- Email: [email protected]
Recurrence relations between kernels of the nonlinear Boltzmann collision integral
European Journal of Mechanics B/Fluids 36 (2012) 17–24
Contents lists available at SciVerse ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Recurrence relations between kernels of the nonlinear Boltzmann collision integral A.Ya. Ender a , I.A. Ender b , L.A. Bakaleinikov a,∗ , E.Yu. Flegontova a a
Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russia
b
St. Petersburg State University, St. Petersburg, 199034, Russia
article
abstract
info
Article history: Received 28 July 2011 Received in revised form 29 February 2012 Accepted 6 March 2012 Available online 17 April 2012
A new approach to the solution of the nonlinear Boltzmann equation based on distribution function expansion in terms of spherical harmonics is considered. Owing to this approach, the complex five-fold collision integral can be replaced by a set of rather simple integral operators. Recurrence relations between nonlinear kernels Gll1 ,l2 (c , c1 , c2 ) of the integral operators are derived on the basis of the collision integral invariance with respect to choice of the reference frame velocity. Recurrence formulae allow one to find kernels with arbitrary indices provided the kernel G00,0 (c , c1 , c2 ) is known. Explicit analytical expressions for kernels G11,0 , G10,1 , and G01,1 for the cases of hard spheres and Maxwellian molecules are presented. © 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Boltzmann equation Distribution function Spherical harmonics Nonlinear kernels Inter-kernel relations Collision integral
3/2
1. Introduction Matrix elements (MEs) of the Boltzmann collision integral resulting from distribution function (DF) expansion in spherical Hermitean polynomials (Burnett’s functions [1]) were investigated in our previous papers [2–4]. Matrix elements Kji,k are defined as Kji,k =
Hi ˆI (MT Hj , MT Hk ) d3 v/gi ,
gi =
MT Hi2 d3 v,
(1)
Hj (c) =
(Θ , ϕ) (c ), √ i = 0, 1; c = α(v − u), α = m/(2kT ), c l Slr+1/2
2
0 Ylm (Θ , ϕ) = Plm (cos Θ ) cos mϕ, 1 Ylm (Θ , ϕ) = Plm (cos Θ ) sin mϕ,
0 ≤ m ≤ l.
(2) (3)
∗ Correspondence to: Department of Applied Mathematics and Mathematical Physics, A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26, Politechnicheskaya Street, St. Petersburg, 194021, Russia. Tel.: +7 812 2927112; fax: +7 812 5504890. E-mail addresses: [email protected], [email protected] (L.A. Bakaleinikov). 0997-7546/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2012.03.004
2
(K0i ,k ) kind. As shown in [2], collision integral invariance with respect to rotation leads to a proportionality between arbitrary MEs r ,l,m,i Kr ,l ,m ,i ;r ,l ,m ,i and the corresponding axisymmetric MEs 1 1
where ˆI is the collision integral. The spherical Hermitean polynomials Hj are defined as i Ylm
Index j includes four indices (r , l, m, i), MT = πα e−α(v−u) is the Maxwellian weight function characterized by temperature T i and average velocity u, Ylm (Θ , ϕ) are the real spherical harmonics, m Pl (cos Θ ) are the associated Legendre functions, and Slr+1/2 (c 2 ) are the Sonine (Laguerre) polynomials. Linear MEs correspond to the linear collision integral and can be of the first (Kji,0 ) and second
r ,l
1 1
2 2
2 2
r ,l,0,0
Kr ,1 ;r ,l = Kr ,1 ,0,0;r ,l ,0,0 , i.e., the equation 1 1 2 2 1 l 2 2 r ,l,m,i
l,m,i
r ,l
Kr1 ,l1 ,m1 ,i1 ,r2 ,l2 ,m2 ,i2 = Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 Kr1 ,l1 ,r2 ,l2
(4) l,m,i
is valid. Here, the universal numerical coefficients Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 that are independent of scattering cross section can easily be expressed through the Clebsch–Gordan coefficients. Therefore, problems with arbitrary DF can be solved if axisymmetric MEs alone are known, even when DF are not axisymmetric. To study properties of axisymmetric MEs, we consider here the axial symmetric case. Spherical harmonics are identical in this case to Legendre polynomials, and the set of four indices j converts into a pair r , l. The distribution function can be expanded in the vicinity of a Maxwellian which may be characterized by arbitrary temperature T and average velocity magnitude u. Collision integral
18
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
invariance with respect to the Maxwellian temperature choice leads to the ‘‘temperature’’ recurrence relationships for MEs
T
d dT
l − R Krr1,,ll1 ,r2 ,l2 = rKrr1−,l11,,lr2 ,l2 − (r1 + 1)Krr1,+ 1,l1 ,r2 ,l2
− (r2 + 1)Krr1,,ll1 ,r2 +1,l2 .
(5)
Here R = r1 + r2 − r + (l1 + l2 − l)/2. Invariance with respect to the average velocity magnitude yields the ‘‘velocity’’ recurrence relationships 1 r −1,l+1 r ,l β(l − 1)Krr1,,l− l1 ,r2 ,l2 + γ (r − 1, l + 1)Kr1 ,l1 ,r2 ,l2 − β(l1 )Kr1 ,l1 +1,r2 ,l2 r ,l l − γ (r1 , l1 )Krr1,+ 1,l1 −1,r2 ,l2 − β(l2 )Kr1 ,l1 ,r2 ,l2 +1
+ γ (r2 , l2 )Krr1,,ll1 ,r2 +1,l2 −1 = 0,
(6)
where
β(l) = −
l+1 2l + 1
,
γ (r , l) =
(r + 1) l . 2l + 1
In [2,4] these relationships were generalized for the case of a mixture of gases with arbitrary molecular masses. Recurrence procedures were developed to obtain arbitrary MEs from the simplest isotropic (l = 0) linear diagonal MEs. To perform the calculations, special high-accuracy arithmetic routines were developed. The recurrence relations allowed both linear and nonlinear MEs with very high indices to be obtained (in our calculations, each index r , l, r1 , l1 , r2 , l2 varied from 0 to 128). ‘‘Temperature’’ recurrence relationships (5) for power potentials proved to be algebraic, and MEs were found by using the recurrence procedures based on (5)–(6). In the case of arbitrary interaction potentials, isotropic linear MEs should be expanded in terms of Ω -integrals and then expansion coefficients for arbitrary MEs can be found through the recurrence procedures [2]. Note that, while calculating all nonlinear MEs, we obtain also linear nonisotropic MEs. Linear MEs differ from the widely known bracket integrals [5] by a numerical factor alone. The bracket integrals are the object of continuous interest. Earlier the bracket integrals were used to calculate kinetic theory transport coefficients (only bracket integrals with l = 1, l = 2 and low r were used). Recently high-accuracy calculations of the bracket integrals with indices l = 1 and l = 2 and high r have been presented [6–10]. To remind, the index r at a fixed l corresponds to the Sonine polynomial degree in the spherical Hermitean polynomial expansion. The possibility of obtaining MEs with high indices [2–4] gave a new impetus to advances in the standard moment method. This method, using a large number of MEs, was applied in [11] to investigate ion DF evolution after triggering a rather strong electric field or crossed electric and magnetic fields. The DF was calculated for velocities of up to ten thermal velocities. However, the requirement for DF expansion convergence in the standard moment method imposes a restriction on DF. This restriction consists in a finiteness of the DF norm in the Hilbert space spanned by the Sonine polynomials mutually orthogonal with respect to a Maxwellian weight function. This limitation will be removed if DF is expanded into spherical harmonics. In this case a complex five-fold collision integral will be replaced by a set of relatively simple integral operators. The kernels Gll1 ,l2 (c , c1 , c2 ) of the operators depend on the velocity magnitudes alone and play the same role as MEs in the standard moment method. The problem of kernel construction is a great challenge. Suffice it to say that explicit expressions even for linear kernels have so far only been found for the cases of hard spheres [12,13] and Maxwellian molecules [14]. Hilbert [12] was the first to transform the Boltzmann collision integral into a Fredholm-type integral with a symmetrical kernel. This transformation was accomplished only for the hard spheres model. Expansion of this kernel in spherical
harmonics was performed by Hecke [13]. Later this result was also obtained in [15]. Use of the kernels that depend on velocity magnitudes alone made the process of solving kinetic equation much simpler. This approach yielded a number of valuable results in rarefied gas dynamics boundary problems [16,17]. It should be emphasized that all these results were obtained for the linearized Boltzmann equation alone. The construction of nonlinear kernels will enable solution of a wide range of problems in gas dynamics and plasma physics where the distribution function is far from equilibrium and the nonlinear character of processes is essential. We showed [2,18] that nonlinear kernels may be expressed in terms of nonlinear MEs. Existence of relations between MEs suggests that there are similar inter-kernel relations that follow from collision integral invariance with respect to the choice of basis. 2. Kernels of the nonlinear collision integral By using (4) we showed in [2,18] that, if DF is expanded in spherical harmonics, the collision integral can always be represented through axisymmetric kernels Gll1 ,l2 (c , c1 , c2 ):
l,m,i ∂ f (c) i Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 Ylm (Θ , ϕ) = ∂t col l1 m1 ,i1 l2 ,m2 ,i2 l,m,i ∞ ∞ i, × Gll1 ,l2 (c , c1 , c2 ) fl1i ,m1 (c1 ) fl2 ,m2 (c2 ) c12 c22 dc1 dc2 ,
0
(7)
0
where Gll1 ,l2 (c , c1 , c2 ) are expressed through MEs as Gll1 ,l2 (c , c1 , c2 ) = M (c )
×
c l Slr+1/2 (c 2 )Krrl1 l1 ,r2 l2
r ,r 1 ,r 2 l r c11 Sl11+1/2
l
r
(c12 ) c22 Sl22+1/2 (c22 )
σr1 l1
σrl = Γ (r + l + 3/2)/(2π 3/2 r !),
σr2 l2
,
(8)
2 M (c ) = e−c /π 3/2 .
In (7), fl1i ,m1 (c ) and fl2i ,m2 (c ) are the coefficients of DF f (c) expanl,m,i
i (Θ , ϕ). Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 are sion in terms of spherical harmonics Ylm the above-mentioned universal numerical coefficients. These coefficients may differ from zero only if |ll − l2 | ≤ l ≤ l1 + l2 , m = |m1 ± m2 |, l + l1 + l2 and i + i1 + i2 are even. If l1 = m1 = 0, then Z = 1 and (l2 , m2 ) = (l, m). If l2 = m2 = 0, then Z = 1 and (l1 , m1 ) = (l, m). In the axisymmetric case, m = m1 = m2 = i = i1 = i2 = 0 and Z = 1. Therefore, kernels, as well as MEs, may be considered for the axisymmetric case without loss of generality [18]. As noted above, spherical harmonics are identical in this case to Legendre polynomials. Since we do not expand DF in terms of Sonine polynomials, the basis is now temperature-independent and is determined by the reference frame velocity alone. Let us use the thermal velocity vT corresponding to some arbitrary temperature T as a unit of velocity and consider DF in the reference frame moving with velocity u along the preferred direction ez . By defining the dimensionless velocity as c = (v − uez )/vT , we can express DF as
f (c) =
fl (c )Pl (x),
x = cos θ .
l
Here, θ is the angle between c and ez . The collision integral is expressed in terms of kernels Gll1 ,l2 as
∞ ∞ ∂ f (c) Gll1 ,l2 (c , c1 , c2 ) fl1 (c1 ) = P ( x ) l ∂t 0 0 col l ,l ,l 1 2
× fl2 (c2 ) c12 c22 dc1 dc2 .
