Analysis of the Relaxation Process Using Nonrelativistic Kinetic Equation

Transcription

1 93 Progress of Theoretical Physics, Vol. 13, No. 5, May 1 Analysis of the Relaxation Process Using Nonrelativistic Kinetic Equation Makoto Takamoto 1 and Shu-ichiro Inutsuka 1 Department of Physics, Kyoto University, Kyoto 66-85, Japan Department of Physics, Nagoya University, Nagoya , Japan (Received December 3, 9; Revised February 6, 1) We study the linearized kinetic equation of relaxation model proposed by Bhatnagar, Gross and Krook [P. L. Bhatnagar, E. P. Gross and M. Krook, Phys. Rev. 94 (1954), 511] (also called BGK model) and solve the dispersion relation. Using the solution of the dispersion relation, we analyze the relaxation of the macroscopic mode and kinetic mode. Since the BGK model is not based on the expansion in the mean free path in contrast to the Chapman-Enskog expansion, the solution can describe the accurate relaxation of initial disturbance with any wavelength. This nonrelativistic analysis gives suggestions for our next work on the relativistic analysis of relaxation. Subject Index: Introduction Macroscopic phenomena of dissipative fluid are in general described using Navier- Stokes equations. However, Navier-Stokes equations cannot provide good descriptions if the characteristic length of the phenomenon is comparable to the mean free path of the constituent particles. For this reason, Navier-Stokes equations cannot describe the shock structure and short-wavelength sound wave propagation 1) 3) that are important in astrophysics and plasma physics. To study short-wavelength phenomena, we have to use kinetic theory description, such as by the Boltzmann equation. Unfortunately, it is quite difficult to solve the Boltzmann equation, and various methods for obtaining approximate solutions have been developed. The most standard method is the Chapman-Enskog expansion 4) that was developed by Chapman, Enskog, and Hilbert. This method begins with an expansion of the distribution function in the mean free path of constituent particles and simplifies the results at a given order by introducing expressions from the previous order terms. This method derives the Euler equations at zeroth order and the Navier-Stokes equations at the first order and determines the transport coefficients that cannot be determined using Navier-Stokes equations. For this reason, it is natural to expect that higher order approximations improve the description. The second-order and third-order approximations were proposed by Burnett. 5) 7) However, these equations provide little or no improvement in describing short-wavelength phenomena. It is because the Chapman-Enskog expansion is a sort of iteration process, and therefore inappropriate for a series that is not convergent. Modification of the Chapman-Enskog expansion has been proposed, for example, by Chen et al. 8) 1) who avoid the recycling of lower order results. This extension of the Downloaded from at Pennsylvania State University on March 5, 16

