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1 Supplement to Molecular Gas TitleBirkhäuser, Boston, 007 Dynamic Version Authors Sone, Yoshio Citation Yoshio Sone. 008 Issue Date URL Right Type Book Textversion author Kyoto University

2 Molecular Gas Dynamics Yoshio Sone Birkhäuser, Boston, 007 Supplementary Notes and Errata Yoshio Sone Version September 008 Bibliography Update of bibliography [19] Arkeryd, L. and A. Nouri [006]: to be published [9] Bardos, C., F. Golse, and Y. Sone [006]: to be published [59] Chen, C.-C., I.-K. Chen, T.-P. Liu, and Y. Sone [006]: [006] [007] 59, to be published 60, [136] Liu, T.-P. and S.-H. Yu [006a]: [006a] [006] [137] Liu, T.-P. and S.-H. Yu [006b]: [006b] [007] 59, to be published 60, The corresponding corrections in the text p. 166, the first line in the third paragraph: [004b, 006a] [004b, 006] p. 166, the last line in the third paragraph: [006b] [007] p. 183, the second line: [006] [007] Errata p. 9, the 7th line specular condition specular reflection 1

3 p. 81, the 4th line in Footnote 7 u igm u igm u Gm n n i or φ egm of φ egm p. 83, the first line in Footnote 14 u igm u igm u Gm n n i p. 504, the first line in Footnote 4: damin domain p. 505, Eq. A.60: 1 sin θ e sin θ c θ c 1 b sin θ c θ c 1 d sin θ e sin θ c dθ c 1 db sin θ c dθ c p. 506, the 13th line [The line next to Eq. M-A.63]: with respect to θ c with respect to θ α p. 639, the 3rd line in Ref. [6]: gs gas Supplementary Notes In the present supplementary notes, the letter M is attached to the labels of sections, equations, etc. in the book Molecular Gas Dynamics to avoid confusion. 1 Chapter M Processes of solution of the systems in Section M-3.7. July 007 The processes of solutions of the fluid-dynamic-type equations derived in Section M are straightforward and may not need explanation. For the equations in Section M-3.7., some explanation may be better to be given. The discussion will be made on the basis of the boundary conditions in Section M for a simple boundary where the shape of the boundary is invariant and its velocity component normal to it is zero.

4 1.1.1 Incompressible Navier Stokes set Consider the initial and boundary-value problem of Eqs. M-3.65 M-3.68, i.e., P S1 = 0, 1 u is1 = 0, a u is1 t + u S1 u is1 x = 1 P S + γ 1 5 τ S1 P S1 + 5 t t u τ S1 S1 = 5γ x 4 u is1 x, b τ S1 x, c u is u is t = 1 = ω S1 t ω S1u is1, 3a u is u is1 + u S1 + u S x x PS3 P S ω S1 + γ 1 γ 1ω S1 u is1 x 3 P S + 3 t u P S S1 where = 5γ 4 τ S x + 5γ 5 4 uis + u S u ks δ i x x 3 x k + γ [ 4 uis1 τ S1 + u ] S1 γ 3 τ S1 x x 3 x, 3b + 5 PS1 u S ω S ω Su S1 + ω S1 u S x x t x τ S1 τ S1 + γ 1 uis1 + u S1, 3c x x x P S1 = ω S1 + τ S1, P S = ω S + ω S1 τ S1 + τ S. 4 From Eq. 1, P S1 is a function of t, i.e., P S1 = f 1 t. 5 In an unbounded-domain problem where the pressure at infinity is specified or the pressure is specified at some point, P S1 = f 1 t is known, but in a boundeddomain problem of a simple boundary, f 1 t is unknown at this moment and is determined later. Let u is1 and τ S1 as well as f 1 t be given at time t in such a way that u is1 satisfies Eq. a. Taking the divergence of Eq. b and using Eq. a, we have P S x i = u S1 u is1 x. 6 3

5 On a simple boundary, the derivative of P S normal to it is found to be expressed with u is1 and its space derivatives by multiplying Eq. b by the normal vector to the boundary. 1 In the unbounded-domain problem, where f 1 t is known, P S is determined by Eq. 6. In the bounded-domain problem, P S is determined by Eq. 6 except for an additive function of t [say, f t]. Anyway, P S / is independent of this ambiguity. From Eq. b, u is1 / t at t is determined, irrespective of f t, in such a way that u is1 / / t = 0 for the above choice of P S. Thus, the solution u is1 of Eqs. a and b is determined by Eq. b with the supplementary condition 6 instead of Eq. a. From Eq. c, 5/τ S1 / t P S1 / t or 5/τ S1 / t df 1 t/d t is determined, i.e., where 5/τ S1 / t df 1 t/d t = Gx i, t, 7 Gx i, t = 5 u τ S1 S1 + 5γ x 4 τ S1 x. 8 Thus, τ S1 is determined in the unbounded-domain problem, but τ S1 has ambiguity owing to f 1 t in the bounded-domain problem. The undetermined function f 1 t is determined in the following way. In the bounded-domain problem whose boundary is a simple boundary, the mass of the gas in the domain is invariant with respect to t. The condition at the leading order is d ω S1 dx = 0, 9 d t V where V indicates the domain or its volume in the later. With the aid of Eq. 4, we have df 1 t V d τ S1 dx = 0. d t d t V 10 On the other hand, from Eq. 7, df 1 t d t V + 5 d d t V τ S1 dx = V Gx i, tdx. 11 From Eqs. 10 and 11, we obtain df 1 t/d t and d V τ S1dx/d t as df 1 t = Gx i, tdx, d t 3V V d τ S1 dx = Gx i, tdx. d t 3 V V 1 That is, f 1 t in the bounded-domain problem [and thus the solution τ S1 of Eq. c] is determined. 1 The time-derivative term vanishes owing to the boundary condition mentioned in the first paragraph of Section