(9)
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
The nonlinear kernels Gll1 ,l2 (c , c1 , c2 ) are nonzero if
(−1)l = (−1)l1 +l2 .
|l1 − l2 | ≤ l ≤ l1 + l2 ,
Operator Ψˆ λ,l is applied to function fl (c ). We obtain from (13) (10)
It is important to note that the collision term in Eq. (9) is given for simple gases, however, the same kernel may be used for a mixture of gases. In the latter case the DF variation rate of particles ‘‘a’’ due to their interaction with particles ‘‘b’’ is
a ∞ ∞ ∂ fa (c) (a,b) l Gl1 ,l2 = Pl (x) (c , c1 , c2 ) ∂t 0 0 ab col l ,l ,l
Ψˆ λ,l χ; c ′ fl (c ) =
1
pλ x′ Pl (x)fl (c )dx′ .
(15)
−1
Here we take into consideration the fact that c depends on integration variable x′ . Hence, the operator Ψˆ λ,l is an integral operator. In the limit χ → 0 the fl (c ) function is independent of x′ and can be taken out of the integral. In this case, the Legendre polynomial arguments are identical, and (15) becomes
Ψˆ λ,l χ; c ′ |χ=0 fl (c ) = δλ,l fl (c ).
1 2
× fl1a (c1 ) fl2b (c2 ) c12 c22 dc1 dc2 .
19
(11)
For the inverse operator Ψˆ l,λ we have from (14) 1
If particle masses are equal and their interaction potentials are identical to that of a simple gas, the kernels in (9) and (11) are the same. Due to symmetry of the product fl1 (c1 )fl2 (c2 ) with respect to permutations c1 c2 and l1 l2 , kernel Gll1 ,l2 (c , c1 , c2 )
Ψˆ l,λ (−χ; c ) Fλ c ′ =
in (11) may be replaced by a symmetrical kernel ˜
In future calculations we will need derivative ∂ Ψˆ λ,l (χ; c ′ )/∂χ|χ=0 .
(c , c1 , c2 ) = 1/2 Gll1 ,l2 (c , c1 , c2 ) + Gll2 ,l1 (c , c2 , c1 ) . However, in this paper, as Gll1 ,l2
well as in our previous publications, we consider the nonsymmetrical kernels that can be used for gas mixtures and also for simple gases. Hereinafter we omit kernel indices a and b. The formula for the linear kernel expansion in Sonine polynomials similar to (8) contains only double summation. The linear kernels for Maxwellian molecules and hard spheres were investigated and constructed by means of a double summation formula in [19,20]. To calculate the infinite sum, we used the asymptotic approach. At high indices, the terms of the sum were substituted by their asymptotic expressions, and summation was replaced by integration. However, the problem of estimating the sum remainder in (8) at high velocities emerged. This was a consequence of a complexity of the Sonine polynomials’ asymptotics at high arguments when the index tended to infinity. 3. Inter-basis transition operator and its derivative Consider the distribution function in two reference frames that differ by the velocity of motion along the preferred direction u. DF representations in these reference frames (bases) are related by a transition operator. To find it, we select two bases with velocities u0 and u1 . The difference between the velocities is denoted by χ, χ = u1 − u0 . Let us express fl (c ) and Fλ c ′ for the DF expansion coefficients in terms of Legendre polynomials in bases u0 and u1 . Magnitudes of velocities c and c ′ in bases u0 and u1 satisfy the relationships c 2 = cz2 + cρ2 ,
2 2 c ′ = c ′ z + cρ2 ,
c ′ z = cz − χ ,
cz = cx,
∞
fl (c )Pl (x) =
∞
c ′ z = c ′ x′ .
Fλ c ′ Pλ x′ .
(12)
Now we calculate the scalar product of this identity and Legendre polynomial divided by the square of its norm, πλ , in basis u1 . Integration of (13) multiplied by pλ x′ = Pλ x′ /πλ over the interval [−1, 1] results in
′
Fλ c
=
∞ l =0
Ψˆ λ,l χ; c fl (c ).
′
−1
Ψˆ l,λ (−χ; c ) |χ=0 Fλ (c ) = δl,λ Fλ (c ).
(16)
Let us differentiate function Ψˆ λ,l χ; c ′ fl (c ) defined by (15) with respect to χ taking into account that c and x depend on c ′ , x′ , and χ . This yields
∂ Ψˆ λ,l χ; c ′ fl (c ) ∂χ χ=0 1 dPl (x) ∂ x ′ = pλ (x ) f ( c ) l dx ∂χ χ=0 −1
dx′
χ=0
1
∂ c pλ (x′ ) Pl (x) ∂χ
+ −1
dc
dfl (c )
χ=0
dx′ .
(17)
χ=0
In the limit χ → 0 neither fl (c ) nor dfl (c )/dc function depends on x′ and, hence, both of them can be taken out of the integral. It follows herefrom that the operator Ψˆ λ,l derivative is a differential operator of the form
∂ (2) d = A(λ,1)l + Aλ, Ψˆ λ,l χ; c ′ l ∂χ dc χ=0
(18)
(2)
(1)
where numerical matrices Aλ,l and Aλ,l are defined by (17). To calculate these matrices, explicit forms of c and x dependences on c ′ and x′ are needed. Eq. (12) gives c 2 = c ′2 + 2c ′ x′ χ + χ 2 ,
∂ c ∂χ
, c ∂ x 1 = − x′ 2 − 1 . ∂χ χ=0 c
= x′ ,
χ=0
c ′ x′ + χ
x=
(1) Aλ,l (2) Aλ,l
=−
1
c
dPl (x′ )
1
pλ (x ) x − 1 ′
′2
−1
dx′
dx
′
,
1
(19)
pλ (x )x Pl (x )dx . ′
=
′
′
′
−1
Using the formulae for Legendre polynomials [21] (13)
λ=0
l =0
Since x|χ=0 = x′ , the first and second integrals in (17) are
Here x = cos θ , x′ = cos θ ′ are cosines of the angles between ez and velocity vectors c and c′ . Distribution function expansions in different bases are identical f (c) =
pl (x)Pλ x′ Fλ c ′ dx,
(14)
(x2 − 1) xPl (x) =
dPl (x) dx l+1
=
2l + 1
(l + 1)l [Pl+1 (x) − Pl−1 (x)], 2l + 1
Pl+1 (x) +
l 2l + 1
Pl−1 (x),
we can find from (18) and (19)
∂ Ψˆ λ,l χ; c ′ ∂χ
χ=0
= δl,λ−1 Bˆ (λ1) (c ) + δl,λ+1 Bˆ (λ2) (c ),
(20)
20
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
where
By transforming the integral over c2 in the same manner, we obtain
λ d λ−1 − , 2λ − 1 dc c d λ+2 λ+1 (2) + . Bˆ λ (c ) = 2λ + 3 dc c
(1)
Bˆ λ (c ) =
(21)
4. Inter-kernel relations in two different bases The collision integral is invariant with respect to the choice of basis. Let us expand it in terms of Legendre polynomials in different bases
∂ f (c) ∂t col
∞ ∂ fl (c ) = Pl (x) ∂t l =0 col ∞ ′ ∂ Fλ c = Pλ x′ . ∂ t λ=0
= col
∞ l =0
∂ fl (c ) ′ ˆ Ψλ,l χ; c . ∂t
(23)
col
(a,b) ∞ ∞ ∂ Fλa c ′ = Gλλ1 ,λ2 c ′ , c1′ , c2′ ∂t 0 λ1 ,λ2 0 col ′ b ′ ′ 2 ′ ′ 2 ′ a × Fλ1 c1 Fλ2 c2 c1 dc1 c2 dc2 , a (a,b) ∞ ∞ ∂ fl (c ) Gll1 ,l2 (c , c1 , c2 ) = ∂t 0 0 col l ,l
(24)
(25)
Using (23), (25) and taking into account that fl (c ) and Fλ c ′
are interrelated by inverse operator fl (c ) =
∞
λ=0
Ψˆ l,λ (−χ; c )
1
Let us consider the integral over c1 . Using definition (16) and 2 equality c ′ 1 dc ′ 1 dx′1 = c12 dc1 dx1 , we can write ∞
(c , c1 , c2 )Ψˆ l1 ,λ1 (−χ; ) ( ) ∞ 1 = Gll1 ,l2 (c , c1 , c2 ) pl1 (x1 )Pλ1 (x′1 ) Fλa1 (c1′ )c1′2 dc1′ dx′1 . Gll1 ,l2
0
Since function Fλ1 c1 Fλ2 c Gλλ1 ,λ2 c ′ , c1′ , c2 =
−1
′
a
c1 Fλ1 c1 c12 dc1
−1
′
Gll1 ,l2 (c , c1 , c2 ) pl1 (x1 ) Pλ1 x1
=
l ,l 1 ,l 2
2
is arbitrary, we have
πλ1 Ψˆ λ,l χ; c ′ Ψˆ λ1 ,l1 χ; c1′ πl1
πλ2 Ψˆ λ ,l χ ; c2′ Gll1 ,l2 (c , c1 , c2 ) . πl2 2 2
(26)
The dependences of all nonlinear kernels on c ′ , c1′ , and c2′ at fixed values of indices l, l1 , and l2 are the same in any basis, and the kernels are independent of χ . Hence, the left-hand side of (26) depends on three variables, c ′ , c1′ , c2′ , while the right-hand side depends on four variables, c ′ , c1′ , c2′ , χ . Differentiation of (26) with respect to χ gives zero on the left-hand side and derivatives of the transition operators on the right-hand side. Therefore, at χ = 0 we have
∂ Ψˆ λ,l χ; c ′ ∂χ l
Glλ1 ,λ2 c , c1′ , c2′
πλ ∂ Ψˆ λ1 ,l1 χ; c1′ 1 + Gλl1 ,λ2 c ′ , c1 , c2′ πl1 ∂χ l1 χ=0 πλ ∂ Ψˆ λ2 ,l2 χ; c2′ 2 + Gλλ1 ,l2 c ′ , c1′ , c2 = 0. πl2 ∂χ l
(27)
χ=0
2
By substituting the expressions for the transition operator derivative (20) and (21) into (27) and replacing all λ by l, we obtain the relation between kernels in one and the same basis (2)
+ Bˆ (l13) (c1 )Gll1 +1,l2 (c , c1 , c2 ) + Bˆ (l14) (c1 )Gll1 −1,l2 (c , c1 , c2 ) + Bˆ (l23) (c2 )Gll1 ,l2 +1 (c , c1 , c2 ) + Bˆ (l24) (c2 )Gll1 ,l2 −1 (c , c1 , c2 ) = 0.
