Dilute granular flows in a medium M. Bisi, G. Spiga ∗,
Abstract Some aspects of the dissipative Boltzmann equation for granular matter diffusing in a background medium are investigated. Both collisions with field particles (supposed to be elastic) and inelastic collisions of the grains between themselves (with non–constant restitution coefficient) are taken into account, leading to the simultaneous presence of two different collision operators. The closure problem for the macroscopic transport equations in collision dominated regime is mainly addressed. Two different approximations, one based on local equilibrium and the other on a Grad–type expansion, are worked out. Results relevant to collision equilibria and to the hydrodynamic limit coincide to first order accuracy in the small parameter (Navier–Stokes level). Explicit expressions for the equilibrium temperature and of the limiting drift–diffusion equation for granular density are given.
Key words: Kinetic theory, Granular flows, Boltzmann equation, Macroscopic closure. AMS(MOS) subject classification: 76P05, 82C40. ∗
Dipartimento di Matematica, Universit`a di Parma, Parco Area delle Scienze 53/A, 43100 Parma.
Corresponding author:
[email protected]. Tel: +33 0521 906988. Fax: +33 0521 906950.
1
1
Introduction
In problems like diffusion of grains, fine powders, or small impurities in the atmosphere or in a given environment there are physical regimes in which kinetic theory [11, 10], in spite of its intrinsic difficulties, seems to be the most appropriate tool of investigation. It allows in fact a deeper description and understanding of the phenomenon, and can provide motivated closures [19] for the set of macroscopic equations, leading to self–contained partial differential equations for the most significant observable fields, needed for practical applications. When the dimensions of both test and field particles are much smaller than the relevant mean free paths, if the former constitute a rarefied gas finely embedded in a much denser background, models and methods of linear kinetic theory can be properly applied [9, 25]. In fact, the process is essentially driven by collisions of the grains with the host medium, whose evolution remains unaffected by the presence of such impurities [12]. However, it is interesting and important also to account for the interactions of test particles between themselves, though they are less frequent than the preceding encounters, in order to predict and quantify their effects at the macroscopic level. Coupling of fluid and kinetic description for particles in a medium were considered for instance in [16, 17, 8] for elastic interactions. But a typical fundamental feature of collisions involving grains is that they are inelastic, while elasticity strongly characterizes the classical Boltzmann equation of gas–dynamics. These collisions dissipate kinetic energy, which is transferred to non– participating degrees of freedom, leading to a progressive loss of thermal energy of the grains themselves. This is the scenario of granular flows, that has been widely and deeply investigated in recent years, from many points of view, including experimental, numerical
2
and theoretical, both physical and mathematical. From the ample literature we may quote for example, without pretending to be exhaustive nor selective, the references [23, 15, 5, 4, 13, 26, 3]. We shall deal thus with a dilute granular flow in a host medium, assuming that grains are tiny and smooth enough to be considered as inelastic rigid bodies colliding as billiard balls between themselves. As regards interaction of grains with the embedding environment, the situation is more delicate, since grains are actually macroscopic objects undergoing repeated instantaneous collisions with the several surrounding molecules of the background gas. This process might be described by suitable complex methods of kinetic theory [10], as typically done in problems of gas–surface interactions [24]. For simplicity we shall stick here to the standard approximation of replacing the actual mechanism by a fictitious one in which one considers tiny grains undergoing binary localized mechanical encounters with separate single partners, representing the target obstacles of the thermal bath in which they are diffusing [22, 27, 20, 25]. Reflections of actual background molecules from a grain may fulfil (or not) an overall kinetic energy conservation, depending on the kind of interaction they are experiencing with its boundary. Accordingly, the relevant fictitious collision model should be taken as elastic or inelastic. Since the elastic case is much easier to deal with, in this first approach to a quite cumbersome scenario we shall assume test particle collisions with field particle targets to be elastic, which of course does not mean that momentum and kinetic energy of the granular gas are conserved by the process. On the other hand, the main feature of the physical problem is competition between energy dissipation because of inelastic scattering, and possible energy supply by collision from the thermal bath, so that the situation is not essentially affected by the presence of an additional inelasticity with the background (as discussed also later in this
3