6 The analysis of the higher-order equations is similar; for example, from Eqs. 3a 3c, u is, τ S, and P S3 are determined in the unbounded-domain problem, but f t, u is, τ S, and P S3, except for an additive function of t in P S3, are determined in the bounded-domain problem. Let u is, τ S, and f t be given at t in such a way that Eq. 3a is satisfied. 3 Taking the divergence of Eq. 3b and using Eq. 3a and the results obtained above, we find that P S3 is governed by the Poisson equation P S3 x i = Inhomogeneous term, 13 where the inhomogeneous term consists of u is, P S, and the functions determined in the preceding analysis. On a simple boundary, the derivative of P S3 normal to it being known, 4 P S3 is determined by this equation, except for an additive function of t [say, f 3 t] in the bounded-domain problem. Then, from Eq. 3b, u is / t at t is determined irrespective of f 3 t. From Eq. 3c, 3P S 5ω S / t [or 5τ S P S / t] at t is determined. Thus, u is and τ S except for the additive function f /5 in the bounded-domain problem [thus, ω S except for the additive function 3f /5] are determined. In the boundeddomain problem, where the boundary is a simple boundary, the condition of invariance of the mass of the gas in the domain at the corresponding order is 5 d ω S dx = d t V With the aid of Eq. 4, df t/d t at t is determined as df 1 t/d t is done. To summarize, the solution u is1, P S1, τ S1, P S of the initial and boundaryvalue problem of Eqs. 1 c is determined, with an additive arbitrary function f t in P S in a bounded-domain problem of a simple boundary, when the initial data of u is1, P S1, τ S1, and P S satisfy Eqs. a and 6. The additive function f t does not affect the other variables. The function f t is determined in the next-order analysis. In other words, the solution u is1, P S1, τ S1 of Eqs. 1 c is determined consistently by Eqs. 1, b, and c with the supplementary condition 6, instead of Eq. a, when the initial data of u is1, P S1, and τ S1 satisfy Eq. a. Naturally, the initial P S is required to satisfy Eq. 6. This process is natural for numerical computation. Note that, with the aid of Eq. 4, the time-derivative term 3 P S/ t 5 ω S/ t in Eq. 3c is transformed into 5 τ S/ t P S / t + 5 ω S1τ S1 / t. 3 The time derivative ω S1 / t is known from τ S1 / t, df 1 t/d t, and Eq Shift the discussion of the boundary condition for P S to the next order. 5 The contribution of the Knudsen-layer correction to the mass in the domain is of a higher order, though it is required to ω S. 5

7 1.1. Ghost-effect equations M-3.75 M-3.78b: Consider the initial and boundary-value problem of Eqs. M-3.75 M-3.78b, i.e., ˆp SB0 = ˆp 0 t, 15 ˆp SB1 = ˆp 1 t, 16 ˆρ SB0 t + ˆρ SB0ˆv isb1 = 0, 17a ˆρ SB0ˆv isb1 t = 1 ˆp SB + 1 ˆp 0 + ˆρ SB0ˆv SB1ˆv isb1 x + 1 x x Γ 7 ˆT SB0 [ Γ 1 ˆT SB0 ˆT SB0 3 ˆρ SB0 ˆTSB0 + 5 ˆρ SB0ˆv isb1 ˆTSB0 = 5 t 4 where ˆp 0 and ˆp 1 depend only on t, and ˆviSB1 x ˆT SB0 x 1 3 ˆv ksb1 δ i + ˆv SB1 3 x k ˆT SB0 δ i x k, Γ ˆT SB0 ˆT SB0 ] 17b, 17c ˆp SB0 = ˆρ SB0 ˆTSB0, ˆp SB1 = ˆρ SB1 ˆTSB0 + ˆρ SB0 ˆTSB1, 18 ˆp SB = ˆρ SB ˆTSB0 + ˆρ SB1 ˆTSB1 + ˆρ SB0 ˆTSB, ˆp SB = ˆp SB + Γ 3 3ˆp 0 x ˆT SB0 ˆT SB0. 19 k x k Let ˆρ, ˆv i, and ˆT thus, ˆp = ˆρ ˆT at time t be given; thus, ˆρ SB0, ˆv isb1, ˆT SB0 ˆp SB0, etc., including ˆp SB, are given. Then ˆρ SB0 / t, ˆρ SB0ˆv isb1 / t, and ˆT SB0 / t at t are given by Eqs. 17a 17c; thus, the future ˆρ SB0, ˆv isb1, and ˆT SB0 also ˆp SB0 are determined. However, the future ˆp SB0, as well as ˆp SB0 at t, is required to be independent of x i owing to Eq. 15. Taking this point into account, we discuss how the solution is determined. For convenience of the discussion, transform Eq. 17c in the form where ˆp SB0 t P = 5 3 ˆp ˆv isb1 SB = P, 0 Γ ˆT SB0 ˆT SB0. 6

8 First, consider the case where ˆp thus, ˆp SB0, ˆp SB1, etc. is specified at some point, e.g., at infinity. Then, from Eq. 15, ˆp 0 t is a given function of t, and ˆp SB0 is determined. The initial value of ˆp SB0 is uniform, i.e., ˆp SB0 = ˆp 0 0. On the other hand, from Eq. 0, the variation of ˆp SB0 / t is also determined by the data of ˆp SB0, ˆTSB0, ˆv isb1, and their space derivatives at t. This must coincide with the corresponding data given by Eq. 15, i.e., ˆp SB0 / t =dˆp 0 /d t. Substituting this relation into Eq. 0, we have ˆp SB0ˆv isb1 Γ ˆT SB0 ˆT SB0 = 3 dˆp 0 5 d t, 1 which requires a relation among ˆp SB0, ˆTSB0, and ˆv isb1 for all t, since dˆp 0 /d t is given. This condition is equivalently replaced by the following two conditions: The initial data of ˆp SB0, ˆTSB0, and ˆv isb1 are required to satisfy Eq. 1, and the time derivative of Eq. 1 has to be satisfied for all t, i.e., t ˆp SB0ˆv isb1 Γ ˆT SB0 ˆT SB0 = 3 d ˆp 0 5 d t. With the aid of Eqs. 17a 17c and 0, the left-hand side of Eq. is expressed in the form without the time-derivative terms, i.e., ˆp SB0 / t, ˆT SB0 / t, and ˆv isb1 / t, as follows: t ˆp SB0ˆv isb1 Γ ˆT SB0 ˆT SB0 = 1 ˆp SB0 1 ˆp SB + fn 1, ˆρ SB0 where fn 1 is a given function of ˆρ SB0, ˆv isb1, ˆTSB0, and their space derivatives. Thus, the condition is reduced to an equation for ˆp SB, i.e., 1 ˆp SB = Fn, 3 ˆρ SB0 where Fn = ˆp 0 fn d ˆp 0. d t The boundary condition for ˆp SB in Eq. 3 on a simple boundary is derived by multiplying Eq. 17b by the normal n i to the boundary. In this process, the contribution of its time-derivative terms vanishes. 6 Thus, ˆp SB or ˆp SB is determined in the present case, where ˆp thus, ˆp SB is specified at some point. The solution ˆp SB of Eq. 3 being substituted into Eq. 17b, Eqs. 17a 17c with the first relation in Eq. 18 are reduced to the equations for ˆρ SB0, ˆT SB0, and ˆv isb1 which naturally determine ˆρ SB0 / t, ˆT SB0 / t, and ˆv isb1 / t. Further, if the initial data of ˆρ SB0, ˆTSB0, and ˆv isb1 being chosen in such a way that ˆρ SB0 ˆTSB0 = ˆp SB0 = ˆp 0 and that Eq. 1 is satisfied, the variation ˆp SB0 / t 6 The discussion is similar to that in Footnote 1. 7

