carlo simulations carried out in Ref. [
32] for IHS.
Figure 1 highlights one of the most characteristic features of granular mixtures (as compared with molecular mixtures): the partial temperatures are different even in homogenous states. We observe that the departure of
from 1 (breakdown of energy equipartition) increases with increasing the disparity in the mass ratio. In general, the temperature of the lighter species is smaller than that of the heavier species. It is also important to remark the excellent agreement found between theory (developed for IMM) and computer simulations (performed for IHS), even for quite strong inelasticity.
4.1. Eigenvalues for Inelastic Maxwell Mixtures
Apart from the partial temperatures, it is worthwhile analyzing the time evolution of the higher-degree velocity moments in the HCS. To obtain this equation, one takes velocity moments in both sides of Equations (
35) and (
36) and obtains the set of coupled equations
In Equation (
42), we have introduced the short-hand notation
. To study the time evolution of the moments
it is convenient to introduce the
scaled moments
where
is a thermal velocity of the mixture. In accordance with Equation (
37), one expects that after a transient regime the dimensionless moments
(scaled with the time-dependent thermal velocity
) reach an asymptotic steady value.
The time evolution of the scaled moments
can be obtained when one takes into account the time evolution Equation (
38) for the temperature
. In that case, from Equations (
38), (
42) and (
43), one simply gets
where
,
and
Since
is an effective collision frequency, the parameter
measures time as the number of (effective) collisions per particle. Here, for the sake of concreteness, we will consider Model B with
. In this case, as in previous works [
14,
22], the effective collision frequency
is
Needless to say, the results derived in this section are independent of the choice of
; they apply for both Models A and B.
According to Equations (
22), (
23) and (
29)–(
34), it is easy to observe that the combination
has the structure
where the terms
are bilinear combinations of moments of degree less than
. Since the first two terms on the right-hand side of Equation (
49) are linear, then the quantities
and
can be considered as the eigenvalues (or
collisional rates) of the linearized collision operators corresponding to the eigenfunctions
. Their explicit forms for velocity moments of degree less than or equal to four are given in the
Appendix B.
As an illustration, the dependence of the eigenvalues (collision rates) associated with the second, third, and fourth degree moments on the (common) coefficient of restitution
is plotted in
Figure 2 and
Figure 3 for a binary mixture constituted by particles of the same mass density. Here,
,
, and
. While the eigenvalues
and
decrease with increasing inelasticity, the other two eigenvalues (
and
) exhibit a non-monotonic dependence on
. The eigenvalues of the second- and third-degree moments associated to cross-collisions (
,
,
, and
) are negative and they increase with decreasing
. A similar behavior can be found for the eigenvalues associated with the fourth-degree moments, as
Figure 3 shows. In general, we can conclude that the influence of inelasticity on those eigenvalues is in general important, especially in the case of the ones associated with the self-collisions (i.e., those of the form
).
4.2. Time Evolution of the Velocity Moments
Let us obtain the dependence of the (scaled) velocity moments
on time. Thus, inserting the expression (
49) into Equation (
44), in matrix form, one finally obtains
where
is the column matrix defined by the set
and the square matrix
is given by
In Equation (
52), we have introduced the quantities
The collision rates
can be considered as
shifted collisional rates associated with the scaled moments
. Moreover, the column matrix
is
The solution of Equation (
50) can be written as
where the asymptotic steady value
is
The long time behavior of
is governed by the smallest eigenvalue
of the matrix
. Given that the eigenvalues
ℓ of the matrix
obey the quadratic equation
the smallest eigenvalue
is
If
, then all the scaled moments of degree
tend asymptotically to finite values. Otherwise, for given values of the parameters of the mixture, if
becomes negative for
smaller than a certain critical value
, then the moments of degree
exponentially grow in time for
. The critical value
can be obtained from the condition
, which implies
6. USF: Tracer Limit
As a second application, we study in this section the USF problem. This state is macroscopically characterized by constant densities
, a uniform temperature
T, and a linear velocity profile
where
a is the constant shear rate. This linear velocity profile assumes no boundary layer near the walls and is generated by the Lees-Edwards boundary conditions [
33], which are simply periodic boundary conditions in the local Lagrange frame moving with the flow velocity. Since
and
T are uniform, then the mass and heat fluxes vanish and the transport of momentum (measured by the pressure tensor) is the relevant phenomenon. At a microscopic level, the USF is characterized by a velocity distribution function that becomes
uniform in the local Lagrangian frame, i.e.,
. In that case, the Boltzmann equation for the binary mixture is given by the set of coupled kinetic equations
Equations (
63) and (
64) are invariant under the changes
and
(
).