Introduction). Variations in the phase space density f (granular distribution function) following free flight trajectories are then expressed by two separate collision integrals with different characteristic properties. The first, which plays the dominant role in the evolution, is linear and elastic, guarantees mass conservation, and describes (conservative) exchange of momentum and kinetic energy with the background. The second, which is taken of the same relevance as the streaming operator in the overall process, is instead nonlinear and inelastic, and accounts for actual dissipation of kinetic energy in the encounters between grains. Mass is the only conserved quantity, and collision equilibrium is possible since energy loss by inelasticity may be balanced by energy supply by scattering with the thermal bath. The dominant operator is also known to fulfil a Lyapunov theorem (H– theorem) predicting relaxation to equilibrium, and even to admit a wide class of entropy functionals [21, 20]. To the authors’ knowledge, the simultaneous presence of two collision terms of different nature like the present ones is scarcely considered in the literature, and makes the analysis much harder than for the purely linear and purely nonlinear frameworks. We shall try to extend to the present physical case the results obtained in [2] for a single linear non–conservative Boltzmann collision integral, moving towards a kinetic theory based hydrodynamic description of dissipative diffusion of impurities [7]. Microscopic details of the encounters will be determined according to the commonly used hard–sphere collision model, which is certainly a considerable simplification of reality, but is preferable to other mathematically treatable easier models since grains do not exert any kind of long distance force. Such test particles are then considered in the usual way as equal rigid spheres of mass m and diameter d, endowed with only translational degrees of
4
freedom. The same for field particles, which will be labelled by a superscript B, and that are supposed to constitute a background in local thermodynamical equilibrium, thus with a Maxwellian distribution versus the velocity variable v, with assigned number density nB , mass velocity uB , and temperature T B . The latter parameters are bound together by Euler equations, which include the special case of steady homogeneous host medium (nB , uB , T B independent of both position x and time t). Dissipative binary encounters are described in terms of the so–called restitution coefficient e, with 0 < e ≤ 1, and e = 1 corresponding to no dissipation (elastic scattering). This coefficient measures the ratio of the normal components after and before collision of the velocity g of the bullet particle in the relative motion with respect to the target. As usual, tangential components are assumed instead to be preserved in the interaction. The same occurs to the center of mass velocity G of the colliding pair, so that momentum is also conserved here, together with mass. The restitution coefficient is often assumed for simplicity to be constant with respect to the impact parameters, but this is simply a working approximation, since there is theoretical and experimental evidence that it should exhibit a dependence on the normal component g · n ˆ (where n ˆ is the unit vector of the apse line) of the impinging velocity [14]. For the sake of generality, we will assume here a power–like dependence with a given exponent δ; it includes, for δ = 0, the constant restitution case, and, for δ = 1/5, the leading term of the model which seems to be most realistic in the literature, the so–called viscoelastic spheres [6]. See [4] for a different kind of non–constant restitution coefficient, depending, in that case, on the local temperature. It should be pointed out that another restitution coefficient could be introduced in order to describe a possible inelasticity in the binary collision between a test and a field particle. Indeed, if such
5
additional coefficient is supposed to be constant, it is known [22, 27, 20] that the relevant collision equilibrium would keep the same Maxwellian structure as in the elastic case, only with a lower but well defined temperature, and that again relaxation to equilibrium would be guaranteed by an H-theorem, with a wide class of Lyapunov functionals. Therefore the whole machinery developed below could be repeated, at the price only of technical changes and heavier manipulations, when the present assumption of elastic scattering of the granular gas with the background is replaced by a more general hypothesis of inelastic binary encounters with constant coefficient of restitution between test and field particles. The paper is organized as follows. Section 2 collects the main definitions and results about the physical problem and its kinetic description, and constructs explicitly both collision integrals in order to derive the pertinent Boltzmann equation, its weak form, and the exact set of non–closed macroscopic balance equations. The following two Sections are devoted to the problem of an approximate closure of such equations in a collision dominated regime, in which the mean free path for collisions with the background is much shorter than the other typical lengths. In both Sections an approximate distribution function is used for the closure, a self–contained set of partial differential equations for the crucial fields n, u, T is obtained, its “collision equilibria” for u and T in terms of n and of the input parameters are determined, and, in the limit when the small parameter tends to zero, an hydrodynamic equation at the Navier–Stokes level is derived. It is here a single equation, since there is only one conserved quantity (number density) for our problem, and turns out to be of drift–diffusion type. In Section 3, the approximating distribution is a local Maxwellian, which corresponds to maximize a typical relative entropy of (elastic) linear kinetic theory, whereas in Section 4 it is a Grad–type [18] polynomial expansion
6
which maximizes a quadratic entropy. In both cases then we are applying a Maximum Entropy Principle with respect to the dominant operator. The two approximations yield of course similar but different sets of balance equations for n, u, T : the problem of selecting the closure that better approximates the kinetic description among these or other possible options is one of our main goals in future investigations. It is comforting to note however that the hydrodynamic limits coincide up to the first order in the small parameter (same “Navier–Stokes” equation).