2 94 M. Takamoto and S. Inutsuka procedures provided improved descriptions of short-wavelength phenomena in the case of the BGK model of kinetic theory, although this procedure cannot describe phenomena whose Knudsen number is close to or larger than unity. Another procedure is the moment method introduced by Grad. 1), 13) This procedure is an expansion of the distribution function not in the mean free path but using Hermite polynomials, and its 13-moment expansion reproduces the Navier- Stokes equations in the leading order. However, this expansion is asymptotic rather than convergent, so that the further development of this expansion does not easily produce very accurate solutions in the short-wavelength limit. 14) 17) In this paper, we derive the dispersion relation of the BGK model 11) of the linearized kinetic equation and analyze the relaxation process using the solutions of the dispersion relation. Since we study the kinetic model equation, the solution of the dispersion relation is correct in the applicability domain of the BGK model. The accuracy of the BGK model has been studied in many papers: the experimental data for phase velocities and attenuation rate 1) indicate that the BGK model is almost accurate for any Knudsen number 11), ) but inadequate for phenomena sensitive to high-order moments. 18) We are interested in the time evolution of initial disturbance so that we solve the dispersion relation in terms of the frequency and obtain all modes of relaxation. Since we use the solution of the dispersion relation of the kinetic equation, the solution accurately describes the relaxation of initial disturbance with any wavelength and can be applied to various problems of the dilute gas dynamics; for example, applications include dilute gas dynamics around an object, 19) study of the atmospheres of cosmic bodies, ) and the motion of a micrometer incident on the Earth s upper atmosphere. 1) In addition, this nonrelativistic analysis may give suggestions for our next work on the relativistic analysis of relaxation. In, we derive dispersion relations of the BGK equation of the linearized kinetic equation. In 3, we solve the dispersion relation obtained in and derive eigenfunctions for all modes. In 4, we study the corresponding physical meaning of each eigenfunction.. Linearized kinetic equations and dispersion relations In this section, we derive dispersion relations of the BGK model for the linearized kinetic equation. 11), 6) The detailed calculation is presented in Appendix A. When there is no external field, the BGK model of kinetic equation is Df(t, x, v) = f(t, x, v) f eq(t, x, v), (.1) Dt ( ) τ D Dt = t + v. (.) Downloaded from at Pennsylvania State University on March 5, 16 In Eq. (.1), τ is the relaxation time and f eq represents the local Maxwell-Boltzmann distribution f eq (t, x, v) = ρ(t, x) (v`u(t;x)) e RT (t,x), (.3) m(πrt(t, x)) 3/

3 Analysis of the Relaxation Process 95 R = k B /m, (.4) and k B is the Boltzmann constant. Equation (.1) is a nonlinear equation for f(t, x, v) because of the nonlinear dependence of f eq on f through the following conditions: (f eq f) ψ μ d 3 v =, (.5) ψ μ = m (1, v, 1 ) (v u). (.6) These conditions are called matching conditions. For obtaining the dispersion relation, we start by expanding the distribution function around an equilibrium state f (v) δf = f f, δf eq = f eq f. (.7) The linearized the BGK model of the kinetic equation becomes the following form: ( ) t + v δf = δf δf eq. (.8) τ We assume the following form of solution: Then, Eq. (.8) is ( 1 τ δf = δ fe iω(t t )+ik x. (.9) ) iω + ik v δf = 1 τ δf eq, (.1) δf eq = f [ δρ ρ + v δu f = ( v + RT ) δt ], (.11) 3 RT T ρ /RT m(πrt ) 3/ e v. (.1) Downloaded from at Pennsylvania State University on March 5, 16 Using the matching conditions, we can rewrite δρ, δu, andδt as the integrals of δf δρ(t, x) = mδfd 3 v, (.13) m δu(t, x) = vδfd 3 v, (.14) ρ ( m v δt(t, x) = 3ρ R 3 ) T δfd 3 v. (.15)

4 96 M. Takamoto and S. Inutsuka Then, Eq. (.8) becomes ( ) 1 iω + ik v δf(v) = f (v) τ τ [ d 3 v m + m ρ ρ v RT v RT + m 3ρ ( v RT 3 )( v RT 3 ) ] δf(v ). (.16) In the following, we take τ as a unit of time and RT as a unit of velocity, ωτ ω, RT τk k, v v. (.17) RT Hereafter, we omit. Finally, the linearized equation of the BGK model is δf(v) = d 3 v K(v, v )δf(v ), (.18) K(v, v ) m(rt ) 3/ ρ [ f (v) 1+v v + ( v 3 )( v 3 )]. 1 iω + ik v 3 (.19) The above equations make sense only when 1 iω +ik v. We explain afterward thecase1 iω + ik v =. Equation (.18) is the homogeneous Fredholm integral equation of the second kind. In particular, the kernel function K(v, v ) can be separated with respect to the variables v and v, and this equation can be solved according to the general procedure. Firstly, we perform integration of Eq. (.18) with respect to v. The equation becomes δρ δu x δt I 11 + I 1 + I 13 =, (.) ρ RT T I 11 = 1 iω πe b Erfc(b), (.1) ( b I 1 =i 1 b ) πe b Erfc(b), (.) [ ( I 13 = b b + ) 1 ] πe b Erfc(b), (.3) Downloaded from at Pennsylvania State University on March 5, 16 b = 1 iω. (.4) k Secondly, we multiply Eq. (.18) by k v and perform integration with respect to v. The equation becomes δρ δu x δt I 1 + I + I 3 =, (.5) ρ RT T

