Re

Ñ Ñ

p T T 1 ( 1 ) T 2 ( 2 ) T = T( ) T = T(, t) t

p(, t)v(, t) = k B T(, t) p(, t) V(, t) (, t) (, t) = ρ(, t) (, t) Ñ V (, t) p(, t) ρ(, t) S =

ρ V 0 m = ş ρ V V 0 BV 0 Φ = ρ, Φ ą 0 V Φ ă 0 V V 0 V 0 Φ = ρ. BV 0 V 0 Φ 1 = B t ż V 0 ρ V.

Φ Φ 1 ż B t ρ V = ρ. V 0 BV 0 V ż V = V BV ż ż B t ρ V = (ρ ) V V 0 V ż 0 ñ [B t ρ + (ρ )] V = 0, V 0 V 0 B t ρ + (ρ ) = 0. (ρ ) = ρ + ρ B t ρ + ρ + ρ = 0 ρ + ρ = 0 t t = B t + t B = B t +.

= ρ B t ρ + = 0 V 0 = p BV ż 0 = p V, V 0 V p V t t + t t p = ρ t / t t t B t t = (x, y, z) t xb x + yb y + zb z = ( ),

= B t t + ( ), t t = B t + ( ) B t + ( ) = p ρ. ρ B t + ( ) = p ρ +. s = ñ s t = 0

/ t s t = B ts + s = 0. B t (ρs) + (ρs ) = 0 ρs S( ) t=0 = ñ S(, t) = @t w w = T s lomon + V p lomon, V = 1/ρ T s = ñ s = 0 w = V p = p ρ. w = 1 ρ p. B t + ( ) = w, B t + ( ) = w +.

ˆ = 0 ( ) = 2 2 ˆ ( ˆ ) (, t) B t ( ˆ ) = ˆ [ ˆ ( ˆ )] ρ = v K = 0 = 0. v K = 0 v 1 K = v2 K = v K. ˆ = 0 g = m = U. s ˆ P ρ 0.

B t = 0. B t ˆ ( ˆ ) = ) (w + 2 2 2 2 ˆ ( ˆ ) = w. 2 2 + w = 1 3 2 1 (t 1 ) 1 (t 2 ) t 1 t 2 t 3 1 (t 3 )

2 2 + w + gz = ρ 2 2 lomon + p lomon + ρgz lomon =. Ñ 0 z h h v = 0 p = 0 z = 0 z = 0 = 0 p = ρgh.

p = 0 2 2 = gh. v v = a 2gh, v z ρ 2 2 + p =, Õ p Œ p 0 p ă p 0

p 0 p 0 p ă p 0 p 0 B t + ( ) = p ρ, B t ρ + (ρ ) = 0. 0 t = B t + ( ) «B t p p 0 ρ ρ ρ 0 ρ = ρ 0 + ξ lomon B x ρ x0 + ξ2 2 B2 +... xρˇˇx0 =0

ρ «ρ 0 ρ 0 B t + p = 0 B t ρ + ρ 0 = 0 x 0 p ρ ρ p = c 2 ρ, ρ 0 c «c p0 ρ 0. x 0 x ρ 0 =. p 0 = «ˆ =. c ñ c =. «.. T [ C] 0 c [ / ] c c pv κ =.

κ κ = c p c v = 1 + 2 f f f = 5 κ 2 = 7/5 «1.4 f = 3 κ 1 = 5/3 p ρ = κ p 0 ρ 0 = c 2 c = c κ p 0 ρ 0 «?.. «., pv n =, n = 0 n = 1 n = κ n = 8 c p ρ ρ 0 B t + c 2 ρ = 0 v t B 2 t ρ + (ρ 0 B t ) = 0, B 2 t ρ = c 2 ρ. p p p B 2 t p c2 ρ B 2 t p = c 2 p.

= Bx 2 = Bx 2 + B 2 y B 2 t p = c 2 B 2 x p, p(x, t) = F 1 (x + ct) + F 2 (x ct) F 1 F 2 t = 0 p = f 1 (x), B t p = f 2 (x) F 1 (x) + F 2 (x) = f 1 (x), F 1 1(x) F 1 2(x) = 1 c f 2(x). F 1,2 (x) = 1 f 1 (x) 1 2 c ż x x 0 f 2 (ξ) ξ. p(x, t) f 1 (x) Ð 1 2 f 1(x) 1 2 f 1(x) Ñ Ð c c Ñ x f 2 0 f 1 (x) c c t = 0

ω = 2π/T ν = ω/(2π) = 1/T F 1 F 2 α β F 1 (x + ct) = b (kx + ωt + β), F 2 (x ct) = a (kx ωt + β). a = b c = ω k = λ T. 0 p 1 p = 0 ñ p =. p = ρ. p 2 p 3 ρ = B x p = B y p = 0 B z p = ρg ñ p = ρgz = = p 0 z = h p = p 0 ñ = p 0 + ρgh ñp = p 0 + ρg (h z).