ˆ (1)
Bl (c ) = (2) Bˆ l (c ) =
(3) Bˆ l (c ) =
By using (15), we can represent the integral over x1 as 1
′
′
l ,l 1 ,l 2
(28)
Here, in accordance with (20), (21) and (27), we have
× Ψˆ l2 ,λ2 (−χ; c2 ) Fλb2 c2′ c12 dc1 c22 dc2 .
0
Ψˆ λ,l χ; c ′
l πλ1 ′ πλ2 ˆ ′ ˆ × Ψλ ,l χ; c1 Ψλ ,l χ; c2 Gl1 ,l2 (c , c1 , c2 ) πl1 1 1 πl2 2 2 2 2 × Fλa1 c1′ Fλb2 c ′ 2 c ′ 1 dc ′ 1 c ′ 2 dc ′ 2 = 0. ′ b ′ a
(1)
λ1 ,λ2
0
1 Bˆ l (c )Gll− (c , c1 , c2 ) + Bˆ l (c )Gll1+,1l2 (c , c1 , c2 ) 1 ,l 2
, we obtain
(a,b) ∞ ∞ ∂ Fλa c ′ = Ψˆ λ,l χ; c ′ Gll1 ,l2 (c , c1 , c2 ) ∂t 0 l,l1 ,l2 0 col × Ψˆ l1 ,λ1 (−χ; c1 ) Fλa c1′
χ=0
1 2
× fl1a (c1 ) fl2b (c2 ) (c1 )2 dc1 (c2 )2 dc2 .
Fλ c
Gλλ1 ,λ2 c ′ , c1′ , c2′ −
5. Inter-kernel relations in one and the same basis
According to (11), collision integral expansion coefficients in bases u1 and u0 can be represented as
′
0
(22)
Now we calculate a scalar product of identity (22) and normalized Legendre polynomial in basis u1 , i.e., multiply (22) by pλ x′ and ′ integrate the product over x . By using (14), we obtain
∂ Fλ c ′ ∂t
λ1 ,λ2
∞
×
col
∞
dx′1
πλ1 Ψˆ λ ,l χ; c ′ Gll1 ,l2 (c , c1 , c2 ) . π l1 1 1
(4) Bˆ l (c ) =
∂ − 2l − 1 ∂ c l+1 ∂ + 2l + 3 ∂ c l+1 ∂ + 2l + 1 ∂ c ∂ l − 2l + 1 ∂ c l
l−1
,
c l+2
,
c l+2
,
c l−1 c
(29)
.
The set of Eqs. (28) is similar to the algebraic relations between MEs obtained in [2].
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
Note that it is possible to derive these relations by using the recurrence formula for MEs (6). To this end, we should l
r
multiply (6) by the product M (c )c l Slr+1/2 (c 2 ) c11 Sl11+1/2 (c12 )/σr1 l1
l
r
c22 Sl22+1/2 (c22 )/σr2 l2 , sum up the result over r , r1 , r2 , and use
the nonlinear kernel representation (8) and relationships between Sonine polynomials M (c )
(c ) =
1
d
l−1
− + M (c ) (c ), dc c 1 d l+2 + M (c )c l+1 Slr+−31/2 (c 2 ), rM (c )c l−1 Slr−1/2 (c 2 ) = c l Slr+1/2
2
2
2
c l Slr+1/2 (c 2 )
σ r ,l (r + 1)
=
1
d
+
dc
l+2
c l−1 Slr−1/2
2
c l+1 Slr+3/2 (c 2 )
2
dc
2
c
(30)
6
λ−1 τ ν−1
λ− 1 τ −1 ν −2
i.e., calculation of the kernels with λ = 1. Suppose the kernel G00,0 (c , c1 , c2 ) is known. Relations (31) at λ = 1 are
(3)
(2)
B0 (c )
G10,1
(34)
(3)
(c , c1 , c2 ) + B0 ( )
= −B1 ( )
c2 G00,0
c1 G01,1
(c , c1 , c2 )
(c , c1 , c2 ) .
It can be shown that the kernels with λ = 1 satisfy boundary conditions G10,1 (0, c1 , c2 ) = 0,
G11,0 (0, c1 , c2 ) = 0.
G01,1 (c , 0, c2 ) = 0, G01,1
G11,0 (c , 0, c2 ) = 0.
(c , c1 , 0) = 0,
G10,1
(35)
(c , c1 , 0) = 0.
In solving (34) under boundary conditions (35) we will use inverse operators, such as
(3)
B0 (c2 )
−1
f =
c2
1 c22
f (t )t 2 dt .
(36)
0
−1
(3)
By applying operator B0 (c2 )
to the first equation of set (34),
we obtain
−1
(3)
G01,1 (c , c1 , c2 ) = − B0 (c2 )
(2)
B0 (c )G11,0 (c , c1 , c2 )
+ B(14) (c1 ) G00,0 (c , c1 , c2 ) .
(c , c1 , c2 )
− Bˆ (l24−) 1 (c2 )Gll1 ,l2 −2 (c , c1 , c2 ) .
(31)
ν = l2
(37)
Similarly, we can express G10,1 (c , c1 , c2 ) in terms of G11,0 (c , c1 , c2 )
To create a recurrence procedure, it is convenient to introduce parameters
τ = (l − l1 + l2 ) /2,
(2)
(4)
1 = −Bˆ (l 1) (c )Gll− (c , c1 , c2 ) − Bˆ (l14) (c1 )Gll1 −1,l2 −1 (c , c1 , c2 ) 1 ,l 2 − 1
(32)
instead of indices l, l1 , and l2 . Conditions (10) under which the kernel Gll1 ,l2 may differ from zero are equivalent to the inequalities 0≤τ ≤ν≤λ
5
λ−1 τ −1 ν−1
= −B(11) (c )G00,0 (c , c1 , c2 )
(2)
λ = (l + l1 + l2 ) /2,
4
λ τ −1 ν−1
(3)
1 Bˆ l2 −1 (c2 )Gll1 ,l2 (c , c1 , c2 ) + Bˆ l (c )Gll+ (c , c1 , c2 ) 1 ,l 2 − 1
+ Bl1 ( )
3
λ τ ν−1
B0 (c2 ) G10,1 (c , c1 , c2 ) + B0 (c1 ) G11,0 (c , c1 , c2 )
By using relation (28), a recurrence procedure for derivation of kernels with arbitrary indices can be constructed. A similar approach was used in [2–4] to construct a procedure for obtaining nonlinear MEs with arbitrary high indices from algebraic relations (6). In constructing MEs all the values of indices r , r1 , and r2 were taken into consideration in addition to all the values of indices l, l1 , and l2 . Following [2–4], we replace l2 + 1 in (28) by l2 . Now we rearrange terms in (28) as
c1 Gll1 +1,l2 −1
2
λ τ ν
= −B(14) (c1 ) G00,0 (c , c1 , c2 )
6. Construction of the recurrence procedure
ˆ (3)
1
λi τi νi
(3)
This is another procedure for derivation of (28)–(29). It can be shown that, vice versa, relations (6) can be obtained from (28)–(29). To this end, it is sufficient to multiply (28) by respective Sonine polynomials and integrate the result over c , c1 , c2 taking into account (8) and (30). Hence, relations (6) and (28) are equivalent.
(3)
i
B0 (c2 )G01,1 (c , c1 , c2 ) + B0 (c ) G11,0 (c , c1 , c2 )
.
σr ,l+1 l −1 r +1 2 (c ) 1 d l − 1 c Sl−1/2 (c ) = . − + σr ,l 2 dc c σr +1,l−1
c l Slr+1/2
Table 1 Parameters λi , τi and νi for the recurrence relation terms (31); i corresponds to the term position in (31).
c
21
(33)
where λ, τ and ν are integers. Parameters λ, τ , and ν for all kernels of (31) are given in Table 1. Note that, if one of the indices λi , τi , νi becomes negative, the corresponding kernel vanishes, and the number of terms in the recurrence relation decreases. To find all relations at fixed λ, it is sufficient to consider all combinations of parameters τ and ν satisfying the inequality 0 ≤ τ ≤ ν ≤ λ + 1 and to ignore the relations containing only zero terms. If all kernels with λ = λ0 − 1 are known, it is possible to find all kernels with λ = λ0 from recurrence relations (31). Note that relations (31) are written in such a way that all the known kernels at each step of the recurrence procedure are assembled on the right side. In this paper we consider the first recursion step,
−1
(3)
G10,1 (c , c1 , c2 ) = − B0 (c2 )
(3)
B0 (c1 ) G11,0 (c , c1 , c2 )
+ B(11) (c )G00,0 (c , c1 , c2 ) .
(38)
By using (37), (38), we obtain from the third equation of (34) G11,0 (c , c1 , c2 ) = −
1 2
(3)
−1
B0 (c1 )
(2)
(3)
B1 (c ) + B0 (c )
−1
−1 −1 (3) × B(14) (c1 ) + B(03) (c1 ) B0 (c ) × B(03) (c2 ) B(14) (c2 ) G00,0 (c , c1 , c2 ) .
(39)
By substituting (39) into (37) and (38) we can get expressions for G01,1 (c , c1 , c2 ) and G10,1 (c , c1 , c2 ). Thus, the kernels for λ = 1 can
be constructed if the kernel G00,0 (c , c1 , c2 ) is known. Note that the analysis performed above is valid for the loss term and gain term of the collision integral individually. It is rather easy to construct the loss term kernels (see [18,22]). The gain term (its kernels will be marked with the upper index +) is much more complicated than the loss term. It was shown in [18] that
22
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
l +l Fig. 1. Domain structure of kernels G+ l1 ,l2 (c , c1 , c2 ) (a) and reduced kernels Ψl1 ,l2 (x, y) (b).
+ r ,l
Table 2 Definition of regions D1–D4. D1
D2
D3
D4
c 2 ≤ c12 + c22 , c1 ≤ c , c2 ≤ c
c1 ≤ c , c2 ≥ c
c 1 ≥ c , c2 ≤ c
c 1 ≥ c , c2 ≥ c
0 G+ 1,1 (c , c1 , c2 ) ≡ 0,
l the nonlinear collision integral kernels G+ l,0 (c , c1 , c2 ) are related to
linear (or linearized) collision integral kernels Ll (c , c1 ; ζ ) by +
L+ l (c , c1 ; ζ ) =
l 2 G+ l,0 (c , c1 , c2 )M (c2 , ζ )c2 dc2 .
