Kinetic and hydrodynamic models of nearly elastic granular flows Giuseppe Toscani Dipartimento di Matematica, Universit´a di Pavia, 27100 Pavia, Italy, e-mail:
[email protected], http://dimat.unipv.it/∼toscani
Abstract We introduce and discuss certain models of dilute granular systems of spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. The dissipation is taken into account by introducing a correction to the Boltzmann collision operator in the form of a nonlinear friction type operator. Using this correction we obtain formally from the Boltzmann equation in a direct way a hydrodynamic description of a system of nearly elastic particles colliding with a variable coefficient of restitution. In one dimension of the velocity variable, the correction reduces to the nonlinear friction operator obtained in [24] as the quasi-elastic limit of a model Boltzmann equation for partially anelastic spheres. The large–time asymptotic of this one–dimensional model can be described in detail.
Key words. Granular gases, nonlinear friction equations, long-time behavior of solutions. AMS(MOS) subject classification. 76P05, 82C40.
1
Introduction
The aim of this paper is to discuss some questions connected with the modelling of granular flows. In the physical literature, granular gases have been introduced to describe the evolution of materials composed of many small discrete grains, in which the mean free path of the grains is much larger than the typical particle size. Once initialized with a certain velocity distribution, granular gases cool down due to inelastic collisions of their particles. Similar as molecular gases, granular gases can be described at a mesoscopic level within the concepts of classical statistical mechanics, by means of methods borrowed from the kinetic theory of rarefied gases [11]. Many recent papers
1
2
Nearly elastic granular flows
(see [2, 7, 21, 13] and the references therein), consider in fact Boltzmann-like equations for partially inelastic rigid spheres. This choice relies in the physical hypothesis that the grains must be cohesionless, which implies the hard-sphere interaction only, and no long-range forces of any kind. The dissipation of kinetic energy causes a series of non–trivial effects, as formation of clusters and other spatial structures [17, 20], non– Maxwellian velocity distributions, anomalous diffusion, and others. These phenomena are difficult to observe in realistic models with few particles [16]. On the other hand, full kinetic collisional models of Boltzmann type are extremely complicated to study. In reason of this, the study of the cooling of a granular gas has recently been attacked within the assumption of weak dissipation, that justifies the introduction of simplified models, typically in one dimension of the velocity variable. When collisions are close to be elastic [24], the Boltzmann collision operator simplifies, and it becomes convenient to use a nonlinear friction operator (see the definition in the next section). This choice leads to various advantages. In fact, despite their relatively simple (with respect to the Boltzmann equation) structure, these nonlinear friction equations exhibit the main properties of any kinetic model with dissipative collisions, like conservation of mass and mean velocity and decay of the temperature. Likewise, the equilibrium state is given by a Dirac mass located at the mean velocity of particles. In addition, on the contrary to the Boltzmann models for granular flows (see the discussion in [5]), these equations exhibit similarity solutions, which are in general of noticeable importance to understand the cooling process of the granular flow, and to construct reasonable macroscopic equations [21]. The study of these similarity solutions showed that the speed of relaxation of the temperature (cooling process) is heavily linked to the dependence of the restitution coefficient on the relative velocity. In this paper we develop a general idea to study weak dissipative granular gases with variable restitution coefficient, by introducing a correction to the classical Boltzmann collision operator. This correction is represented by a nonlinear friction type operator, with a kernel which depends of the variable restitution coefficient. This will be done in Section 3. In section 4 the main properties of the model are discussed, together with a study of the cooling of the granular gas. Section 5 contains a formal derivation of hydrodynamic equations for weakly dissipative granular gases. We generalize in this way previous formal transitions to hydrodynamics [1, 5]. The last section contains a study of the large–time behavior of the solutions to nonlinear friction equations in one dimension of the velocity variable. The result is a consequence of the fact that one can recover an exact equation for the time-evolution of the Wasserstein metric [26], from which one can easily reckon explicit rates of its time-decay. Some problem linked to dissipative equations containing nonlinear friction operators has been addressed before, mostly in one dimension of the velocity variable. The introduction of a nonlinear friction equation goes back to McNamara and Young [21], who called it the test-particle equation. The same equation was derived independently some year later in [2] in a suitable scaling limit from a one-dimensional system of N particles colliding inelastically. This test-particle equation corresponds to set the coefficient of restitution, which characterizes the loss of energy of two colliding grains, equal to a material constant. A few years ago, Benedetto, Caglioti, Carrillo and Pulvirenti
G. Toscani
3
[3] studied the asymptotic behavior of a granular gas in a thermal bath by using the test-particle equation to model the dissipative gas. Almost general nonlinear friction equations have been recently obtained in [24] as the quasi elastic limit of a model Boltzmann equation with dissipative collisions and variable coefficient of restitution. The large–time behavior of these equations was done in general in [10], and later [18] gave a simple proof of the Wasserstein decay of any two solutions in the one dimensional case. As we shall see, the one dimensional model is significant, since the nonlinear friction equation corresponds in this case to a consistent correction of the Boltzmann operator.