5 Analysis of the Relaxation Process 97 ( I 1 = i 1 b ) πe b Erfc(b), (.6) ( I = k +b 1 b ) πe b Erfc(b), (.7) { ( I 3 = ib b b + ) 1 } πe b Erfc(b). (.8) Thirdly, we multiply Eq. (.18) by v and perform integration with respect to v. The equation becomes I δu =, (.9) RT I = k πe b Erfc(b). (.3) Finally, we multiply Eq. (.18) by v and perform integration with respect to v. The equation becomes δρ δu x δt I 31 + I 3 + I 33 =, (.31) ρ RT T I 31 = 1 [ { 3k b (b 1) }] πe b Erfc(b), (.3) [ { I 3 = b b (b 1) } ] πe b Erfc(b) 3, (.33) I 33 = 1 [ { 3k b(1 b )+(b 4 b +1) }] πe b Erfc(b). (.34) If the determinant of the above homogeneous system is set equal to zero, the following dispersion relation is obtained: I 11 I 1 I 13 I 1 I I 3 I 31 I 3 I 33 =. (.35) I I This condition implies either or I 11 I 1 I 13 I 1 I I 3 I 31 I 3 I 33 =, (.36) Downloaded from at Pennsylvania State University on March 5, 16 I =. (.37) Equation (.36) corresponds to longitudinal mode (δu x,δu = ), and Eq. (.37) corresponds to transverse mode (δu x =, δu ).

6 98 M. Takamoto and S. Inutsuka Fig. 1. Decay rate of the thermal conduction mode. Fig.. Decay rate of the shear flow mode. Finally, we obtain the exact solution δf in the following form: δf(v) = [ C n f (v) δρ ωn + v δu ( ω n v + 3 ) ] δt ωn e i(ωnt+k x), 1 iω n n + ik v RT RT (.38) C n is a constant coefficient and δρ ωn, δu ωn, and δt ωn are eigenfunctions obtained using Eq. (.36). We should study the case 1 iω + ik v =.Inthiscase,themodebecomes continuous. According to Eq. (.16), the eigenfunction for this mode satisfies the following equation: = d 3 v f [ (v) m + m v v + m ( v 3 )( v 3 )] δf(v ) τ ρ ρ RT 3ρ RT RT = δf eq. (.39) This mode represents decaying of the moments of f with vanishing δf eq, i.e., δρ = δt =,δu =. 3. Result 3.1. Dispersion relation In this section, we show dispersion relations obtained in the previous section. First, we solve the dispersion relations numerically and the results are shown in Figs.1,,and3. Solid lines represent the decay rate ( Im ω) and dotted lines represent the frequency Re ω. In the long-wavelength part, the solid lines are proportional to k and reproduce the result that can be obtained by the first Chapman-Enskog approximation. Regarding shear waves, we find only the imaginary part of frequency and it indicates that shear waves cannot propagate in fluid when we consider initial value problems. We also find rapidly decaying modes Im ω 1/τ in Figs. 4, 5, and 6. Since Downloaded from at Pennsylvania State University on March 5, 16