The relevant macroscopic balance equation in the USF state is the balance equation for the temperature
. This equation can be easily obtained from Equations (
63) and (
64). In dimensionless form, it can be written as
where
,
,
,
,
is the hydrostatic pressure. Equation (
65) shows that the temperature changes in time due to the competition of two opposite mechanisms: on the one hand, viscous heating (
) and, on the other hand, energy dissipation in collisions (
). The
reduced shear rate
is the nonequilibrium relevant parameter of the USF problem since it measures the departure of the system from the HCS (vanishing shear rate). It is apparent that, except for Model A (
), the (reduced) shear rate
is a function of time. Therefore, for
(model B), after a transient regime, a steady state is achieved in the long time limit when both viscous heating and collisional cooling cancel each other and the mixture autonomously seeks the temperature at which the above balance occurs. In this steady state, the reduced steady shear rate
and the coefficients of restitution
are not independent parameters, since they are related through the
steady state condition
where the subindex st means that the quantities are evaluated in the steady state. However, when
(model A),
,
and so, the reduced shear rate remains in its initial value regardless of the values of the coefficients of restitution
. As a consequence, there is no steady state (unless
takes the specific value given by the condition (
66)) and
and
are
independent parameters in the USF problem. This is one of the main advantages of using Model A instead of Model B in the USF problem.
Before going ahead, it is convenient to write the form of
for arbitrary values of
. Here, although we will mainly consider model A, as in previous works on IMM [
23,
26], we will keep the same form for
, as in model B with
. Thus,
can be written as [
23,
26]
where the value of the quantity
is irrelevant for our calculations. In Equation (
67),
As in the case of the HCS, to determine the hierarchy of moment equations in the USF, we multiply both sides of Equations (
62) and (
63) by
and integrates over
. The result is
Here, we have called
In particular,
Since
is a polynomial of degree
, then the quantity
can be expressed as linear combinations of moments of degree
. This means that the hierarchy of Equations (
69) and (
70) can be exactly solved in a recursive way. This contrasts with the set of coupled equations for the moments in the HCS, where a general solution for them can be formally written.
Due to the technical difficulties involved in the solution of Equations (
69) and (
70) for a general binary mixture, we consider here the limit case where the concentration of one of the species (let us say, species 1) is negligible (
). This is the so-called
tracer limit. In this situation, one can assume that the state of the excess component 2 is not perturbed by the presence of the tracer particles and so, Equation (
70) reduces to
On the other hand, one can also neglect the collisions among tracer particles themselves in Equation (
69) and so, this equations reads
For the sake of convenience, let us introduce the scaled moments
where
is the thermal velocity of the excess species. Note that in the tracer limit
and
. The evolution equations for the scaled moments
can be obtained from Equations (
73) and (
74) when one takes into account the balance equation (
65) for the temperature
. They are given by
where
and
Upon writing Equations (
76) and (
77), use has been made of the identity
and the definition (
47) of
with the replacement
.
As expected, the evolution equations (
77) and (
78) involve the (reduced) shear stress
(second-degree moment). Thus, to determine the time evolution of the high-degree moments in the USF, one has to first obtain the second-degree moments. These moments are the most relevant ones from a rheological point of view.
