/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat

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1 / Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiation and the Continuing Failure of the Bilinear Formalism, Advances in Thermodynamics 3 (1990) 435 Hiromi Saida Two-Temperature Steady State Thermodynamics for a Radiation Field, Physica A356 (2005) 481 ( arxiv: cond-mat/ )

2 > > =

3 1.2 ρ 1.3 g/cm 3 ρ 5.5 g/cm 3 l mfp 2 cm R cm T/R K/cm 10 6 erg/g s quasi-steady process Energy Flux F = (λ rad + λ cd ) T 1g 1s 2 erg/g s

4 / { = BH Disc T F rad = λ rad T

5 = BH Disc

6 { { BH Disc

7 1.3 Flux Bi-Positon / 3 2 Bilinear Form 3 4 Boltzmann

8 2. Bilinear Form Bilinear Form Bilinear Form = [Force] [Flux] Biliniear Form No 2

9 2.2 Navier-Stokes Fourier 16 v [cm/s] ρ [g/cm 3 ] Specific Volume V = 1/ρ [cm 3 /g] u Specific Internal Energy [erg/g] q Heat Flux [erg/cm 2 s] T [erg] k B = 1 P [dyn/cm 2 ] P ij = P ji (ij = 1, 2, 3) P = p U + p v U + P p : v p v : Bulk Viscosity P v : Shear Viscosity TrP ( U Unit tensor ) v = 0 q, p v, P v

10 q = λ T Fourier λ p v = ζ i v i Stokes ζ P v = 2η V Newton η

11 Bilinear Form [ 1] Energy Balance de dt = dq dt + dw dt 1 ( 1 E := ρ u + V (t) 2 ρ v 2 ) dv, V (t) dq dt := q n dσ, Σ(t), n dw dt := Σ(t) Σ(t) P ij n i v j dσ + V (t) ρ F v dv ( F [dyn/g] ) v v + const. 5 T µν ;ν = 0, N µ ;µ = 0

12 Energy Balance + Galilean Invariance dρ : = ρ dt i v i 5 : ρ dvi = dt j P ij + ρ F i : ρ du = dt i q i P ij i v j ( d Lagrange dt = ) t + vi i Stokes + Newton = Navier-Stokes 5 7 2

13 [ 2] Entropy Balance s Specific Entropy [entropy/g] [entropy] = [1] k B = 1 J s Entropy Flux [entropy/cm 2 s] σ s [entropy/cm 3 s] Entropy Balance ρ ds dt = i Js i + σ s 1 Entropy Balance σ s σ s Bilinear Form

14 σ s Bilinear Form 1 ds dt = 1 T dρ dt du dt du dt + p T dv dt, V = 1 ρ [cm3 /g] dρ dt = 1 dv V 2 dt i v j i v j = 1 3 ( k v k ) δ ij + V ij + W ij Velocity Gradient Tensor V ij := (i v j) 1 3 ( k v k ) δ ij : Shear Velocity Tensor (Traceless) W ij := [i v j] : Vorticity Velocity Tensor

15 1 + Entropy Balance ρ ds dt J s = 1 q T = i J i s + σ s σ s = 1 T 2 qi i T 1 T pv i v i 1 T P v ij V ij Bilinear Form = Flux Force Thermodynamic Force 1 T 2 T 1 T i v i 1 T { Dissipative Flux Force Flux q p v P v V

16 Navier-Stokes Fourier 2 σ s Bilinear Form Fourier, Stokes, Newton σ s = λ T 2 ( T ) 2 + ζ T ( ) 2 2η ( V ) 2 div v + 0 T 2 σ s 0 Bilinear Form 2 Onsager

17 2.3 Bilinear Form Extended Irreversible Thermodynamics (EIT) ( Israel ) Navier-Stokes + Fourier Bilinear Form & Bilinear Form SST

18 2.4 Bilinear Form Ref: C. Essex Planck Matter System Photons OK Planck Isolating Wall Matter System A OK Wall Photons Free Streaming Α Isolating Wall Matter System + A A Matter System Free Streaming

19 Σ tot = Σ MS + Σ rad { ΣMS = [ Matter System ] Σ rad = [ ] Σ MS = dx 3 σ s ( x) Bilinear Form MS σ s ( x) x Bilinear Form Σ rad Bilinear Form Photons Matter System Α Isolating Wall