0
(40)
ζ 3/2
Here M (c , ζ ) = π exp(−ζ c 2 ) is the Maxwellian, ζ = TT , B and TB is the background temperature. Eq. (40) is in essence the 2 Laplace transform from c2 to ζ . Analytical expressions for the kernels L+ l (c , c1 ; ζ ) were obtained for hard spheres (HS) in [12, 13] and for pseudo-Maxwellian molecules (PMM) (i.e., for the interaction model in which the cross section is isotropic and inversely proportional to the relative velocity) in [14]. 0 In these cases the kernel G+ 0,0 (c , c1 , c2 ) can be found by using the inverse Laplace transformation, and then the remaining kernels can be constructed through the recurrence relations. It was found that the structures of the kernels obtained are the same in both models under consideration. The entire domain c ≥ 0, c1 ≥ 0, c2 ≥ 0 of kernels is subdivided into five regions (see Fig. 1(a)). Derivatives are discontinuous on the boundaries of these regions. In region D0 (c12 + c22 < c 2 ) all kernels vanish, which is a consequence of the energy conservation law. All remaining regions are presented in Table 2. Analytical formulae for the kernels with λ = 0, 1 for HS model are given in Table 3. Table 3 shows that 1 G+ 1,0 (c , c1 , c2 )
=
1 G+ 0,1 (c , c2 , c1 ).
(42)
0 i.e., the kernel G+ 1,1 is zero in all regions for the PMM model. This property of the PMM model is a consequence of the theorem proved in [2]: in the interaction models involving the cross section inversely proportional to relative velocity (as for Maxwellian molecules) and arbitrarily depending on scattering angle the MEs r ,0 Kr1 ,2q+1,r2 ,2q+1 (where q = 0, 1, 2, . . .) are zero. As a consequence,
∞
+r ,l
the symmetry of MEs Kr1 ,l1 ,r2 ,l2 = Kr2 ,l2 ,r1 ,l1 that takes place for the interaction model of any type in which the cross section is isotropic. The kernels for the PMM model constructed by the recurrence procedure exhibit symmetry (41) (as expected). In addition,
G01,1 = G03,3 = G05,5 = · · · ≡ 0. Identity (42) is a particular case of this general result. The fact that it can be found with the aid of the recurrence procedure is an additional proof of the validity of this procedure. Taking into account the kernel properties given by (41) and (42), it is sufficient to present here the formulae only for the kernels 0 +1 G+ 0,0 and G1,0 . These formulae are given in Table 4. In [18] we have shown that the kernels corresponding to the power potential can be represented as Gll1 ,l2
(c , c1 , c2 ) =
c2
+
c12
+
c22
−3+2µ
Ψll1 ,l2 (x, y) .
Parameter µ is determined by the index of power in the power potential expression. In the case of hard spheres µ = 1/2, while for Maxwellian molecules µ = 0. Variables x and y are related to velocities c , c1 , and c2 by x=
(41)
Note that permutation c1 c2 leads to permutation of the D2 and D3 regions. The symmetry described by (41) is equivalent to
c c 2 + c12 + c22
,
y=
c1 c 2 + c12 + c22
,
l and Ψl+ (x, y) is a reduced kernel. 1 ,l2
Table 3 Kernels for the HS model at λ = 0, 1. Kern
D1
c12 +c22 −c 2
0 G+ 0,0
8π c1 c2
1 G+ 1,0
8π c 2 c12 c2
8π c 2 c1 c22
1 G+ 0,1
c12 (c12 + c22 − c 2 )1/2 − c22 (c12 + c22 − c 2 )1/2 −
0 G+ 1,1
−
8π c12 +c22 −c 2 9cc12 c22
3/2
2 3
D3
8π cc2
8π cc1
(c12 + c22 − c 2 )3/2
8π c1 3c 2 c2
8π c 2 c12
(c12 + c22 − c 2 )3/2
8π c 2 c22
c2 2 3
D2
c2 −
π c1 − 89cc 2 2
2 2 c 3 1
D4 8π c1 c2
c2 −
8π c2 3c 2 c1
2 2 c 3 2
8π c 3c12 c2 8π c 3c1 c22
π c2 − 89cc 2 1
8π c − 9c 2 c2 2
1 2
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
23
Table 4 0 +1 Kernels G+ 0,0 and G1,0 for the PMM model. 0 G+ 0 ,0
1 G+ 1,0
D1
8π cc1 c2
arctan
D2
8π cc1 c2
arctan
D3
8π cc1 c2
arctan
D4
4π 2 cc1 c2
c12 +c22 −c 2
2π c 2 c12 c2
c2
c12
c22
2π c 2 c12 c2
c12
1 − π2 arctan
c12 +c22 −c 2
c2
2π
c12 +c22 −c 2
c12 +c22
c1 c2 2c 2 −c12 −c22 c12 +c22
c 2 c12 c2
√
c2 c1 c12 +c22 −2c 2
c 2 c12 c2
c c12 −c22
c12 +c22
2π
c22
c c22 −c12
√
√ c12 +c22 −c 2 + 2c 2 + c12 − c22 arctan c
+ 2c 2 + c12 − c22 arctan cc21
+ 2c 2 + c12 − c22 arctan cc12
c12 +c22 −c 2 c12 +c22
+ 2c 2 + c12 − c22 arctan √
c
c12 +c22 −c 2
Fig. 2. Reduced kernels Ψ1+,01 (x, y) for pseudo-Maxwellian molecules (a) and hard spheres (b).
Hence, the kernel is determined by two-variable function
l Ψl+ (x, y). Regions D0–D4 in the space of velocity magni1 ,l 2 tudes transform into the D0–D4 regions on the x, y plane (see Fig. 1(b)). Fig. 2 presents reduced kernels Ψ1+,01 (x, y) for Maxwellian
molecules and hard spheres. One can see that these reduced kernels differ from each other only slightly. It can be supposed that the reduced kernels for the intermediate values of µ (between 0 and 1/2) have a similar shape. Note that the kernels of the gain terms of the nonlinear collision integral we obtained satisfy symmetry relations and, together with the kernels of the loss terms found in [22], ensure fulfillment of conservations laws. 7. Summary We have obtained relations between kernels of the nonlinear collision integral on the basis of collision integral invariance with respect to the choice of basis. By using these relations, a recurrence procedure for construction of kernels with ascending indices has been developed. At each recursion step, kernels are found by solving partial differential equations. A similar recurrence procedure was suggested in [2] to calculate MEs but the relations between MEs were algebraic, in contrast to the differential relations derived in this paper. The kernel G00,0 necessary for launching the recurrence procedure can be obtained by using the relation between linear and nonlinear collision integral kernels that can be regarded as Laplace transformation [23]. The first step of the recurrence procedure has been fulfilled, and analytical representations for the kernels G10,1 , G11,0 , and G10,1 for the cases of hard spheres and PMM have been obtained. The recurrence relations we have derived are suitable for finding kernels of the nonlinear collision integral for arbitrary interaction models and arbitrary sets of indexes l, l1 , l2 . This can provide a tool for determining a distribution function in a highly non-equilibrium situation.
Acknowledgment The work was supported by the Russian Foundation for Basic Research (project No 09.08.01017). References [1] D. Burnett, The distribution of molecular velocities and the mean motion in a non-uniform gas, Proc. Lond. Math. Soc. (3) 40 (1936) 382. [2] A.Ya. Ender, I.A. Ender, The collision integral of Boltzmann equation and moment method, St. Petersburg University Publishing House, St. Petersburg, 2003 (in Russian). [3] A.Ya. Ender, I.A. Ender, Polynomial expansions for the isotropic Boltzmann equation and invariance of the collision integral with respect to the choice of basis functions, Phys. Fluids 11 (1999) 2720. [4] A.Ya. Ender, I.A. Ender, Properties of the collision integral in the axisymmetric Boltzmann equation, Transport Theory Statist. Phys. 36 (7) (2007) 563. [5] J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland Publishing Company, Amsterdam, 1972. [6] S.K. Loyalka, E.L. Tipton, R.V. Tompson, Chapman–Enskog solutions to arbitrary order in Sonine polynomials I: simple, rigid-sphere gas, Physica A 379 (2007) 417. [7] E.L. Tipton, R.V. Tompson, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials II: viscosity in a binary, rigid-sphere, gas mixture, Eur. J. Mech. B Fluids 28 (2009) 335. [8] E.L. Tipton, R.V. Tompson, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials III: diffusion, thermal diffusion, and thermal conductivity in a binary, rigid-sphere, gas mixture, Eur. J. Mech. B Fluids 28 (2009) 353. [9] R.V. Tompson, E.L. Tipton, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials IV: summational expressions for the diffusionand thermal conductivity-related bracket integrals, Eur. J. Mech. B Fluids 28 (2009) 695. [10] R.V. Tompson, E.L. Tipton, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials V: summational expressions for the viscosityrelated bracket integrals, Eur. J. Mech. B Fluids 29 (2010) 153. [11] A.Ya. Ender, I.A. Ender, A.B. Gerasimenko, Standard moment method in the problems on ion kinetics in neutral gas, Open Plasma Phys. J. 2 (2009) 24. [12] D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912) 562. [13] E. Hecke, Über die Integralgleichung der kinetischen Gastheorie, Math. Z. 12 (1922) 274.
24
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
[14] A.Ya. Ender, I.A. Ender, Analytical representation of linear kernels in collision integral of Boltzmann equation for Maxwellian molecules, Tech. Phys. Lett. 37 (2011) 197. [15] C.L. Pekeris, Z. Alterman, L. Finkelstein, K. Frankoowski, Propagation of sound in a gas of rigid spheres, Phys. Fluids 5 (1962) 1608. [16] S.K. Loyalka, Temperature jump and thermal creep slip: rigid sphere gas, Phys. Fluids A 1 (1989) 403. [17] R.D.M. Garcia, C.E. Siewert, The viscous-slip, diffusion-slip, and thermal-creep problems for a binary mixture of rigid spheres described by the linearized Boltzmann equation, Eur. J. Mech. B Fluids 26 (2007) 749. [18] A.Ya. Ender, I.A. Ender, L.A. Bakaleinikov, Some general properties of the nonlinear collision integral in the Boltzmann equation, Tech. Phys. 55 (2010) 1400.
[19] L.A. Bakaleinikov, A.Ya. Ender, I.A. Ender, Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case, Tech. Phys. 51 (2006) 1110. [20] L.A. Bakaleinikov, E.Yu. Flegontova, A.Ya. Ender, I.A. Ender, Calculation of the linear kernel of the collision integral for the hard-sphere potential, Tech. Phys. 54 (2009) 182. [21] I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Series and Products, Fizmatlit, Moskow, 1962, (in Russian) p. 1092. Academic, New York, 1965. [22] A.Ya. Ender, I.A. Ender, L.A. Bakaleinikov, E.Yu. Flegontova, Matrix elements and kernels of the collision integral in the Boltzmann equation, Tech. Phys. 56 (2011) 452. [23] A.Ya. Ender, I.A. Ender, L.A. Bakaleinikov, E.Yu. Flegontova, Construction of kernels of the nonlinear collision integral in the Boltzmann equation using Laplace transformation, Tech. Phys. 57 (2012) 735.