z h p 0 0 ρ ω = F r = ρrω 2, U r = 1 2 ρr2 ω 2 = U. ñ (p + U) = 0 U = ρgz 1 2 ρr2 ω 2 = ρg (z r2 ω 2 ). 2g p = = U ñ p + U = ( r 2 ω 2 ô p = ρg 2g ) z +. z 0 r = 0 p

p = 0 ñ 0 = ρgz 0 + ñ = ρgz 0. z h r z z 0 z 0 0 x ( r 2 ω 2 ) p = ρg + z 0 z 2. p = 0 z z 0 = r2 ω 2 2g h r = R ñ h = z z 0 v = ωr h = v2 2g. ρ v2 2 + ρε = +,

ε [ ] B t ρ v2 2 + ρε, ε = T s + p ρ 2 ρ [ ] [ ( )] B t ρ v2 v 2 2 + ρε = ρ 2 + w w = ε + pv = ε + p ρ. V BV V B t ż V ] ż [ρ v2 2 + ρε V = V [ ( )] v 2 ρ 2 + w V. B t ż V ] [ρ v2 2 + ρε V = BV = BV ( ) v 2 ρ 2 + w ( ) v 2 2 + w. V F = BV ( ) ( ) v 2 v 2 ρ 2 + w = 2 + w = ρ v 2 /2 + w w = ε + p/ρ

ż ] B t [ρ v2 2 + ρε V = V BV ( ) v 2 ρ 2 + ε BV ρ, ρ = B t (ρ ) = B t ż V ż ρ dv = V A [ p + ρv 2] V [ = p + ρv 2 ]. p + ρv 2, Γ =, C C

Γ t = t C. δ δ = 2 1 Γ = δ. 1 2 3 Γ t = t δ = v t δ + t δ. t δ = δ t = δ = δ v2 2 1 2 δ v2 2 = 0, Γ t = t δ = t δ. = t = B t + ( ) = w

ñ C ż = ( ˆ ) C t δ = A ż A ( ˆ t ) = 0 / t = w ˆ = 0 ˆ = 0 ñ = 0 t C ñ Γ = =. C δc ˆ ż = ( ˆ ) «( ˆ ) δ δc δa! =. ˆ = 0 ˆ 0 ˆ = 0

ˆ 0. ˆ = 0 δc ż = = ( ˆ ). δc A ˆ = 0 8 = ˆ = 0

a! l l B t + ( ) = w. z(t) = a ωt u(t) = ωa ωt B t u(t) = ω 2 a ωt u = ωa ˇ ˇB t u ˇˇ = ω 2 a z a l u l v B t v u l.

v u v u ñ ( ) u2 l. ω u/a v u B t ωu u2 a. a! l ( )! B t ñ B t» w, B t ( ˆ ) = 0 ñ ˆ =. t = 0 ˆ = 0. 0 Γ = C ż lomon = A ( ˆ ) = 0. 0 ˆ = 0 Φ = Φ,

B t + 2 2 ˆ ( ˆ ) = w ) (B t Φ + 2 2 + w = 0. B t Φ + 2 2 + w = f (t) f (t) w = p/ρ Φ B t Φ = 0, 2 2 + w =, ρ ρ! 1,

ρ p ρ = B p ρˇˇs= p. p ρv 2. c ñ ρ ρ B ρ p s = c 2 ñ ρ ρv2 c 2 v2 c 2! 1 v! c. s/c s τ s c! τ. B t = p ˇ ρ ˇ, v τ p sρ ñ o s τ ρv. ρ ρ p/c 2 ρ sρv τc 2.

B t ρ ρ B t ρ ρ ρ/τ! ρv/s ρ/ρ sv/ ( τc 2)! τv/s τ " s/c ρ» ρ B t + ( ) = p ρ +. ρ = = 0. B t ( ˆ ) = ˆ [ ˆ ( ˆ )]. (p/ρ) w v 2 2 + p + gz =, ρ ( ) ( 2 2 ρ 2 + w = ρ 2 + p ) ρ. ˆ = 0 = 0 = Φ Φ Φ = 0.

v K = 0 v K = v K = B ek Φ e K v 2 2 + p =, ρ p 0 p = p 0 + ρ u2 2. ψ(x, y) v x = B y ψ, v y = +B x ψ, = B x v x + B y v y = 0. ψ B t ( ˆ ) = ˆ [ ˆ ( ˆ )].

ˆ = ˆ x B z v y + ˆ y B z v x + ˆ z ( Bx v y B y v x ). z ˆ = ˆ z ψ, = B 2 x + B 2 y. ψ B t ψ = (B x ψ) B y ψ + ( B y ψ ) B x ψ. v z = 0 x = y v x v y ñ v y x v x y = 0, 1 3 2 v x (ψ) v y (ψ) B x ψ x + B y ψ y = ψ = 0 ñ ψ =, ψ(x, y) v K ż 2 Q = ρ = ρ 1 ż 2 1 ż 2 v K l = ρ ψ. 1 ( vy x + v x y )

x y Q Q = ρ (ψ 2 ψ 1 ). v x = B x ϕ = B y ψ, v y = B y ϕ = +B x ψ, ϕ ψ B x ϕ = B y ψ, B y ϕ = B x ψ. w = ϕ + iψ, z = x + iy w(z) z w z = B xϕ + ib x ψ = v x iv y = w ˇ z ˇ bv = = 2 x + v 2 y = v. w 1 w/ z ϑ w 1 = w z = ve iϑ.