20 Free Streaming Σ rad Σ rad = da S, S = J ν ( r a ) cos θ dω dν A θ θ ν Photon r a A Α J ν ( r a ) = 2k B ν 2 [ c 2 X = c2 I ν ( r a ) 2hν 3 I ν ( r a ) ] X ln X (1 + X) ln(1 + X) Boson r a I ν ( r a ) Boltzmann J ν Bilinear Form r a

21 Photons Σ tot = Σ MS + Σ rad Σ rad Bilinear Form Matter System Free Streaming Α Isolating Wall Bilinear Form

22 3. Ref: H.S. 3.1 { Bilinear Form 2 2 Stationary Energy Flow T out T in ( T in > T out ) { Tin T out ( ex. T in > T out ) T out T in

23 H.S. Physica A356(2005) 481, cond-mat/ T in T (1) out (2) Stationary Energy Flow (3) ( > ) T in T out

24 3.3 1 f( x, p ) 2 f( x, p ) = ω = p c 1 exp [ ω/t ( x, p ) ] 1 T ( x, p in ) = T in T ( x, p out ) = T out T out p out T in x ω p x p in g ( ) in x p in = T in p in = T out

25 3.4 2 e rad ( x ) T out p out T in x p in g ( ) in x e rad ( dp 3 x ) = 2 (2π ) 3 ω f( x, p ) = 4σ [ g c in ( x ) Tin 4 + g out( ] x ) Tout 4 g in ( x ) = 1 dω ] 4 π xp p in g out ( x ) = g in ( p in p out ) Ω xp x p σ = π2 60h 3 c2 Stefan-Boltzmann

26 3.5 1 s rad ( x ) Landau-Lifshitz 1 55 H- s rad ( dp 3 x ) = 2 [ (1 + f) ln(1 + f) f ln f ] (2π ) 3 = 16 σ 3 [ g in ( x ) T 3 in + g out( x ) T 3 out T out ] x p in E tot = E in + E out + dx 3 e rad ( x ) S tot = S in + S out + dx 3 s rad ( x ) p out T in g ( ) in x S tot

27 2 Tin Tout Relaxation Nonequilibrium radiation field (Tin > Tout) Teq Teq Teq Equilibrium radiation field de tot = 0, T in > T out [ ds tot = C in + 16 σ T 3 c in dx 3 g in ( ] ( 1 x ) 1 ) T in T out dt in C in = de in > 0 dt in ds tot 0 = for T in = T out S rad Well-Defined 0 3 OK

28 3.6 2 τ, ψ( x ) τ, ψ τ 0, ψ 0 as T in T out 0 f rad := e rad (P rad, s rad, ψ) T rad s rad τ ψ ψ = f rad τ τ, ψ T rad ( x ) τ = T in T out ψ( x ) = 16σ 3c g in( x ) g out ( x ) T rad ( x ) = g in ( x ) T in g out ( x ) T out ( Tin 3 T out 3 ) T out p out T in x p in g ( ) in x

29 3.7 3 j ( x ) { p tot = x n = p tot j ( x ) = j( x ) n j( dp 3 x ) = 2 (2π ) 3 ω c f( x, p ) cos φ φ n p in/out ( = σ γ in ( x ) Tin 4 γ out( ) x ) Tout 4 T out x p in γ in ( x ) = 1 dω π xp cos φ p in γ out ( x ) = γ in ( p in p out ) p out g in ( x ), g out ( x ) T in g ( ) in x

30 3.8 : e rad ( x ) = g in ( x ) e eq (T in ) + g out ( x ) e eq (T out ) : s rad ( x ) = g in ( x ) s eq (T in ) + g out ( x ) s eq (T out ) : T rad ( x ) = g in ( x ) T in g out ( x ) T out : τ = T in T out : ψ( x ) = g in ( x ) g out ( x ) [ s eq (T in ) s eq (T out ) ] : j( x ) = σ ( γ in ( x ) Tin 4 γ out( x ) Tout 4 ) e eq (T ) T out x p in s eq (T ) p out T in g ( ) in x

31 4. Essex Boltzmann

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Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j