Contents lists available at SciVerse ScienceDirect
European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu
Recurrence relations between kernels of the nonlinear Boltzmann collision integral A.Ya. Ender a , I.A. Ender b , L.A. Bakaleinikov a,∗ , E.Yu. Flegontova a a
Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russia
b
St. Petersburg State University, St. Petersburg, 199034, Russia
article
abstract
info
Article history: Received 28 July 2011 Received in revised form 29 February 2012 Accepted 6 March 2012 Available online 17 April 2012
A new approach to the solution of the nonlinear Boltzmann equation based on distribution function expansion in terms of spherical harmonics is considered. Owing to this approach, the complex five-fold collision integral can be replaced by a set of rather simple integral operators. Recurrence relations between nonlinear kernels Gll1 ,l2 (c , c1 , c2 ) of the integral operators are derived on the basis of the collision integral invariance with respect to choice of the reference frame velocity. Recurrence formulae allow one to find kernels with arbitrary indices provided the kernel G00,0 (c , c1 , c2 ) is known. Explicit analytical expressions for kernels G11,0 , G10,1 , and G01,1 for the cases of hard spheres and Maxwellian molecules are presented. © 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Boltzmann equation Distribution function Spherical harmonics Nonlinear kernels Inter-kernel relations Collision integral
3/2
1. Introduction Matrix elements (MEs) of the Boltzmann collision integral resulting from distribution function (DF) expansion in spherical Hermitean polynomials (Burnett’s functions [1]) were investigated in our previous papers [2–4]. Matrix elements Kji,k are defined as Kji,k =
Hi ˆI (MT Hj , MT Hk ) d3 v/gi ,
gi =
MT Hi2 d3 v,
(1)
Hj (c) =
(Θ , ϕ) (c ), √ i = 0, 1; c = α(v − u), α = m/(2kT ), c l Slr+1/2
2
0 Ylm (Θ , ϕ) = Plm (cos Θ ) cos mϕ, 1 Ylm (Θ , ϕ) = Plm (cos Θ ) sin mϕ,
0 ≤ m ≤ l.
(2) (3)
∗ Correspondence to: Department of Applied Mathematics and Mathematical Physics, A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26, Politechnicheskaya Street, St. Petersburg, 194021, Russia. Tel.: +7 812 2927112; fax: +7 812 5504890. E-mail addresses: [email protected], [email protected] (L.A. Bakaleinikov). 0997-7546/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2012.03.004
2
(K0i ,k ) kind. As shown in [2], collision integral invariance with respect to rotation leads to a proportionality between arbitrary MEs r ,l,m,i Kr ,l ,m ,i ;r ,l ,m ,i and the corresponding axisymmetric MEs 1 1
where ˆI is the collision integral. The spherical Hermitean polynomials Hj are defined as i Ylm
Index j includes four indices (r , l, m, i), MT = πα e−α(v−u) is the Maxwellian weight function characterized by temperature T i and average velocity u, Ylm (Θ , ϕ) are the real spherical harmonics, m Pl (cos Θ ) are the associated Legendre functions, and Slr+1/2 (c 2 ) are the Sonine (Laguerre) polynomials. Linear MEs correspond to the linear collision integral and can be of the first (Kji,0 ) and second
r ,l
1 1
2 2
2 2
r ,l,0,0
Kr ,1 ;r ,l = Kr ,1 ,0,0;r ,l ,0,0 , i.e., the equation 1 1 2 2 1 l 2 2 r ,l,m,i
l,m,i
r ,l
Kr1 ,l1 ,m1 ,i1 ,r2 ,l2 ,m2 ,i2 = Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 Kr1 ,l1 ,r2 ,l2
(4) l,m,i
is valid. Here, the universal numerical coefficients Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 that are independent of scattering cross section can easily be expressed through the Clebsch–Gordan coefficients. Therefore, problems with arbitrary DF can be solved if axisymmetric MEs alone are known, even when DF are not axisymmetric. To study properties of axisymmetric MEs, we consider here the axial symmetric case. Spherical harmonics are identical in this case to Legendre polynomials, and the set of four indices j converts into a pair r , l. The distribution function can be expanded in the vicinity of a Maxwellian which may be characterized by arbitrary temperature T and average velocity magnitude u. Collision integral
18
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
invariance with respect to the Maxwellian temperature choice leads to the ‘‘temperature’’ recurrence relationships for MEs
T
d dT
l − R Krr1,,ll1 ,r2 ,l2 = rKrr1−,l11,,lr2 ,l2 − (r1 + 1)Krr1,+ 1,l1 ,r2 ,l2
− (r2 + 1)Krr1,,ll1 ,r2 +1,l2 .
(5)
Here R = r1 + r2 − r + (l1 + l2 − l)/2. Invariance with respect to the average velocity magnitude yields the ‘‘velocity’’ recurrence relationships 1 r −1,l+1 r ,l β(l − 1)Krr1,,l− l1 ,r2 ,l2 + γ (r − 1, l + 1)Kr1 ,l1 ,r2 ,l2 − β(l1 )Kr1 ,l1 +1,r2 ,l2 r ,l l − γ (r1 , l1 )Krr1,+ 1,l1 −1,r2 ,l2 − β(l2 )Kr1 ,l1 ,r2 ,l2 +1
+ γ (r2 , l2 )Krr1,,ll1 ,r2 +1,l2 −1 = 0,
(6)
where
β(l) = −
l+1 2l + 1
,
γ (r , l) =
(r + 1) l . 2l + 1
In [2,4] these relationships were generalized for the case of a mixture of gases with arbitrary molecular masses. Recurrence procedures were developed to obtain arbitrary MEs from the simplest isotropic (l = 0) linear diagonal MEs. To perform the calculations, special high-accuracy arithmetic routines were developed. The recurrence relations allowed both linear and nonlinear MEs with very high indices to be obtained (in our calculations, each index r , l, r1 , l1 , r2 , l2 varied from 0 to 128). ‘‘Temperature’’ recurrence relationships (5) for power potentials proved to be algebraic, and MEs were found by using the recurrence procedures based on (5)–(6). In the case of arbitrary interaction potentials, isotropic linear MEs should be expanded in terms of Ω -integrals and then expansion coefficients for arbitrary MEs can be found through the recurrence procedures [2]. Note that, while calculating all nonlinear MEs, we obtain also linear nonisotropic MEs. Linear MEs differ from the widely known bracket integrals [5] by a numerical factor alone. The bracket integrals are the object of continuous interest. Earlier the bracket integrals were used to calculate kinetic theory transport coefficients (only bracket integrals with l = 1, l = 2 and low r were used). Recently high-accuracy calculations of the bracket integrals with indices l = 1 and l = 2 and high r have been presented [6–10]. To remind, the index r at a fixed l corresponds to the Sonine polynomial degree in the spherical Hermitean polynomial expansion. The possibility of obtaining MEs with high indices [2–4] gave a new impetus to advances in the standard moment method. This method, using a large number of MEs, was applied in [11] to investigate ion DF evolution after triggering a rather strong electric field or crossed electric and magnetic fields. The DF was calculated for velocities of up to ten thermal velocities. However, the requirement for DF expansion convergence in the standard moment method imposes a restriction on DF. This restriction consists in a finiteness of the DF norm in the Hilbert space spanned by the Sonine polynomials mutually orthogonal with respect to a Maxwellian weight function. This limitation will be removed if DF is expanded into spherical harmonics. In this case a complex five-fold collision integral will be replaced by a set of relatively simple integral operators. The kernels Gll1 ,l2 (c , c1 , c2 ) of the operators depend on the velocity magnitudes alone and play the same role as MEs in the standard moment method. The problem of kernel construction is a great challenge. Suffice it to say that explicit expressions even for linear kernels have so far only been found for the cases of hard spheres [12,13] and Maxwellian molecules [14]. Hilbert [12] was the first to transform the Boltzmann collision integral into a Fredholm-type integral with a symmetrical kernel. This transformation was accomplished only for the hard spheres model. Expansion of this kernel in spherical
harmonics was performed by Hecke [13]. Later this result was also obtained in [15]. Use of the kernels that depend on velocity magnitudes alone made the process of solving kinetic equation much simpler. This approach yielded a number of valuable results in rarefied gas dynamics boundary problems [16,17]. It should be emphasized that all these results were obtained for the linearized Boltzmann equation alone. The construction of nonlinear kernels will enable solution of a wide range of problems in gas dynamics and plasma physics where the distribution function is far from equilibrium and the nonlinear character of processes is essential. We showed [2,18] that nonlinear kernels may be expressed in terms of nonlinear MEs. Existence of relations between MEs suggests that there are similar inter-kernel relations that follow from collision integral invariance with respect to the choice of basis. 2. Kernels of the nonlinear collision integral By using (4) we showed in [2,18] that, if DF is expanded in spherical harmonics, the collision integral can always be represented through axisymmetric kernels Gll1 ,l2 (c , c1 , c2 ):
l,m,i ∂ f (c) i Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 Ylm (Θ , ϕ) = ∂t col l1 m1 ,i1 l2 ,m2 ,i2 l,m,i ∞ ∞ i, × Gll1 ,l2 (c , c1 , c2 ) fl1i ,m1 (c1 ) fl2 ,m2 (c2 ) c12 c22 dc1 dc2 ,
0
(7)
0
where Gll1 ,l2 (c , c1 , c2 ) are expressed through MEs as Gll1 ,l2 (c , c1 , c2 ) = M (c )
×
c l Slr+1/2 (c 2 )Krrl1 l1 ,r2 l2
r ,r 1 ,r 2 l r c11 Sl11+1/2
l
r
(c12 ) c22 Sl22+1/2 (c22 )
σr1 l1
σrl = Γ (r + l + 3/2)/(2π 3/2 r !),
σr2 l2
,
(8)
2 M (c ) = e−c /π 3/2 .
In (7), fl1i ,m1 (c ) and fl2i ,m2 (c ) are the coefficients of DF f (c) expanl,m,i
i (Θ , ϕ). Zl1 ,m1 ,i1 ,l2 ,m2 ,i2 are sion in terms of spherical harmonics Ylm the above-mentioned universal numerical coefficients. These coefficients may differ from zero only if |ll − l2 | ≤ l ≤ l1 + l2 , m = |m1 ± m2 |, l + l1 + l2 and i + i1 + i2 are even. If l1 = m1 = 0, then Z = 1 and (l2 , m2 ) = (l, m). If l2 = m2 = 0, then Z = 1 and (l1 , m1 ) = (l, m). In the axisymmetric case, m = m1 = m2 = i = i1 = i2 = 0 and Z = 1. Therefore, kernels, as well as MEs, may be considered for the axisymmetric case without loss of generality [18]. As noted above, spherical harmonics are identical in this case to Legendre polynomials. Since we do not expand DF in terms of Sonine polynomials, the basis is now temperature-independent and is determined by the reference frame velocity alone. Let us use the thermal velocity vT corresponding to some arbitrary temperature T as a unit of velocity and consider DF in the reference frame moving with velocity u along the preferred direction ez . By defining the dimensionless velocity as c = (v − uez )/vT , we can express DF as
f (c) =
fl (c )Pl (x),
x = cos θ .
l
Here, θ is the angle between c and ez . The collision integral is expressed in terms of kernels Gll1 ,l2 as
∞ ∞ ∂ f (c) Gll1 ,l2 (c , c1 , c2 ) fl1 (c1 ) = P ( x ) l ∂t 0 0 col l ,l ,l 1 2
× fl2 (c2 ) c12 c22 dc1 dc2 .
