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Macroscopic limit for an evaporation–condensation problem
European Journal of Mechanics B/Fluids 63 (2017) 106–112
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Macroscopic limit for an evaporation–condensation problem Hans Babovsky Technische Universität Ilmenau, Institute of Mathematics, Weimarer Str. 25, D-98693 Ilmenau, Germany
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Article history: Received 16 March 2016 Received in revised form 4 August 2016 Accepted 23 January 2017 Available online 7 February 2017 Keywords: Evaporation and condensation Discrete kinetic model Macroscopic limit
abstract We investigate the hydrodynamic limit of a vapor–noncondensable gas mixture in a pressure gradient between two walls. Earlier papers based on conventional asymptotic analysis techniques predict the emergence of a boundary layer of noncondensables which completely blocks the vapor flow (Takata and Aoki, 2001; Aoki et al., 2003). This ‘‘ghost effect’’ (Sone, 2007) contradicts physical intuition. In the present paper we reveal the bifurcation structure of the underlying transport operator and combine it with an appropriate macroscopic scaling. As a result, the hydrodynamic limit describes the coexistence of a streaming mode of vapor with the other component at rest thus avoiding the ghost effect. For sake of clarity, the paper restricts to a simplified setting (discrete velocity model, mechanically identical particles). However, the results also apply in more general situations. © 2017 Elsevier Masson SAS. All rights reserved.
1. Introduction The paper deals with the kinetic modeling of gas mixtures in the fluid dynamic limit. Both problems, the kinetics of mixtures (and in particular: vapor–noncondensable mixtures) and questions concerning the fluid dynamic limit of rarefied flows, have been recently investigated in a couple of papers (see, e.g. [1–3] for the first, and [2,4] with the literature cited there for the latter aspects). One particular problem is that of boundary layers for mixtures. Due to the scaling used at small Mach numbers, it seems sufficient to study planar problems like half space or slab geometries. Once this is completely understood (see [1] for the evaporation–condensation problem discussed below, and [3] for the discrete velocity case), such boundary layers can be used to couple boundaries to hydrodynamic flow fields. For the fluid dynamic limit there exist well-established and widely accepted techniques like the Chapman–Enskog or the Hilbert expansion. Matching these flow fields to boundary layers designed e.g. to adapt the flow to diffusive reflection laws, seems to create sufficiently good solutions in a number of problems. The situation changes if a local matching of the boundary or interface conditions is not sufficient, since they serve as global control mechanisms. Such a situation occurs in the case of a binary gas mixture consisting of vapor and of a noncondensable between parallel walls. In the case of a pressure difference between the walls, vapor starts to move from one wall (evaporation) to the other (condensation) thus introducing a flux. The noncondensable
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.euromechflu.2017.01.012 0997-7546/© 2017 Elsevier Masson SAS. All rights reserved.
follows this flux and creates a barrier at one of the walls slowing down the vapor flux. Following the classical tools from asymptotic analysis, this barrier has to be modeled and matched as a boundary layer. This leads to a result in the hydrodynamic limit which is in literature known as a ‘‘ghost effect’’ and which obviously does not yield an appropriate description of the hydrodynamic limit. The aim of the paper is to provide an alternative description to this situation. Based on a detailed view on the collision operator and its algebraic structure, and combined with an appropriate scaling it yields a solution to the evaporation–condensation problem which is much more intuitive from a physical point of view. Due to the lack of data possibly used for comparisons and benchmarks (experimental data like those in [5,6] are not applicable; reliable numerical results are very hard to obtain due to stiffness problems) it is a question of plausibility whether to accept the presented ansatz or to look for an alternative. In our opinion, it provides a reliable perspective, since the bifurcation property (which is the keystone of the investigations) is based on a wellknown property of the algebraic structure of the involved operator (this property is not worked out in investigations like those in [3], which are thus restricted to small perturbations), and the choice of diffusive scaling has proven useful in a variety of problems for the derivation of diffusion phenomena [7–9]. This scaling gets rid of the ghost effect; furthermore it also seems to be the appropriate way to solve a couple of other problems related to the connections of the Navier–Stokes and the Boltzmann equations (like the problem discussed in [10]). Consider a gas mixture composed of two species confined between two parallel walls (Fig. 1). Species A (‘‘vapor’’) is emitted and adsorbed at the boundaries according to a prescribed pressure
H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
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Fig. 1. Schematic view on density profiles for mixture (upper line) and noncondensable (lower line) without (left) and with vapor flux (right).
difference. Species B (‘‘noncondensable’’) is totally reflected when hitting the walls. If there is a pressure decay from the wall at x = a to the wall at x = b, then a flow of vapor is induced from a to b. At the same time one expects the noncondensable to follow the flow and form a boundary layer at b which slows down the vapor flow. This problem has been studied in a couple of papers in recent years and in particular the fluid dynamic limit was of interest (see, e.g. [11,12]). It turned out that the application of standard asymptotic analysis methods for the fluid dynamic limit leads to a curious situation. In the limit both A and B exhibit the same Maxwellian profile with an infinitesimally small bulk velocity v (in fact, v = 0), and a thin boundary layer of noncondensable is formed at b completely suppressing the vapor flow. This phenomenon contradicts physical intuition and is known in the literature as ghost effect [13]. In the present work we propose a different macroscopic limit, based on a scaling (‘‘diffusive scaling’’) which in the past has been applied in a variety of problems for the derivation of diffusion phenomena (see, e.g. [7–9]). It turns out that this kind of scaling leads in the limit to a boundary layer of well-defined thickness for the noncondensable which slows down but does not stop the vapor flow. The results are based on a careful investigation of a bifurcation phenomenon of the governing transport operator in the presence of a small drift. In case of zero drift, its nullspace has geometric dimension one (related to mass flow conservation) but algebraic multiplicity two. At the emergence of a drift, the twodimensional nullspace splits up into two simple eigenspaces giving rise to a new nonzero eigenvalue. In this paper we are not interested at all in any existence theorem for some boundary value problems. The only intention is to provide a scenario which allows in the hydrodynamic limit a gas mixture with one component at rest while the second one is streaming. This is the main result, and it reveals the major difference to the classical asymptotic expansions for which such a result is not possible (see [11,12]). In order not to hide this behind a bunch of technical details, we restrict to the following simplified setting. However, due to the similarities of the structure of the underlying transport operators (see, e.g. [14,15] in the case of the continuous and the discrete Boltzmann collision operator), we are convinced, that the present results can be generalized. We investigate the problem in the framework of Discrete Velocity Models (DVM) on the basis of two-particle collisions (see [16]). We consider the steady spatially one-dimensional problem in the slab [0, 1] in the simplest possible case of mechanically identical species A and B. This means that both are driven by the same Boltzmann collision operator. The only difference is the wall interaction. Denote by g the distribution of A and by h that of B. Then the sum f = g + h is governed by a nonlinear one-species Boltzmann collision operator J. If f is known, then g and h evolve according to a linear transport operator. We restrict to the case of f being a fixed global Maxwellian. In the case of zero flow between 0 and 1, f is a centered Maxwellian with zero bulk velocity. The corresponding transport operator L0 exhibits a typical structure concerning the algebraic nullspace which in a
similar situation has been observed in a couple of papers (e.g. [14] for the continuum case, [17,15] for DVM). For our investigation we require the DVM to satisfy four assumptions (see (2.3), (2.4), (3.2), (3.8) below), two of them being crucial. The first one is a symmetry condition and requires the velocity grid and the collision model to be invariant under a change of sign of the velocity components perpendicular to the walls. This leads to a linear ODE system with a matrix having a special antisymmetric block structure which is essential. (In the paper we exclude the case of zero normal velocities which would lead to a DAE rather than an ODE system. However, numerical experiments indicate that this condition can be weakened.) The second assumption concerns the existence of a maximal number of pairwise different nonzero eigenvectors. This in particular prohibits the existence of artificial invariants of the transport operator. (A discussion of this point may be found in [17,15].) 2. The evaporation–condensation problem 2.1. The model Consider a gas mixture confined in the slab [0, 1]. The two components of the gas are species A (‘‘vapor’’) with density function g(t , x, v) and species B (‘‘noncondensable’’) with density function h(t , x, v). The are represented in the form v = (vx , v⊥ ), with vx the component pointing in x-direction, and v⊥ the orthogonal complement. Concerning the gas particle interaction, both types are mechanically identical in the sense that both are governed by the same Boltzmann collision operator. The only difference lies in the gas–wall interaction. While species A may pass through the walls in both directions (condensation, evaporation), species B is totally reflected. As a consequence, there may be a total nonzero mass flux of A through the walls while the mass flux of B is zero. We write f = g + h and let the governing equations for g and h be the nonlinear two-species Boltzmann equation
(∂t + vx ∂x )g = J [f, g] (∂t + vx ∂x )h = J [f, h]
(2.1) (2.2)
with the collision operator J [., .] to be specified below. Since J [., .] is bilinear, a consequence of (2.1), (2.2) is that f solves the nonlinear Boltzmann equation
(∂t + vx ∂x )f = J [f, f].
(2.3)
In most of the paper we restrict to the steady variant of the system,
vx ∂x g = J [f, g] vx ∂x h = J [f, h].
(2.4) (2.5)
In order to extend the equations to a well-posed boundary value problem, they have to be supplemented with boundary conditions either in the form of reflection laws or by prescribing the flows into the domain [0, 1]. For our purposes such a detailed description is not necessary.
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Definition 2.1. We call a pair (g, h) a solution to the evaporation–condensation problem, if Eqs. (2.4), (2.5) are satisfied with f = g + h, and if h has zero mass flux, i.e.