ψ = 0 w(z) C 2πi C w z z = w 1 z = 2πi ÿ A k, k C C A k w 1 C w 1 ( ) z = vx iv y ( x + i y) C ( = vx x + v y y ) +i looooooooomooooooooon C =:Γ C ( vx y v y x ). Γ C ρ C = 0 Γ = 2πi ÿ k A k. A k Γ

a B t v + vb r v = 1 ρ B rp v r v ă 0 B t ρ = 0 ñ = 1 r 2 B ( r r 2 ) v r = 0, r 2 v F(t) B r v = 0 B t v = F 1 (t)/r 2 F 1 (t) r 2 + vb r v = 1 ρ B rp. r R(t) ď a 8 a F1 (t) R + V2 2 = p 0 ρ V = R(t)/ t p 0 R Ñ 8 R Ñ 8 = 0 r 2 v = F(t) R 2 (t)v(t) = F(t), F 1 (t) = 2R lomon R 1 V + R 2 V t = 2RV2 + R 2 V t =V

p 0 ρ = 2RV2 R R V t + V2 2 = 3 2 V2 R V R R lomon t = 3 2 V2 R V 2 2 R. =V d V = R ( ) t = (+) 2p 0 a 3 3ρ R 3 1 ñ t = c 2p 0 3ρ R ( a 3 R 3 1 ), V = 0 R = a τ τ = = ż τ d ż 3ρ R t = b 2p 0 ( a ) 3 0 0 R 1 d 3a 2 c ρπ Γ( 5 /6) ρ 2p 0 Γ( 1 /3) «. a. p 0 a =. p 0 ˆ ˆ = = ˆ = ˆ = ˆ τ «. ˆ «. ρ = = τ a? ρ 1/? p 0

ν = c g k, ν = ν(k) = ν(λ) k = 2π/λ v ą c α α = c v. 2α x y α + c c A(x, y, t) = A 0 e i(kx ωt) e ky, k = 2π/λ ω = 2π/T α v = λ/t = ω/k A 0 A ω k

v φ e iφ v φ = kx ωt k x ω t = 0 v = x t = ω k. u v u = ω k. v λ k ω = vk ñ ω = v k ñ ω k = v u. ω = v k + k v = v k + k v k k ñ u = v + k v k, u = ω/ k k = 2π/λ k/ λ = 2π/λ 2 v k = v λ λ k = v λ 2π ñ k v v = λ k λ, λ 2 λ = h p = mu h u = mλ h

u = v λ v λ v λ = 0 ñ u = v v λ ą 0 ñ u ă v ă v λ ă 0 ñ u ą v h " λ v = c gλ 2π, v λ = 1 2λ v h! λ u = v 1 2 v = 1 2 v ă 0. v = a gh B t Φ + v2 2 + 1 (ρ + U) = F(t) ρ = Φ F(t) p 0 F(t) 1 ñ B t Φ = u ρ = ρgy ρ = gy.

v λ λ σ σ B t Φ + p ρ = 0. v = c σ ρ 2π λ, λ a gh v v b v 1 = gλ 2π λ 0 v 2 = b σρ 2π λ λ λ = λ 0 λ ă λ 0 λ ą λ 0

c σ 2π looomooon ρ λ 0 = c gλ0 2π loomoon ñ λ0 2 = σ (2π)2 ρg c σ ñ λ 0 = 2π ρg. v 2 = v 2 1 + v2 2 v 1 = v 2 ñ v 2 = 2v 2 1 = 2v 2 2 d c σg ñ v = 2 ρ. ρ = = ˆ g =. σ =. ˆ. ˆ = c σ λ 0 = 2π ρg «. ˆ =., d c b σg v = 2 ρ = ˆ?.. «. =. λ = λ.. λ ă λ 0

= B t ρ + (ρ ) = 0. η η ą 0 ζ ζ ą 0 η ζ ρ T η ζ ( looooooooooooooomooooooooooooooon ρ [B t + ( ) ] = p +η + ζ + η ) ( ). 3 ζ = 0 B t + ( ) = p ρ + η. ρ

η ρ η [η] = = ν = η ρ [ν] = η [ ] ν [ ˆ ] /. ˆ........ η ν η pv = ν 9 V 9 1 p. = ˆ B t ( ˆ ) = ˆ ( ˆ ( ˆ )) + ν ( ˆ ) looooomooooon =0. ˆ ( ˆ ) = ( ) ( ) + ( ) ( ), ( ) = [( ˆ ) ], ( ) = ( ) ( ˆ ), ( ) = 0 ( ˆ ) = 0, ( ) = 0 = 0.