2
Inelastic Boltzmann equation
As stated in the Introduction, granular test particles (tp) undergo hard–sphere elastic scattering against field particles (f p) of a given background, with a Maxwellian distribution function f B = nB M B (v), where M B (v) = M0B (v − uB ) and M0B is the normalized isotropic Gaussian
µ M0B (v)
=
mB 2πKT B
¶ 32
·
¸ mB 2 exp − v . 2KT B
(1)
Momentum and kinetic energy are conserved in the collision, and their exchange is crucially determined by the mass ratio parameter α=
mB m + mB
(2)
with 0 < α < 1 (the singular limiting cases of Lorentz and Rayleigh gas are excluded). The post–collision velocities (labelled by a prime) corresponding to an encounter between a pair (tp , f p) with incoming velocities (v, w) are v0 = v − 2 α(g · n ˆ )ˆ n
w0 = w + 2(1 − α)(g · n ˆ )ˆ n
7
(3)
where g = v − w is the relative velocity and n ˆ the unit vector of the apse line. In addition, test particles undergo inelastic scattering between themselves, in which momentum is preserved but kinetic energy of relative motion is dissipated, namely g∗ · n ˆ = − e(g · n ˆ)
g∗ − (g∗ · n ˆ )ˆ n = g − (g · n ˆ )ˆ n,
(4)
where e is the restitution coefficient, with 0 < e ≤ 1. Post–collision values are labelled now by a star superscript. In the literature the parameter e is usually supposed to be constant [5, 13, 26], however experimental works [23] show that the restitution coefficient may also depend on the relative velocity, and precisely the degree of inelasticity increases when the normal relative speed becomes bigger. An assumption that turns out to be in good agreement with experimental data is given by the so–called viscoelastic spheres, whose restitution coefficient can be expressed, to the leading order, by the law 1 − e = 2 β γ(|g · n ˆ |)
where
γ(r) = rδ ,
(5)
with the particular option δ = 1/5 [6]. For this reason we shall assume here that e has the form (5), with general parameter δ ≥ 0; the case δ = 0 corresponds to a constant restitution coefficient. In (5), the parameter β > 0 represents the degree of inelasticity, and the classical elastic case e = 1 is recovered as β = 0. From (4), it easily follows v∗ = v −
1+e (g · n ˆ )ˆ n 2
w∗ = w +
1+e (g · n ˆ )ˆ n. 2
(6)
The inelastic kinetic equation to be dealt with for the unknown (tp) distribution function f (x, v, t) reads then as ∂f + v · ∇x f = JEL (f ) + JIN (f, f ) ∂t
8
(7)
where JEL denotes the linear elastic collision operator taking into account scattering with background particles, while JIN is the dissipative integral operator, quadratic in the unknown f , describing the rate of change due to collisions of granular particles between themselves. Omitting (for brevity) in the notation all dependences on x and t, they read as 1 JEL (f ) = nB 2
µ
d + dB 2
¶2 Z
1 JIN (f, f ) = d2 2
Z R3
Z
Z R3
h i 0 B 0 B |g · n ˆ | f (v )M (w ) − f (v)M (w) d3 w d2 n ˆ,
(8)
S2
h i |g · n ˆ | χ f (v∗ )f (w∗ ) − f (v)f (w) d3 w d2 n ˆ
(9)
S2
where χ is the Jacobian of the pre-post–collision transformation, whose complicated expression is not necessary here and will not be reported. The same for v∗ and w∗ , which denote the pre–collision velocities associated to the pair (v, w). As usual, it is convenient to make (7) dimensionless by introducing suitable scalings: standard manipulations single out in a spontaneous way the (tp) mean free paths corresponding to their possible collision partners, namely " λB = n ¯B π
µ
d + dB 2
¶2 #−1 ,
³ ´−1 λ= n ¯ π d2 .