Second-Degree Moments of the Excess Species—Model A
In the case of the excess component, the set of coupled equations for the moments
and
can be easily obtained from Equation (
76):
where in the tracer limit
In the hydrodynamic regime (which holds for times longer than the mean free time), the dependence of the (scaled) moments
and
on the dimensionless time
is via the time-dependence of the reduced shear rate
. Therefore, in Model A,
(since
) and the scaled moments
and
achieve stationary values, which are nonlinear functions of
and
. Their expressions are [
27]
where
is the real root of the cubic equation
namely,
Here,
7. Second and Third Degree Moments of the Tracer Species—Model A
In this section, we study the time evolution of the second and third-degree moments of the tracer species within the context of Model A. In particular, to obtain the time evolution of the scaled second-degree moments
of the tracer species, we assume that the scaled moments
have reached their stationary values. Therefore, from Equation (
77), one obtains the set of coupled equations
where
is the temperature ratio,
is the mass ratio,
and
Upon writing Equations (
87)–(
89), we have taken into account the relationship
. Note that the moments associated with the tracer species are proportional to
. For this reason, the right hand side of Equations (
87)–(
89) are proportional to
.
For long times, in the case of Model A,
and so, the solution to Equations (
87) and (
88) can be written as
where
In terms of
, the expression of
is
As expected, from Equations (
92)–(
94), it is straightforward to verify the constraint
Equations (
92)–(
94) are consistent with the results obtained in the Appendix C of Ref. [
26].
It still remains to determine the temperature ratio
. This quantity can be obtained by combining the balance equations for the temperatures
and
. In the case of Model A,
is determined by numerically solving the equation
where
,
, and
For mechanically equivalent particles,
and the condition (
97) yields
for any value of both the shear rate and the coefficients of restitution. This is the expected result. Moreover, when
and
, one recovers the results obtained in the tracer limit of the HCS. To illustrate the shear-rate dependence of the temperature ratio, we plot in
Figure 9 the ratio
versus the (reduced) shear rate
for
,
,
, and three different values of the mass ratio. Here,
is the value of the temperature ratio in the HCS. We observe first that the influence of
on
is significant since the ratio
clearly differs from 1. In addition, in contrast to the results obtained in the HCS [see
Figure 1], the temperature ratio
exhibits a non-monotonic dependence on
regardless of the mass ratio considered. To complement
Figure 9,
Figure 10 shows the shear-rate dependence of the scaled moments
and
for the same systems as that of
Figure 9. While the first moment is related with the tracer contribution to the shear stress, the second moment is a measure of the normal stress differences. It is quite apparent that the non-Newtonian effects on tracer species increase as increasing the shear rate, as expected. In addition, the departure from equilibrium becomes more significant as the tracer species is lighter than the excess species.
Third-Degree Moments—Model A
We consider now the time evolution of the (scaled) third-degree moments in the context of Model A. Let us assume first that the scaled second-degree moments have achieved their stationary values given by Equations (
83) for the excess species and Equations (
92), (
93) and (
95) for the tracer species. Moreover, as shown in Ref. [
27], all the scaled third-degree moments of the excess species vanish for long times in the USF. Here, as in the analysis of the second-degree moments of the tracer species, we also assume that the scaled moments
and
have reached their steady values (and so, they vanish). In what follows, for the sake of simplicity, we will particularize to a two-dimensional system (
).
In a two-dimensional mixture, there are four independent third-degree moments for the tracer species. Here, we take the scaled moments
After some algebra, the time evolution of the moments (
99) is given by
where
and
In the absence of shear rate (), the eigenvalues associated with the moments and are and , respectively. This result agrees with the ones obtained in the HCS in the tracer limit when one assumes that the (scaled) third-degree moments of the excess component vanish. Thus, in the HCS, the moments and are divergent if and are negative, respectively.
When
, the eigenvalues
ℓ associated with the time behavior of the third-degree moments (
99) are the roots of the characteristic quartic equation
The long time behavior of the moments (
99) is governed by the eigenvalue
with the smallest real part. If
becomes negative then the third-degree moments of the tracer species can be divergent.