9 of ˆp SB0 = ˆρ SB0 ˆTSB0 given by these equations is consistent with Eq. 15, since Eq. 3 or with the condition 1 at the initial state guarantees Eq. 1, i.e., ˆp SB0 / t = dˆp 0 /d t, for all t. Equations 15 and 17a 17c with Eqs. 18 and 19 determine ˆρ SB0, ˆT SB0, ˆp SB0, ˆv isb1, and ˆp SB consistently for appropriately chosen initial data. However, these equations are the leading-order set of equations derived by the asymptotic analysis of the Boltzmann equation. In the above system, ˆp SB is determined. On the other hand, the variation ˆp SB / t is determined independently by the counterpart of Eq. 0 at the order after next. The situation is similar to that at the leading order, where Eqs. 15, with a given ˆp 0, and 0 determine ˆp SB0 independently. The analysis can be carried out in a similar way. Let ˆp SB determined by Eq. 3 be indicated by ˆp SB 0 and the equation for ˆp SB / t, or the counterpart of Eq. 0 at the order after next, be put in the form ˆp SB t = P, 4 where P is a given function of ˆρ SBm, ˆv isbm+1 ˆTSBm m, and their space derivatives. For the consistency, ˆp SB 0 / t is substituted for ˆp SB / t in Eq. 4, i.e., P = ˆp SB 0, 5 t where ˆp SB 0 / t is known. This requires a relation among ˆρ SBm, ˆv isbm+1 ˆT SBm m, and their space derivatives. This condition is equivalently replaced by the following two conditions: Equation 5 is applied only for the initial state, and the time derivative of Eq. 5, i.e., P t = ˆp SB 0 t, has to be satisfied for all t. The ˆρ SBm / t, ˆv isbm+1 / t, ˆT SBm / t m in P / t being replaced by the counterparts of Eqs. 17a 17c and 0 at the corresponding order, an equation for ˆp SB4 for all t is derived. 7 The conclusion is that an additional initial condition and the condition for ˆp SB4 are introduced and, instead, that the condition 3 for ˆp SB is required only for the initial data. The higher-order consideration does not affect the determination of the solution ˆρ SB0, ˆT SB0, and ˆv isb1 thus also ˆp SB0. In this way, the solution of Eqs. 15, 17a 19 is determined consistently by Eqs. 17a 19 with the aid of the supplementary condition 3, instead of Eq. 15, when the initial data of ˆρ SB0, ˆTSB0, and ˆv isb1 satisfy Eqs. 15 and 1, where ˆp 0 t is a known function of t from the boundary condition. Secondly, consider a bounded-domain problem of a simple boundary. In contrast to the first case, dˆp 0 /d t is unknown because no condition is imposed on ˆp SB0 on a simple boundary. However, in a bounded-domain problem of a 7 The conditions on the odd-order ˆp SBn+1 s are derived by the analysis starting from the condition 16 that ˆp SB1 is independent of x i. 8

10 simple boundary, the mass of the gas in the domain is invariant with respect to t, i.e., at the leading order, d ˆρ SB0 dx V = 0, 6 d t where V indicates the domain under consideration. Using the first relation of Eq. 18, i.e., ˆρ SB0 = ˆp 0 / ˆT SB0, in Eq. 6, we have dˆp 0 d t V 1 ˆT SB0 dx = ˆp 0 V 1 ˆT SB0 ˆT SB0 dx. 7 t Using Eq. 17c for ˆT SB0 / t in Eq. 7, we find that the variation dˆp 0 /d t is expressed with ˆp 0, ˆT SB0, and ˆv isb1 as follows: where P t = ˆp 0 V V dˆp 0 d t = P t, 8 [ 1 5 Γ ˆT SB0 6ˆρ SB0 x ˆT SB0 ˆT SB0 5 i 3 ˆv ˆT ] SB0 isb1 dx 1 1 dx. 9 ˆT SB0 With this expression of dˆp 0 /d t, we can carry out the analysis in a similar way to that in the first case. The variation dˆp 0 /d t or ˆp SB0 / t is also determined by Eq. 0. The two ˆp SB0 / t s given by Eq. 8 with Eq. 9 and Eq. 0 have to be consistent. Thus, substituting Eq. 8 with Eq. 9 into ˆp SB0 / t in Eq. 0, we have ˆp SB0ˆv isb1 Γ ˆT SB0 ˆT SB0 = 3 5 P t, 30 where P t is given by Eq. 9. This must hold for all t for consistency. This condition is equivalently replaced by the following two conditions: The initial data of ˆp SB0, ˆT SB0, ˆv isb1 are required to satisfy Eq. 30, and the time derivative of Eq. 30 has to be satisfied for all t, i.e., t ˆp SB0ˆv isb1 Γ ˆT SB0 ˆT SB0 = 3 5 dp t. 31 d t Using Eqs. 17a, 17b, and 0 for the time derivatives ˆρ SB0 / t, ˆv isb1 / t, and ˆp SB0 / t in Eq. 31, we find that ˆp SB at t is determined by the equation 1 ˆp SB ˆp + L SB = Fn, 3 ˆρ SB0 9

11 where Fn is a given functional of ˆρ SB0, ˆv isb1, ˆT SB0, and their space derivatives, and L ˆp SB / is a given linear functional of ˆp SB /, i.e., ˆp L SB = 1ˆp0 V 1 ˆT SB0 ˆT SB0 ˆp SB dx V 1 1 dx. ˆT SB0 On a simple boundary, the derivative of ˆp SB normal to the boundary is specified. Thus, ˆp SB is determined except for an additive function of t. The solution ˆp SB of Eq. 3 being substituted into Eq. 17b, the result is independent of the additive function. Thus, Eqs. 17a 17c with the first relation in Eq. 18 and the above ˆp SB substituted are reduced to those for ˆρ SB0, ˆTSB0, and ˆv isb1, which naturally determine ˆρ SB0 / t, ˆT SB0 / t, and ˆv isb1 / t. Further, if the initial data of ˆρ SB0, ˆT SB0, and ˆv isb1 being chosen in such a way that ˆρ SB0 ˆTSB0 = ˆp SB0 = ˆp 0 and that Eq. 30 is satisfied, the variation ˆp SB0 / t of ˆp SB0 = ˆρ SB0 ˆTSB0 given by these equations is consistent with Eq. 15, since Eq. 3 or 31 with the condition 30 at the initial state guarantees Eq. 30, i.e., ˆp SB0 / t = dˆp 0 /d t, for all t. Equations 15 and 17a 17c with Eqs. 18 and 3 determine ˆρ SB0, ˆT SB0, ˆp SB0, ˆv isb1, and ˆp SB, except for an additive function of t in ˆp SB, consistently for appropriately chosen initial data. However, these equations are the leading-order set of equations derived by the asymptotic analysis of the Boltzmann equation. The analysis of the higher-order equations not shown here is carried out in a similar way. First, the undetermined additive function in ˆp SB is determined by the condition of invariance of the mass of the gas in the domain at the order after next as dˆp 0 /d t is determined. 8 The ˆp SB / t or ˆp SB determined in this way is indicated by ˆp SB 0 / t or ˆp SB 0. On the other hand, the variation ˆp SB / t is determined independently by Eq. 4 or the counterpart of Eq. 0 at the order after next. The two results must coincide. The discussion from here is the same as that given from the sentence starting from Eq. 4 to the end of the paragraph. The results are that an additional initial condition and the condition for ˆp SB4 are introduced, and that the condition 3 for ˆp SB is required only for the initial data. The higher-order consideration does not affect the determination of the solution ˆρ SB0, ˆT SB0, and ˆv isb1 thus also ˆp SB0. In this way, the solution of Eqs. 15, 17a 17c is determined consistently by Eqs. 17a 17c with the aid of the supplementary condition 3, instead of Eq. 15, when the initial data of ˆρ SB0, ˆT SB0, and ˆv isb1 satisfy Eqs. 15 and Notes on basic equations in classical fluid dynamics 1..1 Euler and Navier Stokes sets For the convenience of discussions, the basic equations in the classical fluid dynamics are summarized here. 8 The Knudsen-layer correction to ˆρ SB1, already determined see Footnote 7, contributes to the mass at this order. 10