2
Two–particle interaction of dissipative particles
In a granular gas, the microscopic dynamics of grains is governed by the restitution coefficient e which relates the normal components of the particle velocities before and after a collision. If we assume that the grains are identical perfect spheres of diameter σ > 0, (x, v) and (x − σn, w) are their states before a collision, where n ∈ S 2 is the unit vector along the center of both spheres, the post collisional velocities (v ∗ , w∗ ) are such that (v ∗ − w∗ ) · n = −e((v − w) · n) . (1) Thanks to (1), and assuming the conservation of momentum, one finds the change of velocity for the colliding particles as 1 v ∗ = v − (1 + e)((v − w) · n)n , 2
1 w∗ = w + (1 + e)((v − w) · n)n. 2
(2)
For elastic collisions one has e = 1, while for inelastic collisions e decreases with increasing degree of inelasticity. In the literature, it is frequently assumed that the restitution coefficient is a physical constant. A constant restitution coefficient however leads to well-known unrealistic physical states in some applications, while it does not agree even with a dimensional analysis [23]. In fact, the restitution coefficient may depend on the relative velocity in such a way that collisions with small relative velocity are close to be elastic. The simplest physically correct description of dissipative collisions is based on the assumption that the spheres are composed by viscoelastic material, which is in good agreement with experimental data. The velocity–dependent restitution coefficient for viscoelastic spheres of diameter σ > 0 and mass m reads
with
e = 1 − C1 Aα2/5 |(v − w) · n|1/5 + C2 A2 α4/5 |(v − w) · n|2/5 ± . . .
(3)
√ √ σY 3 3 α= , 2 m(1 − ν 2 )
(4)
where Y is the Young modulus, ν is the Poisson ratio, and A depends on dissipative parameters of the material. The constant C1 and C2 can be explicitly computed. It is to remark that formula (3) refers to the case of pure viscoelastic interactions, i.e. it holds when the relative velocity (v − w) · n belongs to a certain interval (a, b) ∈ R+
4
Nearly elastic granular flows
with a bounded away from zero (to neglect surface effects) and b 0. The velocities (v ∗∗ , w∗∗ ) are the pre collisional velocities of the so–called inverse collision, which results with (v, w) as post collisional velocities. The factor χ in the gain term appears respectively from the Jacobian of the transformation dv ∗∗ dw∗∗ into dvdw and from the lengths of the collisional cylinders e|q ∗∗ · n| = |q · n|. For a constant restitution coefficient, χ = e−2 . To avoid the presence of the function χ, and to study approximation to the granular operator (8) it is extremely convenient to write the operator (8) in weak form. More precisely, let us define with < · , · > the inner product in L1 (R3 ). For all smooth functions ϕ(v), it holds Z 2 ¯ ¯ f )(v) dv = < ϕ , Q(f, f ) > = 4σ ϕ(v)Q(f, 3 R Z Z Z 2 4σ q · n (ϕ(v ∗ ) − ϕ(v)) f (v)f (w)dv dw dn = 3 R3 S R Z Z Z + 2σ 2 |q · n| (ϕ(v ∗ ) − ϕ(v)) f (v)f (w)dv dw dn.. (9)
R3 R3
S2
The last equality follows since the integral over the hemisphere S+ can be extended to the entire sphere S 2 , provided the factor 1/2 is inserted in front of the integral itself. In fact changing n into −n does not change the integrand. Let (v 0 , w0 ) be the post collisional velocities in a elastic collision with (v, w) as incoming velocities, v 0 = v − (q · n)n ,
w0 = w + (q · n)n.
(10)
Using (2) and (10) one obtains 1 1 v ∗ = v 0 + (1 − e)(q · n)n , w∗ = w0 − (1 − e)(q · n)n. 2 2 If we assume that the coefficient of restitution satisfies (5), v ∗ − v 0 = βγ (|q · n|) (q · n)n.
