Macroscopic description of rarefied gas flows in the transition regime
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(117) ; . ! & F. 12 * ( 6 0 0 # & -- ±vW & # = # 0 . 1 2 * ! 5 6 0 & & # ; 1 2 # 12 * - 6 0& U D - & 0 G - θW θW . x2 = 0 ! & F . F . # - # ( 6 0 0 & G - 273 K ±100 m/s ( - 0 $< 1 2 ( 1 2 . ! & F<. FD. # - # # 5 6 0 0 ˜ 1 = 0.2355 " - # Kn = 0.072 1 # G. 2 Kn = 0.15 1 2 Kn = 0.4 1 2 Kn = 1.0 1 5 2 ! Kn = 0.072 - 0 ( 1 2 - ! & FD. F@. = # 4 # 6 0. 6 0 0 - - 8 5 35! 0 " 5 - 1 &5 2 8 5 35! 0 5 - 1 2 & <5 0 5 1 . 3 2 - 3 7. 1 2 MF D F FN # χ = {0.5, 0.75, 1.0} * 0 5 - - 8! 6 0 & FB ! & F@. # 6 ; q˜1 /τ v˜1 /τ 0 # - 6 0 $< 1 2 0 # # = 0 χ = 1 = 4 7 - 3 1 2 # $# MFN ! 3 # - 3 - 0 √ 2 1 0 2 Kn = 0.088> 1 . 3 2. Kn = 0.177> 1 0 2 Kn = 0.353> 1 5 . 3. 2 Kn = 0.530 . FC.
(118) ;. ! & FB. 6 0 MT /τ Kn - 6 0 $< 1 2 -
(119) 17 2 # $# MF F N - # 6 0 : χ 0 - * - 6 0 6 0 7 & " # 0 Kn → ∞ Kn → 0 . ! & FF. F. & 6 0 ET /τ Kn - 7 6 0 $< 1 2 -
(120) 1 2 # $# MF N - # & 6 0 : χ 0 - * - 6 0 & 6 0 & " # 0 Kn → ∞ Kn → 0 . ! & C. F. 12 * ( 6 0 6 0 0 - - - θW - H ; # - - 1 2 * - & 6 0 - 7 - & # -- * - 5 6 0 . ! & C . C . * 0 # = ( 6 0 & - # 4 5 3 ω t˜ = {π, π/2}. # 8 5 35! & <5 % - ( . ' 4 1- 6 02 0 8 5 35! - # $< % ! & C<. CF. ( - # - " # - & 6 0 - 0 8! $< # 0 = E 1ω˜ = 1 Kn = 0.12 1ω˜ = 8 Kn = 0.32 - 0 0 = ω ˜ t = 0 ω ˜ t = π/2. 0 # . CC.
(121) ;. ! & CD. - - # # 4 - & 6 0 # 8! $< "& 0 # % ω˜ = {1, 8} # χ = 1 # & π/2 # & Kn # 8! $< ! & 5# % 6 0 ω ˜ = 8 - . -- - # # Kn ≈ 0.2 ! & . C. ( 6 0 & ( 6 0 & " &- 0 0 = - - . ! & . # - E . D. 8! 0 " 5. > &5 8! 0 5 . > . 3 $< 0 5 7. $ # 1 - - 2 # 6 ; 1 - 2 1 - 2 0 0 # = χ = 1 0 # & . - < - ! 7 = # 4 - 8! 0 - - 5 - 0 # & 0 $< - ! 4 8! 0 $< - ! & <. # - E . 8! 0 " 5. > &5 8! 0 5 . > . 3 $< 0 5 . $ # 1 - - 2 # 6 ; 1 - 2 1 - 2 # Kn = 0.1 0 # & . - < - ! = # # # - 8! 0 - - 5. - 0 # & 0 $< - ! 4 8! 0 $< - D.