(9)
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
The nonlinear kernels Gll1 ,l2 (c , c1 , c2 ) are nonzero if
(−1)l = (−1)l1 +l2 .
|l1 − l2 | ≤ l ≤ l1 + l2 ,
Operator Ψˆ λ,l is applied to function fl (c ). We obtain from (13) (10)
It is important to note that the collision term in Eq. (9) is given for simple gases, however, the same kernel may be used for a mixture of gases. In the latter case the DF variation rate of particles ‘‘a’’ due to their interaction with particles ‘‘b’’ is
a ∞ ∞ ∂ fa (c) (a,b) l Gl1 ,l2 = Pl (x) (c , c1 , c2 ) ∂t 0 0 ab col l ,l ,l
Ψˆ λ,l χ; c ′ fl (c ) =
1
pλ x′ Pl (x)fl (c )dx′ .
(15)
−1
Here we take into consideration the fact that c depends on integration variable x′ . Hence, the operator Ψˆ λ,l is an integral operator. In the limit χ → 0 the fl (c ) function is independent of x′ and can be taken out of the integral. In this case, the Legendre polynomial arguments are identical, and (15) becomes
Ψˆ λ,l χ; c ′ |χ=0 fl (c ) = δλ,l fl (c ).
1 2
× fl1a (c1 ) fl2b (c2 ) c12 c22 dc1 dc2 .
19
(11)
For the inverse operator Ψˆ l,λ we have from (14) 1
If particle masses are equal and their interaction potentials are identical to that of a simple gas, the kernels in (9) and (11) are the same. Due to symmetry of the product fl1 (c1 )fl2 (c2 ) with respect to permutations c1 c2 and l1 l2 , kernel Gll1 ,l2 (c , c1 , c2 )
Ψˆ l,λ (−χ; c ) Fλ c ′ =
in (11) may be replaced by a symmetrical kernel ˜
In future calculations we will need derivative ∂ Ψˆ λ,l (χ; c ′ )/∂χ|χ=0 .
(c , c1 , c2 ) = 1/2 Gll1 ,l2 (c , c1 , c2 ) + Gll2 ,l1 (c , c2 , c1 ) . However, in this paper, as Gll1 ,l2
well as in our previous publications, we consider the nonsymmetrical kernels that can be used for gas mixtures and also for simple gases. Hereinafter we omit kernel indices a and b. The formula for the linear kernel expansion in Sonine polynomials similar to (8) contains only double summation. The linear kernels for Maxwellian molecules and hard spheres were investigated and constructed by means of a double summation formula in [19,20]. To calculate the infinite sum, we used the asymptotic approach. At high indices, the terms of the sum were substituted by their asymptotic expressions, and summation was replaced by integration. However, the problem of estimating the sum remainder in (8) at high velocities emerged. This was a consequence of a complexity of the Sonine polynomials’ asymptotics at high arguments when the index tended to infinity. 3. Inter-basis transition operator and its derivative Consider the distribution function in two reference frames that differ by the velocity of motion along the preferred direction u. DF representations in these reference frames (bases) are related by a transition operator. To find it, we select two bases with velocities u0 and u1 . The difference between the velocities is denoted by χ, χ = u1 − u0 . Let us express fl (c ) and Fλ c ′ for the DF expansion coefficients in terms of Legendre polynomials in bases u0 and u1 . Magnitudes of velocities c and c ′ in bases u0 and u1 satisfy the relationships c 2 = cz2 + cρ2 ,
2 2 c ′ = c ′ z + cρ2 ,
c ′ z = cz − χ ,
cz = cx,
∞
fl (c )Pl (x) =
∞
c ′ z = c ′ x′ .
Fλ c ′ Pλ x′ .
(12)
Now we calculate the scalar product of this identity and Legendre polynomial divided by the square of its norm, πλ , in basis u1 . Integration of (13) multiplied by pλ x′ = Pλ x′ /πλ over the interval [−1, 1] results in
′
Fλ c
=
∞ l =0
Ψˆ λ,l χ; c fl (c ).
′
−1
Ψˆ l,λ (−χ; c ) |χ=0 Fλ (c ) = δl,λ Fλ (c ).
(16)
Let us differentiate function Ψˆ λ,l χ; c ′ fl (c ) defined by (15) with respect to χ taking into account that c and x depend on c ′ , x′ , and χ . This yields
∂ Ψˆ λ,l χ; c ′ fl (c ) ∂χ χ=0 1 dPl (x) ∂ x ′ = pλ (x ) f ( c ) l dx ∂χ χ=0 −1
dx′
χ=0
1
∂ c pλ (x′ ) Pl (x) ∂χ
+ −1
dc
dfl (c )
χ=0
dx′ .
(17)
χ=0
In the limit χ → 0 neither fl (c ) nor dfl (c )/dc function depends on x′ and, hence, both of them can be taken out of the integral. It follows herefrom that the operator Ψˆ λ,l derivative is a differential operator of the form
∂ (2) d = A(λ,1)l + Aλ, Ψˆ λ,l χ; c ′ l ∂χ dc χ=0
(18)
(2)
(1)
where numerical matrices Aλ,l and Aλ,l are defined by (17). To calculate these matrices, explicit forms of c and x dependences on c ′ and x′ are needed. Eq. (12) gives c 2 = c ′2 + 2c ′ x′ χ + χ 2 ,
∂ c ∂χ
, c ∂ x 1 = − x′ 2 − 1 . ∂χ χ=0 c
= x′ ,
χ=0
c ′ x′ + χ
x=
(1) Aλ,l (2) Aλ,l
=−
1
c
dPl (x′ )
1
pλ (x ) x − 1 ′
′2
−1
dx′
dx
′
,
1
(19)
pλ (x )x Pl (x )dx . ′
=
′
′
′
−1
Using the formulae for Legendre polynomials [21] (13)
λ=0
l =0
Since x|χ=0 = x′ , the first and second integrals in (17) are
Here x = cos θ , x′ = cos θ ′ are cosines of the angles between ez and velocity vectors c and c′ . Distribution function expansions in different bases are identical f (c) =
pl (x)Pλ x′ Fλ c ′ dx,
(14)
(x2 − 1) xPl (x) =
dPl (x) dx l+1
=
2l + 1
(l + 1)l [Pl+1 (x) − Pl−1 (x)], 2l + 1
Pl+1 (x) +
l 2l + 1
Pl−1 (x),
we can find from (18) and (19)
∂ Ψˆ λ,l χ; c ′ ∂χ
χ=0
= δl,λ−1 Bˆ (λ1) (c ) + δl,λ+1 Bˆ (λ2) (c ),
(20)
20
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
where
By transforming the integral over c2 in the same manner, we obtain
λ d λ−1 − , 2λ − 1 dc c d λ+2 λ+1 (2) + . Bˆ λ (c ) = 2λ + 3 dc c
(1)
Bˆ λ (c ) =
(21)
4. Inter-kernel relations in two different bases The collision integral is invariant with respect to the choice of basis. Let us expand it in terms of Legendre polynomials in different bases
∂ f (c) ∂t col
∞ ∂ fl (c ) = Pl (x) ∂t l =0 col ∞ ′ ∂ Fλ c = Pλ x′ . ∂ t λ=0
= col
∞ l =0
∂ fl (c ) ′ ˆ Ψλ,l χ; c . ∂t
(23)
col
(a,b) ∞ ∞ ∂ Fλa c ′ = Gλλ1 ,λ2 c ′ , c1′ , c2′ ∂t 0 λ1 ,λ2 0 col ′ b ′ ′ 2 ′ ′ 2 ′ a × Fλ1 c1 Fλ2 c2 c1 dc1 c2 dc2 , a (a,b) ∞ ∞ ∂ fl (c ) Gll1 ,l2 (c , c1 , c2 ) = ∂t 0 0 col l ,l
(24)
(25)
Using (23), (25) and taking into account that fl (c ) and Fλ c ′
are interrelated by inverse operator fl (c ) =
∞
λ=0
Ψˆ l,λ (−χ; c )
1
Let us consider the integral over c1 . Using definition (16) and 2 equality c ′ 1 dc ′ 1 dx′1 = c12 dc1 dx1 , we can write ∞
(c , c1 , c2 )Ψˆ l1 ,λ1 (−χ; ) ( ) ∞ 1 = Gll1 ,l2 (c , c1 , c2 ) pl1 (x1 )Pλ1 (x′1 ) Fλa1 (c1′ )c1′2 dc1′ dx′1 . Gll1 ,l2
0
Since function Fλ1 c1 Fλ2 c Gλλ1 ,λ2 c ′ , c1′ , c2 =
−1
′
a
c1 Fλ1 c1 c12 dc1
−1
′
Gll1 ,l2 (c , c1 , c2 ) pl1 (x1 ) Pλ1 x1
=
l ,l 1 ,l 2
2
is arbitrary, we have
πλ1 Ψˆ λ,l χ; c ′ Ψˆ λ1 ,l1 χ; c1′ πl1
πλ2 Ψˆ λ ,l χ ; c2′ Gll1 ,l2 (c , c1 , c2 ) . πl2 2 2
(26)
The dependences of all nonlinear kernels on c ′ , c1′ , and c2′ at fixed values of indices l, l1 , and l2 are the same in any basis, and the kernels are independent of χ . Hence, the left-hand side of (26) depends on three variables, c ′ , c1′ , c2′ , while the right-hand side depends on four variables, c ′ , c1′ , c2′ , χ . Differentiation of (26) with respect to χ gives zero on the left-hand side and derivatives of the transition operators on the right-hand side. Therefore, at χ = 0 we have
∂ Ψˆ λ,l χ; c ′ ∂χ l
Glλ1 ,λ2 c , c1′ , c2′
πλ ∂ Ψˆ λ1 ,l1 χ; c1′ 1 + Gλl1 ,λ2 c ′ , c1 , c2′ πl1 ∂χ l1 χ=0 πλ ∂ Ψˆ λ2 ,l2 χ; c2′ 2 + Gλλ1 ,l2 c ′ , c1′ , c2 = 0. πl2 ∂χ l
(27)
χ=0
2
By substituting the expressions for the transition operator derivative (20) and (21) into (27) and replacing all λ by l, we obtain the relation between kernels in one and the same basis (2)
+ Bˆ (l13) (c1 )Gll1 +1,l2 (c , c1 , c2 ) + Bˆ (l14) (c1 )Gll1 −1,l2 (c , c1 , c2 ) + Bˆ (l23) (c2 )Gll1 ,l2 +1 (c , c1 , c2 ) + Bˆ (l24) (c2 )Gll1 ,l2 −1 (c , c1 , c2 ) = 0.