7 Analysis of the Relaxation Process 99 Fig. 3. Dispersion relation of the sound wave mode. The solid line represents the decay rate ( Im ω) and the dotted line represents the frequency Re ω. Fig. 5. Dispersion relation of the longitudinal kinetic mode. The solid line represents the decay rate ( Im ω) and the dotted line represents the frequency Re ω. Fig. 4. Dispersion relation of the transverse kinetic mode. The solid line represents the decay rate ( Im ω) and the dotted line represents the frequency Re ω. Fig. 6. Another dispersion relation of the longitudinal kinetic mode. The solid line represents the decay rate ( Im ω)andthedotted line represents the frequency Re ω. ( Im ω) is1/τ at k =, these modes represent kinetic modes that do not appear in macroscopic descriptions. We discuss kinetic modes in the following section. Figures 7 and 8 plot the inverse of phase velocity and decay rates of sound wave modes that are obtained by solving Eq. (.36) for k as a function of given ω. These reproduce the results of Bhatnagar et al. 11) and are analogous to the results of Sirovich and Thurber, ) in which they have performed linear analysis of the Boltzmann equation for various forms of collision term. We show only the sound wave mode for k but have found other modes, for example, thermal wave modes. 3.. Eigenfunctions In this section, we show eigenfunctions to study the physical picture of each mode. By substituting dispersion relations obtained in 3.1 for Eqs. (.), (.5), and (.31), each mode is plotted with respect to the fluctuations δρ, δu x,andδt that are normalized in δρ/ρ + δu x / (RT )+δt/t =1. In Fig. 9, δt is dominant and δu x is very small for a long-wavelength part, Downloaded from at Pennsylvania State University on March 5, 16

8 91 M. Takamoto and S. Inutsuka Fig. 7. Phase velocity of sound wave mode in units of the adiabatic sound speed, γrt. The dispersion relation is compared with the experiment of Meyer and Sessler. 1) Fig. 9. Eigenfunction of thermal conduction mode. Fig. 8. Decay rates of sound wave mode in units of the adiabatic sound speed, γrt. The dispersion relation is compared with the experiment of Meyer and Sessler. 1) Fig. 1. Eigenfunction of sound wave mode. which indicates that this mode represents thermal conduction. Interestingly, δu x increases with decreasing wavelength and it indicates that the picture of pure thermal conduction cannot be applied to the short-wavelength part of this mode. In Fig. 1, δρ and δu x are dominant for large wavelength and this mode can be regarded as sound wave mode. In the short-wavelength regime, δt dominates the others and it represents that this mode is oscillation between δt and δu x in this regime. Figures 11 and 1 indicate that these two modes show similar relaxation. Both modes contain small δρ and large δt and δu x for any wavelength. However, the timescale of decay of these modes is shorter than relaxation time τ, and these modes cannot propagate and do not appear on a macroscopic scale. Downloaded from at Pennsylvania State University on March 5, 16

9 Analysis of the Relaxation Process 911 Fig. 11. Eigenfunction of the longitudinal kinetic mode Kinetic decay 4. Discussion Fig. 1. Eigenfunction of the second longitudinal kinetic mode. In general, relaxation is achieved by collisions so that decay rate ( Im ω) isnot simply expected to be larger than 1/τ. However, Figs. and 3 indicate that it is contrary. In this section, we discuss why this happens. In a short-wavelength regime, the relaxation rate ( Im ω) ofthismodeislarger than 1/τ and it contradicts our understanding that relaxation is achieved through collision. In the BGK model adopted in this paper, however, τ is the average relaxation time; namely, τ is independent of particle velocity v. Thus, we should consider that the physical system becomes collisionless gas when k is larger. For obtaining the physical description of relaxation in the collisionless regime, we consider the macroscopic momentum equation. From kinetic theory, the linearized momentum equation is ρ t δu = δp P, (4.1) P is traceless stress tensor. We consider only the longitudinal mode. We multiply Eq. (.5) by ω and use the ideal gas law: p = ρrt. Then, Eq (.5) can be cast into the following form: [ ωk δu + k (δρ + δt)+ k (δρ + δt)+ωδρ ( 1 b { πe b Erfc(b)+iωb b ( i + b ( b + 1 πe b Erfc(b) ) k ω )}) ] δt =. (4.) Downloaded from at Pennsylvania State University on March 5, 16 Figure 13 plots the ratio of magnitude of each term of the momentum equation in sound wave mode Eq. (4.). This figure indicates that the viscosity is small for a long-wavelength and becomes larger for a short-wavelength as in the macro description. Figure 14 is the same plot for kinetic modes. These modes are purely kinetic and are not found in the Chapman-Enskog approximation. Figure 14 indicates that