ν = η/ρ B t ( ˆ ) + ( ) ˆ [( ˆ ) ] = ν ( ˆ ). = 0 p = ρ (B k v i ) (B i v k ) = ρb k B i (v i v k ) ψ(x, y) v x = B y ψ, v y = +B x ψ, = B x v x + B y v y = 0 B z v z = 0. B t ψ (B x ψ) ( B y ψ ) + ( B y ψ ) (B x ψ) ν ψ = 0. = 0 v K = 0 v = 0 v K = 0 v K = v = 0

E k = ρ 2 ż ñ B t E k = ż v 2 V B t ρv 2 2 V = ż ρv i B t v i V. B t v i = v k B k v i 1 ρ B i p + 1 ρ B k lomon = η ρ v σ 1 ik σik 1 = η [B kv i + B i v k ] t E k = η 2 3ÿ ż i=1 V [B k v i + B i v k ] 2 V, k = 1, 2, 3. t E k ă 0. η ą 0 t E k 9 η.

Q ρ ρ! 1, ρ «, B tρ = 0. ż R Q = 2πρ rv r, 0 x y z = vˆ z. B x v x + B y v y = 0. B t = 0, = vb z R 2πr r 0 t = B t + ( ) = 0, ρ [B t + ( ) ] = 0 = p + η. ñ p = η = η vˆ z, z p = p(z) z x

y v = 1 p η z = 1 δp η l δp l δz = ô 1 r ( r v r r ( ) = v(r) ) = δp ηl. v(r) = δp 4ηl r2 + a (r/r) + b. r = 0 a = 0 b BV = 0 R r 2πr r v(r) = 0 0 r = R v = δp 4ηl R2 + b ñ b = δp 4ηl R2 ñ v(r) = δp ( R2 r 2). 4ηl R λ 0 p 4ηl v(r)

R 2πr r ρv2πr r Q ż R Q = 2πρ rv r. 0 Q = 2πρδp 4ηl = πρδp 2ηl = πδp 8νl R4 ż R 0 r ( R 2 r 2) r ( 1 2 R2 R 2 ρ [ 1 4 r4 ] R 0 Q = πδpρ R 4 8ηl ) η = νρ η B t + ( ) = p ρ + η ρ B t ρ + (ρ ) = 0 = 0 R e Q 9 R 4

v v R 1 R 2 α R 1 R 2 = αr 1, x 2 = αx 1, y 2 = αy 1, z 2 = αz 1 R 2 v 2 = βv 1. [v] = / α/β t 2 = α β t 1.

ρ 2 = γρ 1. [ρ] = / γ 3 α m 2 = γα 3 m 1. ν = η/ρ ν 2 = δν 1, p 2 = ϵp 1 ϵ α β γ Ñ B t + ( ) = p ρ + η ρ R 2 = αr 1 v 2 = βv 1 β 2 /α v 2 = βv 1 ν 2 = δν 1 δβ/α 2 R 2 = αr 1 ρ 2 = γρ 1 p 2 = ϵp 1 ϵ/ (γα) = 1 β 2 α : δ β α 2 : 1 ϵ γ α = 1 : 1 : 1 ñ βα γ = 1 ϵ γβ 2 = 1. v 1 R 1 ν 1 = v 2R 2 ν 2 p 1 ρ 1 v 2 1 = p 2 ρ 2 v 2 2,

Re = vr ν ρvr η = ρ ( ), η s = p ρv 2. R Ñ l Re = ( ) ρvl η. Re Re Re Re Re «, Re «, Re ą Re

ν = η/ρ η vl ρ vl Re Re! 1 ( ) = 1 ρ p + η ρ. ρ ( ) η 9 Re, Re! 1 η p = 0, = 0 ( ˆ ) = 0.

= 6πRη, F 9 R, η, u u R Re! 1 «( ) r " R ( ) Ñ ( ) ( ) = 1 p + ν. ρ ( ) Re = ul/ν ( = 6πη R 1 + 3Re ) 8 l R l " R

0 ˆ = 0 η η ˆ = x ( ) = 1 p + ν. ρ r " R [ urϑ2 ] v r (ϑ) = Fr 4πρνr 4ν. + v r F r 4πρvr ϑ Φ Φ = 0, = Φ

Φ 9 1/r v 9 1/r 2 Φ = 1 [ F x + F y φ ϑ ] 4πρur 2, z z = 0 : v r = 0, v φ = ωr, v z = 0 z = 8 : v r = 0, v φ = 0, v z =. v z z=8 z φ ω r

v r = rωf(z 1 ), v φ = rωg(z 1 ), v z =? νωh(z 1 ) c ω p = ρνωp(z 1 ) z 1 = ν z. φ r v z ( ) = P ρ ñ F 2 G 2 + F 1 H = F 2 2FG + G 1 H = G 2 HH 1 = P 1 + H 2, + ν 1 z 1 B z z 1 = c ω ν. (ρ ) 0 = 1 r B r (rv r ) + 1 r B φv φ + B z v z = 2ωF +? c ω νω ν H1 = 2ωF + ωh 1. z 1 = 0 : F = 0, G = 1, H = 0 z 1 = 8 : F = 0, G = 0.