φ[h](x) = ⟨vx h(x)⟩ = 0.
(2.6)
In the rest of the paper we simplify the problem by considering only solutions of the evaporation–condensation problem for which f is a known global equilibrium function of the Boltzmann collision operator. In this case, Eqs. (2.4), (2.5) turn into a system of linear transport equations which can be solved by analyzing the corresponding transport operator. Furthermore, it is sufficient to construct h, since h is a solution of the transport equation if and only if g = f − h is.
Let V = {v1 , . . . , vN } ⊂ Rd (d ≥ 2) be a finite velocity set, (i) (i) vi = (vx , v⊥ ), and f = (fi )Ni=1 ∈ RN+ a distribution function on V . A single collision event describes an exchange between two velocity pairs,
rα [f, f] (Jα [f, f])m = −rα [f, f]
if if else.
0
m ∈ {i, l} m ∈ {j, k}
γα Jα [f, f].
(2.7)
α∈R
A linear version of J [f, f] is needed when considering the dynamics of a test particle (with distribution g) in a given scattering field with distribution f. The corresponding linear transport operator J [f]g is given by the matrix
γα Jα [f]
α∈R
with
0.5(fj gk + fk gj ) − fl gi 0.5(fi gl + fl gi ) − fk gj (Jα [f]g)m = 0.5(fi gl + fl gi ) − fj gk 0.5(fj gk + fk gj ) − fi gl 0
if if if if else.
m=i m=j m=k m=l
Its matrix representation is
−fl 0.5fl Jα [f] = Pα 0.5fl 0
0.5fk −fk 0 0.5fk
0.5fj 0 −fj 0.5fj
0 0.5fi T P 0.5fi α −fi
with the N × 4-Matrix Pα defined in column representation as Pα = (ei , ej , ek , el ) (em the mth canonical unit vector). It is well-known that Maxwellians, i.e. functions of the form f(v) = exp −|v − v| /2Θ
2
We end up with the compact formulation (2.9)
with F = diag(fi , i = 1 . . . N ), and the symmetric matrix
C =
α∈R
−1 0.5 πα Pα 0.5 0
0.5 −1 0 0.5
0.5 0 −1 0.5
=:Γ
0 0.5 T P 0.5 α −1
(2.10)
and πα = γα φα . Define
1 := (1 . . . 1)T ∈ RN and f−1 := (fi−1 , i = 1 . . . N )T = F −1 1.
From physical considerations (momentum and energy conservation) we consider only elementary collisions for which vi vl and vj vk are the diagonals of a rectangle in Rd . We denote by R ⊂ {1, . . . , N }4 all α = (i, j, k, l) representing a non-degenerate rectangle in the above sense. With this we can now choose collision frequencies γα ≥ 0 to define the Boltzmann collision operator on V as
J [f] =
0.5(fj gk + fk gj ) − fl gi = φα 0.5(fj−1 gj + fk−1 gk ) − fi−1 gi .
For short we write α = (i, j, k, l) and rα [f, f] = fj fk − fi fl . The above collision is described by the elementary collision operator
(2.8)
Thus we can rewrite the terms of the transport operator by extracting the factor φα , e.g.
(vj , vk ).
↔
J [f, f] =
fi fl = fj fk =: φα .
J [f] = CF −1
2.2. The discrete system
(vi , vl )
are equilibrium solutions of the nonlinear collision operator (and they are the only ones, if a sufficient number of rectangles α with γα > 0 appears in the sum (2.7), i.e. if the model is regular in the sense of [18]). In this case, a special situation arises. One easily checks that α = (i, j, k, l) describes a rectangle with vi vl and vj vk as diagonals if and only if
The following result follows immediately from inspection of the matrix Γ . Lemma 2.2. (a) C conserves the total mass, i.e. 1T · C = 0. (b) 1 ∈ ker(C ). The first model assumption requires that the number of collisions is large enough to prohibit artificial invariants. Model assumption 2.3. C RN = 1⊥ . Equivalent to this assumption is that the restriction C : 1⊥ → ⊥ 1 is bijective. (When writing about the inverse C −1 of C we mean in the following the restriction C −1 : 1⊥ → 1⊥ .) An essential part of the following considerations are symmetry arguments. Therefore we have to ensure that the velocity space and the collision model are symmetric with respect to reflections about the x-axes in the following sense. Model assumption 2.4. (i) If v = (vx , v⊥ ) ∈ V , then vx ̸= 0, and the reflected velocity Tx v := (−vx , v⊥ ) ∈ V . (ii) The collision frequencies γα are Tx -invariant. This means: If α = (i, j, k, l) and α ′ = (i′ , j′ , k′ , l′ ) are such that the corresponding velocities vm and vm′ satisfy vm′ = Tx vm for m ∈ {i, j, k, l}, then γα ′ = γα . From this follows that N is even, N = 2n. We choose a (i) numbering of V such that vx > 0 for i = 1 . . . n, and vi+n = Tx vi . Notice that due to the Assumption 2.4 C is symmetric and has the block matrix structure A∗ B∗
C =
B∗ A∗
(see the discussion in [17]). Since V contains no velocities v with vx = 0, the matrix Vx = diag(vx(i) , i = 1 . . . N ) is regular, and the system (2.5) can be rewritten as the ODE system
∂x h = Lh
(2.11)
H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
where due to (2.9) L takes the form L = Vx−1 CF −1 .
(2.12) ⊥
We easily find the following properties, denoting by vx the (i)
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Model assumption 3.2. L0 has n − 1 pairwise different strictly positive eigenvalues λi , i = 1 . . . n − 1. We collect the results concerning the spectrum of L0 . A central property is the solvability of the equation
hyperplane perpendicular to vx = (vx , i = 1 . . . N )T .
L0 r0 = f0
Lemma 2.5. (a) ker(L) = span(f) and L(RN ) = v⊥ x . (b) The equation Lg = h is solvable if and only if h ∈ v⊥ x . (c) Any eigenvector t to an eigenvalue λ ̸= 0 is orthogonal to vx .
which follows from the lemma below. We call a vector even if it is of the form (p, p)T , and odd if it is (p, −p)T . The subset of even resp. odd vectors is denoted by RNeven resp. RNodd . Furthermore we call an operator M even if it maps even into even and odd into odd; we call M odd if it maps even into odd and odd into even.
Proof. (a) follows from the corresponding properties of C , see Lemma 2.2(b) and Assumption 2.3. (b) is an application of Fredholm’s alternative. (c) follows from (b) and Lt = λt ⊥ vx . ⃝ As a consequence, the restriction L : (f Its inverse
) → vx is bijective.
−1 ⊥
−1 ⊥ L−1 : v⊥ ) , x → (f
⊥
(3.2)
Lemma 3.3. (a) L0 is odd. (b) L0 (RNodd ) = RNeven . 1 N (c) There exists a unique solution r0 = L− 0 f0 ∈ Rodd to Eq. (3.2). N N N (d) L0 (Reven ) ( L0 (Reven ) ⊕ span{r0 } = Rodd
Proof. (a) By Assumption 2.4 and the numbering of the velocity space V , C and F0 are even and Vx−1 is odd.
L−1 = FC −1 Vx
is shortly called the inverse L−1 of L. If h ⊥ vx , then g = L−1 h is the unique solution of Lg = h orthogonal to f−1 .
(b) follows from (a) and N ⊥ v⊥ = L0 (RNodd ) ∪ L0 (RNeven ) x = Reven ∪ Rodd ∩ vx
(c) follows from f0 ∈ Reven ⊥ vx and Lemma 2.5.
3. The steady transport operator
We consider the transport operator L (now denoted as L0 ) given by (2.12) with f taking the form of a centered Maxwellian,
T (d) Let be t+ i = (pi , qi ) be eigenvectors for the positive eigenvalues λi and denote t− = (qi , pi )T as the corresponding eigenveci − tors vor −λi , i = 1 . . . n − 1. Then s+ = t+ ∈ RNeven and i i + ti − + − N si = ti − ti ∈ Rodd span (n − 1)-dimensional subspaces of RNeven 2 − 2 resp. RNodd , and L20 s− i = λi si and L0 r0 = 0. Thus
f0 (v) = exp(−|v|2 /2Θ ).
N r0 ̸∈ span(s− i , i = 1 . . . n − 1) = L0 (Reven ).
3.1. Zero bulk velocity
Due to assumption 2.4 and the chosen numbering, the corresponding diagonal matrix F0 takes the block diagonal structure F0 = diag(f0 (vi ), i = 1 . . . N )
= =:
diag(f0 (vi ), i = 1 . . . n) 0 (1/2)
F0
0
0
(1/2)
F0
N = diag (Λ, −Λ, N0 ) with Λ = diag (λi , i = 1 . . . n − 1),
N0 =
.
This equips the operator L0 = Vx CF0 (1/2) −1
with the block structure (1/2)
(Vx(1/2) )−1 A∗ (F0 ) (1/2) −(Vx(1/2) )−1 B∗ (F0 )−1 A B =: . −B −A
(Vx(1/2) )−1 B∗ (F0 )−1 (1/2) −(Vx(1/2) )−1 A∗ (F0 )−1
L0 =
(3.1)
Lemma 3.1. λ > 0 is an eigenvalue of L0 if and only if −λ is eigenvalue. Proof. Define t+ := (p, q)T and t− := (q, p)T . Then we find easily L0 t
= λt
⇔
L0 t
−
= −λt . −
± −1 with t± i = ti (0) as in the proof of Lemma 3.3(d), and r0 = L0 f0 .
Corollary 3.5. The general solution h of ∂x h = L0 h in the slab [0, 1] takes the form
The spectrum of matrices of this form was studied in [17,15]. Simple to prove is the following result.