(11)
(12)
Let us consider a Taylor expansion of ϕ(v ∗ ) around ϕ(v 0 ). Thanks to (12) we get ϕ(v ∗ ) = ϕ(v 0 ) + β∇ϕ(v 0 ) · γ (|q · n|) (q · n)n + 1 2 X ∂ 2 ϕ(v 0 ) 2 β γ (|q · n|) (q · n)2 ni nj + . . . 2 ∂vi0 ∂vj0
(13)
i,j
If the collisions are nearly elastic, β = 2σ |q · n| × 3 R3 S 2 R ¡ ¢ ϕ(v 0 ) − ϕ(v) + β∇ϕ(v 0 ) · γ (|q · n|) (q · n)n f (v)f (w)dv dw dn = < ϕ , Q(f, f ) > +β < ϕ , I(f, f ) > .
(14)
6
Nearly elastic granular flows
It is a simple matter to recognize that in (14) Q(f, f ) is the classical Boltzmann collision operator for elastic hard–spheres molecules [11], Z Z © ª Q(f, f )(v) = 2σ 2 |q · n| f (v 0 )f (w0 ) − f (v)f (w) dw dn. (15)
R3
S2
In fact, the velocity v 0 into (14) is obtained from (v, w) through the elastic collision (10). Let us now study in more detail the second contribution to the inner product (14). Using the properties of the transformation (10), we obtain < ϕ , I(f, f ) > = Z Z Z 2 ∇ϕ(v 0 ) · n(q · n)|q · n|γ (|q · n|) f (v)f (w)dv dw dn = 2σ 3 2 3 RZ RZ SZ 2 ∇ϕ(v) · n(q · n)|q · n|γ (|q · n|) f (v 0 )f (w0 )dv dw dn = −2σ 2 3 3 Z R R S Z Z 2 2σ dv ϕ(v)divv n(q · n)|q · n|γ (|q · n|) f (v 0 )f (w0 )dw dn.
R3
R3
(16)
S2
In fact, the transformation dv dw into dv 0 dw0 given by (10) is such that q 0 · n = −q · n, while its Jacobian is equal to unity. The last equality follows from the divergence theorem. This shows that the granular correction is the nonlinear friction operator βI(f, f )(v), where Z Z I(f, f )(v) = 2σ 2 divv n(q · n)|q · n|γ (|q · n|) f (v 0 )f (w0 )dw dn. (17)
R3
S2
Finally, for nearly elastic granular collisions, with a restitution coefficient satisfying (5), the Enskog–Boltzmann equation can be modelled at the leading order as ∂f + v · ∇x f = G(ρ)Q(f, f )(x, v, t) + G(ρ)βI(f, f )(x, v, t), (18) ∂t where Q is the classical elastic Boltzmann collision operator, and I is a dissipative nonlinear friction operator which is based on elastic collisions between particles. Remark 3.1 The model Boltzmann equation we obtained is valid, at least formally, for weakly anelastic dissipative collisions, i. e. when the value of β is sufficiently small. Other moderately dissipative regimes require the consideration of higher order terms in the expansion (13). The second–order term in this expansion gives a diffusive correction to the Boltzmann equation. Numerical computations [22] showed that for a one-dimensional (in the velocity space) dissipative Boltzmann equation the results relative to the full equation are in good agreement with those relative to the equation with a second–order correction, even in regimes of high inelasticity. This is not true if we cut the expansion at the first–order, except in nearly elastic regimes, where the approximation is essentially correct. The same analysis showed that higher–order corrections introduce problems in the numerical spectral approximation, without essential improvements in the accuracy of the solution. This suggests that, while in nearly elastic regimes the first–correction is the dominant one, in other regimes one has to cut the expansion at the second–order, considering both the drift and diffusive corrections.
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Remark 3.2 A small inelasticity expansion, analogous to (14) has been recently performed in [9] to compute a small inelasticity approximation of the steady state of a driven granular media in which particles interact inelastically, while they follow Brownian dynamics in between collisions. However, the model used in [9] is a Boltzmanntype kinetic equation with a pseudo-Maxwellian interaction kernel, with a constant coefficient of restitution.
4
Nonlinear friction and the cooling of a granular gas
The main purpose of this section is to study in some detail the main properties of the nonlinear friction operator I. To simplify notations, we set from now on σ = 1. Choosing ϕ = 1, v, v 2 /2 into (16) shows at once that both mass and bulk velocity are conserved, while 1 < v 2 , I(f, f ) > = Z2 Z Z 2 v 0 · n(q · n)|q · n|γ (|q · n|) f (v)f (w)dv dw dn = RZ3 RZ3 SZ2 −2 w0 · n(q · n)|q · n|γ (|q · n|) f (v)f (w)dv dw dn = 3 3 2 Z ZR ZR S (v 0 − w0 ) · n(q · n)|q · n|γ (|q · n|) f (v)f (w)dv dw dn = 3 3 2 RZ RZ SZ − |q · n|3 γ (|q · n|) f (v)f (w)dv dw dn
R3 R3
(19)
S2
The multiple integral on (19) can be simplified by evaluating first the surface integral. One obtains in this way Z Z Z |q · n|3 γ (|q · n|) f (v)f (w)dv dw dn = 3 3 2 ZR ZR S A(|q|)f (v)f (w)dv dw , (20)
R3 R3
where 4π A(|q|) = |q|
Z
|q|
r3 γ(r) dr.