v z (8) =? νωh(z 1 Ñ 8) =.? νω. 1 G H. 1 F 2 3 z 1 = a ω ν z σ zφ = η (B z v φ ) z=0 = ηrω B z G(z 1 ) z=0 = ηrωb z z 1 B z G(z 1 ) z=0 c ω = ηrω ν G1 (0) = rρ? νω 3 G 1 (0), σ zφ 9 ρ,? νω 3 ż R M = 2 2πr 2 σ zφ r 0 = πr 4 ρ? νω 3 G 1 (0). M =. R 4 ρ? νω 3. M 9 R 4, ρ,? ν

Re = vd ν = ρvd η ą Re Re ď Re ď B t = ( ) p ρ + ν +. d v v 2 E d E v 2 v 2 (r) 9 r 2 /3.

2 /3 Re Re v(r) = δp ( R2 r 2) 4ηl ñ v (r = 0) = δp 4ηl R2 = 2 v,

v = Rş 0 rv(r) r ş r r = δp 8ηl R2. r R v 0 v R v v(r) v Re «140 Re «2300

r = 0 0 ω =..... u φ (r) u φ (r) r ω

T T T T T f 1 T ω Re f 1 f 2 f 1 f 2 f 1 f 2

0 ( ) 1 (, t) = 0 + 1 p = p 0 + p 1 B t + ( ) = p ρ = 0. + ν B t 0 = 0 ( 0 ) 0 = p 0 ρ + ν 0 0 = 0, 1 1! 0 B t 1 + ( 0 ) 1 + ( 1 ) 0 = p 1 ρ + ν 1 1 = 0. 1 = 0 1 1 1 (t) 9 e iωt ω ω P C ω = ω 1 + iγ 1 γ 1 ą 0 e iωt γ 1 = Im(ω) ă 0 ω Re ą Re Re 10 ď Re ď

100 Re = vd ν = ρvd η, Re «. Re ă Re ω = ω 1 + iγ 1 γ 1 ă 0 Re = Re Dω γ 1 = 0 γ 1 (Re ) = 0 Re ą Re γ 1 ą 0 γ 1! ω 1 Re «Re γ 1 ą 0 Re ą Re 1 1 (, t) = A(t) ( ) A(t) = e iωt A(t) = e γ 1t e iω 1t t ě 0. d A(t) Re «Re A 2 = 2 e 2γ 1t ˇ ˇe iω1t e +iω 1t ˇ. loooooomoooooon =1 t A 2 = 2γ 1 A 2. A t

t A 2 = 2γ 1 A 2 α A 4, α t 1 A 2 = α 2γ 1 + e 2γ 1t. t Ñ 8 A 2 A 2 = 2γ 1 α. γ 1 γ 1 (Re ) = 0 Re γ 1 = (Re Re ) c 2 ñ A «(Re Re ) 1/2. α t A 2 = 2γ 1 A 2 α A 4 β A 6 α ă 0 β ą 0 t Ñ 8 A 2 = α [ α 2.. 2 + 2 α ] 1/2 β γ 1. A ( ) 1/2 2γ1 α Re A Re = Re A = α /β Re 1 ă Re ă Re A ă α. A ą α. Re ă Re 1 Re ą Re Re

A 2 ( ) 2γ1 α A 2 Re 1 Re Re A(t) / / 3

T = 6

Re = vl ν = ρvl η η ν v K = v = 0 v K = 0 0 = 0 δ l = 5l? Re 9? ν, δ t =. 5 cν l4 4. l = Re = ˆ δ l =. v K = v = 0

p + ρ u2 = 2 ñ 1 p ρ x = u u x, z y x v y! v x Bxv 2 c! B 2 yv x. v x B x v x + v y B y v x νb 2 yv x = u u x B x v x + B y v y = 0 v x = v y = 0

[ ] [ ( )] ρv 2 v 2 B t 2 + ρϵ = ρ 2 + w =, ϵ w = ϵ + p/ρ 1 = σ 1, j 1 k = v iσ 1 ik. T T T

= 0 ô T = 0 «κ T κ ą 0 T [ v 2 j = ρ 2 + w ] loooooomoooooon = σ 1 κ T loomoon =, [ ] ρv 2 B t 2 + ρϵ =, [ ] ρv 2 B t 2 + ρϵ = v2 2 B tρ + ρ B t + ρb t ϵ + ϵb t ρ, B t ρ B t B t ϵ ϵ = T s p V = T s + p ρ 2 ρ, ñ B t ϵ = TB t s + p ρ 2 B tρ ρt lomon B t s + lomon s = σik 1 loomoon B kv i + (κ T) loooomoooon. s t = 0. σ 1 ik B kv i = ηb k v i [ Bk v i + B i v k 2 3 δ ikb l v l ]

ż S = ρs V v! c T B t s = (B T ) p B t T, s = (B T s) p T, (B T s) p = C p /T C p = T (B T s) p ñ B t s = C p B t T, T s = C p T. ρc p [B t T + T] = σ 1 ik B kv i + (κ T). = 0, T η κ C p ρc p B t T + T = χ T + ν 2C p [B k v i + B i v k ] 2 ν = η/ρ χ = κ/ ( ρc p )

