housoku.dvi

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れいなさくいし
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1 : 1 :, : 3 13 :

2 : A : 12 B 14 C 15 ( ) ( ) ( ),, (,, ) 2,,

3 : (t),, ρv 0 (1) t (t) ( A ) ( ) ρ (t) t +ρ v V 0 ρ t +ρ v 0 (2),, v ρ + (ρv) 0, (3), ρv 12 : : v(x)ρv t σ n (x)s + F b (x)v (4), v(x), v(x)ρv (5)

4 : 1 4 (4), F b (x), σ n (x) (Euler s first law of motion ) ρ v t V σ n (x)s + F b (x)v σ ij σ n (x) ij σ ij (x)e i (e j,n) ij σ ij (x)n j e i (6) ( : ) ρ v t V σ ij n j e i S + F b (x)v σ ij e i V + F b (x)v x j, ( ρ v i t σ ) ij F bi V 0 x j ρ v i t σ ij x j F bi 0 ρ v i t σ ij x j +F bi (7) (ρv i ) + x j (ρv i v j σ ij ) F bi (8)

5 : : : jρv x σ n (x)s + x F b (x)v (9) t j, j x v(x)+s (10) s (nonpolar material), j x v(x) (11) (9) (Euler s first law of motion ) σ ij σ n (x) ij σ ij (x)e i (e j,n) ij σ ij (x)n j e i (12) ( : ), j i ρv t ǫ ijk x j σ kl n l S + ǫ ijk ǫ ijk x j F bk V, Gauss j i t ρv (ǫ ijk x j σ kl )V + ǫ ijk x j F bk V, x l, j i t t (ǫ x j ijkx j v k +s i ) ǫ ijk t v v k k +ǫ ijk x j t + s i t v k ǫ ijk v j v k +ǫ ijk x j t + s i t ǫ v k ijkx j t + s i t

6 : x l (ǫ ijk x j σ kl ) ǫ ijk x j x l σ kl +ǫ ijk x j σ kl x l ǫ ijk δ jl σ kl +ǫ ijk x j σ kl x l ǫ ijk σ kj +ǫ ijk x j σ kl x l ( ǫ ijk x j ρ v k t σ ) kl F bk V + ρ s i x l t V ǫ ijk σ kj V (4) (ρ s i t ǫ ijkσ kj )V 0 ρ s i t ǫ ijkσ kj 0 (13), ǫ ijk σ kj 0, (14), 14,,,, 141

7 : 1 7, ( ) 1 ρ t 2 v2 i +ε V σ ik v i n ks + F bi v i V q k n ks 1 ε, q i 2 Gauss ρ ( 1 t 2 v2 i )V +ε q k (v i σ ik )V + F bi v i V V x k x k ρ ( ) 1 t 2 v2 i +ε (v i σ ik )+F bi v i q k x k x k [ ( )] 1 ρ 2 v2 i +ε + [ ( ) ] 1 ρ x k 2 v2 i +ε v k σ ik v i +q k F bi v i (15), Φ F b ρ Φ, (16) 1, F i v i ρ Φ v i ρ Φ x i t (ρφ) (ρφv) 1 2 q T, q ra,, qi T T κ ij x j 2, 1,?

8 : 1 8 e tot + (e tot v k v i σ ik +q k ) 0, (17) x k ( ) 1 e tot ρ 2 v2 i +ε+φ (18) e tot, F k e tot v k v i σ ik +q k,,, 142 (15), (7) ρv i i ( 1 2 ρv2 i ) + { ( } 1 v i v k x k 2 ρv2 i ) v i σ ik F bi v i σ ik (19) x k 143 (19) (15) (ρε)+ (ρεv) σ v i ik q (20) x k 144 (20) h h ε+ p ρ (20) (ρε)+ (ρεv) { ( ρ h p )} { ( + ρ h p ) } v ρ ρ