(10)
In our problem we have necessarily n ¯ > d) the ratios λB /λ and n ¯ /¯ nB are of the same order of magnitude and both very small. The ratios of λB and λ to the macroscopic characteristic length L constitute the so–called Knudsen numbers [10]. Now of course different scalings may be devised. According to our assumptions, the dominant role in the process is played by JEL , while gradient and inelastic operators share the same level of importance. This implies that we have λ ≈ L >> λB , and therefore we shall choose L = λ and introduce the small
9
parameter ε=
λB λB = = KnB +∇x · < v ϕ, f >= < ϕ, QEL (f ) > + < ϕ, QIN (f, f ) > ∂t ε
(12)
with < ϕ, QEL (f ) >
nB = 2π
< ϕ, QIN (f, f ) > =
1 2π
Z
Z
Z
h i 0 |g · n ˆ |f (v)M (w) ϕ(v ) − ϕ(v) d3 v d3 w d2 n ˆ B
Z
R3
R3
Z
R3
R3
Z
S2
h i |g · n ˆ |f (v)f (w) ϕ(v∗ ) − ϕ(v) d3 v d3 w d2 n ˆ, S2
(13)
where the effects of collisions are accounted for only by the test function evaluated at the post–collision velocity, either v0 or v∗ . The symbol < ·, · > stands for the usual dual product with respect to the v variable. The background distribution M B (w) is the dimensionless version of M0B (w − uB ) as given by (1), and its macroscopic parameters nB , uB , T B fulfil the Euler equations for a perfect fluid in thermodynamical equilibrium ∂nB ∂(nB uB k) + =0 ∂t ∂xk 1 ∂(nB T B ) ∂uB ∂uB i i + uB = − k ∂t ∂xk nB mB ∂xi
(14)
2 B ∂uB ∂T B ∂T B k + uB = − T . k ∂t ∂xk 3 ∂xk Equations (14) are reported in indicial notation, and Einstein convention on repeated indices will be used throughout the paper. As well known, the dominant operator QEL is pushing the distribution function to-
10
wards the Maxwellian shape M (v) = M0 (v − uB ), where ³ m ´ 32 h m 2i M0 (v) = exp − B v , 2πT B 2T
(15)
at the same local drift velocity and temperature of the background. This distribution is approached after a fast initial transient, and apart from boundary layers, both of order ε, so that in the hydrodynamic evolution taking place in such bulk region the distribution function is not far from the shape defined by M (v) [10]. Transport equations for the five fundamental macroscopic fields n, u, T are clearly the specialization of (12) to ϕ = 1, v, v 2 . In fact, the corresponding moments of f are n, n u, µ ¶ 3T 2 and n u + , respectively. It is easy to see that ϕ(v) = 1 is collision invariant for m the whole collision operator, and then obviously n is a conserved quantity. As concerns ϕ = v, v 2 , the relevant power moments of QEL and QIN are not amenable merely to moments of f , because of the structure of the collision integrals. The elastic collision contributions may be reduced to a simpler form by the same manipulations performed in detail in [2] to which the reader is referred. Only the final result will be written down later. We focus our attention on the inelastic terms. Bearing (5) and (6) in mind, the differences involved in the second of (13) are v∗ − v
=
(v ∗ )2 − v 2 =
h i 1 − β γ(|g · n ˆ |) (g · n ˆ )ˆ n h i h i 1−2β γ(|g · n ˆ |) + β 2 γ 2 (|g · n ˆ |) (g · n ˆ )2 − 2 1−β γ(|g · n ˆ |) (g · n ˆ )(v · n ˆ ). (16)