ˆ (1)
Bl (c ) = (2) Bˆ l (c ) =
(3) Bˆ l (c ) =
By using (15), we can represent the integral over x1 as 1
′
′
l ,l 1 ,l 2
(28)
Here, in accordance with (20), (21) and (27), we have
× Ψˆ l2 ,λ2 (−χ; c2 ) Fλb2 c2′ c12 dc1 c22 dc2 .
0
Ψˆ λ,l χ; c ′
l πλ1 ′ πλ2 ˆ ′ ˆ × Ψλ ,l χ; c1 Ψλ ,l χ; c2 Gl1 ,l2 (c , c1 , c2 ) πl1 1 1 πl2 2 2 2 2 × Fλa1 c1′ Fλb2 c ′ 2 c ′ 1 dc ′ 1 c ′ 2 dc ′ 2 = 0. ′ b ′ a
(1)
λ1 ,λ2
0
1 Bˆ l (c )Gll− (c , c1 , c2 ) + Bˆ l (c )Gll1+,1l2 (c , c1 , c2 ) 1 ,l 2
, we obtain
(a,b) ∞ ∞ ∂ Fλa c ′ = Ψˆ λ,l χ; c ′ Gll1 ,l2 (c , c1 , c2 ) ∂t 0 l,l1 ,l2 0 col × Ψˆ l1 ,λ1 (−χ; c1 ) Fλa c1′
χ=0
1 2
× fl1a (c1 ) fl2b (c2 ) (c1 )2 dc1 (c2 )2 dc2 .
Fλ c
Gλλ1 ,λ2 c ′ , c1′ , c2′ −
5. Inter-kernel relations in one and the same basis
According to (11), collision integral expansion coefficients in bases u1 and u0 can be represented as
′
0
(22)
Now we calculate a scalar product of identity (22) and normalized Legendre polynomial in basis u1 , i.e., multiply (22) by pλ x′ and ′ integrate the product over x . By using (14), we obtain
∂ Fλ c ′ ∂t
λ1 ,λ2
∞
×
col
∞
dx′1
πλ1 Ψˆ λ ,l χ; c ′ Gll1 ,l2 (c , c1 , c2 ) . π l1 1 1
(4) Bˆ l (c ) =
∂ − 2l − 1 ∂ c l+1 ∂ + 2l + 3 ∂ c l+1 ∂ + 2l + 1 ∂ c ∂ l − 2l + 1 ∂ c l
l−1
,
c l+2
,
c l+2
,
c l−1 c
(29)
.
The set of Eqs. (28) is similar to the algebraic relations between MEs obtained in [2].
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
Note that it is possible to derive these relations by using the recurrence formula for MEs (6). To this end, we should l
r
multiply (6) by the product M (c )c l Slr+1/2 (c 2 ) c11 Sl11+1/2 (c12 )/σr1 l1
l
r
c22 Sl22+1/2 (c22 )/σr2 l2 , sum up the result over r , r1 , r2 , and use
the nonlinear kernel representation (8) and relationships between Sonine polynomials M (c )
(c ) =
1
d
l−1
− + M (c ) (c ), dc c 1 d l+2 + M (c )c l+1 Slr+−31/2 (c 2 ), rM (c )c l−1 Slr−1/2 (c 2 ) = c l Slr+1/2
2
2
2
c l Slr+1/2 (c 2 )
σ r ,l (r + 1)
=
1
d
+
dc
l+2
c l−1 Slr−1/2
2
c l+1 Slr+3/2 (c 2 )
2
dc
2
c
(30)
6
λ−1 τ ν−1
λ− 1 τ −1 ν −2
i.e., calculation of the kernels with λ = 1. Suppose the kernel G00,0 (c , c1 , c2 ) is known. Relations (31) at λ = 1 are
(3)
(2)
B0 (c )
G10,1
(34)
(3)
(c , c1 , c2 ) + B0 ( )
= −B1 ( )
c2 G00,0
c1 G01,1
(c , c1 , c2 )
(c , c1 , c2 ) .
It can be shown that the kernels with λ = 1 satisfy boundary conditions G10,1 (0, c1 , c2 ) = 0,
G11,0 (0, c1 , c2 ) = 0.
G01,1 (c , 0, c2 ) = 0, G01,1
G11,0 (c , 0, c2 ) = 0.
(c , c1 , 0) = 0,
G10,1
(35)
(c , c1 , 0) = 0.
In solving (34) under boundary conditions (35) we will use inverse operators, such as
(3)
B0 (c2 )
−1
f =
c2
1 c22
f (t )t 2 dt .
(36)
0
−1
(3)
By applying operator B0 (c2 )
to the first equation of set (34),
we obtain
−1
(3)
G01,1 (c , c1 , c2 ) = − B0 (c2 )
(2)
B0 (c )G11,0 (c , c1 , c2 )
+ B(14) (c1 ) G00,0 (c , c1 , c2 ) .
(c , c1 , c2 )
− Bˆ (l24−) 1 (c2 )Gll1 ,l2 −2 (c , c1 , c2 ) .
(31)
ν = l2
(37)
Similarly, we can express G10,1 (c , c1 , c2 ) in terms of G11,0 (c , c1 , c2 )
To create a recurrence procedure, it is convenient to introduce parameters
τ = (l − l1 + l2 ) /2,
(2)
(4)
1 = −Bˆ (l 1) (c )Gll− (c , c1 , c2 ) − Bˆ (l14) (c1 )Gll1 −1,l2 −1 (c , c1 , c2 ) 1 ,l 2 − 1
(32)
instead of indices l, l1 , and l2 . Conditions (10) under which the kernel Gll1 ,l2 may differ from zero are equivalent to the inequalities 0≤τ ≤ν≤λ
5
λ−1 τ −1 ν−1
= −B(11) (c )G00,0 (c , c1 , c2 )
(2)
λ = (l + l1 + l2 ) /2,
4
λ τ −1 ν−1
(3)
1 Bˆ l2 −1 (c2 )Gll1 ,l2 (c , c1 , c2 ) + Bˆ l (c )Gll+ (c , c1 , c2 ) 1 ,l 2 − 1
+ Bl1 ( )
3
λ τ ν−1
B0 (c2 ) G10,1 (c , c1 , c2 ) + B0 (c1 ) G11,0 (c , c1 , c2 )
By using relation (28), a recurrence procedure for derivation of kernels with arbitrary indices can be constructed. A similar approach was used in [2–4] to construct a procedure for obtaining nonlinear MEs with arbitrary high indices from algebraic relations (6). In constructing MEs all the values of indices r , r1 , and r2 were taken into consideration in addition to all the values of indices l, l1 , and l2 . Following [2–4], we replace l2 + 1 in (28) by l2 . Now we rearrange terms in (28) as
c1 Gll1 +1,l2 −1
2
λ τ ν
= −B(14) (c1 ) G00,0 (c , c1 , c2 )
6. Construction of the recurrence procedure
ˆ (3)
1
λi τi νi
(3)
This is another procedure for derivation of (28)–(29). It can be shown that, vice versa, relations (6) can be obtained from (28)–(29). To this end, it is sufficient to multiply (28) by respective Sonine polynomials and integrate the result over c , c1 , c2 taking into account (8) and (30). Hence, relations (6) and (28) are equivalent.
(3)
i
B0 (c2 )G01,1 (c , c1 , c2 ) + B0 (c ) G11,0 (c , c1 , c2 )
.
σr ,l+1 l −1 r +1 2 (c ) 1 d l − 1 c Sl−1/2 (c ) = . − + σr ,l 2 dc c σr +1,l−1
c l Slr+1/2
Table 1 Parameters λi , τi and νi for the recurrence relation terms (31); i corresponds to the term position in (31).
c
21
(33)
where λ, τ and ν are integers. Parameters λ, τ , and ν for all kernels of (31) are given in Table 1. Note that, if one of the indices λi , τi , νi becomes negative, the corresponding kernel vanishes, and the number of terms in the recurrence relation decreases. To find all relations at fixed λ, it is sufficient to consider all combinations of parameters τ and ν satisfying the inequality 0 ≤ τ ≤ ν ≤ λ + 1 and to ignore the relations containing only zero terms. If all kernels with λ = λ0 − 1 are known, it is possible to find all kernels with λ = λ0 from recurrence relations (31). Note that relations (31) are written in such a way that all the known kernels at each step of the recurrence procedure are assembled on the right side. In this paper we consider the first recursion step,
−1
(3)
G10,1 (c , c1 , c2 ) = − B0 (c2 )
(3)
B0 (c1 ) G11,0 (c , c1 , c2 )
+ B(11) (c )G00,0 (c , c1 , c2 ) .
(38)
By using (37), (38), we obtain from the third equation of (34) G11,0 (c , c1 , c2 ) = −
1 2
(3)
−1
B0 (c1 )
(2)
(3)
B1 (c ) + B0 (c )
−1
−1 −1 (3) × B(14) (c1 ) + B(03) (c1 ) B0 (c ) × B(03) (c2 ) B(14) (c2 ) G00,0 (c , c1 , c2 ) .
(39)
By substituting (39) into (37) and (38) we can get expressions for G01,1 (c , c1 , c2 ) and G10,1 (c , c1 , c2 ). Thus, the kernels for λ = 1 can
be constructed if the kernel G00,0 (c , c1 , c2 ) is known. Note that the analysis performed above is valid for the loss term and gain term of the collision integral individually. It is rather easy to construct the loss term kernels (see [18,22]). The gain term (its kernels will be marked with the upper index +) is much more complicated than the loss term. It was shown in [18] that
22
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
l +l Fig. 1. Domain structure of kernels G+ l1 ,l2 (c , c1 , c2 ) (a) and reduced kernels Ψl1 ,l2 (x, y) (b).
+ r ,l
Table 2 Definition of regions D1–D4. D1
D2
D3
D4
c 2 ≤ c12 + c22 , c1 ≤ c , c2 ≤ c
c1 ≤ c , c2 ≥ c
c 1 ≥ c , c2 ≤ c
c 1 ≥ c , c2 ≥ c
0 G+ 1,1 (c , c1 , c2 ) ≡ 0,
l the nonlinear collision integral kernels G+ l,0 (c , c1 , c2 ) are related to
linear (or linearized) collision integral kernels Ll (c , c1 ; ζ ) by +
L+ l (c , c1 ; ζ ) =
l 2 G+ l,0 (c , c1 , c2 )M (c2 , ζ )c2 dc2 .