+
1 . 0
+ − − T (0) = {t+ 1 , . . . tn−1 , t1 , . . . tn−1 , f0 , r0 }
Vx = diag(Vx(1/2) , −Vx(1/2) ). −1
0 0
(b) A corresponding transformation matrix is
For the same reasons we can decompose Vx into
+
This result yields a complete description of the eigenspace structure of L0 and proves the following theorem. Theorem 3.4. (a) L0 is similar to the Jordan normal form
0 diag(f0 (vi ), i = 1 . . . n)
−1
⃝
⃝
The following model assumption has been motivated for a similar situation in [17,15] and results in a structure for L0 which is well-suited for our purpose.
h(x) =
n −1
γi− exp(−λi x)t− i +
n −1
i =1
γi+ exp(−λi (1 − x))t+ i
i =1
+ (γn + γr · x)f0 + γr r0 with free parameters γi± , γn and γr . The first two sums represent boundary layers at x = 0 and x = 1 which are used eventually to model prescribed inflow conditions at the boundaries. Remark 3.6. Assumption 2.4 is quite restrictive since it prohibits zero x-components of the velocities. It turns out that this can be weakened. In all numerical experiments we performed as far, the system (2.5) turned out to represent an index-1 differential algebraic system which could be transformed into an ODE system with a matrix which has precisely the same Jordan structure as that given in the theorem.
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H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
3.2. Shifted Maxwellian We replace the centered Maxwellian f0 with a new one shifted in x-direction,
1 2 fv (v) = exp − |v − v| = f0 (v) + v · 1f(v) 2Θ v = (v, 0)T . 1f can be chosen continuous in v = 0, with f′ := 1f(0) = lim (fv (v) − f0 (v))/v = v→0
1
Θ
· Vx f0 .
Prˆ g = g − vTx g ·
This introduces an analytic change of all operators introduced above. These are in particular continuously differentiable with respect to v . The prime indicates in the following the derivative at v = 0. The transport operator is affected by the change in two ways. The operator C0 changes into Cv = C0 + v · 1C (v) with 1C continuous,
1C (0) = C ′ =
φα′ =
α∈R (i)
γα φα′ Pα Γ PαT ,
vx + vx(l) · f0 (vi )f0 (vl ), Θ
α = (i, j, k, l)
(F
) := 1F
−1
(0) = −
1
Θ
· F0 Vx .
1
Θ
vTx rˆ
.
The solution of (3.4) is equivalent to finding a fixed point of the mapping 1 rˆ → c · L− v Prˆ fv
(3.7)
(with c normalizing constant). Since = O (v) and since r0 is fixed point for v = 0, it follows that for v small the mapping (3.7) is a contraction with a unique fixed point. The continuity of the mapping follows from the continuity of simple eigenvectors of analytically perturbed operators (see [19]). ⃝ vTx fv
Main theorem 3.10. (a) For |v| ̸= 0 sufficiently small there exists a new eigenvalue λ(v) = v · 1λ(v) depending analytically on v with 1λ(0) = λ′ ̸= 0 and a corresponding eigenvector of the form t = rv + λ(v)−1 · fv . rv ⊥ vx is the unique solution of Lv rv = fv + λ(v)rv . (b) Lv is similar to the diagonal matrix
−2
Finally, L0 = Vx−1 CF0−1 changes into Lv = L0 + v 1L(v) with
1L(0) = L′ = Vx−1 C ′ F0−1 −
rˆ
This leads to the proof of the main result of this section.
and F0−1 is to be replaced with Fv−1 = F0−1 + v ∆F −1 (v) with −1 ′
Proof. A necessary condition for the continuity of 1λ and 1r at v = 0 is obtained inserting the ansatz (3.3) into Eq. (3.4) and taking the limit v → 0. This leads to Eq. (3.6). From Fredholm’s alternative, this equation is solvable if and only if the right hand side is orthogonal to vx . Thus f′ − L′ r0 +λ′ · r0 has to be the projection of f′ − L′ r0 along r0 onto vTx . From this follows (3.5). Given r ̸= 0, define its corresponding normalized vector rˆ = ∥r∥−1 r and the projection Prˆ along r onto vTx ,
· L0 · F0−1 Vx .
+ − − N = diag (λ+ 1 . . . λn−1 , −λ1 · · · − λn−1 , 0, λ(v)).
′
It is important to mention that L is an even operator. We list some of the main properties. Lemma 3.7. For small v there exist positive eigenvalues λ+ i (v) and (v) depending analytically on v with λ± negative eigenvalues −λ− i → i λi for v → 0. Proof. This is a standard result from perturbation theory since all ±λi are simple eigenvalues of L0 (see Kato). ⃝ Crucial for the following is how the algebraic nullspace is affected by the change. If there were a solution of Lv rv = fv then its Jordan normal form remained the same as before. The following (generic) model assumption prohibits this. Recall that RNodd is + − spanned by s− i = ti − ti , i = 1 . . . n − 1, and r0 (Lemma 3.3(d)).
Proof. In order to prove (a) we remark that the Jordan block J0 of the nullspace of L0 changes into
Jv =
0 0
1
λ(v)
which is similar to diag(0, λ(v)). (b) follows then immediately.
⃝
4. A macroscopic limit In order to derive a meaningful macroscopic limit we introduce the diffusive scaling (see, e.g. [7–9]) for the equation
(∂t + vx ∂x )g = J [f]g.
Model assumption 3.8. f′ + L′ r0 ̸∈ {s− i , i = 1 . . . n − 1}.
It consists in replacing the macroscopic variables t and x with ϵ −2 t and ϵ −1 x and leads to the rescaled equation
Under this assumption the two-dimensional nullspace splits up into two simple eigenspaces as is shown now.
(∂t + ϵ −1 vx ∂x )g = ϵ −2 J [f]g.
Lemma 3.9. In a neighborhood U0 = (−v 0 , v 0 ) of v = 0 there exists a continuous mapping v → (1λ(v), 1r(v)) such that the pair
(λ(v), rv ) = (v · 1λ(v), r0 + v · 1r(v))
(3.3)
Formally this is equivalent to replacing the space V of microscopic velocities with ϵ −1 V and scaling up the collision frequency by a factor ϵ −2 . This is the approach which we take here. Replacing vi with wi = ϵ −1 vi requires to change the Maxwellians
solves Lv rv = fv + λv · rv 1 λ− v fv
λ := 1λ(0) = ′
fv = (exp(−|vi − v|2 /2Θ ), i = 1 . . . N )T to (ϵ) fv = (exp(−|wi − v|2 /2Θ ), i = 1 . . . N )T = (exp(−|v − ϵ v|2 /2ϵ 2 Θ ), i = 1 . . . N )T
(3.4)
+ rv is eigenvector with eigenvalue λv . Furthermore, vTx (L′ r0 − f′ ) vTx r0
(3.5)
L0 r′ = f′ − L′ r0 + λ′ · r0 .
(leaving the macroscopic bulk velocity v unchanged) which itself makes only sense if we rescale the temperature as T = ϵ 2 Θ . From now on we define (ϵ)
(1)
fv = fϵv = (exp(−|v − ϵ v|2 /2T ), i = 1 . . . N )T
and r′ := 1r(0) is solution of (3.6)
(4.1)
(4.2)
with T > 0 constant. For convenience we assume in the following λ′ v > 0.
H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
111
(ϵ)
Remark 4.1. Associated to fv are the moments
= ⟨1f(01) ⟩ + O (ϵ 2 )
(ϵ)
density ρv (ϵ)
flux φv
= ⟨wx fv(ϵ) ⟩ = (v/T ) · ⟨vx2 f(01) ⟩ + O (ϵ 2 ).
(ϵ)
(ϵ)
Fv is the diagonal matrix with the coefficients of fv as entries, (ϵ)
Fv
(1) = Fϵv = diag(fv(ϵ) ) = F0 I +
ϵv T
Vx
+ O (ϵ 2 ).
(4.3)
Fig. 2. A 12- and a 9-velocity model.
evaporation–condensation problem (in the sense of Definition 2.1). (ϵ) Then hv is of the form
The steady version of (4.1) is
vx ∂x g = ϵ
−1
(ϵ)
J [f]g.
(4.4)
hv
+
(ϵ)
(1) = Cϵv =
α∈R
πα(ϵv) Pα Γ PαT
T
(ϵ) −1 (ϵ) + γr exp −λ(ϵ) ( 1 − x ) · (r(ϵ) v v + (λv ) fv )
(4.7)
± (fv(ϵ) − hv(ϵ) , h(ϵ) v )ϵ>0 of solutions asymptotically bounded, if γi , γr are bounded for ϵ ↘ 0. In this case the moments of the solutions are well-defined in the limit ϵ ↘ 0 and given as follows. (ϵ)
1
′ λ(ϵ) v → λ v > 0 for ϵ → 0
with λ given by (3.5). The following results are easy to prove from the Main theorem 3.10. (ϵ)
Corollary 4.2. (a) Lv is similar to the diagonal matrix −1 + diag (ϵ −1 λ+ λn−1 (ϵv), −ϵ −1 λ− 1 (ϵv), . . . , ϵ 1 (ϵv), . . . , (ϵ) − ϵ −1 λ− n−1 (ϵv), 0, λv ).