As expected, an analysis of the solutions of the quartic Equation (
104) shows that
may be negative, especially when the diameter of the tracer species is smaller than that of the excess species. Moreover, surprisingly, in most of the cases studied, we have found that the main effect of shear rate on
is to reduce its magnitude so that it becomes positive for shear rates larger than a certain critical value. As an illustration,
Figure 11 shows the dependence of
on
for
,
,
, and two values of the (common) coefficient of the restitution. We observe that
is a non-monotonic function of the shear rate; it becomes positive for sufficiently large values of
. To complement
Figure 11,
Figure 12 shows the phase diagram associated with the singular behavior of the third-degree moments for the case
,
, and
. Here, as in
Figure 11, we have assumed that
. The curve
splits the parameter space in two regions: the region above the curve corresponds to states
with finite (zero) values of these moments (i.e.,
); the region below the curve provides states where those moments diverge in time. Thus, at a given value of
, there exists a critical value
, such that the moments are convergent for
. In particular, we observe that
(and so, the moments become convergent) for sufficiently large values of the (reduced) shear rate
.
8. Discussion
It is well known that for molecular gases (i.e., particles colliding elastically), the model of Maxwell molecules (namely, when the collision rate of two colliding particles is independent of their relative velocity) is a very useful starting point to obtain exactly transport properties in far from equilibrium states [
2,
3]. On the other hand, when the collisions are inelastic and characterized by a constant coefficient of normal restitution
, one can also introduce the inelastic version of the Maxwell model (IMM). In this model, the form of the Boltzmann collision operator can be obtained from its corresponding form for IHS by replacing the collision rate of hard spheres by an effective collision rate independent of the relative velocity. Thanks to this property, the collisional moments of the Boltzmann operator for IMM can be exactly written in terms of the velocity moments of the distributions
and
without explicitly knowing these distributions. This mathematical property of IMM opens up the possibility of obtaining exact results (the elastic limit
is a special limit) for granular flows, such as the Navier–Stokes transport coefficients [
13,
14] and/or the rheological properties of sheared granular gases [
22,
23,
27].
In the case of monocomponent granular gases, the choice of the Ikenberry polynomials
of degree
allows one to express the corresponding collisional moment
as an eigenvalue
times the velocity moment
plus a bilinear combination of moments of degree less than
. All the third and fourth degree collisional moments of IMM for monocomponent granular gases were evaluated in Ref. [
19]. We have extended in this paper the above results to the interesting case of binary granular mixtures. Due to the intricacy of the general problem, we have considered here situations where diffusion processes are absent. This means that the mean flow velocities
of each species are equal to the mean flow velocity
of the mixture (
). Apart from this simplification, the results reported in this paper for the third and fourth degree collisional moments are exact for arbitrary values of the masses
, diameters
, concentrations
, and coefficients of restitution
. In addition, all the derived expressions apply for any dimensionality
d. Known results for three-dimensional molecular gases [
2,
3] and for
d-dimensional monocomponent granular gases [
19] are recovered. In the one-dimensional case (
) for binary granular mixtures, our results for the (isotropic) collisional moments
and
agree with the ones obtained by Marconi and Puglisi [
16]. This shows the consistency of our general results with those previously reported in some particular limits.
As for monocomponent granular gases [
19], we have observed that some of the eigenvalues
exhibit a non-monotonic dependence on the coefficients of restitution
at given values of the mass and diameter ratios and the concentration. We have also observed that the impact of the inelasticity in collisions on the eigenvalues is in general important, especially in the case of the eigenvalues associated with the self-collision terms. Although the above observations are restricted to the moments of degree
, we expect that they extend to moments of higher degree.