12 The mass, momentum, and energy-conservation equations of fluid flow are given by ρ t + ρv i = 0, 33 X i t ρv i + ρv i v + p i = 0, 34 X [ ρ e + 1 ] t v i + [ρv e + 1 ] X v i + v i p i + q = 0, 35 where ρ is the density, v i is the flow velocity, e is the internal energy per unit mass, p i, which is symmetric with respect to i and, is the stress tenor, and q i is the heat-flow vector. The pressure p and the internal energy e are given by the equations of state as functions of T and ρ, i.e., Especially, for a perfect gas, p = pt, ρ, e = et, ρ. 36 p = RρT, e = et. 37 Equations 34 and 35 are rewritten with the aid of Eq. 33 in the form ρ v i t + ρv v i + p i = 0, 38 X X ρ e + 1 t v i + ρv e + 1 X v i + v i p i + q = X The operator /t+v /x, which expresses the time variation along the fluid particle, is denoted by D/Dt, i.e., D Dt = t + v. X Multiplying Eq. 38 by v i we obtain the equation for the variation of kinetic energy as ρ D 1 p i Dt v i = v i. 40 X Another form of Eq. 35, where Eq. 40 is subtracted from Eq. 39, is given as Noting the thermodynamic relation ρ De Dt = p v i i q. 41 X X De Dt = T Ds Dt + p Dρ ρ Dt, 4 11

13 where s is the entropy per unit mass, and Eq. 33, Eq. 41 is rewritten as ρ Ds Dt = 1 [ p i pδ i v i + q ]. 43 T X X Equation 43 expresses the variation of the entropy of a fluid particle. Equations contain more variables than the number of equations. Thus, in the classical fluid dynamics, the stress tensor p i and the heat-flow vector q i are assumed in some ways. The Navier Stokes set of equations or the Navier Stokes equations is Eqs where p i and q i are given by vi p i = pδ i µ + v X X i 3 v k δ i X k µ B v k X k δ i, 44 q i = λ T X i, 45 where µ, µ B, and λ are, respectively, called the viscosity, bulk viscosity, and thermal conductivity of the fluid. They are functions of T and ρ. The Euler set of equations or the Euler equations is Eqs where p i and q i are given by p i = pδ i, q i = 0, 46 or the Navier Stokes equations with µ = µ B = λ = 0. For the Navier Stokes equations, in view of the relations 44 and 45, the entropy variation is expressed in the form 9 ρ Ds Dt = 1 T [ µ vi + v X v k vk δ i + µ B + X i 3 X k X k X i λ T ]. X i 47 For the Euler equations, for which p i and q i are given by Eq.46, the entropy of a fluid particle is invariant, i.e., ρ Ds = Dt 9 Note the following transformation: v i vi + v «v k δ i X X X i 3 X k = 1 vi + v v k δ i + «v k vi δ i + v «v k δ i X X i 3 X k 3 X k X X i 3 X k = 1 vi + v «v k δ i + 1 v l vi δ i + v «v k δ i. X X i 3 X k 3 X l X X i 3 X k The second term in the last expression is easily seen to vanish. 1

14 For an incompressible fluid, the first relation of Eq. 36 is replaced by 10 Dρ Dt = 0 or ρ t + v ρ = X Thus, from Eqs. 33 and 49, v i X i = Equation 44 for the Navier Stokes-stress tensor reduces to vi p i = pδ i µ + v. 51 X X i The first term on the right-hand side of Eq. 41 reduces to [ v i vi p i = pδ i µ + v ] vi X X X i X = µ vi + v. X X i Thus, Eq. 41 reduces to ρ De Dt = µ vi + v + λ T. 5 X X i X X To summarize, the Navier Stokes equations for incompressible fluid are v i = 0, 53a X i ρ v i t + ρv v i = p + [ vi µ + v ], 53b X X i X X X i ρ e t + ρv e = µ vi + v + λ T, 53c X X X i X X with the incompressible condition 49 being supplemented, i.e., ρ t + v ρ = X 10 The density is invariant along fluid-particle paths. If ρ is of uniform value ρ 0 initially, it is a constant, i.e., ρ = ρ 0. In a time-independent or steady problem, the density is constant along streamlines. 13

15 1.. Boundary condition for the Euler set In Section M-3.5, we discussed the asymptotic behavior for small Knudsen numbers of a gas around its condensed phase where evaporation or condensation with a finite Mach number is taking place, and derived the Euler equations and their boundary conditions that describe the overall behavior of the gas in the limit that the Knudsen number tends to zero. The number of boundary conditions on the evaporating condensed phase is different from that on the condensing one. We will try to understand the structure of the Euler equations giving the non-symmetric feature of the boundary conditions by a simple but nontrivial case. Consider, as a simple case, the two-dimensional boundary-value problem of the time-independent Euler equations in a bounded domain for an incompressible ideal fluid of uniform density. The mass and momentun-conservation equations of the Euler set are u x + v = 0, 55 u u x + v u = 1 p ρ x, 56 u v x + v v = 1 p ρ, 57 where ρ is the density, which is uniform, u, v is the flow velocity, and p is the pressure. Owing to Eq. 55, the stream function Ψ can be introduced as u = Ψ, v = Ψ x. 58 Eliminating p from Eqs. 56 and 57, we have 11 where Ω is the vorticity, i.e., From Eqs. 58 and 59, u Ω x + v Ω = 0, 59 Ω = u v x = Ψ x + Ψ. 60 Ψ Ω x Ψ x 11 The following equation is formed from them: Ω Eq. 56/ Eq. 57/x = 0. =