10 91 M. Takamoto and S. Inutsuka Fig. 13. Relative magnitude of each term in the momentum equation for sound wave mode. Fig. 14. Relative magnitude of each term in the momentum equation for longitudinal kinetic mode. viscosity dominates pressure and momentum density for a long-wavelength. This is obviously different from the viscosity description using the Chapman-Enskog approximation in which viscosity is a small correction to macroscopic variables. This implies that these modes are purely kinetic modes. 4.. Asymptotic analysis of dispersion relation For studying the property of the dispersion relation, we take the long-wavelength limit. First, we consider the shear flow mode. As in, the dispersion relation of the shear flow is ρ δu = m d 3 vv δf [ = m d 3 f eq δρ vv + v δu ( v + 3 ) ] δt. (4.3) 1 i ω + i k v x ρ RT RT T We expand the integrand in powers of d = k/(1 i ω) and neglect the terms higher than second order on the right-hand side. Then, the above equation reduces to ρ δu d 3 1 [ vv 1 id vx d v ] x δfeq. (4.4) 1 i ω Rewriting the above equation, we obtain ρ δu = 1 1 i ω d 3 vv (1 d v x)δf eq = 1 1 i ω (1 α 1d )ρ δu, (4.5) α 1 = d 3 v mv xv ρ (RT ) f eq. (4.6) Neglecting ω, we obtain Downloaded from at Pennsylvania State University on March 5, 16 ω = iα 1 d iα 1 k. (4.7)

11 Analysis of the Relaxation Process 913 Next, we study the long-wavelength limit of the thermal conduction mode and sound wave mode. As in, the conservation of particle number is δρ = m d 3 vδf [ = m d 3 f eq δρ v + v δu ( v + 3 ) ] δt. (4.8) 1 i ω + i k v x ρ RT RT T The same as for the shear flow mode, we expand the integrand in powers of d = k/(1 i ω) and neglect the terms higher than second order on the right-hand side. Then, the above equation reduces to δρ m d 3 v(1 id v x d v 1 i ω x)δf eq = 1 [ δu x δρ idρ ρ d 1 i ω RT δt T ], (4.9) weuse ( δρ d 3 vmvxδf eq = ρ RT + δt ). (4.1) ρ T Rewriting the above equation, we obtain ) (i ω d δρ id δu x d δt =. (4.11) ρ RT T Similarly, the conservation of the momentum and energy reduces to id δρ +(iω α 4 d δu x ) id δt =, (4.1) ρ RT T [ ] 3 δρ i ω α d iα d δu [ ( x 3 + i ω d α 3 3 )] δt ρ RT α =, (4.13) T α = α 3 = α 4 = d 3 v mv vx ρ (RT ) f eq = 5 4, (4.14) d 3 v mv4 vx ρ (RT ) 3 f eq = 35 8, (4.15) mv 4 x d 3 v ρ (RT ) f eq = 3 4. (4.16) From the above equations, we obtain the dispersion relation in the following forms: 3i ω3 + 19d ω ( id + 35 ) 8 id4 ω 5d4 8 15d6 16 =. (4.17) We obtain three roots given by ω T and ω S± that are accurate to the second order in k, ω T = id i k, (4.18) Downloaded from at Pennsylvania State University on March 5, 16