(ϵ)
(ϵ)
The limits of the associated moments are
t− i (ϵv)
(1)
density ⟨1 fv ⟩ → ⟨1 f0 ⟩ (ϵ)
(1)
flux ⟨wx fv ⟩ → v · ⟨vx2 f0 ⟩/T . The concentration profile of the noncondensable, i.e. the layer at the wall point x = 1 is given as noncondensable concentration ′ ⟨1 hv(ϵ) ⟩/⟨1 f(ϵ) v ⟩ → γH exp(−λ v(1 − x)),
i.e. is given as a solution of the differential equation
The corresponding eigenvectors are
(i = 1 . . . n − 1),
γH = H (λ′ v)2 (exp(λ′ v) − 1)−1 = H · λ′ v + O (v 2 ). (ϵ)
′
(i = 1 . . . n − 1),
(ϵ)
rv + (λv )−1 fv .
(b) The general solution of the rescaled system (4.4) is
+ γi+ exp −ϵ −1 λ+ i (ϵv)(1 − x) · ti (ϵv)
i=1 n−1
⟨1 h(ϵ) v (x)⟩dx = H = const.
(1)
where rϵv = r0 + ϵv 1r(ϵv) is given as in Lemma 3.9, and
+
(ϵ)
h0 = γH exp(−λ′ v(1 − x)) · f0 ,
−1 λ(ϵ) v = ϵ λ(ϵv)
n −1
0
(ϵ)
Then it converges for ϵ ↘ 0 pointwise in x to (f0 − h0 , h0 ) given by
(ϵ)
rv = ϵ rϵv
(ϵ)
(4.9)
Corollary 4.3. Suppose (fv − hv , hv )ϵ>0 is an asymptotically bounded family of solutions to the evaporation–condensation problem with a prescribed amount of noncondensable,
(1)
fv ,
− γi− exp −ϵ −1 λ− i (ϵv)(1 − x) · ti (ϵv)
with coefficients γi± and γr depending on ϵ . We call a family of pairs
Lv is identical to the operator Lv investigated in the previous section. Finally define
t+ i (ϵv)
n −1
(4.6)
and
πα(ϵv) = γα f(ϵv1) (vi )f(ϵv1) (vl ) ϵv = πα(0) · 1 + · (vx(i) + vx(l) ) + O (ϵ 2 )
+ γi+ exp ϵ −1 λ+ i (ϵv)x · ti (ϵv)
i =1
(4.5)
with Cv
n −1 i =1
Thus we have to study the rescaled transport operator (ϵ) (ϵ) (ϵ) (1) Lv = ϵ −1 Vx−1 Cv (Fv )−1 = ϵ −1 Lϵv
=
− γi− exp −ϵ −1 λ− i (ϵv)x · ti (ϵv)
i=1
(ϵ) (ϵ) (ϵ) −1 (ϵ) + γn f(ϵ) v + γr exp −λv (1 − x) · (rv + (λv ) fv ). (4.8) Recall that the only term in (4.8) with nonzero flux is that related (ϵ) (ϵ) to the zero eigenvector fv (see Lemma 2.5(c)), and that ⟨wx fv ⟩ converges to a nonzero value for ϵ ↘ 0 (Remark 4.1). Suppose the (ϵ) (ϵ) (ϵ) (ϵ) (ϵ) pair (gv , hv ) = (fv − hv , hv ) is a solution of the rescaled steady
χ ′ = λ′ v · χ .
(4.10)
It is Eq. (4.10) which marks the crucial difference to the theory leading to the ghost effect. This equation states that the emergence of a blocking wall of noncondensable is proportional to the vapor flux. Thus in the case of v = 0, there is no blocking wall at all, and this also holds in the macroscopic limit. The size of the barrier depends besides the velocity v (which is implicitly given by the boundary conditions) also on λ′ , which is a constant depending on the collision model, and on the total amount of noncondensable. Thus the situation of an infinitesimal amount in the order of the Knudsen number, which was discussed in [12], leads again to a barrier which vanishes in the limit. We finish these investigations with two examples. Example 4.4. Consider the 12-velocity model as indicated in Fig. 2. (1) (1) Its vx -components are given by the vector vx = (vx , vx , (1) (1) T (1) T −vx , −vx ) with vx = (1, 1, 3) , and centered Maxwellians (1) (1) (1) (1) (1) are of the form f0 = (f0 , f0 , f0 , f0 )T with f0 = (τ 10 , τ 2 , τ 10 ),
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H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
τ = exp(−1/2T ). We include the collisions (2, 3, 5, 6), (5, 4, 11, 10), (8, 9, 11, 12) and (2, 1, 8, 7) with collision frequency γ1 and π1 = fi fl γ1 = τ 12 γ1 , and the collisions (1, 3, 8, 5), (2, 6, 11, 4), (8, 5, 12, 10), and (2, 7, 11, 9) with frequency γ2 and π2 = τ 12 γ2 . Assuming a density which is symmetric with respect to the yaxis of the velocity space, then we can identify the velocities and densities with numbers n = 4, 5, 6, 10, 11, 12 with those with numbers n − 3. Thus the collision model can be compressed to a 6 × 6-collision matrix Cˆ =
C11 C12
C12 C11
with
−2(π1 + π2 ) π1 π1 −3π1 − 4π2 = π2 π1 + π2 π1 π2 0 = π2 π1 + 2π2 0 .
π2 π1 + π2 , −π1 − 2π2
C11
C12
0
0
0 (1)
(1)
(1)
(1)
We have n0 = f0 and n′ = Vx f0 = (n1 , n1 , −n1 , −n1 )T with (1)
(1)
(1)
(1)
(1)
n1 = (τ 10 , τ 2 , 3τ 10 )T . The vector u0 = (u0 , u0 , −u0 , −u0 )T is determined from the linear system
(C11 − C12 )u(01) = f(01) and finally λ′ from the orthogonality relation
References
n′ + λ′ u0 ⊥ vx . Example 4.5. We finish with the 9-velocity model indicated by Fig. 2. Since vx = (1, 1, 1 − 1, −1, −1, 0, 0, 0)T , the describing equations form a DAE system rather than an ODE system. In its simplest form the collision model contains the collisions (1, 3, 7, 9), (2, 3, 8, 9), (5, 6, 8, 9) and (4, 6, 7, 9) with given π1 > 0, and (3, 7, 8, 6) with π2 > 0. Assuming again y-symmetry of the density functions and thus identifying the velocities 2, 5 and 8 with 1, 4 and 7, the corresponding DAE system is of the form
diag(1, 1, −1, −1, 0, 0)∂x φ =
C11 0 C31
0 C11 C31
C13 C13 C33
with
π1 , −4π1 − 2π2 −4π1 − 2π2 2π1 = , 4π1 −8π1
C11 = C33
−2π1 2π1
π1 2π2
0 2π1
π1
π2 . 2π1
C13 = C31 =
0
,
The algebraic part reads C31 (φ (1) + φ (2) ) + C33 φ (3) = 0. Inserting into the differential part yields the ODE system
∂x (φ (1) , φ (2) )T =
A −B
B −A
component at rest. Crucial for this result is the diffusive scaling which in an appropriate manner takes into account the bifurcation structure at hand. By way of contrast, the classical Chapman–Enskog expansion leads in lowest order to a Maxwellian at rest for both components and is not capable of covering such a scenario. Let me finally comment shortly on the main model restrictions of the paper. Some of them are crucial, others have been introduced for simplicity and transparency of the arguments. First of all, symmetry of the model (grid and collisions) is essential, since the main proof concerning the algebraic properties of the problem are built on symmetry arguments. On the other hand, the restriction to 2D models is not necessary, the theory applies also in the 3D case. The assumption of mechanically identical species has been crucial here since it simplifies theory considerably. Though we are convinced that a generalization is possible, the proofs for such an extension are not completely evident. The assumption of nonvanishing x-velocity components avoids the necessity of handling a DAE system. However, the last example shows a line along which the theory can be developed. Finally, the assumption of a fixed global equilibrium (which is certainly very strong) and the restriction to the steady case can be certainly weakened, at least in the framework of small perturbations. The aim of the paper was the deduction of an alternative theory; the investigation of its limitations is open for future.
−1 −1 with A = C11 − C13 C33 C31 and B = −C13 C33 C31 which has the same structure as the systems described in the paper.
5. Conclusions and final comments We have performed the hydrodynamic limit in a way which allows the coexistence of a streaming component with a
[1] S. Taguchi, K. Aoki, V. Latocha, Vapor flows along a plane condensed phase with weak condensation in the presence of a noncondensable gas, J. Stat. Phys. 124 (2006) 321–369. [2] S. Takata, F. Golse, Half-space problem of the nonlinear Boltzmann equation for weak evaporation and condensation of a binary mixture of vapors, Eur. J. Mech. B Fluids 26 (2007) 105–131. [3] N. Bernhoff, Half-space problem for the discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas, J. Stat. Phys. 147 (2012) 1156–1181. [4] I.V. Karlin, S.S. Chikatamarla, M. Kooshkbaghi, Non-perturbative hydrodynamic limits: A case study, Physica A 403 (2014) 189–194. [5] P.N. Shankar, M.D. Deshpande, Observations of temperature jumps in liquidvapour phase change, Pranama-J.- Phys. 31 (1988) L337–L341. [6] P.N. Shankar, M.D. Deshpande, On the temperature distribution in liquid-vapor phase change between liquid surfaces, Phys. Fluids A 2 (1990) 1030–1038. [7] H. Babovsky, On Knudsen flows within thin tubes, J. Stat. Phys. 44 (1986) 865–878. [8] H. Babovsky, C. Bardos, und T. Płatkowski, Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary, Asymptot. Anal. 3 (1991) 265–289. [9] H. Babovsky, P. Kowalczyk, Diffusion limits for discrete velocity models in a thin gap, Multiscale Model. Simul. 6 (2007) 631–655. [10] K. Aoki, F. Golse, S. Kosuge, The steady Boltzmann and Navier Stokes equations, Bull. Inst. Math. Acad. Sin. 10 (2015) 205–257. [11] S. Takata, K. Aoki, The ghost effect in the continuum limit for a vapor–gas mixture around condensed phases: Asymptotic analysis of the Boltzmann equation, Transport Theory Statist. Phys. 30 (2001) 205–237. [12] K. Aoki, S. Takata, S. Taguchi, Vapor flows with evaporation and condensation in the continuum limit: effect of a trace of noncondensable gas, Eur. J. Mech. B/Fluids 22 (2003) 51–71. [13] Y. Sone, Molecular Gas Dynamics, Birkhäuser, Boston, 2007. [14] C. Bardos, R. Caflish, B. Nicolaenko, The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas, Comm. Pure Appl. Math. 39 (1986) 323–352. [15] H. Babovsky, T. Płatkowski, Kinetic boundary layers for the Boltzmann equation on discrete velocity Lattices, Arch. Mech. 60 (2008) 87–116. [16] H. Babovsky, A numerical model for the Boltzmann equation with applications to micro flows, Comput. Math. Appl. 58 (2009) 791–804. [17] H. Babovsky, Kinetic boundary layers: on the adequate discretization of the Boltzmann collision operator, J. Comput. Appl. Math. 110 (1999) 225–239. [18] H. Babovsky, Discrete kinetic models in the fluid dynamic limit, Comput. Math. Appl. 67 (2014) 256–271. [19] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995.