16 This equation shows that Ω is a function of Ψ, 1 i.e., Ω = F Ψ. 6 This functional relation between Ω and Ψ is a local relation, and therefore F may be a multivalued function of Ψ. From Eqs. 60 and 6, Ψ x + Ψ = F Ψ. 63 Consider a boundary-value problem in a simply-connected bounded domain, where Ψ is given on the boundary Ψ = Ψ B. Introduce a coordinate s 0 s < S along the boundary in the direction encircling the domain counterclockwise. Then, the fluid flows into the domain on the boundary where Ψ B /s < 0, and the fluid flows out from the domain on the boundary where Ψ B /s > 0. When F is given, the problem is a standard boundary-value problem. In the present problem, we have a freedom to choose F on the part where Ψ B /s < 0 or Ψ B /s > 0. For example, take the case where Ψ B /s < 0 for 0 < s < S m and Ψ B /s > 0 for S m < s < S, and choose the distribution Ω B s of Ω along the boundary for the part 0 < s < S m. By the choice of Ω B, the function F Ψ is determined in the following way. Inverting the relation Ψ = Ψ B s between 1 This can be seen with the aid of theorems on implicit functions see References M-[47, 48, 67]. The proof is outlined here. The Ω and Ψ are functions of x and y : Solving the second equation with respect to x, we have Ω = Ωx, y, Ψ = Ψx, y. With this relation into Eq., x = ˆxΨ, y. Ω = ΩˆxΨ, y, y = ˆΩΨ, y, Ψ = ΨˆxΨ, y, y = ˆΨΨ, y. That is, Ω is expressed as a function of Ψ and y. From Eqs. a and b, a b On the other hand, ˆΩΨ, y ˆΨΨ, y ˆΨΨ, y = ΩˆxΨ, y, y = Ωx, y ˆxΨ, y + x Ωx, y, a = 0. b = ΨˆxΨ, y, y = Ψx, y ˆxΨ, y + x Ψx, y. Thus, Ψx, y ˆxΨ, y Ψx, y + = 0. x From Eqs. 61, a and, we have ˆΩΨ, y = 0, or Ω = ˆΩΨ. 15

17 Ψ and s on the part 0 < s < S m, i.e., sψ, and noting the relation 6, we find that F is given by F Ψ = Ω B sψ. 64 Then, the boundary-value problem is fixed. That is, Eq. 63 is fixed as 13 Ψ x + Ψ = Ω BsΨ, 65 and the boundary condition is given as Ψ = Ψ B s. This system is a standard from the point of counting of the number of boundary conditions. Obviously, from Eq. 60, the solution of the above system automatically satisfies condition Ω = Ω B s along the boundary for 0 < s < S m. We cannot choose the distribution of Ω on the boundary for S m < s < S. The energy-conservation equation of the incompressible Euler set is given by Eq. 5 with µ = λ = 0, i.e., u e x + v e = 0, or Ψ e x Ψ e x = 0, 66 where e is the internal energy. Thus, e is a function of Ψ, i.e., e = F 1 Ψ. 67 In the above boundary-value problem, therefore, e can be specified on the the part 0 < s < S m of the boundary, but no condition can be specified on other part S m < s < S and vice versa. 14 To summarize, we can specify three conditions for Ψ, Ω, and e on the part Ψ B /s < 0 Ψ B /s > 0 of boundary but one condition for Ψ on the other part Ψ B /s > 0 Ψ B /s < 0. The number of the boundary conditions is not symmetric and consistent with that derived by the asymptotic theory Ambiguity of pressure in the incompressible Navier Stokes system It may be better to note ambiguity of the solution of the initial and boundaryvalue problem of the incompressible Navier Stokes equations in a bounded domain of simple boundaries. 13 There is still some ambiguity. The case where there is a region with closed stream lines Ψx, y = const inside the domain is not excluded. 14 From the second relation on e of Eq. 36 and the uniform-density condition, the condition on e can be replaced by the condition on the temperature T. 16

18 Consider the Navier Stokes equations for an incompressible fluid, i.e., v i = 0, ρ v i t + ρv v i = p + x ρ e t + ρv e = µ X ρ t + v ρ i = 0, vi µ x vi X + v X i 68a + v, 68b x + λ T, 68c X X 68d where e, µ, and λ are functions of T and ρ. Consider the initial and boundary-value problem of Eqs. 68a 68d in a bounded domain D on the boundary D of which v i and T are specified as v i = v wi and T = T w v wi and T w are, respectively, the surface velocity and temperature of the boundary satisfying D v win i ds = 0, n i : the unit normal vector to the boundary and no condition is imposed on ρ and p. Let v s i, ρ s, T s, p s be a solution of the initial and boundary-value problem. Let P a be an arbitrary function of t, independent of x i, that vanishes at initial time t = 0, i.e., P a = ft with f0 = 0. Put v i, ρ, T, p = v s i, ρ s, T s, p s + P a. Then, e, µ, and λ corresponding to the new v i, ρ, T, p are equal to e s, µ s, and λ s respectively, because they are determined by ρ and T. The new v i, ρ, T, p satisfy the equations 68a 68d and the initial and boundary conditions Equations derived from the compressible Navier Stokes set when the Mach number and the temperature variation are small It is widely said that the set of equations derived from the compressible Navier Stokes set when the Mach number and the temperature variation are small is the incompressible Navier Stokes set. This statement should be made precise. The difference is briefly explained in the book Molecular Gas Dynamics in connection with the equations derived by the S expansion from the Boltzmann equation in Sections M-3..4 and M Here, we explicitly show the process of analysis from the compressible Navier Stokes set. The resulting set of equations no longer has ambiguity of pressure in contrast to the incompressible Navier Stokes set. Take a monatomic perfect gas, for which the internal energy per unit mass is 3RT/. The corresponding Navier Stokes set of equations is written in the nondimensional variables introduced by Eq. M-1.74 in Section M-1.10 as 17

19 follows: Sh ω ˆt ωu i = 0, 69 Sh 1 + ωu i ωu i u + 1 ˆt x P i = 0, 70 Sh [ ] ω ˆt 1 + τ + u i + [ ] ωu x 1 + τ + u i + u i δ i + P i + Q = The nondimensional stress tensor P i, and heat-flow vector Q i are expressed as 15 P i = P δ i µ 0RT 0 1/ ui 1 + µ + u u k δ i, p 0 L x 3 x k 7a λ 0 T 0 τ Q i = 1 + λ. Lp 0 RT 0 1/ 7b Here, µ and λ are, respectively, the nondimensional perturbed viscosity and thermal conductivity defined by µ = µ µ, λ = λ λ, where µ 0 and λ 0 are, respectively, the values of the viscosity µ and the thermal conductivity λ at the reference state. The µ and λ are functions of τ and ω. The first relation of the equation of state [Eq. 37] is expressed as thus, P = ω + τ + ωτ. 73 Take a small parameter ε, and consider the case where u i = Oε, ω = Oε, τ = Oε, Sh = Oε, 74a µ 0 RT 0 1/ = γ 1 ε, p 0 L λ 0 T 0 Lp 0 RT 0 1/ = 5 4 γ ε, P = Oε, µ = Oε, λ = Oε. 74b According to the definition of u i in Eq. M-1.74, ε is of the order of the Mach number. In view of this and the definition of the Prandtl number Pr = 5Rµ/λ see Section M-3.1.9, γ 1 and γ are, respectively, of the orders of 1/Re and 1/PrRe Re : the Reynolds number. According to Eq. M-1.48a, the condition Sh = Oε in Eq. 74a means that the time scale t 0 of the variation of variables is of the order of L/RT 0 1/ ε, which is of the order of time scale of viscous 15 For a monatomic gas, the bulk viscosity vanishes, i.e., µ B = 0. 18