12 914 M. Takamoto and S. Inutsuka 5 ω S± = ± 6 d 4id 3. (4.19) We substitute d = k/(1 i ω) intoeq.(4.19) and solve the equation with respect to ω. Then, we obtain 5 ω S± ± 6 k i k. (4.) The above result shows that the decay rates (imaginary part) of the thermal conduction Eq. (4.18) and sound wave Eq. (4.) are the same in the long-wavelength limit. It corresponds to the fact that the value of the Prandtl number is unity in the BGK model. (The Prandtl number is the ratio of the viscosity times C p to the thermal conductivity.) 4.3. Mathematical structure Discrete spectrum In, we have derived the linearized equation of the BGK model Eq. (.18). However, as is indicated in the last part of, we assume 1 iω + ik v when we derived Eq. (.18). When we assume 1 iω + ik v, we can formally consider the linearized equation of the BGK model, Eq. (.18), as the following integral equation: δf(v) =λ d 3 v K(v, v )δf λ (v ), (4.1) λ =1, K(v, v ) m(rt ) 3/ ρ [ f (v) 1+v v + ( v 3 )( v 3 1 iω + ik v 3 (4.) )]. (4.3) This is the homogeneous Fredholm equation of the second kind with eigenvalue λ =1, and obtaining frequency ω is equivalent to deriving the degenerated eigenfunction δf λω. Mathematically, discrete modes that include both macroscopic modes and kinetic modes are degenerated solutions of integral equation Eq. (4.1), and the number of discrete modes is equal to the dimensions of eigenspace of eigenvalue λ = 1. In Eq. (4.1), K(v, v ) is the compact integral operator. This indicates that the number of discrete modes is not infinite but finite because of the Fredholm alternative theorem Continuous spectrum In, we obtained continuous spectrum 1 iω + ik v = in addition to the discrete modes. In this section, we discuss the eigenfunction of this mode. We assume a situation that a beam of particles of constant velocity v is imposed under the initial condition. Then, δf becomes as follows: δf = δf 1 + Dδ D (v v ), (4.4) Downloaded from at Pennsylvania State University on March 5, 16

13 Analysis of the Relaxation Process 915 D is a constant coefficient and δ D is the Dirac delta function. Dδ D (v v ) represents the particle flux distribution of v = v. δf 1 represents the perturbed distribution function of gas. We substitute this δf into Eq. (.16), then it becomes ( ) ( ) 1 1 iω + ik v δf 1 + τ τ iω + ik v D = d 3 v f [ (v) m + m v v + m ( v 3 )( v 3 )] δf 1 (v ) τ ρ ρ RT 3ρ RT RT + f [ (v) m + m v v + m ( v 3 )( v 3 )] D. (4.5) τ ρ ρ RT 3ρ RT RT Then, we assume the next continuous spectrum, ω = i τ + k v. (4.6) We substitute this ω into Eq. (4.5). Then, Eq. (4.5) becomes δf 1 = d 3 v [ f (v) m + m iτk (v v ) ρ ρ [ f (v) m + + m iτk (v v ) ρ ρ v v + m RT 3ρ v RT v + m 3ρ ( v 3 RT ( v 3 RT T )( v 3 RT )( v 3 RT )] δf 1 (v ) )] D. (4.7) The solution of this integral equation can be obtained in the same manner as in. We explain the case of shear flow for simplicity. We multiply Eq. (4.7) by v and perform integration with v. The equation becomes { k } πe b Erfc(b) δu 1y = πe b Erfc(b) Dv y, (4.8) b = ik v = iv x. (4.9) k Finally, we obtain eigenfunction δu 1y in the following forms: [ ] 1 k δu 1y = πe vxerfc( iv x ) 1 Dv y. (4.3) Downloaded from at Pennsylvania State University on March 5, 16 δρ 1, δu 1,andδT 1 can be obtained by a similar procedure General solution The continuous and discrete modes we discussed in the previous sections seem to cover all spectra of Eq. (4.1), and the general solution of the linearized the BGK equation would be as follows: δf(v) = C n K(v, v )δf ωn (v )+ d 3 v C(v ) ˆK(v, v )δf v (v ), (4.31) n ˆK(v, v ) (1 iω + ik v)k(v, v ). (4.3)