The knowledge of the second, third, and fourth degree collisional moments for inelastic Maxwell mixtures opens up the possibility of studying specific nonequilibrium situations. We have analyzed in this paper two different problems. First, we have studied the time evolution of the moments of degree equal to or less than 4 in the HCS. In this state, given that the granular temperature
T decreases in time, one has to scale the moments with the thermal speed
to reach steady values in the long time limit. Our analysis shows that while all the second degree moments tend towards finite values for long times, the third degree moments
(which are related to the heat flux) can diverge in a region of the parameter space of the mixture. This sort of divergence also appears in all (isotropic and anisotropic) fourth degree moments. The above conclusions contrast with the ones achieved for monocomponent granular gases [
19], where all the moments of degree
are convergent for
. The singular behavior of the third degree moments is consistent with an algebraic high velocity tail of the form
, where
when the moments
are divergent. We plan to explore this possibility in a forthcoming work.
As a second application, we have analyzed the time evolution of the second and third degree moments of a
sheared granular binary mixture where one of the species is present in tracer concentration. In this situation, given that the dynamic properties of the excess species coincide with those previously obtained for simple granular gases [
27], the study is focused on the tracer species. In particular, in contrast to the findings of monocomponent granular gases of IMM [
27], our results demonstrate that the (scaled) third-degree moments of the tracer species can diverge in time for given values of the parameters of the mixture. This is the expected result according to the analysis made in the HCS. However, it is quite apparent that in general those moments become convergent for sufficiently large values of the (reduced) shear rate. Thus, one can conclude that the main effect of the shear rate on the third-degree moments of tracer species is to increase the size of the region where those moments are convergent.
One of the limitations of the results derived in this paper is its restriction to non-equilibrium situations where the flow velocities of both species are equal (
). This yields a vanishing mass flux (
). The extension to situations where
is possible but the determination of these new terms (coupling
with other moments) in the corresponding collisional moments involves a quite long and tedious calculation. A previous work [
14] on IMM has accounted for these new contributions for the collisional moments
,
,
, and
. We plan to extend the present expressions for the collisional moments
,
,
, and
for non-vanishing mass fluxes in the near future. This will allow us to obtain the collisional moments of the second, third and fourth degree in a granular binary mixture of IMM without any kind of restriction.
The fact that the third and fourth degree moments in the HCS may be divergent have important physical consequences on the transport coefficients, given that the HCS plays the role of the reference state in the Chapman–Enskog perturbative solution [
34] to the Boltzmann equation. In particular, as Brey et al. [
35] pointed out in the monodisperse case, the transport coefficients associated with the heat flux can be divergent for values of
(
at
and
at
). These authors [
35] found that below the critical value
, one of the kinetic modes (the one associated with the heat flux) decays more slowly than the hydrodynamic mode associated with the granular temperature. They concluded that a hydrodynamic description is not possible for values of
. A similar behavior is expected for granular mixtures, although the values of
will have a complex dependence on the concentration and the mass and diameter ratios. Regarding the above point, it is interesting to remark that a slightly different view to the one offered in Ref. [
35] on the singular behavior of the heat flux transport coefficients has been provided in Ref. [
36]. According to this work, the origin of the above divergence could be also associated with the possible high-velocity tail of the first-order distributions
of the Chapman–Enskog solution. Thus, although
could be well defined for any value of the coefficients of restitution, its third-order velocity moments (such as the heat flux) might diverge due to the high-velocity tail of this distribution. In any case and according to the results reported in the present paper for the velocity moments in the HCS for granular mixtures, given that the critical values
are generally small, the possible breakdown of granular hydrodynamics has no important consequences for practical purposes.
The explicit results provided in this paper can be employed to analyze different nonequilibrium problems. As mentioned before, one of them is to extend our analysis to binary mixtures with arbitrary values of the concentration. In the USF problem, apart from the rheological properties [
22,
23], it would be interesting to study the time evolution of the fourth degree velocity moments towards their steady values and investigate whether these moments can be divergent, as occurs for elastic collisions [
37]. Another interesting application of the present results is to determine some of the generalized transport coefficients characterizing small perturbations around the simple shear flow problem [
26,
38]. Work along these lines will be carried out in the near future.