20 diffusion. Thus, we are considering the case where the Mach number is small, the Reynolds and Prandtl numbers are of the order of unity, and the time scale of variation of the system is of the order of the time scale of viscous diffusion. We can take t 0 = L/RT 0 1/ ε without loss of generality. Then, Sh = ε. 75 Corresponding to the above situation, u i, ω, P, and τ are expanded in power series of ε, i.e., u i = u i1 ε + u i ε +, ω = ω 1 ε + ω ε +, P = P 1 ε + P ε +, τ = τ 1 ε + τ ε +, 76a 76b 76c 76d µ = µ 1 ε + µ ε +, 76e λ = λ 1 ε + λ ε +, 76f P i = P 1 δ i ε + P i ε +, Q i = Q i ε +. 76g 76h Substituting Eqs. 76a 76h with Eqs. 74b and 75 into Eqs with Eqs. 7a and 7b, and arranging the same-order terms of ε, we have u i1 = 0, P 1 = 0, u i1 = 0, ω 1 ˆt + ω 1u i1 + u i = 0, u i1 + u i1u P γ 1 ui1 + u 1 ˆt x x x 3 3 P 1 ˆt + 5 x u + 5 P 1u γ τ 1 = 0, x u k1 δ i = 0, x k and so on. At the leading order, the equations derived from Eqs. 69 and 71 degenerate into the same equation u i1 / = 0. Owing to this degeneracy, in order to solve the variables from the lowest order successively, the equations should be rearranged by combination of equations of staggered orders. Thus, we rearrange the equations as follows: P 1 = 0, 77 19

21 u i1 = 0, 78a u i1 ˆt + u 1 u i1 x = 1 5 τ 1 ˆt P 1 ˆt + 5 u τ 1 i1 P + γ 1 = 5 4 γ u i1 x, 78b τ 1 x, 78c where u i = ω 1 ˆt ω 1u i1, 79a u i u i u i1 + u 1 + u ˆt x x = 1 P3 P ω 1 + γ 1 ui + u u k δ i x x 3 x k 3 P γ 1ω 1 u i1 x + γ 1 ˆt + 3 u P x = 5γ 4 τ + λ 1 τ 1 [ ui1 µ 1 + u 1 x x u P 1 ], 79b ω x ˆt ω 1u + ω u 1 x + γ 1 ui1 + u 1, 79c x P 1 = ω 1 + τ 1, P = ω + τ + ω 1 τ These equations are very similar to Eqs. M-3.65 M-3.68 [or Eqs. 1 4] obtained by the S expansion of the Boltzmann equation in Section M-3.7. or Section The solution is determined in the same way as the solution of the S-expansion system is done in Section What should be noted is the determination of P 1, P, in a bounded-domain problem. They are determined by the condition of invariance of the mass of the gas in the domain with the aid of higher-order equations in the same way as P S1, P S, in the S-expansion system see Section In order to compare Eqs. 78a 78c and 80 with the incompressible Navier Stokes equations 68a 68d, we will rewrite the latter equations for the situation where the former equations are derived. The starting equations are Eqs. 69 7b 16 and the nondimensional form of Eq. 49, i.e., Sh ω ˆt + u ω i = 0, 81 instead of Eq The analysis is carried out in a similar way and the 16 As the internal energy e, 3RT/ [= 3RT τ/] is chosen for consistency. 17 From Eqs. 69 and 81, we have u i / = 0. 0

22 equations corresponding to Eqs. 78a 78c are 18 u i1 = 0, 8a u i1 ˆt + u 1 u i1 x = 1 P + γ 1 u i1 x, 8b 3 τ 1 ˆt + 3 u τ 1 i1 = 5 4 γ τ 1 x. 8c Equations 8a and 8b are, respectively, of the same form as Eqs. 78a and 78b. Equation 78c is rewritten with the aid of Eqs. 77 and 80 as 3 τ 1 ˆt + 3 u τ 1 i1 ω1 ˆt + u i1 The difference of Eq. 78c or 83 from Eq. 8c is ω 1 ˆt + u ω 1 i1, ω 1 = 5 4 γ τ 1 x. 83 which vanishes for an incompressible fluid. The work W done on unit volume of fluid by pressure, given by p 0 RT 0 1/ 1 + P u i /, is transformed with the aid of Eq. 79a in the following way: W p 0 RT 0 = 1 + P u i 1/ = u i1 ε = u i ε + x i ω1 = ˆt + u i1 u i1 P 1 + u i ε + ω 1 ε +. The work vanishes up to the order considered here for an incompressible fluid, because u i / = 0 and P 1 / = 0 see Footnote 18. That is, Eq. 78c differs from Eq. 8c by the amount of the work done by pressure. To summarize, the mass and momentum-conservation equations of the set derived from the compressible Navier Stokes set when the Mach number and the temperature variation are small are of the same form as the corresponding equations for the incompressible Navier Stokes set, but the energy-conservation equation differs by the work done by pressure. 19 Incidentally, we have already 18 We also obtain P 1 / = When the density ρ is uniform initially, for which ρ is a constant for an incompressible fluid, the viscosity and thermal conductivity are constants, and heat production by viscosity is neglected, Eqs. 8a 8c can be compared directly with Eqs. 78a 78c and 80, without carrying expansion, and the same results are obtained. 1

23 seen that the pressure in the incompressible Navier Stokes set has ambiguity of an additive function of time irrespective of the size of the parameters in contrast to the pressure in the former set derived from the compressible Navier Stokes set. Chapter M-9.1 Processes of solution of the equations with the ghost effect of infinitesimal curvature July 007 The way in which Eqs. M-9.33 M-9.39b or Eqs. M-9.49a M-9.50e, including the time-dependent case with the additional time-derivative terms given by Eq. M-9.4 or the mathematical expressions next to Eq. M-9.59, contain the pressure terms, ˆp S0, ˆp S or P 01, P 0, P 0, is different from the way in which the Navier Stokes equations M-3.65 M-3.66c do the pressure terms, P S1, P S. In Section M-9.4, we consider the time-independent solution of Eqs. M-9.49a M-9.50e [Eqs. M-9.56 M-9.57d] that is uniform with respect to χ. Here, it may be better to explain how a solution of Eqs. M-9.33 M-9.39b or Eqs. M-9.49a M-9.50e in a general case or a time-dependent solution that depends on χ or χ is obtained. Incidentally, the boundary conditions for the time-dependent case are derived in the same way as in Section M Naturally from the derivation of the equations, the domain of a gas is in a straight pipe or channel of infinite length whose axis is in the x or χ direction..1.1 Equations M-9.33 M-9.39b: Take Eqs. M-9.33 M-9.39b with the additional time-derivative terms given by Eq. M-9.4, i.e., 0 ˆp S0 = ˆp S0 = 0, 84 ˆρ S0 ˆt + ˆρ S0ˆv xs0 χ + ˆρ S0ˆv ys1 + ˆρ S0ˆv zs1 = 0, 85 ˆρ S0 ˆv xs0 ˆt ˆv xs0 + ˆρ S0 ˆv xs0 χ + ˆv ˆv xs0 ys1 = 1 ˆp S0 χ + 1 Γ 1 ˆv xs0 0 Equation M-9.33 is replaced by its equivalent form 84. ˆv xs0 + ˆv zs1 + 1 ˆv xs0 Γ 1, 86