0
Let ρ denote the (constant) number density of the granular gas, and T (t) the temperature. If the (constant) bulk velocity is taken equal to zero, we can immediately find the equation for the evolution of the temperature for the spatially homogeneous Boltzmann equation of the nearly elastic granular gas. Thanks to (19) and (20), d 1 ¯ f ) >= ρ dT (t) = −βG(ρ) < v 2 , Q(f, dt 2 dt
Z
Z
R3 R3
A(|q|)f (v)f (w)dv dw .
(21)
8
Nearly elastic granular flows
The previous equation follows considering that the temperature (internal energy) is a collisional invariant for the elastic Boltzmann collision operator Q. For particular choices of the function γ, one can easily reckon upper bounds for the time decay of the temperature. This is the case, for example, if γ(r) = rp , with p ≥ 0. In this case, since the bulk velocity is equal to zero, Z Z Z 1 2 2 |q| f (v)f (w)dv dw = 4ρ v f (v)dv . R3 2 R3 R3 On the other hand, by Cauchy-Schwarz inequality, ¶(3+p)/2 µZ Z Z 2 1+p |q| f (v)f (w)dv dw ≤ρ
R R
R R 3
Z 3
3
|q|3+p f (v)f (w)dv dw . 3
Hence, we obtain for T (t) the differential inequality dT (t) 25+p π ≤ −βG(ρ)ρ(1−p)/2 T (t)(3+p)/2 . dt p+4 which can be solved to give the bound #−2/(1+p) "µ ¶ 4+p π 1 (1+p)/2 2 t T (t) ≤ + (1 + p)βG(ρ)ρ(1−p)/2 . T0 p+4
(22)
(23)
For a constant coefficient of restitution we recover in this way that the typical decay is of the order t−2 , while we obtain a decay of order t−5/3 for the physically relevant case of the viscoelastic spheres. A second interesting property of the collisional integral I is that it leads to exact computations in correspondence to a locally Maxwellian function µ ¶ (v − u(x, t))2 ρ(x, t) exp − . (24) M (x, v, t) = 2T (x, t) (2πT (x, t))3/2 By (19)
Z Z 1 < v 2 , I(M, M ) > = − A(|q|)M (v)M (w)dv dw. (25) 2 R3 R3 Let us first remark that the integral on the right–hand side is independent of the bulk velocity u(x, t). This follows considering that q = v − w = (v − u) − (w − u), so that the transformation v, w into v − u, w − u leaves the integral unchanged. Next, since ³ q2 q ´2 v 2 + w2 = v 2 + (v − q)2 = +2 v− , 2 2 we obtain Z Z A(|q|)M (v)M (w)dv dw = R3Z R3 ¶Z ¶ µ µ 1 1 q2 (v − q/2)2 2 dv = ρ dqA(|q|) exp − exp − 4T T 3 3 (4πT )3/2 (πT )3/2 R R ¶ µ Z 1 q2 2 ρ dqA(|q|) . (26) exp − 4T (4πT )3/2 R3
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Let γ(r) = rp , with p ≥ 0. Then A(|q|) = 4π|q|3+p /(4 + p), and one gets Z Z 1 2 4π < v , I(M, M ) > = − |q|3+p M (v)M (w)dv dw = 2 4 + p R3 R3 µ ¶ Z 4πρ2 1 q2 3+p − dq|q| exp − = 4 + p R3 4T (4πT )3/2 Z ¡ ¢ 1 4πρ2 √ 3+p dq|q|3+p 3/2 exp −q 2 = − (2 T ) 4+p 3 π √R 7+p Γ(2 + p/2)2 π 2 (3+p)/2 − ρ T . (4 + p)
5
(27)
Hydrodynamic limit and the Euler equations.