14 916 M. Takamoto and S. Inutsuka Because of the Fredholm alternative theorem, the number of discrete modes is finite. We may conclude that most of the degrees of freedom of δf belong to the continuous spectrum and only their finite parts belong to the discrete modes. 5. Conclusion We have solved the linearized kinetic equation of the BGK model and obtained the exact solution for the relaxation of initial disturbances that describe the kinetic modes in addition to macroscopic modes that can be obtained by Chapman-Enskog expansion. Using those solutions, we have obtained the eigenfunction and analyzed the relaxation process corresponding to those solutions. Since we use the solution of the dispersion relation of the kinetic equation, the solution describes the accurate relaxation of initial disturbance with any wavelength. In the next paper, we will apply this method to the relativistic kinetic equation. Acknowledgements We wish to thank Takayuki Muto and Takayuki Muranushi for fruitful discussions. We would also like to thank Professor Yoshio Tsutsumi for the very helpful discussion on the mathematical structure of the linearized the BGK model. This work was supported by the Grant-in-Aid for the Global COE Program The Next Generation of Physics, Spun from Universality and Emergence from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. Appendix A Detailed Calculation of Fluid Equations In, we obtained the BGK model of linearized kinetic equation in the following forms: δf(v) = d 3 v K(v, v )δf(v ), (A.1) K(v, v ) m(rt ) 3/ ρ [ f (v) 1+v v + ( v 3 )( v 3 )], 1 iω + ik v 3 (A.) f is the unperturbed state. We use the spherical coordinates in velocity space setting k as the z-axis. Then, we have to relate v δu in δf eq to k. We define θ, θ,andθ as the angles measuring between k and v, k and δu, andv and δu, andφ as the angle between v and δu measured on the plane projected perpendicular to k. Then, we have the following relation: cos Θ =cosθ cos θ +sinθ sin θ cos φ. (A.3) In this section, we calculate the momentum equation of k direction as an example. We set the x- and the y-axis in the direction of k and δu that is the projection Downloaded from at Pennsylvania State University on March 5, 16

15 Analysis of the Relaxation Process 917 of δu on the plane perpendicular to k. Then, δu x becomes kδu x = k d 3 vmvδf 1 =πm 1 v dv 1 vμδf eq dμ 1 iω + ikvμ, (A.4) μ =cosθ. Then, we replace δf eq with Eq. (.11) and replace a dimensional variables with dimensionless form as in Eq. (.17), kρ 1 τ μv 3 {( e v dv dμ v 3 ) δt + δρ + vμk } π 1 iω + ikvμ k δu. (A.5) We remove μ of odd index and the equation becomes kρ 1 { ( dv dμ μv3 e v πτ k b + v μ ivμ v 3 = 4ρ dv πτ 1 1 dμ v μ + b b b + v μ v e v Then, we use the next integral formula 1 ) δt ivμδρ +b μv { ( i v 3 ) δt iδρ + b } k k δu } k k δu, (A.6) b = 1 iω. (A.7) k 1 dμ a + b μ = 1 ab arctan b a, (A.8) and the equation can be integrated, 4ρ ( dvv e v 1 b v ){ ( πτ v tan 1 i v 3 ) δt iδρ + b } b k k δu = 4ρ [ ( dve v v { i v 3 ) δt iδρ + b } πτ k k δu b tan 1 v { i ( v 3 + v v + 3 ) b v δtivδρ + bv }] k k δu. (A.9) Using the identity we perform partial integration 4ρ dv πτ 1 + b b + v d dv (e v v l )=(lv l 1 v l+1 )e v, dμe v [ v { i ( v 3 ) δt iδρ + b } k k δu (A.1) { ( v i 1 ) δt + i 4 δρ bk }] k δu. (A.11) Downloaded from at Pennsylvania State University on March 5, 16