Contents lists available at ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Macroscopic limit for an evaporation–condensation problem Hans Babovsky Technische Universität Ilmenau, Institute of Mathematics, Weimarer Str. 25, D-98693 Ilmenau, Germany
article
info
Article history: Received 16 March 2016 Received in revised form 4 August 2016 Accepted 23 January 2017 Available online 7 February 2017 Keywords: Evaporation and condensation Discrete kinetic model Macroscopic limit
abstract We investigate the hydrodynamic limit of a vapor–noncondensable gas mixture in a pressure gradient between two walls. Earlier papers based on conventional asymptotic analysis techniques predict the emergence of a boundary layer of noncondensables which completely blocks the vapor flow (Takata and Aoki, 2001; Aoki et al., 2003). This ‘‘ghost effect’’ (Sone, 2007) contradicts physical intuition. In the present paper we reveal the bifurcation structure of the underlying transport operator and combine it with an appropriate macroscopic scaling. As a result, the hydrodynamic limit describes the coexistence of a streaming mode of vapor with the other component at rest thus avoiding the ghost effect. For sake of clarity, the paper restricts to a simplified setting (discrete velocity model, mechanically identical particles). However, the results also apply in more general situations. © 2017 Elsevier Masson SAS. All rights reserved.
1. Introduction The paper deals with the kinetic modeling of gas mixtures in the fluid dynamic limit. Both problems, the kinetics of mixtures (and in particular: vapor–noncondensable mixtures) and questions concerning the fluid dynamic limit of rarefied flows, have been recently investigated in a couple of papers (see, e.g. [1–3] for the first, and [2,4] with the literature cited there for the latter aspects). One particular problem is that of boundary layers for mixtures. Due to the scaling used at small Mach numbers, it seems sufficient to study planar problems like half space or slab geometries. Once this is completely understood (see [1] for the evaporation–condensation problem discussed below, and [3] for the discrete velocity case), such boundary layers can be used to couple boundaries to hydrodynamic flow fields. For the fluid dynamic limit there exist well-established and widely accepted techniques like the Chapman–Enskog or the Hilbert expansion. Matching these flow fields to boundary layers designed e.g. to adapt the flow to diffusive reflection laws, seems to create sufficiently good solutions in a number of problems. The situation changes if a local matching of the boundary or interface conditions is not sufficient, since they serve as global control mechanisms. Such a situation occurs in the case of a binary gas mixture consisting of vapor and of a noncondensable between parallel walls. In the case of a pressure difference between the walls, vapor starts to move from one wall (evaporation) to the other (condensation) thus introducing a flux. The noncondensable
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.euromechflu.2017.01.012 0997-7546/© 2017 Elsevier Masson SAS. All rights reserved.
follows this flux and creates a barrier at one of the walls slowing down the vapor flux. Following the classical tools from asymptotic analysis, this barrier has to be modeled and matched as a boundary layer. This leads to a result in the hydrodynamic limit which is in literature known as a ‘‘ghost effect’’ and which obviously does not yield an appropriate description of the hydrodynamic limit. The aim of the paper is to provide an alternative description to this situation. Based on a detailed view on the collision operator and its algebraic structure, and combined with an appropriate scaling it yields a solution to the evaporation–condensation problem which is much more intuitive from a physical point of view. Due to the lack of data possibly used for comparisons and benchmarks (experimental data like those in [5,6] are not applicable; reliable numerical results are very hard to obtain due to stiffness problems) it is a question of plausibility whether to accept the presented ansatz or to look for an alternative. In our opinion, it provides a reliable perspective, since the bifurcation property (which is the keystone of the investigations) is based on a wellknown property of the algebraic structure of the involved operator (this property is not worked out in investigations like those in [3], which are thus restricted to small perturbations), and the choice of diffusive scaling has proven useful in a variety of problems for the derivation of diffusion phenomena [7–9]. This scaling gets rid of the ghost effect; furthermore it also seems to be the appropriate way to solve a couple of other problems related to the connections of the Navier–Stokes and the Boltzmann equations (like the problem discussed in [10]). Consider a gas mixture composed of two species confined between two parallel walls (Fig. 1). Species A (‘‘vapor’’) is emitted and adsorbed at the boundaries according to a prescribed pressure
H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
107
Fig. 1. Schematic view on density profiles for mixture (upper line) and noncondensable (lower line) without (left) and with vapor flux (right).
difference. Species B (‘‘noncondensable’’) is totally reflected when hitting the walls. If there is a pressure decay from the wall at x = a to the wall at x = b, then a flow of vapor is induced from a to b. At the same time one expects the noncondensable to follow the flow and form a boundary layer at b which slows down the vapor flow. This problem has been studied in a couple of papers in recent years and in particular the fluid dynamic limit was of interest (see, e.g. [11,12]). It turned out that the application of standard asymptotic analysis methods for the fluid dynamic limit leads to a curious situation. In the limit both A and B exhibit the same Maxwellian profile with an infinitesimally small bulk velocity v (in fact, v = 0), and a thin boundary layer of noncondensable is formed at b completely suppressing the vapor flow. This phenomenon contradicts physical intuition and is known in the literature as ghost effect [13]. In the present work we propose a different macroscopic limit, based on a scaling (‘‘diffusive scaling’’) which in the past has been applied in a variety of problems for the derivation of diffusion phenomena (see, e.g. [7–9]). It turns out that this kind of scaling leads in the limit to a boundary layer of well-defined thickness for the noncondensable which slows down but does not stop the vapor flow. The results are based on a careful investigation of a bifurcation phenomenon of the governing transport operator in the presence of a small drift. In case of zero drift, its nullspace has geometric dimension one (related to mass flow conservation) but algebraic multiplicity two. At the emergence of a drift, the twodimensional nullspace splits up into two simple eigenspaces giving rise to a new nonzero eigenvalue. In this paper we are not interested at all in any existence theorem for some boundary value problems. The only intention is to provide a scenario which allows in the hydrodynamic limit a gas mixture with one component at rest while the second one is streaming. This is the main result, and it reveals the major difference to the classical asymptotic expansions for which such a result is not possible (see [11,12]). In order not to hide this behind a bunch of technical details, we restrict to the following simplified setting. However, due to the similarities of the structure of the underlying transport operators (see, e.g. [14,15] in the case of the continuous and the discrete Boltzmann collision operator), we are convinced, that the present results can be generalized. We investigate the problem in the framework of Discrete Velocity Models (DVM) on the basis of two-particle collisions (see [16]). We consider the steady spatially one-dimensional problem in the slab [0, 1] in the simplest possible case of mechanically identical species A and B. This means that both are driven by the same Boltzmann collision operator. The only difference is the wall interaction. Denote by g the distribution of A and by h that of B. Then the sum f = g + h is governed by a nonlinear one-species Boltzmann collision operator J. If f is known, then g and h evolve according to a linear transport operator. We restrict to the case of f being a fixed global Maxwellian. In the case of zero flow between 0 and 1, f is a centered Maxwellian with zero bulk velocity. The corresponding transport operator L0 exhibits a typical structure concerning the algebraic nullspace which in a
similar situation has been observed in a couple of papers (e.g. [14] for the continuum case, [17,15] for DVM). For our investigation we require the DVM to satisfy four assumptions (see (2.3), (2.4), (3.2), (3.8) below), two of them being crucial. The first one is a symmetry condition and requires the velocity grid and the collision model to be invariant under a change of sign of the velocity components perpendicular to the walls. This leads to a linear ODE system with a matrix having a special antisymmetric block structure which is essential. (In the paper we exclude the case of zero normal velocities which would lead to a DAE rather than an ODE system. However, numerical experiments indicate that this condition can be weakened.) The second assumption concerns the existence of a maximal number of pairwise different nonzero eigenvectors. This in particular prohibits the existence of artificial invariants of the transport operator. (A discussion of this point may be found in [17,15].) 2. The evaporation–condensation problem 2.1. The model Consider a gas mixture confined in the slab [0, 1]. The two components of the gas are species A (‘‘vapor’’) with density function g(t , x, v) and species B (‘‘noncondensable’’) with density function h(t , x, v). The are represented in the form v = (vx , v⊥ ), with vx the component pointing in x-direction, and v⊥ the orthogonal complement. Concerning the gas particle interaction, both types are mechanically identical in the sense that both are governed by the same Boltzmann collision operator. The only difference lies in the gas–wall interaction. While species A may pass through the walls in both directions (condensation, evaporation), species B is totally reflected. As a consequence, there may be a total nonzero mass flux of A through the walls while the mass flux of B is zero. We write f = g + h and let the governing equations for g and h be the nonlinear two-species Boltzmann equation
(∂t + vx ∂x )g = J [f, g] (∂t + vx ∂x )h = J [f, h]
(2.1) (2.2)
with the collision operator J [., .] to be specified below. Since J [., .] is bilinear, a consequence of (2.1), (2.2) is that f solves the nonlinear Boltzmann equation
(∂t + vx ∂x )f = J [f, f].