Since we assume γ(r) = rδ , all required angular integrations may be performed explic-
11
itly as
Z |g · n ˆ |(g · n ˆ) n ˆ d2 n ˆ
= πgg
S2
Z |g · n ˆ |δ+1 (g · n ˆ) n ˆ d2 n ˆ = S2
4 π δ+1 g g 4+δ
Z |g · n ˆ |(g · n ˆ )2 d2 n ˆ
(17)
= π g3
S2
Z |g · n ˆ |δ+1 (g · n ˆ )2 d2 n ˆ
=
S2
Z |g · n ˆ |2δ+1 (g · n ˆ )2 d2 n ˆ = S2
4 π δ+3 g 4+δ 4π g 2δ+3 , 4 + 2δ
where g = |g|. At this point, after some algebra, by putting together streaming, elastic and inelastic terms, the macroscopic exact (non–closed) equations for the first five moments may be cast in convective form as ¢ ∂n ∂ ¡ + n uk = 0 , ∂t ∂xk ∂ui ∂ui 1 ∂nT 1 ∂pij nB n + nuk + + =− α ∂t ∂xk m ∂xi m ∂xj ε
Z Z g gi f (v) M B (w) d3 v d3 w , R3
R3
∂T 2 ∂uk 2 ∂uh 2 ∂qk ∂T + nuk + nT + phk + = ∂t ∂xk 3 ∂xk 3 ∂xk 3 ∂xk ½Z Z ¾ Z Z nB 2m B 3 B =− α ggi (vi −ui )f (v) M (w) d3 v d3 w − α g f (v) M (w) d3 v d3 w ε 3 R3 R3 R3 R3 ¸ Z Z · 2 1 β δ+3 2δ+3 − mβ g − g f (v)f (w) d3 v d3 w . 3 4 + 2δ R3 R3 4 + δ (18) n
As anticipated, only the first one, the continuity equation, represents a conservation law, and the complete set is not closed also because of the presence in the streaming terms of global pressure tensor P = p + nT I (with p denoting its viscous part and I the unit tensor) and heat flux q, defined as usual by Z P=m
1 q= m 2
(v −u)⊗(v −u)f (v) d3 v , R3
12
Z (v −u)2 (v −u)f (v) d3 v . (19) R3
Moreover, collision contributions involve complicated integrals of the distribution function itself. The three collision integrals affected by the parameter α describe exchange of momentum and energy with the background and are O(1/ε), whereas the two O(1) contributions affected by the dissipation parameter β appear as expected only in the balance equation for thermal energy. It is easy to see that, under the conditions on the parameters β, δ, and T B for which the representation (5) is consistent in our physical situation, the content of the square brackets in the last integral in (18) makes the integral itself positive, so that the whole inelastic contribution to the temperature trend is negative and represents an actual dissipation. In the next Sections we shall deal with the closure problem and the asymptotic limit for the moment equations (18) in the frame of the considered collision dominated regime.
3
Local equilibrium approximation
A typical approximation of kinetic theory consists in replacing the actual distribution function by a local Maxwellian (the classical elastic collision equilibrium), namely by a Gaussian sharing with f the exact moments n, u, T f¯(v) = n
³ m ´ 32 h m i 2 exp − (v − u) . 2πT 2T
(20)
As pointed out in [2], this approximation maximizes the so–called relative entropy ·
¸ f (v) H[f ] = − f (v) log d3 v M (v) R3 Z
(21)
which is often used in linear kinetic theory. The form (20) makes viscous stress and heat flux vanish, so that higher order moments in streaming terms of equations (18) are simply
13
replaced by Pij = nT δij ,
qi = 0 .