(122) ; . ! & D. $ # = # 0 Kn = 0.447 4 " &- L = r − r 8 5. 3 $< - ( 1 2 # $# MBN @ ! & @. $ # Kn = 0.08 0 = #7 4 " # 8 5 3 $< - ( 1 2 # $# MCN " 0 1 2 & " 0 = # 4 B. ! & B. # - - E . 8! 0 . " 7 > &7 8! 0 5 . > . 3 $< 0 7 . $ # 580 1" 02 6 ; 1 02 - 1 - 2 0 ! - = # = # # 4 - C ! & 6 0 - 6 0 7 0 0 12 0 & & - - 1 2 * # - 5 6 0 - = 0 L & * 7 - 6 0 - & ; z 5 -- 0 0 - # 0 & - & 6 0 # 0 0 - G - 7 & -- 0 - 6 0 * & - - & & . - & 6 0 -- .
(123) ; . ! & $ # 6 ; 14 2 6 0 # Kn = {0.07, 0.14, 0.35} # = # χ = 1 $ # 8 5 35! 0 " 5 - . 18! > &5 . 2 8 5 35! 0 5 -. 18! > 2 & <5 0 5. 1$<> . 3 2 - 3 . 1 2
(124) A4 3 # $# M D @N
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(126) A4 3 1 2 # $# M@BN - - # 4 # = # χ = 1 - " - 6 0 & 0 Kn < 0.1 . ! & D = # 4 # 6 0 6 0 6 0 & 8 5 35 ! 0 " 5 - 18! > &5 2 8 5 35! 0 5 - 18! > 2 . & <5 0 5 1$<> 3 2 - 3 1 2 # χ = {0.6, 0.8, 1.0} +# & - . -
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(128) . -
(129) A4 # χ = 1 0 # $# M@BN 0 & 3 # MC C<N
(130) . # χ = 1 0 . 3 . # MBN MBFN - # $# MCN * 0 5 - - 8! 7 6 0 & * - 1&2 1 2 # χ = 1 - D.
(131) ; ;. ! & @ $ # 6 ; - 6 0 # Kn = {0.07, 0.14, 0.35} # = # χ = 1 $ # 8 5. 35! 18!> 2 & <5 1$<> 3 2 - 4 1 2
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(133) . # MB BFN 8 # 6 ; 8! * - 6 0 " 5 5 - # 8! % B ! & B = # 4 # & 6 0 - 6 0 . & $ # 8 5 35! 18!> 2 & 7 <5 1$<> 3 2 - 3 17 2 # χ = {0.6, 0.8, 1.0} 12 12 - - 0
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(135) . 3 # MCN MBN - * - 6 0 " 5 5 - # 8! % # - 6 # : F ! & F / # '& ? - - - 6 0 ; # 8 5 35! 18!> 2 & <5 1$<> 3 2 % ' # χ = {1.0, 0.8, 0.6} 4 - +
(136) 4 1 2 # MC BN * 8! '& ? # 4 ) 0 $< '& # Kn < 0.2 C ! & C / # '& ? - - 5- 6 0 0 #. & <5 % ' # χ = {1.0, 0.75, 0.5} 4 -
(137) 1 2 # MF FN. .
(138) ;;. ! & 0 50 " - 6 0 0 - 5 6 0 # - 5 6 0 6 0 -- 6 0 * - 12 - & = # 4 0 * - 1 2 4 "; = - & ;
(139) - 0 # # = 0 χ = 1 < ! & ;- # - = γ # $< % # = : - 3 1 2 # $# MCN < ! & # - - #. ( 6 0 0 & G - 273 K ±100 m/s ( - 0 $< 1 2 (. 1 2 <F ! & # - - # # 5 6 0 ˜ 1 = 0.2355 " - # Kn = 0.072 0 # G. 1 2 Kn = 0.15 1 2 Kn = 0.4 1 2 Kn = 1.0 1 5 2 ! Kn = 0.072 - 0 ( 1 2 - <F ! & < *6 # 4 - - " # 5 6 0 - - - # 3 - L - # & 4 ( # Kn = 0.072 ˜ 1 = 0.2355 <C G. ! & D # - {˜σ11 /τ 2 , σ˜22 /τ 2 } ˜ 2 ρ˜/τ 2 0 # - 7 - θ/τ. 6 0 5 $< 0 # # = 0 χ = 1 = 4 0 8 . % & 0 - # % 1 2 Kn = 0.088> 1 2 Kn = 0.177> 1 2 Kn = 0.353> 1 5 2 Kn = 0.530 D .