0
(40)
ζ 3/2
Here M (c , ζ ) = π exp(−ζ c 2 ) is the Maxwellian, ζ = TT , B and TB is the background temperature. Eq. (40) is in essence the 2 Laplace transform from c2 to ζ . Analytical expressions for the kernels L+ l (c , c1 ; ζ ) were obtained for hard spheres (HS) in [12, 13] and for pseudo-Maxwellian molecules (PMM) (i.e., for the interaction model in which the cross section is isotropic and inversely proportional to the relative velocity) in [14]. 0 In these cases the kernel G+ 0,0 (c , c1 , c2 ) can be found by using the inverse Laplace transformation, and then the remaining kernels can be constructed through the recurrence relations. It was found that the structures of the kernels obtained are the same in both models under consideration. The entire domain c ≥ 0, c1 ≥ 0, c2 ≥ 0 of kernels is subdivided into five regions (see Fig. 1(a)). Derivatives are discontinuous on the boundaries of these regions. In region D0 (c12 + c22 < c 2 ) all kernels vanish, which is a consequence of the energy conservation law. All remaining regions are presented in Table 2. Analytical formulae for the kernels with λ = 0, 1 for HS model are given in Table 3. Table 3 shows that 1 G+ 1,0 (c , c1 , c2 )
=
1 G+ 0,1 (c , c2 , c1 ).
(42)
0 i.e., the kernel G+ 1,1 is zero in all regions for the PMM model. This property of the PMM model is a consequence of the theorem proved in [2]: in the interaction models involving the cross section inversely proportional to relative velocity (as for Maxwellian molecules) and arbitrarily depending on scattering angle the MEs r ,0 Kr1 ,2q+1,r2 ,2q+1 (where q = 0, 1, 2, . . .) are zero. As a consequence,
∞
+r ,l
the symmetry of MEs Kr1 ,l1 ,r2 ,l2 = Kr2 ,l2 ,r1 ,l1 that takes place for the interaction model of any type in which the cross section is isotropic. The kernels for the PMM model constructed by the recurrence procedure exhibit symmetry (41) (as expected). In addition,
G01,1 = G03,3 = G05,5 = · · · ≡ 0. Identity (42) is a particular case of this general result. The fact that it can be found with the aid of the recurrence procedure is an additional proof of the validity of this procedure. Taking into account the kernel properties given by (41) and (42), it is sufficient to present here the formulae only for the kernels 0 +1 G+ 0,0 and G1,0 . These formulae are given in Table 4. In [18] we have shown that the kernels corresponding to the power potential can be represented as Gll1 ,l2
(c , c1 , c2 ) =
c2
+
c12
+
c22
−3+2µ
Ψll1 ,l2 (x, y) .
Parameter µ is determined by the index of power in the power potential expression. In the case of hard spheres µ = 1/2, while for Maxwellian molecules µ = 0. Variables x and y are related to velocities c , c1 , and c2 by x=
(41)
Note that permutation c1 c2 leads to permutation of the D2 and D3 regions. The symmetry described by (41) is equivalent to
c c 2 + c12 + c22
,
y=
c1 c 2 + c12 + c22
,
l and Ψl+ (x, y) is a reduced kernel. 1 ,l2
Table 3 Kernels for the HS model at λ = 0, 1. Kern
D1
c12 +c22 −c 2
0 G+ 0,0
8π c1 c2
1 G+ 1,0
8π c 2 c12 c2
8π c 2 c1 c22
1 G+ 0,1
c12 (c12 + c22 − c 2 )1/2 − c22 (c12 + c22 − c 2 )1/2 −
0 G+ 1,1
−
8π c12 +c22 −c 2 9cc12 c22
3/2
2 3
D3
8π cc2
8π cc1
(c12 + c22 − c 2 )3/2
8π c1 3c 2 c2
8π c 2 c12
(c12 + c22 − c 2 )3/2
8π c 2 c22
c2 2 3
D2
c2 −
π c1 − 89cc 2 2
2 2 c 3 1
D4 8π c1 c2
c2 −
8π c2 3c 2 c1
2 2 c 3 2
8π c 3c12 c2 8π c 3c1 c22
π c2 − 89cc 2 1
8π c − 9c 2 c2 2
1 2
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
23
Table 4 0 +1 Kernels G+ 0,0 and G1,0 for the PMM model. 0 G+ 0 ,0
1 G+ 1,0
D1
8π cc1 c2
arctan
D2
8π cc1 c2
arctan
D3
8π cc1 c2
arctan
D4
4π 2 cc1 c2
c12 +c22 −c 2
2π c 2 c12 c2
c2
c12
c22
2π c 2 c12 c2
c12
1 − π2 arctan
c12 +c22 −c 2
c2
2π
c12 +c22 −c 2
c12 +c22
c1 c2 2c 2 −c12 −c22 c12 +c22
c 2 c12 c2
√
c2 c1 c12 +c22 −2c 2
c 2 c12 c2
c c12 −c22
c12 +c22
2π
c22
c c22 −c12
√
√ c12 +c22 −c 2 + 2c 2 + c12 − c22 arctan c
+ 2c 2 + c12 − c22 arctan cc21
+ 2c 2 + c12 − c22 arctan cc12
c12 +c22 −c 2 c12 +c22
+ 2c 2 + c12 − c22 arctan √
c
c12 +c22 −c 2
Fig. 2. Reduced kernels Ψ1+,01 (x, y) for pseudo-Maxwellian molecules (a) and hard spheres (b).
Hence, the kernel is determined by two-variable function
l Ψl+ (x, y). Regions D0–D4 in the space of velocity magni1 ,l 2 tudes transform into the D0–D4 regions on the x, y plane (see Fig. 1(b)). Fig. 2 presents reduced kernels Ψ1+,01 (x, y) for Maxwellian
molecules and hard spheres. One can see that these reduced kernels differ from each other only slightly. It can be supposed that the reduced kernels for the intermediate values of µ (between 0 and 1/2) have a similar shape. Note that the kernels of the gain terms of the nonlinear collision integral we obtained satisfy symmetry relations and, together with the kernels of the loss terms found in [22], ensure fulfillment of conservations laws. 7. Summary We have obtained relations between kernels of the nonlinear collision integral on the basis of collision integral invariance with respect to the choice of basis. By using these relations, a recurrence procedure for construction of kernels with ascending indices has been developed. At each recursion step, kernels are found by solving partial differential equations. A similar recurrence procedure was suggested in [2] to calculate MEs but the relations between MEs were algebraic, in contrast to the differential relations derived in this paper. The kernel G00,0 necessary for launching the recurrence procedure can be obtained by using the relation between linear and nonlinear collision integral kernels that can be regarded as Laplace transformation [23]. The first step of the recurrence procedure has been fulfilled, and analytical representations for the kernels G10,1 , G11,0 , and G10,1 for the cases of hard spheres and PMM have been obtained. The recurrence relations we have derived are suitable for finding kernels of the nonlinear collision integral for arbitrary interaction models and arbitrary sets of indexes l, l1 , l2 . This can provide a tool for determining a distribution function in a highly non-equilibrium situation.
Acknowledgment The work was supported by the Russian Foundation for Basic Research (project No 09.08.01017). References [1] D. Burnett, The distribution of molecular velocities and the mean motion in a non-uniform gas, Proc. Lond. Math. Soc. (3) 40 (1936) 382. [2] A.Ya. Ender, I.A. Ender, The collision integral of Boltzmann equation and moment method, St. Petersburg University Publishing House, St. Petersburg, 2003 (in Russian). [3] A.Ya. Ender, I.A. Ender, Polynomial expansions for the isotropic Boltzmann equation and invariance of the collision integral with respect to the choice of basis functions, Phys. Fluids 11 (1999) 2720. [4] A.Ya. Ender, I.A. Ender, Properties of the collision integral in the axisymmetric Boltzmann equation, Transport Theory Statist. Phys. 36 (7) (2007) 563. [5] J.H. Ferziger, H.G. Kaper, Mathematical Theory of Transport Processes in Gases, North-Holland Publishing Company, Amsterdam, 1972. [6] S.K. Loyalka, E.L. Tipton, R.V. Tompson, Chapman–Enskog solutions to arbitrary order in Sonine polynomials I: simple, rigid-sphere gas, Physica A 379 (2007) 417. [7] E.L. Tipton, R.V. Tompson, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials II: viscosity in a binary, rigid-sphere, gas mixture, Eur. J. Mech. B Fluids 28 (2009) 335. [8] E.L. Tipton, R.V. Tompson, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials III: diffusion, thermal diffusion, and thermal conductivity in a binary, rigid-sphere, gas mixture, Eur. J. Mech. B Fluids 28 (2009) 353. [9] R.V. Tompson, E.L. Tipton, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials IV: summational expressions for the diffusionand thermal conductivity-related bracket integrals, Eur. J. Mech. B Fluids 28 (2009) 695. [10] R.V. Tompson, E.L. Tipton, S.K. Loyalka, Chapman–Enskog solutions to arbitrary order in Sonine polynomials V: summational expressions for the viscosityrelated bracket integrals, Eur. J. Mech. B Fluids 29 (2010) 153. [11] A.Ya. Ender, I.A. Ender, A.B. Gerasimenko, Standard moment method in the problems on ion kinetics in neutral gas, Open Plasma Phys. J. 2 (2009) 24. [12] D. Hilbert, Begründung der kinetischen Gastheorie, Math. Ann. 72 (1912) 562. [13] E. Hecke, Über die Integralgleichung der kinetischen Gastheorie, Math. Z. 12 (1922) 274.
24
A.Ya. Ender et al. / European Journal of Mechanics B/Fluids 36 (2012) 17–24
[14] A.Ya. Ender, I.A. Ender, Analytical representation of linear kernels in collision integral of Boltzmann equation for Maxwellian molecules, Tech. Phys. Lett. 37 (2011) 197. [15] C.L. Pekeris, Z. Alterman, L. Finkelstein, K. Frankoowski, Propagation of sound in a gas of rigid spheres, Phys. Fluids 5 (1962) 1608. [16] S.K. Loyalka, Temperature jump and thermal creep slip: rigid sphere gas, Phys. Fluids A 1 (1989) 403. [17] R.D.M. Garcia, C.E. Siewert, The viscous-slip, diffusion-slip, and thermal-creep problems for a binary mixture of rigid spheres described by the linearized Boltzmann equation, Eur. J. Mech. B Fluids 26 (2007) 749. [18] A.Ya. Ender, I.A. Ender, L.A. Bakaleinikov, Some general properties of the nonlinear collision integral in the Boltzmann equation, Tech. Phys. 55 (2010) 1400.
[19] L.A. Bakaleinikov, A.Ya. Ender, I.A. Ender, Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case, Tech. Phys. 51 (2006) 1110. [20] L.A. Bakaleinikov, E.Yu. Flegontova, A.Ya. Ender, I.A. Ender, Calculation of the linear kernel of the collision integral for the hard-sphere potential, Tech. Phys. 54 (2009) 182. [21] I.S. Gradshtein, I.M. Ryzhik, Tables of Integrals, Series and Products, Fizmatlit, Moskow, 1962, (in Russian) p. 1092. Academic, New York, 1965. [22] A.Ya. Ender, I.A. Ender, L.A. Bakaleinikov, E.Yu. Flegontova, Matrix elements and kernels of the collision integral in the Boltzmann equation, Tech. Phys. 56 (2011) 452. [23] A.Ya. Ender, I.A. Ender, L.A. Bakaleinikov, E.Yu. Flegontova, Construction of kernels of the nonlinear collision integral in the Boltzmann equation using Laplace transformation, Tech. Phys. 57 (2012) 735.