(22)
For the elastic scattering contributions, the reader is referred again to [2], where, for a similar problem, the explicit step by step calculation of the relevant integrals is given. The final result, reported later in this paper, is obtained following the same procedure. As concerns the inelastic terms, we have to evaluate the integrals Z Z g y f¯(v) f¯(w) d3 v d3 w R3
R3
corresponding to y = δ + 3 and y = 2 δ + 3. To this aim, it is convenient to use as integration variables the relative velocity g = v − w and the center of mass velocity G = 12 (v + w). Then, taking into account that for each k, η ≥ 0 it holds Z k
− η x2
x e R3
µ ¶ µ ¶k+3 k+3 1 2 Γ , d3 x = 2 π η 2
(23)
where Γ is the Euler gamma function [1], we have Z ³ m ´3 Z m 2 m 2 g y − 4T d3 g g e e− T G d3 G = 2πT R3 R3 y µ ¶ µ ¶ 2y+1 y+3 T 2 = √ Γ n2 . 2 m π
Z Z
g f¯(v) f¯(w) d3 v d3 w = n2 y
R3 R3
(24)
Now, after easy manipulations, and recalling the recurrence formula Γ(α + 1) = α Γ(α), the inelastic contribution appearing in the temperature balance equation in (18) becomes ·
¸ 1 β δ+3 2δ+3 ¯ g − g f (v)f¯(w) d3 v d3 w = 4 + δ 4 + 2 δ 3 3 R R " # ¶ µ ¶ δ+3 µ µ ¶ 2δ+3 2 2 T 16 δ T +2 = √ m n2 β 2δ Γ − β 22δ Γ(δ + 2) , 2 m m 3 π 2 mβ 3
Z Z
(25)
and exhibits two different power dependences on temperature T , one with exponent 2δ + 32 and the other with exponent δ + 32 , which are distinct for any δ 6= 0. By summing it to 14
the elastic terms evaluated in [2] the closed set of evolution equations for the five fields n, u, T reads as ∂n ∂ + (nuk ) = 0 , ∂t ∂xk r r ∂ui 1 ∂ α4 2 αm ∂ui B n + nuk + (nT ) = − nn ∂t ∂xk m ∂xi ε 3 π (1 − α)T B + αT ¸ · (1 − α)T B + αT 1 B 2 i = 1, 2, 3 (ui − uB · (u − u ) + 2 i ), 5 αm r r ∂T ∂T 2 ∂uk α2 2 αm n + n uk + nT = m n nB ∂t ∂xk 3 ∂xk ε 3 π (1 − α)T B + αT ( · ¸ 1 1 αT · α+ (u − uB )4 B 5 3 (1 − α)T + αT · ¸ (1 − α)T B + αT 1 αT +4 α− (u − uB )2 αm 3 (1 − α)T B + αT · ¸2 · ¸) (1 − α)T B + αT αT +8 α− αm (1 − α)T B + αT " # µ ¶ µ ¶ δ+3 µ ¶ 2δ+3 2 2 δ T T 16 +2 − β 22δ Γ(δ + 2) . − √ m n2 β 2δ Γ 2 m m 3 π
(26)
These equations correspond to a simplified description of the granular gas in the medium, in which the distribution function f is replaced by the five dimensional vector (n, u, T ). Its evolution is determined by collision contributions that are now simply of algebraic nature, and that include the small parameter ε. It is worth searching for “collision equilibria” for these macroscopic equations. The first one is already a conservation equation. The second yields at once the equilibrium value u] = uB , which, substituted into the third, leads to the algebraic equation £
(1 − α)T B + α T
¤ 21
(T B − T ) = " µ ¶ µ ¶ 2δ µ ¶δ # 3 1 δ T T n δ 2δ =ε√ β 2 Γ +2 − β 2 Γ(δ + 2) T 2. B 2 m m 2α(1 − α) n
15
(27)
This equation does not allow an explicit solution for a general δ. Therefore, since the parameter ε is small, we look for an asymptotic power series solution of the form T ] = T0 + ε T1 + ε2 T2 + . . . . Ordering in powers of ε, to the leading order we get T0 = T B , hence equilibrium temperature coincides with the background temperature apart from O(ε) corrections. Taking into account this result, next order (O(ε)) terms provide " µ B ¶δ # µ ¶ µ B ¶ 2δ n T1 1 T δ T 2δ δ − β 2 Γ(δ + 2) . =− √ β 2 Γ +2 B B T 2 m m 2α(1 − α) n
(28)