(2.3)
In most of the paper we restrict to the steady variant of the system,
vx ∂x g = J [f, g] vx ∂x h = J [f, h].
(2.4) (2.5)
In order to extend the equations to a well-posed boundary value problem, they have to be supplemented with boundary conditions either in the form of reflection laws or by prescribing the flows into the domain [0, 1]. For our purposes such a detailed description is not necessary.
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H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
Definition 2.1. We call a pair (g, h) a solution to the evaporation–condensation problem, if Eqs. (2.4), (2.5) are satisfied with f = g + h, and if h has zero mass flux, i.e.
φ[h](x) = ⟨vx h(x)⟩ = 0.
(2.6)
In the rest of the paper we simplify the problem by considering only solutions of the evaporation–condensation problem for which f is a known global equilibrium function of the Boltzmann collision operator. In this case, Eqs. (2.4), (2.5) turn into a system of linear transport equations which can be solved by analyzing the corresponding transport operator. Furthermore, it is sufficient to construct h, since h is a solution of the transport equation if and only if g = f − h is.
Let V = {v1 , . . . , vN } ⊂ Rd (d ≥ 2) be a finite velocity set, (i) (i) vi = (vx , v⊥ ), and f = (fi )Ni=1 ∈ RN+ a distribution function on V . A single collision event describes an exchange between two velocity pairs,
rα [f, f] (Jα [f, f])m = −rα [f, f]
if if else.
0
m ∈ {i, l} m ∈ {j, k}
γα Jα [f, f].
(2.7)
α∈R
A linear version of J [f, f] is needed when considering the dynamics of a test particle (with distribution g) in a given scattering field with distribution f. The corresponding linear transport operator J [f]g is given by the matrix
γα Jα [f]
α∈R
with
0.5(fj gk + fk gj ) − fl gi 0.5(fi gl + fl gi ) − fk gj (Jα [f]g)m = 0.5(fi gl + fl gi ) − fj gk 0.5(fj gk + fk gj ) − fi gl 0
if if if if else.
m=i m=j m=k m=l
Its matrix representation is
−fl 0.5fl Jα [f] = Pα 0.5fl 0
0.5fk −fk 0 0.5fk
0.5fj 0 −fj 0.5fj
0 0.5fi T P 0.5fi α −fi
with the N × 4-Matrix Pα defined in column representation as Pα = (ei , ej , ek , el ) (em the mth canonical unit vector). It is well-known that Maxwellians, i.e. functions of the form f(v) = exp −|v − v| /2Θ
2
We end up with the compact formulation (2.9)
with F = diag(fi , i = 1 . . . N ), and the symmetric matrix
C =
α∈R
−1 0.5 πα Pα 0.5 0
0.5 −1 0 0.5
0.5 0 −1 0.5
=:Γ
0 0.5 T P 0.5 α −1
(2.10)
and πα = γα φα . Define
1 := (1 . . . 1)T ∈ RN and f−1 := (fi−1 , i = 1 . . . N )T = F −1 1.
From physical considerations (momentum and energy conservation) we consider only elementary collisions for which vi vl and vj vk are the diagonals of a rectangle in Rd . We denote by R ⊂ {1, . . . , N }4 all α = (i, j, k, l) representing a non-degenerate rectangle in the above sense. With this we can now choose collision frequencies γα ≥ 0 to define the Boltzmann collision operator on V as
J [f] =
0.5(fj gk + fk gj ) − fl gi = φα 0.5(fj−1 gj + fk−1 gk ) − fi−1 gi .
For short we write α = (i, j, k, l) and rα [f, f] = fj fk − fi fl . The above collision is described by the elementary collision operator
(2.8)
Thus we can rewrite the terms of the transport operator by extracting the factor φα , e.g.
(vj , vk ).
↔
J [f, f] =
fi fl = fj fk =: φα .
J [f] = CF −1
2.2. The discrete system
(vi , vl )
are equilibrium solutions of the nonlinear collision operator (and they are the only ones, if a sufficient number of rectangles α with γα > 0 appears in the sum (2.7), i.e. if the model is regular in the sense of [18]). In this case, a special situation arises. One easily checks that α = (i, j, k, l) describes a rectangle with vi vl and vj vk as diagonals if and only if
The following result follows immediately from inspection of the matrix Γ . Lemma 2.2. (a) C conserves the total mass, i.e. 1T · C = 0. (b) 1 ∈ ker(C ). The first model assumption requires that the number of collisions is large enough to prohibit artificial invariants. Model assumption 2.3. C RN = 1⊥ . Equivalent to this assumption is that the restriction C : 1⊥ → ⊥ 1 is bijective. (When writing about the inverse C −1 of C we mean in the following the restriction C −1 : 1⊥ → 1⊥ .) An essential part of the following considerations are symmetry arguments. Therefore we have to ensure that the velocity space and the collision model are symmetric with respect to reflections about the x-axes in the following sense. Model assumption 2.4. (i) If v = (vx , v⊥ ) ∈ V , then vx ̸= 0, and the reflected velocity Tx v := (−vx , v⊥ ) ∈ V . (ii) The collision frequencies γα are Tx -invariant. This means: If α = (i, j, k, l) and α ′ = (i′ , j′ , k′ , l′ ) are such that the corresponding velocities vm and vm′ satisfy vm′ = Tx vm for m ∈ {i, j, k, l}, then γα ′ = γα . From this follows that N is even, N = 2n. We choose a (i) numbering of V such that vx > 0 for i = 1 . . . n, and vi+n = Tx vi . Notice that due to the Assumption 2.4 C is symmetric and has the block matrix structure A∗ B∗
C =
B∗ A∗
(see the discussion in [17]). Since V contains no velocities v with vx = 0, the matrix Vx = diag(vx(i) , i = 1 . . . N ) is regular, and the system (2.5) can be rewritten as the ODE system
∂x h = Lh
(2.11)
H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
where due to (2.9) L takes the form L = Vx−1 CF −1 .
(2.12) ⊥
We easily find the following properties, denoting by vx the (i)
109
Model assumption 3.2. L0 has n − 1 pairwise different strictly positive eigenvalues λi , i = 1 . . . n − 1. We collect the results concerning the spectrum of L0 . A central property is the solvability of the equation
hyperplane perpendicular to vx = (vx , i = 1 . . . N )T .
L0 r0 = f0
Lemma 2.5. (a) ker(L) = span(f) and L(RN ) = v⊥ x . (b) The equation Lg = h is solvable if and only if h ∈ v⊥ x . (c) Any eigenvector t to an eigenvalue λ ̸= 0 is orthogonal to vx .
which follows from the lemma below. We call a vector even if it is of the form (p, p)T , and odd if it is (p, −p)T . The subset of even resp. odd vectors is denoted by RNeven resp. RNodd . Furthermore we call an operator M even if it maps even into even and odd into odd; we call M odd if it maps even into odd and odd into even.
Proof. (a) follows from the corresponding properties of C , see Lemma 2.2(b) and Assumption 2.3. (b) is an application of Fredholm’s alternative. (c) follows from (b) and Lt = λt ⊥ vx . ⃝ As a consequence, the restriction L : (f Its inverse
) → vx is bijective.
−1 ⊥
−1 ⊥ L−1 : v⊥ ) , x → (f
⊥
(3.2)
Lemma 3.3. (a) L0 is odd. (b) L0 (RNodd ) = RNeven . 1 N (c) There exists a unique solution r0 = L− 0 f0 ∈ Rodd to Eq. (3.2). N N N (d) L0 (Reven ) ( L0 (Reven ) ⊕ span{r0 } = Rodd
Proof. (a) By Assumption 2.4 and the numbering of the velocity space V , C and F0 are even and Vx−1 is odd.
L−1 = FC −1 Vx
is shortly called the inverse L−1 of L. If h ⊥ vx , then g = L−1 h is the unique solution of Lg = h orthogonal to f−1 .
(b) follows from (a) and N ⊥ v⊥ = L0 (RNodd ) ∪ L0 (RNeven ) x = Reven ∪ Rodd ∩ vx
(c) follows from f0 ∈ Reven ⊥ vx and Lemma 2.5.
3. The steady transport operator
We consider the transport operator L (now denoted as L0 ) given by (2.12) with f taking the form of a centered Maxwellian,
T (d) Let be t+ i = (pi , qi ) be eigenvectors for the positive eigenvalues λi and denote t− = (qi , pi )T as the corresponding eigenveci − tors vor −λi , i = 1 . . . n − 1. Then s+ = t+ ∈ RNeven and i i + ti − + − N si = ti − ti ∈ Rodd span (n − 1)-dimensional subspaces of RNeven 2 − 2 resp. RNodd , and L20 s− i = λi si and L0 r0 = 0. Thus
f0 (v) = exp(−|v|2 /2Θ ).