24 ˆv ys1 ˆv ys1 ˆρ S0 + ˆρ S0 ˆv xs0 ˆt χ ˆρ S0 ˆv zs1 ˆt 5ˆρ S0 ˆT S0 ˆt + ˆv ˆv ys1 ys1 + ˆv zs1 ˆv ys1 = 1 ˆp c S + 1 ˆv xs0 Γ 1 χ + ˆv ys1 Γ [ ˆvyS1 Γ Γ ˆT S0 ˆp S0 7 + Γ ˆT S0 7 { [ + 1 ] ˆvxS0 Γ 8 + ˆv xs0 Γ 8 ˆp S0 ˆv zs1 + ˆρ S0 ˆv xs0 χ + ˆv ˆv zs1 ys1 = 1 ˆp c S + 1 χ + 1 [ ˆvyS1 Γ ˆp S0 Γ 1 ˆv xs0 Γ 7 ˆT S0 { + 1 ˆv xs0 Γ 8 ˆp S0 + 5 ˆρ S0 ˆv xs0 ˆT S0 χ + ˆv zs1 ˆT S0 + ˆv ] zs1 ˆv zs1 + ˆv zs1 1 c ˆv xs0 ˆT S0 ] + ˆv zs1 Γ 1 + Γ 7 ˆT S0 } ˆv xs0, 87 [ ˆv xs0 + ]} ˆvxS0 Γ 8, 88 + ˆv ˆT S0 ys1 + ˆv ˆT S0 zs1 ˆp S0 ˆp S0 ˆv xs0 ˆt χ = 5 Γ ˆT S0 + 5 Γ ˆT [ S0 ˆvxS0 + Γ and the subsidiary relations ˆvxS0 ], 89 ˆp S0 χ, ˆt = ˆρ S0 ˆTS0, 90a 3

25 ˆp c S = ˆp S + Γ 1 ˆvxS0 3 χ + ˆv ys1 + ˆv zs1 + Γ 7 ˆT S0 + ˆT S0 3ˆp S0 [ + Γ ˆT S0 3 + Γ ˆT ] S0 3 3ˆp S0 [ Γ ] 9 ˆvxS0 ˆvxS0 +, 90b 3ˆp S0 where Γ 1, Γ, Γ 3, Γ 7, Γ 8, and Γ 9 are short forms of the functions Γ 1 ˆT S0, Γ ˆT S0,..., Γ 9 ˆT S0 of ˆT S0 defined in Section M-A..9. Consider the solution of the initial and boundary-value problem of Eqs b. Let ˆρ, ˆv i, and ˆT thus, ˆp = ˆρ ˆT at time ˆt be given; thus, ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆT S0 ˆp S0 etc., including ˆp S, are given. Then ˆρ S0 /ˆt, ˆv xs0 /ˆt, ˆv ys1 /ˆt, ˆv zs1 /ˆt, and ˆT S0 /ˆt at ˆt are given by Eqs b; thus, the future ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, and ˆT S0 also ˆp S0 are determined. However, the future ˆp S0 is required to be independent of y and z, as well as ˆp S0 at ˆt, owing to Eq. 84. Taking this into account, we will discuss how the solution is obtained by this system consistently. First, transform Eq. 89 with the aid of Eqs. 85 and 90a in the following form: ˆp S0 = P, 91 ˆt where P = 5 3 ˆp ˆvxS0 S0 χ + ˆv ys1 + ˆv zs1 ˆp S0 ˆv xs0 χ [ Γ ˆT S0 + Γ ˆT S0 ] [ + ˆvxS0 3 Γ 1 + ˆvxS0 ]. 9 For ˆp S0 to be independent of y and z [see Eq. 84], P as well as the initial data of ˆp S0 is required to be independent of y and z. Noting that ˆp S0 is independent of y and z, and taking the average of Eq. 9 over the cross section S of the pipe or channel, 1 we have another expression P of P, explicitly uniform with 1 i In a channel, where the gas extends from z = to z =, the integral R S Adydz per unit length in z, per a period in z, etc. should be considered. Otherwise, it can be infinite. ii Note that ˆv ys1 n y + ˆv zs1 n z = 0 on a simple boundary where n i = 0, n y, n z is the normal to the boundary. 4

26 respect to y and z, i.e., where P = 5 ˆv xs0 3 χ ˆp ˆp S0 S0 ˆv xs0 χ [ + ] ˆvxS0 3 Γ ˆvxS0 1 + A = S [ Γ ˆT S0 + Γ ˆT S0 ], 93 / Adydz dydz. S The expression 93 is noted to be independent of ˆv ys1 and ˆv zs1. The two expressions 9 and 93 must give the same result, i.e., P = P, or 5 3 ˆp ˆvxS0 S0 χ + ˆv ys1 + ˆv zs1 ˆp S0 ˆv xs0 χ [ + 5 Γ ˆT S0 + Γ ˆT ] [ S0 + ˆvxS0 6 3 Γ 1 + ˆvxS0 = P, 94 ] when Eq. 84 holds, and vice versa. The condition 94 for all ˆt is equivalently replaced by the two conditions that the initial data of ˆp S0, ˆT S0, ˆv xs0, ˆv ys1, and ˆv zs1 satisfy Eqs. 84 and 94 and that the time derivative of Eq. 94 holds for all ˆt, i.e., P ˆt = P ˆt. 95 Using Eqs and 91 for ˆρ S0 /ˆt, ˆv xs0 /ˆt, ˆv ys1 /ˆt, ˆv zs1 /ˆt, and ˆp S0 /ˆt ˆρ S0 ˆT S0 /ˆt = ˆp S0 /ˆt ˆT S0 ˆρ S0 /ˆt in P/ˆt derived from Eq. 9, we find that P/ˆt is expressed with ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆp S0, and ˆp c S in the form P ˆt = 5 [ 1 6 ˆp ˆp c S S0 + 1 ˆp c ] S + Fn 1, 96 ˆρ S0 ˆρ S0 where Fn 1 is a given function of ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆp S0, and their space derivatives. The expression 93 of P being independent of ˆv ys1 and ˆv zs1, its time derivative P/ˆt does not contain ˆv ys1 /ˆt and ˆv zs1 /ˆt. Therefore, with the aid of Eqs. 85, 86, and 89, P/ˆt is expressed with ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆp S0, and their space derivatives, i.e., P ˆt = Fn ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆp S0, and their space derivatives, 97 5