16 918 M. Takamoto and S. Inutsuka We use the next integral formulas dxe a x 1 x + b = π b ea b Erfc(ab), (A.1) (a >, b>) Erfc(x) = π 1/ e t dt, (A.13) x dxe a x x n (n 1)!! π = n+1 a n+1. (A.14) (a >, b>)(a >) Finally, the equation can be integrated as [ { ( ρ ib b b + 1 } ( πe b τ ) Erfc(b) δt i δρ + ib ) ( k k δu 1 b ) ] πe b Erfc(b). (A.15) On the other hand, integration with respect to δf eq is clearly k dvmvδf eq = ρ τ k δu. (A.16) In the end, the momentum equation is [ ( k δu + ib b b + ) 1 ] πe b Erfc(b) δt ( i δρ + ib ) ( k k δu 1 b πe Erfc(b)) b =. (A.17) The energy equation can be performed in the same way. The shear flow equation can be calculated as follows. δu is given by ρ δu = d 3 vmv δf [( = m d 3 f eq vv v 3 ) ] δt + δρ +v δu 1 iω + ikv x = m d 3 f eq v v 1 iω + ikv δu. (A.18) x Then, after the same calculation of the momentum equation, we obtain Eq. (.37). Downloaded from at Pennsylvania State University on March 5, 16 Appendix B Analytic Continuation Equation (A.1) can be used only when b> and we have to regard b as b ; if b has an imaginary part, we have to replace b with b in Eq. (A.1). However, in this paper we perform analytic continuation and remove these restrictions. In this section, we explain the analytic continuation.

17 Analysis of the Relaxation Process 919 For simplicity, we explain by using the transverse shear flow mode (δu ). In Eq. (A.18), we consider the circular cylindrical coordinates (v,φ,v x ) and perform φ integral. Changing to dimensionless form, the equation reduces to v 3 dv x dv x v i{kv x i(ω + i)} e v. (B.1) By performing v integral, the equation becomes 1 e v x dv x i{kv x i(ω + i)}. (B.) This integrand may have a first-order pole on the real axis. However, this singularity is removed to the upper half-plane if Im ω> 1 and this results from the collision term of the BGK model Eq. (.1). This fulfills the demand of the Landau method, that is, this problem has to be considered as the Cauchy problem. For this reason, the integration path has to be distorted below the singularity. Concerning the above equation, the restriction Re b> is not satisfied when Im ω < 1; thus, analytic continuation has to be performed. The equation has first-order pole; thus, we use the principal integral f(z)dz z i = P f(z)dz z + iπf(). (B.3) The integration without analytic continuation corresponds to the integration path distorted above the singularity when Re b<: thus, the principal integral is as follows: b π k ( b) e( b) Erfc( b) = 1 P e v x In the end, Eq. (B.) is as follows when Re b<, 1 e v x dv x i{kv x i(ω + i)} = ( b k dv x i(kv x i(ω + i)) iπ ik e (ib). (B.4) π b e( b) Erfc( b)+ πi ) ik e (ib) + πi ik e (ib) = π k eb ( Erfc( b)) = π k eb Erfc(b). (B.5) Downloaded from at Pennsylvania State University on March 5, 16 This indicates that analytic continuation permits us to use Eq. (.37) even when Re b<. References 1) E. Meyer and G. Sessler, Z. Phys. 149 (1957), 15. ) M. Greenspan, J. Acoust. Soc. Am. 8 (1956), ) R. Schotter, Phys. Fluids 17 (1974), ) S. Chapman and T. G. Cowling, in The Mathematical Theory of Non-Uniform Gases, 3rd ed. (The University Press, Cambridge, 1991). 5) D. Burnett, Proc. London Math. Soc. 39 (1935), 385.

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