9 : 1 9 p (ρh) + (ρhv) (pv) p (ρh)+ (ρhv) t p v t (ρh)+ (ρhv) v i σ ik q + p x k t (21) σ ik σ ik+pδ ik pδ ik 145 s ε Ts+ p ρ2ρ (20), ( ) (ρε)+ (ρεv) ρ +v ε ( ) ρt +v s+ p ( ) +v ρ ρ ( ) ρt +v s p v ( ) ρt +v s σ v i ik q (22) x k ( ) (22) ρt +v s, 1 σ ik v i, x k 2 (3), (7), (15) 5, 5 ( ), ( ) :, :

10 : A, A(x,t)V F ns + Q[A](x, t)v, n F A (flux ensity), A Q[A] A (source, sink), Gauss ( ) A + F V Q[A]V A + F Q[A] (23), F Av, F A + (Av +F ) Q[A] (24)

11 : A, ( ) ρ, F ρv, Q[ρ] 0, ρ + (ρv) 0 (25) 23 A i ρv i,,,, (F i ) k Π ik ρv i v k σ ik, Q[ρv i ] ρ Φ x i,, (ρv i)+ x k (ρv i v k σ ik ) ρ Φ x i (26) 24 ) A 1 e tot ρ( 2 v2 +ε+φ, F i σ ijv j +q i, Q[e tot ] 0 0, e tot + x i (e tot v i σ ij v j +q i ) 0 (27)

12 : A : 12 A : A, A(x, t)v (28) t (x,t), (Reynols transport theorem) 2, x 1 ξ (ξ,η,ζ) t (x,t)a(x,t)v A(ξ,t) x t ξ ξ ξηζ, ξ ξ A(ξ,t) x t ξ ξ ξηζ ξ ( A(ξ,t) x ) t ξ ξηζ, Lagrange x ξ,t ξ ξ (x,t) (x,t) ( A(ξ,t) x ) t ξ ξηζ A(ξ, t) x t ξ ξηζ + A(ξ,t) ξ t (x,t) + (x,t) (x,t) (x,t) A(x, t) xyz + t A(x, t) xyz + t A t xyz ( (ẋ,y,z) A(x, t) (ξ,η,ζ) ξ ( ) x ξηζ ξ ( (ẋ,y,z) A(ξ, t) (ξ,η,ζ) + (x,ẏ,z) (ξ,η,ζ) + (x,y,ż) ) ξηζ (ξ,η,ζ) ( (ẋ,y,z) A(x, t) (ξ,η,ζ) + (x,ẏ,z) (ξ,η,ζ) + (x,y,ż) (ξ,η,ζ) (x,t) ) ξ x xyz (ξ,η,ζ) (x,y,z) + (x,ẏ,z) (ξ,η,ζ) (ξ,η,ζ) (x,y,z) + (x,y,ż) ) (ξ,η,ζ) xyz (ξ,η,ζ) (x,y,z) ( A t xyz + vx A (x,t) x + v y y + v ) z xyz z ( ) A t +A v xyz 2 1, t t 0

13 : A : 13 (6) ( ) A A(x,t)V t (x,t) (x,t) t +A v V (29) (Reynols transport theorem) (29) A(x, t)v t (x,t) (x,t) (x,t) (x,t) ( ) A t +A v V { } A + (Av) V A V + Av ns (x,t) (Libniz rule) s, A ρs ρs(x, t)v t (x,t) (x,t) { (x,t) (x,t) ( ) (ρs) +ρs v V t ( ) ρ s t +ρ v ρ s t V +ρ s } V t

14 : B 14 B Batchelor,GK, :,, 614pp Lanau,L, Lifshitz,EM,, 1970 : 1,, 280pp, 1973 : ( ),, 428pp Glansorff,P,Prigogine,I,,, 1977 :, 297pp, 1998 :,, 538pp

15 : C 15 C , ( ), ( ), / : ( ) ( ) ( ), c (Y-Y Hayashi an S Takehiro)

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x