N r0 ̸∈ span(s− i , i = 1 . . . n − 1) = L0 (Reven ).
3.1. Zero bulk velocity
Due to assumption 2.4 and the chosen numbering, the corresponding diagonal matrix F0 takes the block diagonal structure F0 = diag(f0 (vi ), i = 1 . . . N )
= =:
diag(f0 (vi ), i = 1 . . . n) 0 (1/2)
F0
0
0
(1/2)
F0
N = diag (Λ, −Λ, N0 ) with Λ = diag (λi , i = 1 . . . n − 1),
N0 =
.
This equips the operator L0 = Vx CF0 (1/2) −1
with the block structure (1/2)
(Vx(1/2) )−1 A∗ (F0 ) (1/2) −(Vx(1/2) )−1 B∗ (F0 )−1 A B =: . −B −A
(Vx(1/2) )−1 B∗ (F0 )−1 (1/2) −(Vx(1/2) )−1 A∗ (F0 )−1
L0 =
(3.1)
Lemma 3.1. λ > 0 is an eigenvalue of L0 if and only if −λ is eigenvalue. Proof. Define t+ := (p, q)T and t− := (q, p)T . Then we find easily L0 t
= λt
⇔
L0 t
−
= −λt . −
± −1 with t± i = ti (0) as in the proof of Lemma 3.3(d), and r0 = L0 f0 .
Corollary 3.5. The general solution h of ∂x h = L0 h in the slab [0, 1] takes the form
The spectrum of matrices of this form was studied in [17,15]. Simple to prove is the following result.
+
1 . 0
+ − − T (0) = {t+ 1 , . . . tn−1 , t1 , . . . tn−1 , f0 , r0 }
Vx = diag(Vx(1/2) , −Vx(1/2) ). −1
0 0
(b) A corresponding transformation matrix is
For the same reasons we can decompose Vx into
+
This result yields a complete description of the eigenspace structure of L0 and proves the following theorem. Theorem 3.4. (a) L0 is similar to the Jordan normal form
0 diag(f0 (vi ), i = 1 . . . n)
−1
⃝
⃝
The following model assumption has been motivated for a similar situation in [17,15] and results in a structure for L0 which is well-suited for our purpose.
h(x) =
n −1
γi− exp(−λi x)t− i +
n −1
i =1
γi+ exp(−λi (1 − x))t+ i
i =1
+ (γn + γr · x)f0 + γr r0 with free parameters γi± , γn and γr . The first two sums represent boundary layers at x = 0 and x = 1 which are used eventually to model prescribed inflow conditions at the boundaries. Remark 3.6. Assumption 2.4 is quite restrictive since it prohibits zero x-components of the velocities. It turns out that this can be weakened. In all numerical experiments we performed as far, the system (2.5) turned out to represent an index-1 differential algebraic system which could be transformed into an ODE system with a matrix which has precisely the same Jordan structure as that given in the theorem.
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H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
3.2. Shifted Maxwellian We replace the centered Maxwellian f0 with a new one shifted in x-direction,
1 2 fv (v) = exp − |v − v| = f0 (v) + v · 1f(v) 2Θ v = (v, 0)T . 1f can be chosen continuous in v = 0, with f′ := 1f(0) = lim (fv (v) − f0 (v))/v = v→0
1
Θ
· Vx f0 .
Prˆ g = g − vTx g ·
This introduces an analytic change of all operators introduced above. These are in particular continuously differentiable with respect to v . The prime indicates in the following the derivative at v = 0. The transport operator is affected by the change in two ways. The operator C0 changes into Cv = C0 + v · 1C (v) with 1C continuous,
1C (0) = C ′ =
φα′ =
α∈R (i)
γα φα′ Pα Γ PαT ,
vx + vx(l) · f0 (vi )f0 (vl ), Θ
α = (i, j, k, l)
(F
) := 1F
−1
(0) = −
1
Θ
· F0 Vx .
1
Θ
vTx rˆ
.
The solution of (3.4) is equivalent to finding a fixed point of the mapping 1 rˆ → c · L− v Prˆ fv
(3.7)
(with c normalizing constant). Since = O (v) and since r0 is fixed point for v = 0, it follows that for v small the mapping (3.7) is a contraction with a unique fixed point. The continuity of the mapping follows from the continuity of simple eigenvectors of analytically perturbed operators (see [19]). ⃝ vTx fv
Main theorem 3.10. (a) For |v| ̸= 0 sufficiently small there exists a new eigenvalue λ(v) = v · 1λ(v) depending analytically on v with 1λ(0) = λ′ ̸= 0 and a corresponding eigenvector of the form t = rv + λ(v)−1 · fv . rv ⊥ vx is the unique solution of Lv rv = fv + λ(v)rv . (b) Lv is similar to the diagonal matrix
−2
Finally, L0 = Vx−1 CF0−1 changes into Lv = L0 + v 1L(v) with
1L(0) = L′ = Vx−1 C ′ F0−1 −
rˆ
This leads to the proof of the main result of this section.
and F0−1 is to be replaced with Fv−1 = F0−1 + v ∆F −1 (v) with −1 ′
Proof. A necessary condition for the continuity of 1λ and 1r at v = 0 is obtained inserting the ansatz (3.3) into Eq. (3.4) and taking the limit v → 0. This leads to Eq. (3.6). From Fredholm’s alternative, this equation is solvable if and only if the right hand side is orthogonal to vx . Thus f′ − L′ r0 +λ′ · r0 has to be the projection of f′ − L′ r0 along r0 onto vTx . From this follows (3.5). Given r ̸= 0, define its corresponding normalized vector rˆ = ∥r∥−1 r and the projection Prˆ along r onto vTx ,
· L0 · F0−1 Vx .
+ − − N = diag (λ+ 1 . . . λn−1 , −λ1 · · · − λn−1 , 0, λ(v)).
′
It is important to mention that L is an even operator. We list some of the main properties. Lemma 3.7. For small v there exist positive eigenvalues λ+ i (v) and (v) depending analytically on v with λ± negative eigenvalues −λ− i → i λi for v → 0. Proof. This is a standard result from perturbation theory since all ±λi are simple eigenvalues of L0 (see Kato). ⃝ Crucial for the following is how the algebraic nullspace is affected by the change. If there were a solution of Lv rv = fv then its Jordan normal form remained the same as before. The following (generic) model assumption prohibits this. Recall that RNodd is + − spanned by s− i = ti − ti , i = 1 . . . n − 1, and r0 (Lemma 3.3(d)).
Proof. In order to prove (a) we remark that the Jordan block J0 of the nullspace of L0 changes into
Jv =
0 0
1
λ(v)
which is similar to diag(0, λ(v)). (b) follows then immediately.
⃝
4. A macroscopic limit In order to derive a meaningful macroscopic limit we introduce the diffusive scaling (see, e.g. [7–9]) for the equation
(∂t + vx ∂x )g = J [f]g.
Model assumption 3.8. f′ + L′ r0 ̸∈ {s− i , i = 1 . . . n − 1}.
It consists in replacing the macroscopic variables t and x with ϵ −2 t and ϵ −1 x and leads to the rescaled equation
Under this assumption the two-dimensional nullspace splits up into two simple eigenspaces as is shown now.
(∂t + ϵ −1 vx ∂x )g = ϵ −2 J [f]g.
Lemma 3.9. In a neighborhood U0 = (−v 0 , v 0 ) of v = 0 there exists a continuous mapping v → (1λ(v), 1r(v)) such that the pair
(λ(v), rv ) = (v · 1λ(v), r0 + v · 1r(v))
(3.3)
Formally this is equivalent to replacing the space V of microscopic velocities with ϵ −1 V and scaling up the collision frequency by a factor ϵ −2 . This is the approach which we take here. Replacing vi with wi = ϵ −1 vi requires to change the Maxwellians
solves Lv rv = fv + λv · rv 1 λ− v fv
λ := 1λ(0) = ′
fv = (exp(−|vi − v|2 /2Θ ), i = 1 . . . N )T to (ϵ) fv = (exp(−|wi − v|2 /2Θ ), i = 1 . . . N )T = (exp(−|v − ϵ v|2 /2ϵ 2 Θ ), i = 1 . . . N )T
(3.4)
+ rv is eigenvector with eigenvalue λv . Furthermore, vTx (L′ r0 − f′ ) vTx r0
(3.5)
L0 r′ = f′ − L′ r0 + λ′ · r0 .
(leaving the macroscopic bulk velocity v unchanged) which itself makes only sense if we rescale the temperature as T = ϵ 2 Θ . From now on we define (ϵ)
(1)
fv = fϵv = (exp(−|v − ϵ v|2 /2T ), i = 1 . . . N )T
and r′ := 1r(0) is solution of (3.6)
(4.1)
(4.2)
with T > 0 constant. For convenience we assume in the following λ′ v > 0.
H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
111
(ϵ)
Remark 4.1. Associated to fv are the moments
= ⟨1f(01) ⟩ + O (ϵ 2 )
(ϵ)
density ρv (ϵ)
flux φv
= ⟨wx fv(ϵ) ⟩ = (v/T ) · ⟨vx2 f(01) ⟩ + O (ϵ 2 ).
(ϵ)
(ϵ)
Fv is the diagonal matrix with the coefficients of fv as entries, (ϵ)
Fv
(1) = Fϵv = diag(fv(ϵ) ) = F0 I +
ϵv T
Vx
+ O (ϵ 2 ).
(4.3)
Fig. 2. A 12- and a 9-velocity model.
evaporation–condensation problem (in the sense of Definition 2.1). (ϵ) Then hv is of the form
The steady version of (4.1) is
vx ∂x g = ϵ
−1
(ϵ)
J [f]g.