B t T = χ T, ρc p B t T = q = κ T, T κ T = 0. κ (κ T) = 0. Q Q = Q(, t), ρc p B t T = κ T + Q.

t = 0 T = T 0 (x, y, z). T(, t ą 0) ż T(, t) = T (t)e i 3 k (2π) 3 T (t) = ş T(, t)e i 3 x T e i, B t T = χ T : T t T + k 2 χt = 0. T = T 0 e k2 χt, T = T 0 ( ) t = 0 ż T 0 = ż ñ T(, t) = T 0 ( 1 )e i 1 3 x 1 T 0 ( 1 )e k2χt e i ( 1) 3 x 1 3 k (2π) 3. 3 k +8 ż 8 ( π ) 1/2 e αξ2 βξ ξ = e β 2 /(4α), α ξ

T 0 ż [ ] 1 T(, t) = 8 (πχt) 3/2 T 0 ( 1 ) ( 1 ) 2 3 x 1. 4χt T 0 T 0 = T 0 (x) y 1 z 1 ż [ ] 1 T(, t) = 2 (πχt) 1/2 T 0 ( 1 ) (x x1 ) 2 x 1, 4χt δ T 0 ( ) = δ( ) T(, t) = 2 (πχt) 1/2 e r2 /(4χt). r = 0 t 3/2 σ =? 2χt Γ =? 8 2σ l 9? t T 6 5 4 t = 1 8 τ 9 l2 χ, l 3 2 1 t = 1 4 t = 1 2 t = 1 δ 0 1 2 r? γ

T " T, χ T " ν 2C p (B k v i + B i v k ) 2 B t T + T = χ T χ = κ ρc p, κ B t T = 0 T = χ T ( ) = p + ν ρ = 0 T p/ρ ν χ T 1 T 0 T

[ν] = [χ] = 2 ; [u] = ; [l] = ; [T 1 T 2 ] =. Re = ul ν Pr = ν χ = T 1 Pr.... 20 Re Pr Pe Re Pr = ul χ. Re Pr T T ( 0 ) = f T T 1 l, Re, Pr. Re χ Pr ( ) u = l, Re. α α = q T 1 T 0. q = = κ T T 1 T 0

Nu αl κ = f (Re, Pr).

c m 1 M, M = m 1 + m 2 m 1 2 ρc c = m 1 /m B t (ρc) + (ρc ) = 0. B t ż ρc V = ρc. ρ (1 c) B t (ρc) + (ρc ) =,

B t ż ρc V = ρc. c [ ] ρv 2 B t 2 + ρϵ = ϵ = T s + p ρ + µ c ρ 2 w = T s + 1 ρ p + µ c µ ρb t ϵ ρµb t c p ρµ c [ ] ρv 2 B t 2 + ρϵ = j + ρt [B t s + s] σik 1 B kv i + µ. ρt [B t s + s] = σ 1 ik B kv i ( µ ) i µ µ = ( µ ) + i µ µ T = α µ β T = δ µ γ T + µ T 9 E = 3 2 k BT

[ B t c = D c + k ] T T T B t T kt C p (B c µ) p,t B t c = χ T, D k T D D = α ρ (B cµ) p,t k T D = αt ρ (B Tµ) c,p + β. k T D Ñ 0 B t c = D c B t T = χ T, T Ñ c, χ Ñ D. δ t = 0 c(r, t) = M 8ρ (πdt) 3/2 e r2 /(4Dt) M σ =? 2Dt Γ =? 8 2σ

c t = 1 8 t = 1 4 t = 1 2 t = 1 1 2 r? D t = 0 B t W = D W, ] M W(r, t) = [ r2 3/2 8ρ (πdt) 4Dt, t [r, r + r] w(r, t) r M/ρ 1

w(r, t) r = ] 1 [ 2? πd 3 t r2 r 2 r. 3 4Dt t ş 8 0 e u2 u =? π/2 r 2 = 8ż 0 r 2 w(r, t) r = 6Dt, w(r, t) Γ =? 8 2σ b r 2 9? t r D b /b b = = b, b = 6πηR, b = = 1 6πηR F i = a ik v k, a ik b a 1 a 2 a 3 a ik b = 1 ( 1 + 1 + 1 ). 3 a 1 a 2 a 3

b D D = Tb. = ρd c + ρcb. ρc = ρcb µ = ρd µ + ρcb (B c µ) T,p µ = T c + ψ(p, T) ñ = ρdc µ + ρcb. T = 0 µ + U =, U µ = U =, = 0 0 = ρdc + ρcb T D = Tb k B = 1 D = T 6πηR,

u =. u = E L = / 208 + 208 E =? s = [ 2u 2 + 2E L u ] 1/2 «17.3 u = 79 + /? s = =.. / 197 D = RT N 1 6πηd R «. N =. ˆ d R k B =. ˆ k B = R/N 1 =