(140) ;;. ! & . $& ( {x, y, z} 7 {r, ϕ, z} @ . ! & (. - # & # # # ;- & 6 0 - vr = 0 * ; 6 0 0 vz = 0 vϕ = 0 - F .
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(179) D. 5 # % ;- - & 0. # & . ( - @ % - # 0 & - # & 5 . 5 - - # & -- - 6 - * 0. 0 5 -. L - - 8 5 35! = & ( - B ( - # - # = & - # - & 8! $< - 3 7 #
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(187) & & # 1- 2 0 80 & 3 % % # - 1# 6 & 2 1 2 * ; # - 6
(188) % & & 3 % 0 # 0 - & -
(189) % - < 4 Kn 3 - & * " # # - λ & & 0 .
(190) F. & # 6 0 L Kn =. λ . L. 1 2. 4 & # & # 8 % 7 1 # 2 = & 6 0 0 4 : & -- 0 # - & # 0 & 6 0 0 & L & # 5 6 0 ! 4 0 & : 0 - 0 ) & 0 - - - xi t * & 0 4 & 3 # : & % % - 0 $ " 0 3 -- ! & 0 " # & 6 0 & 4 7 -
(191) % & # 4 # 1 2 6 0 & - # 6 0 * & # 4 - - % . 8 5 35! % 7 -
(192) % % - # & . % -- # 6 0 0 Kn 0.001 0 & 8 5 35! % = & # 03 % 6 0 Kn 0.1 ! Kn 0.01 & 5 # 8! % # % = 0 & &5 # * & 0 0.1 Kn 10 8! % # - # 6 0 % ! 0 4 % 3 -- ; - - % 1; 2.
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(195) C. Collisionless Boltzmann Equation. Boltzmann Kinetic Equation Euler Equations Navier-Stokes-Fourier Equations No Velocity Slip and No Temperature Jump. Velocity Slip and Temperature Jump st. 1 Order. nd. 2 Order. Extended Hydrodynamics Equations. Kn. 0.001. 0.01. í. Hydrodynamics Regime. 10. 0.1. 100. Transition Regime. Slip Flow Regime. Kn í. 8. 0. Free Molecular Flow Regime. ! & E ( " # & 6 0 & 4 Kn 0 --. # 4 . - & # 6 0 & . - G
(196) % 6 0 & 6 % -- 4 ! 4
(197) % - 0 - # ; % . . * 0 % - -7 - - # 1 2 # - % 7 # & 0 ! & 0 & # 1& 2 V # # "; S - # & ( vi (xk , t) 0 xk t ( - vi. vi. ni. S. dS. V. ni. dS. ! & E - # 6 0 - # V S ' # dS # 6 0 vi - - # 0- 1 2 0 ni # #
(198) .
(199) . * % # M Ji & E M=. ρ dV,. V. Ji =. ρ vi dV,. V. E=. 1 2. ρ e dV. V. ) ρ vi e - # # &. - # & e & u 1 e = u + v2 . 2. 1 <2. * & 0 # - 1 2 # # - & 1 2 - & X=. 1 D2. ρΦ dV, V. . . 0 Φ = 1, vi , u + v 2 /2 X = {M, Ji , E} & . 0 - & # % & . dt % 6 ; # % & - . & - # % X - & dS & dt ρΦvi ni dS dt 0 vi dS dt - & dS & dt 0 # # dS ni vi ni dS dt - # vi dS dt & ;- # 0 & # d dt. . ρΦ dV V. =−. S. ρΦvi ni dS −. S. Υi ni dS +. ρP dV +. V. ρS dV, V. 1 @2. 0 " & 5 5 6 ; 1Υi 5 6 ; - 2 - # - # Φ V & - . -- # Φ V & . -- # 5 0 3 - # # ni 0 A ? -- & # & V. ∂ (ρΦ) ∂ (ρΦvi + Υi ) + − ρ (P + S ) dV = 0. ∂t ∂xi. 1 B2.