Under the same consistency conditions mentioned about equations (18), it is not difficult to check that for all physically meaningful values of δ (say, δ ≤ 1) the correction T1 is actually negative, as expected because of inelasticity. The evaluation of the second order correction T2 is a bit more complicated: O(ε2 ) terms in (27) yield the equality 1 (T1 )2 B 12 − α 1 − (T ) T2 = B 2 (T ) 2 " µ ¶ µ B¶2δ µ B¶δ # 1 n δ δ + 3 T 2 δ + 3 T 1 − β 22δ Γ(δ + 2) (T B ) 2 T1 =√ β 2δ Γ +2 B 2 2 m 2 m 2α(1 − α) n from which, taking into account the expression (28), we get T2 1 n = −√ β T1 2α(1 − α) nB " µ ¶ µ B ¶ 2δ µ B ¶δ # δ T 2 δ + 3 − α T δ + 3 − α − β 22δ . · 2δ Γ +2 Γ(δ + 2) 2 2 m 2 m (29) Since ε is small, these results provide a good estimate of the local equilibrium states, which become actual global equilibria in the space homogeneous case with a steady background. It is also interesting to investigate the hydrodynamic limit of equations (26), namely the structure they assume in the asymptotic limit when ε → 0. To this end, in order to achieve an hydrodynamic equation at the Navier–Stokes level for the unique hydrodynamic
16
variable n, we shall perform an asymptotic analysis of Chapman–Enskog type, following the main steps outlined in [11, 10] and already applied in [3, 2] in different (dissipative) physical contexts. Keeping n unexpanded, we consider first order expansions for u and T u = u0 + ε u1 ,
T = T0 + εT1
(30)
and we insert them into (26). We have to close the continuity equation ∂n ∂ ∂ + (n u0k ) + ε (n u1k ) = 0 , ∂t ∂xk ∂xk
(31)
hence we have to determine both u0 and u1 . Leading order of momentum equation immediately yields u0 = uB .
(32)
Then, taking into account this result, by equating the coefficients of power ε0 we get 1 3 u =− α 8 n nB 1
r r · ¸ π αm ∂uB 1 B B 0 n + n u · ∇x u + ∇x (nT ) ; 2 (1 − α)T B + αT 0 ∂t m (33)
therefore we need also the leading order temperature T 0 , that, however, from the third equation in (26), is easily given by T 0 = T B.
(34)
Finally, bearing in mind that the host medium fulfils the Euler equations (14), the velocity of granular test particles turns out to be, up to O(ε) accuracy, 3 1 u=u −ε√ α 8 nB B
r
mπ 2T B
·
¸ 1 1 B B B ∇x (nT ) − B B ∇x (n T ) . nm n m
(35)
By inserting this expression into the continuity equation (31), we find the hydrodynamic “Navier–Stokes” equation for the number density of inelastic hard spheres diffusing in a
17
background medium 3 ∂n + ∇x · (nuB ) = ε ∂t 8
r
½ · ¸¾ 1 1 n mπ B B B √ ∇x · ∇x (nT ) − B B ∇x (n T ) . (36) 2α n m nB T B m
This is a drift–diffusion equation, with a convective term determined by the background velocity uB , and a diffusive part depending on the background temperature T B (not constant, in general) and involving second order space derivatives of n itself. Drift– diffusion like equations were obtained also in [8] for a fluid–particle interaction model based on a linear Fokker–Planck elastic operator. To complete our knowledge of the macroscopic fields in the frame of this hydrodynamic limit, we may proceed one step further in the temperature equation and get from the O(ε0 ) terms r B B ∂T B ∂T ∂u 2αTB 1 2 16 k B B n + nuB + nT = − n n (1 − α) T k ∂t ∂xk 3 ∂xk 3 mπ " # µ ¶ µ B ¶ δ+3 µ B ¶ 2δ+3 2 2 δ T T 16 +2 − β 22δ Γ(δ + 2) . − √ m n2 β 2δ Γ 2 m m 3 π
(37)
Since the left hand side vanishes thanks to the Euler equations (14) for the background, the coefficient T 1 turns out to coincide with the first order correction T1 of the equilibrium temperature given in (28). We may conclude that the granular medium drifts with respect to the background with a O(ε) diffusion velocity depending on ∇x n, whereas granular temperature evolves, to O(ε) accuracy, in local collision equilibrium. We may collect the main results following from the present local equilibrium ansatz for a collision dominated regime (ε = KnB