(4.4)
hv
+
(ϵ)
(1) = Cϵv =
α∈R
πα(ϵv) Pα Γ PαT
T
(ϵ) −1 (ϵ) + γr exp −λ(ϵ) ( 1 − x ) · (r(ϵ) v v + (λv ) fv )
(4.7)
± (fv(ϵ) − hv(ϵ) , h(ϵ) v )ϵ>0 of solutions asymptotically bounded, if γi , γr are bounded for ϵ ↘ 0. In this case the moments of the solutions are well-defined in the limit ϵ ↘ 0 and given as follows. (ϵ)
1
′ λ(ϵ) v → λ v > 0 for ϵ → 0
with λ given by (3.5). The following results are easy to prove from the Main theorem 3.10. (ϵ)
Corollary 4.2. (a) Lv is similar to the diagonal matrix −1 + diag (ϵ −1 λ+ λn−1 (ϵv), −ϵ −1 λ− 1 (ϵv), . . . , ϵ 1 (ϵv), . . . , (ϵ) − ϵ −1 λ− n−1 (ϵv), 0, λv ).
(ϵ)
(ϵ)
The limits of the associated moments are
t− i (ϵv)
(1)
density ⟨1 fv ⟩ → ⟨1 f0 ⟩ (ϵ)
(1)
flux ⟨wx fv ⟩ → v · ⟨vx2 f0 ⟩/T . The concentration profile of the noncondensable, i.e. the layer at the wall point x = 1 is given as noncondensable concentration ′ ⟨1 hv(ϵ) ⟩/⟨1 f(ϵ) v ⟩ → γH exp(−λ v(1 − x)),
i.e. is given as a solution of the differential equation
The corresponding eigenvectors are
(i = 1 . . . n − 1),
γH = H (λ′ v)2 (exp(λ′ v) − 1)−1 = H · λ′ v + O (v 2 ). (ϵ)
′
(i = 1 . . . n − 1),
(ϵ)
rv + (λv )−1 fv .
(b) The general solution of the rescaled system (4.4) is
+ γi+ exp −ϵ −1 λ+ i (ϵv)(1 − x) · ti (ϵv)
i=1 n−1
⟨1 h(ϵ) v (x)⟩dx = H = const.
(1)
where rϵv = r0 + ϵv 1r(ϵv) is given as in Lemma 3.9, and
+
(ϵ)
h0 = γH exp(−λ′ v(1 − x)) · f0 ,
−1 λ(ϵ) v = ϵ λ(ϵv)
n −1
0
(ϵ)
Then it converges for ϵ ↘ 0 pointwise in x to (f0 − h0 , h0 ) given by
(ϵ)
rv = ϵ rϵv
(ϵ)
(4.9)
Corollary 4.3. Suppose (fv − hv , hv )ϵ>0 is an asymptotically bounded family of solutions to the evaporation–condensation problem with a prescribed amount of noncondensable,
(1)
fv ,
− γi− exp −ϵ −1 λ− i (ϵv)(1 − x) · ti (ϵv)
with coefficients γi± and γr depending on ϵ . We call a family of pairs
Lv is identical to the operator Lv investigated in the previous section. Finally define
t+ i (ϵv)
n −1
(4.6)
and
πα(ϵv) = γα f(ϵv1) (vi )f(ϵv1) (vl ) ϵv = πα(0) · 1 + · (vx(i) + vx(l) ) + O (ϵ 2 )
+ γi+ exp ϵ −1 λ+ i (ϵv)x · ti (ϵv)
i =1
(4.5)
with Cv
n −1 i =1
Thus we have to study the rescaled transport operator (ϵ) (ϵ) (ϵ) (1) Lv = ϵ −1 Vx−1 Cv (Fv )−1 = ϵ −1 Lϵv
=
− γi− exp −ϵ −1 λ− i (ϵv)x · ti (ϵv)
i=1
(ϵ) (ϵ) (ϵ) −1 (ϵ) + γn f(ϵ) v + γr exp −λv (1 − x) · (rv + (λv ) fv ). (4.8) Recall that the only term in (4.8) with nonzero flux is that related (ϵ) (ϵ) to the zero eigenvector fv (see Lemma 2.5(c)), and that ⟨wx fv ⟩ converges to a nonzero value for ϵ ↘ 0 (Remark 4.1). Suppose the (ϵ) (ϵ) (ϵ) (ϵ) (ϵ) pair (gv , hv ) = (fv − hv , hv ) is a solution of the rescaled steady
χ ′ = λ′ v · χ .
(4.10)
It is Eq. (4.10) which marks the crucial difference to the theory leading to the ghost effect. This equation states that the emergence of a blocking wall of noncondensable is proportional to the vapor flux. Thus in the case of v = 0, there is no blocking wall at all, and this also holds in the macroscopic limit. The size of the barrier depends besides the velocity v (which is implicitly given by the boundary conditions) also on λ′ , which is a constant depending on the collision model, and on the total amount of noncondensable. Thus the situation of an infinitesimal amount in the order of the Knudsen number, which was discussed in [12], leads again to a barrier which vanishes in the limit. We finish these investigations with two examples. Example 4.4. Consider the 12-velocity model as indicated in Fig. 2. (1) (1) Its vx -components are given by the vector vx = (vx , vx , (1) (1) T (1) T −vx , −vx ) with vx = (1, 1, 3) , and centered Maxwellians (1) (1) (1) (1) (1) are of the form f0 = (f0 , f0 , f0 , f0 )T with f0 = (τ 10 , τ 2 , τ 10 ),
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H. Babovsky / European Journal of Mechanics B/Fluids 63 (2017) 106–112
τ = exp(−1/2T ). We include the collisions (2, 3, 5, 6), (5, 4, 11, 10), (8, 9, 11, 12) and (2, 1, 8, 7) with collision frequency γ1 and π1 = fi fl γ1 = τ 12 γ1 , and the collisions (1, 3, 8, 5), (2, 6, 11, 4), (8, 5, 12, 10), and (2, 7, 11, 9) with frequency γ2 and π2 = τ 12 γ2 . Assuming a density which is symmetric with respect to the yaxis of the velocity space, then we can identify the velocities and densities with numbers n = 4, 5, 6, 10, 11, 12 with those with numbers n − 3. Thus the collision model can be compressed to a 6 × 6-collision matrix Cˆ =
C11 C12
C12 C11
with
−2(π1 + π2 ) π1 π1 −3π1 − 4π2 = π2 π1 + π2 π1 π2 0 = π2 π1 + 2π2 0 .
π2 π1 + π2 , −π1 − 2π2
C11
C12
0
0
0 (1)
(1)
(1)
(1)
We have n0 = f0 and n′ = Vx f0 = (n1 , n1 , −n1 , −n1 )T with (1)
(1)
(1)
(1)
(1)
n1 = (τ 10 , τ 2 , 3τ 10 )T . The vector u0 = (u0 , u0 , −u0 , −u0 )T is determined from the linear system
(C11 − C12 )u(01) = f(01) and finally λ′ from the orthogonality relation
References
n′ + λ′ u0 ⊥ vx . Example 4.5. We finish with the 9-velocity model indicated by Fig. 2. Since vx = (1, 1, 1 − 1, −1, −1, 0, 0, 0)T , the describing equations form a DAE system rather than an ODE system. In its simplest form the collision model contains the collisions (1, 3, 7, 9), (2, 3, 8, 9), (5, 6, 8, 9) and (4, 6, 7, 9) with given π1 > 0, and (3, 7, 8, 6) with π2 > 0. Assuming again y-symmetry of the density functions and thus identifying the velocities 2, 5 and 8 with 1, 4 and 7, the corresponding DAE system is of the form
diag(1, 1, −1, −1, 0, 0)∂x φ =
C11 0 C31
0 C11 C31
C13 C13 C33
with
π1 , −4π1 − 2π2 −4π1 − 2π2 2π1 = , 4π1 −8π1
C11 = C33
−2π1 2π1
π1 2π2
0 2π1
π1
π2 . 2π1
C13 = C31 =
0
,
The algebraic part reads C31 (φ (1) + φ (2) ) + C33 φ (3) = 0. Inserting into the differential part yields the ODE system
∂x (φ (1) , φ (2) )T =
A −B
B −A
component at rest. Crucial for this result is the diffusive scaling which in an appropriate manner takes into account the bifurcation structure at hand. By way of contrast, the classical Chapman–Enskog expansion leads in lowest order to a Maxwellian at rest for both components and is not capable of covering such a scenario. Let me finally comment shortly on the main model restrictions of the paper. Some of them are crucial, others have been introduced for simplicity and transparency of the arguments. First of all, symmetry of the model (grid and collisions) is essential, since the main proof concerning the algebraic properties of the problem are built on symmetry arguments. On the other hand, the restriction to 2D models is not necessary, the theory applies also in the 3D case. The assumption of mechanically identical species has been crucial here since it simplifies theory considerably. Though we are convinced that a generalization is possible, the proofs for such an extension are not completely evident. The assumption of nonvanishing x-velocity components avoids the necessity of handling a DAE system. However, the last example shows a line along which the theory can be developed. Finally, the assumption of a fixed global equilibrium (which is certainly very strong) and the restriction to the steady case can be certainly weakened, at least in the framework of small perturbations. The aim of the paper was the deduction of an alternative theory; the investigation of its limitations is open for future.
−1 −1 with A = C11 − C13 C33 C31 and B = −C13 C33 C31 which has the same structure as the systems described in the paper.
5. Conclusions and final comments We have performed the hydrodynamic limit in a way which allows the coexistence of a streaming component with a
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