(200) . % # 1 - 2 & . ∂ (ρΦ) ∂ (ρΦvi + Υi ) + − ρ (P + S ) = 0. ∂t ∂xi. 1 F2. - - % 0 # 7 & - # 0 # % - E # - - 0 V. ρΦ dV M Ji E. ( ! ; (Φ) . 8 5 ! ; (Υi ) . vi u + v 2 /2. pij pij vj + qi. (P) . -- (S ) Gi Gj vj + e˙ . G 0 - P = 0 % pij Gi qi ( - # - ; 5# 56 ; . - & -- e˙ - # 80 6 pij = pδij + σij ,. 1 C2. 0 σij - - 1 % - 2 # - δij 4 3 p = 13 pkk - 0 ; . # . . - -- # ρGi - 0 # & ρGj vj & .% 1 C2 0 # & ∂ρ ∂(ρvi ) + = 0, ∂t ∂xi ∂(ρvj ) ∂(ρvi vj + pδij + σij ) + = ρGj , ∂t ∂xi ∂(ρe) ∂(ρevi + pij vj + qi ) + = ρ(vj Gj + e˙ ). ∂t ∂xi. 1 2 1 2 1 2. * % 1 2 6 ; Πij = ρvi vj + pδij + σij ,. 1 2. 0 ρvi vj - * - 6 0 = 0 -- .
(201) . σij . # & & 6 ; Θi = ρevi + pij vj + qi ,. 1 2. 0 56 ; qi - . & # ! " - 0 - % % # % * 0 & 0 p = ρθ u = 32 θ Q R Q R % # - % θ - & θ = RT 0 T - R & R = k /m 0 k m
(202) # & - ! 6 0 σij = qi = 0 .% 1 251 2 6 0 3 0 % 6 0 * 0 # % ! % = # # % % 0 #.
(203)
(204) * & 8 5 3 ! ? 18!2 0.
(205). σij. =μ. ∂vj ∂vi + ∂xj ∂xi. qi
(206) = −κ. −. 2 ∂vk μ δij , 3 ∂xk. ∂θ , ∂xi. 1 2 1 <2. 0 μ κ : - : # - # 6 # 6 0 - - # 6 ;- 9 3 -- - : M# .% 1 2N 0 - .
(207) σij = −2μ. ∂vi , ∂xj. 1 D2. 0 ∂vi /∂xj 5# - # & ∂vi 1 = ∂xj 2. . ∂vi ∂vj + ∂xj ∂xi. ! " k = 1.38066 × 10. −23. −. 1 ∂vk δij . 3 ∂xk. J/K. . 1 @2.
(208) . * 0 σij 5# - & . 3 #. -- 0 .% 1 251 2 0 # 8 5 3 ! % M.% 1 2 1 <2N - - % # - 6 0 = # # & # 0 3 - % # 5% 6 0 * 0 # # % " #$
(209) #%
(210) 18!2 % * 8 5 35! % 6 ; . - & = 0 8 5 35! " & 5 * 8! 0 -- - xi - # - 1& # v θ 02 ) 0 " & & = - 0 # - & - & & # #. 0 & 0 & 8! & 5 ; % . M<DN 8! & % % > & & - = % 0 % - .
(211) * 3 & " # & - * 7 - 0 # - - # & -- - 3 % % - & # & - " # - - xk ck # - 3 0 t * 5 ( 0 - " & & - # ;5 - - {xk , ck } 0 3 3 % # f (xk , t, ck ) # & " NV =. ck xk. 1 B2. f (xk , t, ck ) dc dx,. 0 NV # - - "; Ω =. . dc dx # - - .
(212) <. t 3 # 3 % - # # # 6 & # -
(213) % 3 % # & ∂f ∂f ∂f + ck + Gk = Q (f, f ) . ∂t ∂xk ∂ck. 1 F2.