27 where Fn is a given functional of its arguments. From Eqs. 95, 96, and 97, we have 1 ˆp c S + 1 ˆp c S = Fn, 98 ˆρ S0 ˆρ S0 where Fn = 6Fn Fn 1 /5ˆp S0, and therefore, Fn is a given functional of ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆp S0, and their space derivatives. This is the equation for ˆp c S over a cross section of the pipe or channel. The boundary condition for ˆp c S on a simple boundary is obtained by multiplying Eqs by the normal n i = 0, n y, n z to the boundary; In this process, the contribution of their time-derivative terms vanishes because ˆv ys1 n y + ˆv zs1 n z = 0; Then, the n y ˆp c S /+ n z ˆp c S / is imposed as the boundary condition. Thus, ˆp c S is determined by Eq. 98 except for an additive function of ˆt and χ. With this ˆp c S substituted into Eqs. 87 and 88, ˆρ S0/ˆt, ˆv xs0 /ˆt, ˆv ys1 /ˆt, ˆv zs1 /ˆt, and ˆp S0 /ˆt are determined by Eqs b independently of the additive function in ˆp c S in such a way that ˆp S0//ˆt = ˆp S0 //ˆt = 0 and P//ˆt = P//ˆt = 0. That is, the solution ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆT S0 of Eqs b is determined by Eqs b with the aid of the supplementary condition 98, instead of Eq. 84, when the initial condition for ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, and ˆT S0 is given in such a way that ˆp S0 = ˆρ S0 ˆTS0 and P are independent of y and z. Equations 84 90b are the leading-order set of equations derived by the asymptotic analysis of the Boltzmann equation. The analysis of the higherorder equations not shown here is carried out in a similar way. The equation for ˆp S /ˆt, corresponding to Eq. 91, is derived at the order after next. However, owing to the consistency of ˆp S0, ˆp S is already determined by Eq. 98 except for an additive function of χ and ˆt. The situation is similar to that at the leading order. That is, ˆp S0 and ˆp S are, respectively, determined by Eqs. 84 and 98, each with an additive function of χ and ˆt and also by Eqs. 91 and the counterpart of Eq. 91 at the order after next. Thus, the higher-order analysis can be carried out in a similar way. The results are that an additional initial condition and an equation for ˆp S4, the counter part of Eq. 98, are introduced and that the condition 98 is required only for the initial data. The higherorder consideration does not affect the determination of the solution ˆρ S0, ˆT S0, ˆv xs0, ˆv ys1, and ˆv zs1 thus also ˆp S0. To summarize, the solution ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, ˆT S0 of Eqs b is determined by Eqs b with the aid of the supplementary condition 98, instead of Eq. 84, when the initial data of ˆρ S0, ˆv xs0, ˆv ys1, ˆv zs1, and ˆT S0 are given in such a way that ˆp S0 = ˆρ S0 ˆTS0 and P are independent of y and z. The results are not affected by the higher-order analysis. If P is independent of y and z, P = P by definition. 6

28 .1. Equations M-9.49a M-9.50e: Take Eqs. M-9.49a M-9.50e with the additional time-derivative terms given in the first mathematical expressions after Eq. M-9.59, i.e., P 01 χ = P 01 = P 01 = 0, P 01 = ω + τ, 99a P 0 = P 0 = 0, 99b u x χ + u y + u z = 0, u x ˆt + u u x x χ + u u x y + u u x z = 1 P 0 χ + γ 1 u y ˆt + u u y x χ + u u y y + u u y z u x C = 1 P 0 + γ 1 u z ˆt + u u z x χ + u u z y + u u z z = 1 P 0 τ ˆt P 01 τ + u x 5 ˆt χ + u τ y + u τ z = γ + γ 1 u z τ + τ u x + u x u y 100a, 100b + u y, 100c, 100d + u z. 100e The qualitative difference of this set of equations from the set 84 90b is the absence of the time-derivative term in Eq. 100a that corresponds to Eq. 85. Consider the solution of the initial and boundary-value problem of Eqs. 99a 100e. Let u x, u y, u z, and τ at ˆt be given in such a way that Eq. 100a is satisfied. [ Integrating Eq. 100a over the cross section of the channel or pipe S Eq. 100adydz], we find that S u xdydz depends only on ˆt, 3 i.e., u x / χdydz = 0, 101 S where S indicates the cross section. Applying Eqs. 99b, 100a, and 101 to the equation Eq. 100bdydz/ χ, we have P 0 / χ as P 0 χ = [ ] u x χ χ + γ u x 1 + u x, 10 where A = S / Adydz dydz. S Thus, P 0 / χ and P 0 are determined if they are specified at a point in the gas. Here, we consider this case. 4 Using Eq. 100a in the sum of [Eqs. 100b]/ χ, 3 See Footnote 1, with ˆv ys1 and ˆv zs1 being replaced by u y and u z. 4 i Imagine the case of the Poiseuille flow. ii Here, P thus, P 01 is specified at some point. Then, P 01 is a given function of ˆt. 7

29 [Eqs. 100c]/, and [Eqs. 100d]/, we obtain the equation for P 0 in the form P 0 + P 0 = Fnu x, u y, u z, and their space derivatives, 103 where Fn is a given functional of the variables in the parentheses, and the time derivatives are absent owing to Eq. 100a. Thus, the right-hand side of Eq. 103 is known. This equation is the Poisson equation for P 0 over the cross section S. Its boundary condition is obtained in a way similar to how the condition for ˆp c S in Eq. 98 is derived. Thus, P 0 over each cross section is determined except for an additive function of ˆt and χ. This ambiguity does not influence P 0 / and P 0 /. With P 0 and P 0 prepared above into Eqs. 100b 100e, the time derivatives u x /ˆt, u y /ˆt, u z /ˆt, and τ/ˆt are determined in such a way that u x / χ + u y / + u z //ˆt = 0 owing to the above choice of P 0. 5 Thus, the solution u x, u y, u z, τ of Eqs. 99b, 100a 100e is determined by Eqs. 100b 100e with the aid of the supplementary conditions 10 and 103 for P 0 and P 0, instead of Eqs. 99b and 100a. This process is natural for numerical computation. The undetermined additive function of χ and ˆt in P 0, which does not affect the solution u x, u y, u z, τ, is determined by the higherorder equation derived from that for ˆv xs /ˆt see Section.1.1, in a way similar to that in which P 0 is determined by Eq. 100b. In the higher-order equation, P 0 plays the same role as P 0 in Eq. 100b; Equation 103 corresponds to Eq. 99b, and P 0 and P 0 are determined by these equations, each with an additive function of χ and ˆt. 5 Note that P 01 is known Footnote 4. 8