. / 208 82 +. / 208 82? s =. N 1 Z 1 N 2 Z 2? s b d(v) = d 0 b 1 v2 c 2 ϵ «. 3 «ˆ «E av =? s u (A 1 + A 2 )

s b b y = 1 2 E + p E p = p E «ϑ 2 η ϑ η R = R(y, t) B t R = 1 τ y B y [( y y ) R ] + B 2 y [ Dy R ] y = 0 τ y D y D y 9 T τ y T ( ) [? ] 1 y + y e t/τ y b R(y, t) = 2πτy (t) 2 2σ y(t) 2 ( ) y y e t/τ y b + 2σ y(t) 2

( σy(t) 2 = D y τ y [1 2t )] = T [ ( 1 2t )] τ y 2k τ y k y, t ą t 2 Γ (t 2 ) =? 8 2σ y (t 2 ) «y 1 y = y t Ñ 8 R(y, t) δ t = 0 t Ñ 8 y b y b y

c c T αβ, α, β, γ = 0, 1, 2, 3; i, k, l = 1, 2, 3; x 0 = ct; x 1 ; x 2 ; x 3. T 00 = T 00 ct 0i T0i c = T0i c T ik f k = p f i ñ T ik pδ ik. T0i c

T 00 ϵ ϵ t αβ = p p p u α u 0 = 1 u i = 0 T αβ = wu α u β pg αβ. 1 g αβ = g αβ = 1 1 1 w = ϵ + p ϵ T αβ = t αβ u 0 = 1 u i = 0 T ik = wvi v k c 2 [ 1 v2 c 2 ] + pδ ik, T 0i wv i = [ ], c 1 v2 c 2 T 00 = w 1 v2 c 2 p = ϵ + p v2 c 2 1 v2 c 2. v! c ϵ Nmc 2 N n

mn Ñ v!c ρ c 1 v2 c 2 «ρ ρv2 2c 2 ñ mnc 2 Ñ ρc 2 ρv2 2 ρ = m V ϵ! ρc 2 p! ρc 2 ρc 2 T 00 T 00 = ρc 2 + ϵ + ρv2 2, T ik = ρv iv k + pδ ik. c 2 T0i c = ct 0i c 2 T 0i /c B β T β α = 0 T αβ = wu α u β pg αβ w = ϵ + p

n α n 0 n i n α = nu α u ( u α = γ, γ c ) γ = 1/ 1 v2 c c 2 B α (nu α ) = 0. T αβ = wu α u β pg αβ B β Tα β = u α B β (wu β) + wu β B β u α + B α p =! 0 u α u α u α = 1 u α B β u α B β (wu β) u β B β p = 0. wu β nu β (w/n) [ nu β w B β n 1 ] n B βp = 0. w = T s + p ñ w = T s + p ( w ) ( s ) ñ = T + 1 p, n n n 1/n s ( s ) u β B β = 0, n

B β (su β) = 0, su β wu β B β u α = B α p u α u β B β p, γw = h b γ = 1/ 1 v2 «1 + 1 v 2 c 2 2 v! c w = ϵ + p c 2

λ λ = u σρ. ˆ =. ˆ «. ˆ! R @ =. ˆ ρ «. ˆ u =. ˆ σ «= ˆ

B t ρ + B x (ρu) = 0 B t u + ub x u = 1 ρ B xp, p(ρ)

p = p(ρ) Φ := B ρ p Λ := (ρ/ρ 0 ) ñ B x Λ = (B x ρ) /ρ, B t Λ = (B t ρ) /ρ. B t Λ + ub x Λ = B x u B t u + ub x u = ΦB x Λ, Φ 1/2 ( B t u + u Φ 1/2) [ ( B x u = Φ 1/2 B t Λ + u Φ 1/2) ] B x Λ. ( Φ 1/2) U = u Φ 1/2. Φ = B ρ p p(ρ) Φ ą 0 ( ) x = u (B ρ p) 1/2, t

(x, t) c S ż x p t = v. ρc S p = κρ n c S = (B ρ p) 1/2 = n = 1 + 2/ f c p /c V ( ) κp 1/2 ρ u u u u p 0 p n = c p /c V ( ) x = u 2c S t n 1, u = c s,0 ( 1 n 1 n + 1 [ ( ) ρ (n 1)/2 1]). ρ 0

= 0 n = n = 5/3 ρ 2 ρ 1 = (n + 1) p 2 + (n 1) p 1 (n 1) p 2 + (n + 1) p 1, p 1 ρ 1 p 2 ρ 2 v 1 v 2 v 2 = (ρ 1 /ρ 2 ) v 1 ρ 1 v 2 1 + p 1 = ρ 2 v 2 2 + p 2 p 2! p 1 ρ 2 Ñ n + 1 ρ 1 n 1. n = 5/3 ρ 2 /ρ 1 = 4 n n = 4/3 ñ ρ 2 /ρ 1 = 7 1 Ñ 1 Ñ ρ

ñ Ω p

y x Ω p ub x + vb y u + f v = c2 S Σ B xσ B x Φ ub x v + vb y v f u = c2 S Σ B yσ B y Φ B x (Σu) + B y (Σv) = 0. u v B x v = f Φ 0 Φ(x, y) «Φ 0 + 1 2 ( B 2 x Φ ) x 2 + 1 ( ) B 2 2 yφ y 2 ub x u + f v = c2 S Σ (B 2 x 2 Φ ) 0 x2, ΣB x u + ub x Σ = 0.