(214) % # f (xk , t, ck ) #5 #
(215) % = # - ? # 6 & * # f 0 - - {xk , t, ck } 0 Gk ; 5# M 5# 0 -- .% 1 2N ' & 5 # .% 1 F2 Q (f, f ) # f 1 2 # - . ! & & # & - # 0 &
(216)
(217) & # MCN Q (f, f ) = . φk ck∗.
(218) f f∗ − f f∗ B (ξk , φk ) dΛ (φ) dc∗ ,. 1 C2. . 0 {f, f∗ } {f , f∗ } - 5 - 5 # 0 & - - = 0 - # ξk = ck − ck∗ φk # # 0 & > dΛ 5& # φk > B (ξk , φk ) 5& # & & # # 3 #
(219) % : # B & - - ? - * ! & < # . = # - 0 r d - 7 0 # 0 & - - . - - 12 & # 0 F → 0 ) 0 03 # 0 & & - # # & & - ; # 0 - " * - 1 2 # - # 0 - 0 - - " * ;0 # & - 12 - & - 0 # 0 - " .
(220) D. - # # - 12 * - - - - # 5- # M BN F. F. d. Repulsion. Repulsion. d. Attraction. r Attraction. r Sphere of influence. (a). d. (b). F. Other molecule. F. d. Repulsion. Repulsion. Molecule. d. r. Attraction. Attraction. r. (d). (c). ! & <E # # # 0 # 0 0 d # 12 - !. r < d # & & - # ' - 0 - " 1 2 3 0 0 - 0 - 12 & - - - ;0 # - 0 - - 12 . - 5- 0 # " - # & ; #
(221) % # - - # & *# 0 0 ; # " Gk = 0 % - ∂f /∂t = ∂f /∂xk = 0 . 0 & .% 1 F2 1 C2
(222) % # # . . f f∗ − f f∗ = 0 0 . 2. (ck − vk ) f = √ 3 exp − 2θ 2πθ ρ. .. 1 2. #
(223) % #. ;0 # .
(224) @. . !
(225) " # !"
(226) . #
(227) % & # f (xk , t, ck ) # - ! & & -- 0 - % & - - 56 ; - % ;- # f 0 & & # # .
(228)
(229)
(230) #. f " - ci . Ci - . Miα1 i2 ··· in Miα1 i2 ··· in. 0 c=. = . =. c21 + c22 + c33. c2α ci1 ci2. ···. cin f dc,. 1 2. Cin f dc,. 1 2. C 2α Ci1 Ci2. ···. . C12 + C22 + C33 .. C=. - = 0 - # & vi 1 2. Ci = ci − vi .. & # # 0 " 5# Mα i1 i2. ··· in. = Miα1 i2. ··· in . =. C 2α Ci1 Ci2. ···. Cin f dc,. 1 <2. % Wα =. . C 2α (f − f ) dc.. 1 D2. * .% 1 <2 & . 3 5# - # 5# - # ; Aij Aij = (Aij + Aji )/2 − Akk δij /3 #. .% 1 @2 # 5# - # & ; # 0 & . 3 ;- # 1 2 0 & . 3 0 $# M B -- N * % W α & .% 1 D2 f .
(231) B. ;0 1% 2 # .% 1 2 # 0 & 0 53 0 - % # # 0. 0. . ρ = M = M = f dc, 1 1 vi = Mi0 = ci f dc, ρ ρ 1 1 M1 = c2 f dc, e= 2ρ 2ρ 1 1 1 M = C 2 f dc, u= 2ρ 2ρ 1 1 1 M = C 2 f dc, θ= 3ρ 3ρ 1 1 1 p= M = C 2 f dc, 3 3 pij = Mij0 = Ci Cj f dc, σij = M0ij = Ci Cj f dc, 1 1 1 qi = Mi = C 2 Ci f dc. 2 2. 1 @2 1 @ 2 1 @2 1 @2 1 @2 1 @#2 1 @&2 1 @ 2 1 @ 2. & & 6 0 0 - & - - - % ) 0 & .% 1 2 1 2 5 & 5 0 - - % 7 0 - - # & # 1 251 D2 0. 0 . $ % & &
(232) & #
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