In this section, we will consider mainly coefficient of restitutions for which γ(r) = rp . This choice will include both the constant coefficient of restitution and the physically relevant case of the viscoelastic spheres. On the basis of the results of the previous section, provided the mean free path 1 G(ρ) = g(ρ) ² is small enough, we can formally derive the fluid dynamical equations in the regime of small inelasticity. To this aim, considering that Q is the classical elastic Boltzmann collision operator, from (18) we obtain ¶ µ Z β ∂f + v · ∇x f − g(ρ) I(f, f )(x, v, t) dv = ψ(v) ² R3 Z ∂t 1 g(ρ) ψ(v)Q(f, f )(x, v, t) dv = 0, (28) ² R3 provided ψ is a collisional invariant, i.e. ψ = 1, v, 21 v 2 . It is well-known that system (28) for the moments of f , which is in general not closed, can be closed by assuming f to be the locally Maxwellian function (24) [11]. Since the dissipative operator I is such that ψ = 1, v are collisional invariants, substituting f = M into (28), and using (27) we obtain the following dissipative Euler equations for density ρ(x, t), bulk velocity u(x, t) and temperature T (x, t) ∂ρ + div(ρu) = 0 ∂t ∂u 1 + (u · ∇)u + ∇p = 0 ∂t ρ ∂T 2 β + (u · ∇)T + T divu = − Cp g(ρ)ρT (3+p)/2 ∂t 3 ² where p = ρT , and
√ Γ(2 + p/2)27+p π Cp = , 3(4 + p)
(29)
(30)
10
Nearly elastic granular flows
This approximation is valid when both ² 0 and all ϕ ∈ C 2 (R), and such that for all ϕ ∈ C 2 (R) Z Z lim ϕ(v)dF (v, t) = ϕ(v)F0 (v). t→0
R
(40)
R
On Mp , p ≥ 2 one can consider several types of metrics. One of these metrics is known as the Wasserstein distance [26]. Given any measure dF (v) ∈ Mp , let us define, R for v ∈ R, F (v) = w≤v dF (w). Let F −1 (w) = inf{v : F (v) > w} denote the pseudo inverse function of the distribution function F (v). The Wasserstein distance between F and G can be rewritten as the L2 -distance of the pseudo inverse functions F −1 and R −1 G , where G(v) = w≤v dG(w) µZ d(F, G) =
1
¶1/2 [F −1 (ρ) − G−1 (ρ)]2 dρ .
(41)
0
Another definition (for functions with the same momentum) is the following 1 [d(F, G)] = 2
Z
2
Z
1
1
dρ 0
dp [|H(ρ, p)| − |K(ρ, p)|]2 .
0
where H(ρ, p) =: F −1 (ρ) − F −1 (p),
K(ρ, p) =: G−1 (ρ) − G−1 (p)
The result of [18] can be resumed as follows Theorem 6.1 Let F (v, t), G(v, t) ∈ C 1 (R+ t , M2 ) be two solutions to the initial value problem for equation (36), corresponding to the initial distributions F0 (v), G0 (v) ∈ M2 , respectively. Then, if 0 < τ < 2, the Wasserstein distance of F (v, t) and G(v, t) is monotonically decreasing with time, and the following decay holds σ2 d d(F (t), G(t)) ≤ − τ −1 d(F (t), G(t))1+τ . dt 2
(42)
Moreover, if −1 < τ < 0 and the initial distribution F0 (v) has bounded support, Supp(F0 ) = L < ∞, the support of the solution decays to zero in finite time, and the following bound holds · Supp(F (v, t)) ≤ Supp(F0 )
|τ |
σ 2 |τ | − |τ |−1 t 2
¸ |τ1|
.
(43)
+
Furthermore, if both the initial distributions F0 (v), G0 (v) ∈ M2 , have bounded supports, the Wasserstein distance of F (v, t) and G(v, t) decays to zero at finite time, and the following time-decay holds ·
¸1 2σ 2 |τ | 2 d(F (t), G(t)) ≤ d(F0 , G0 ) 1 − t (2L)|τ | + where L denotes the maximum of the supports.
1 −1 |τ |
,
(44)
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13
The proofs in [18] are a consequence of a detailed analysis of the evolution equation for the Wasserstein metric, which reads d dt =
Z
Z
1
1
dρ 0
Z
dp (|H(ρ, p)| − |K(ρ, p)|)2
0
Z
1
−2
dρ 0
1
dp (|H(ρ, p)| − |K(ρ, p)|)(|H(ρ, p)|1+γ − |K(ρ, p)|1+γ ).
0
Theorem 6.1 shows that we can expect a finite time extinction of the solution to the nonlinear friction only in the case of largely anomalous granular gases (see the comment after (6)), that is when particles are close to be inelastic for small velocities. Real physical situations reveal a time decay like t−p , where p