1 ( u2 c 2 ) S Bx u = f v Φ 2 u 0x B x v = f.

0 4 0 ñ 3 1 /2 ñ. 4 λ. 4 λ 10 6 10 6 4 λ λ s ą. =. ă. λ 3 T À 10 3 4 4. ˆ 3 λ

c p 20 12 4 T 4 λ =. 2 0 2 T T λ [ ] λ T p = T λ T p = T s s 0 Q = C T C = C Ñ 8 s s 0 λ T λ ą T ą 0

s p ă p c p = p c p ą p c T T 0 T c η

s n n ρs s q = ρts n ˆ s = 0 s n v s = ρ s s + ρ n n. ρ s ρ n ρ = ρ s + ρ n. T Ñ 0 4 ρ n Ñ 0 T ď T λ ρ s Ñ 0 B t ρ + = 0,

Π ik B t j i + B k Π ik = 0. ρs n B t (ρs) + (ρs n ) = 0. s ˆ s = 0 s ( ) v 2 B t s + s 2 + µ = 0 µ Π ik µ B t E + = 0, s n s v s 0 = ρ s + 0 E = ρv2 s 2 + 0 s + E 0 = E s + v2 s 2 0 + Π 0 s + 0 Π ik = ρv si v sk + v si j 0k + v sk j 0i + Π 0ik E 0 = µ ρ + T (ρs) + ( n s ) 0 p = E 0 + Tρs + µρ + ρ n ( n s ) 2 p µ E Q

= Π ik = ( ) µ + v2 s + Tρs n + ρ n n [ n ( n s )] 2 ρ n v ki v nk + ρ s v si v sk loooooooooomoooooooooon =ρv i v k +pδ ik. ρ s ρ n µ s p T w 2 = ( n s ) 2 v n v s ρ n ρ s s(p, T, ) «s(p, T) + w2 2 B ρ u T ρ ρ(p, T, ) «ρ(p, T) + ρ2 w 2 2 B ρ u p ρ n n = 0 nk s n n η ξ 1 ξ 2 ξ 3 κ η n η κ ξ ξ 1 ξ 2 ξ 3

ρ p s B t ρ + = 0, B t (ρs) + ρs n = 0 ρs n B t + p = 0, B t s + µ = 0. B 2 t ρ = p, B 2 t s = ρ ss 2 ρ n T. ρ s = 0 u 2 = (B ρ p) s, [ u 4 u 2 (B ρ p) s + ρ sts 2 ] + ρ sts 2 (B ρ p) ρ n c V ρ n c T = 0 V u 1 = a B ρ p, u 2 = d Ts 2 ρ s cρ n, c» c p» c V. u 1 u 2 T ρ s λ λ c p c V u 2 = d Ts 2 ρ s c p ρ.

c = 3s, ρ n = ctρ 3u 2, ρ n «ρ 1 d Ts ñ u 2 = 2 ρ s c 2 Tρ 3u2 1 = s c u 1?3 = u 1?. 3 T Ñ 0 u 1 u 2 Ñ? 3. v n «v s

B t (ρs) + (ρs ) = 0. B 2 t p = c 2 B 2 x p, p(x, t) = F 1 (x + ct) + F 2 (x ct) F 1 F 2 p = ρg, z p = p(z) ω = U r = 1 2 ρr2 ω 2

a = 0 Φ = Φ B t Φ = gy, h! λ λ = 2π/k σ B t Φ + p ρ = 0 v = v(λ) B t + ( ) = p ρ + η ρ ( ) Ñ v(r) v(r) l δp v(r) = 0 v = R

+ = [ urϑ2 ] v r (ϑ) = Fr 4πρνr 4ν, ( ) = 1 p + ν. ρ Φ = 0 = Φ α ă 0 β ą 0 A(t) γ 1 ě 0 t A 2 = 2γ 1 A 2 α A 4 β A 6. t Ñ 8 B t T = χ T, T χ ż T(, t) = T 0 ( 1 )e k2χt e i ( 1) 3 x 1 3 k (2π) 3

T 0 ( ) = δ( ) Ñ r T(, t) Ñ T(r, t) 3 x e i = + i +8 ż 8 ξ ( π ) 1/2 e αξ2 βξ ξ = e β 2 /(4α), α T(, t) = 8 (πχt) 3/2 e r2 /(4χt) τ d «2R r 2 = 6Dt d τ = d2 6D = 2R2 3D, D = k BT 6πηR R τ = 4πηR3 k B T T αβ = wu α u β pg αβ

w = ϵ + p ϵ p u α g αβ v! c n mnc 2 mnc 2 s ( s ) u β B β = 0 n u β = (γ, γ /c) n su β B β (su β) = 0.