Elementi di fluidodinamica: Un'introduzione per l'Ingegneria (UNITEXT Ingegneria) Italian

G. Riccardi, D. Durante Elementi di fluidodinamica Un’introduzione per l’Ingegneria 13 GIORGIO RICCARDI II Universi...

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G. Riccardi, D. Durante

Elementi di fluidodinamica Un’introduzione per l’Ingegneria

13

GIORGIO RICCARDI II Università degli Studi di Napoli Dipartimento di Ingegneria Aerospaziale e Meccanica Aversa (CE) [email protected] DANILO DURANTE II Università degli Studi di Napoli Dipartimento di Ingegneria Aerospaziale e Meccanica Aversa (CE) [email protected]

Springer-Verlag fa parte di Springer Science+Business Media springer.com © Springer-Verlag Italia, Milano 2006 ISBN 10 88-470-0483-7 ISBN 13 978-88-470-0483-2

Quest’opera è protetta dalla legge sul diritto d’autore. Tutti i diritti, in particolare quelli relativi alla traduzione, alla ristampa, all’uso di figure e tabelle, alla citazione orale, alla trasmissione radiofonica o televisiva, alla riproduzione su microfilm o in database, alla diversa riproduzione in qualsiasi altra forma (stampa o elettronica) rimangono riservati anche nel caso di utilizzo parziale. Una riproduzione di quest’opera, oppure di parte di questa, è anche nel caso specifico solo ammessa nei limiti stabiliti dalla legge sul diritto d’autore, ed è soggetta all’autorizzazione dell’Editore. La violazione delle norme comporta sanzioni previste dalla legge. L’utilizzo di denominazioni generiche, nomi commerciali, marchi registrati, ecc., in quest’opera, anche in assenza di particolare indicazione, non consente di considerare tali denominazioni o marchi liberamente utilizzabili da chiunque ai sensi della legge sul marchio. Riprodotto da copia camera-ready fornita dagli Autori Progetto grafico della copertina: Simona Colombo, Milano Stampato in Italia: Signum, Bollate (Mi)

II

I

V

XIII

VI

XII

≡ =: ... M L T ...

IRn n = 2 3 u(ξ, t) t=0

ξ t x

u · dx

II f (x, t)

III

IV

V

VI

V II

IX

X

RAN S

LES

................................................

2D

......................................

............................................

..........................

......

............................................

.........................................

.....................

.......................

R

E .....................

Γ Π Γijlm + Πijlm

... DN S

......................... 1D

=1

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........................................ ................................................

IRn n = 2

3

u(x)

x

u ∇u ∇u

u

IRn n = 2

A0 ξ ∈ A0

t0 < t ≤ T

3

t0

Ft

IRn t Ft A0 A0

∂A0 ∂A0

Ft : IRn → IRn At = Ft A0 A0 A0 t0

t 0 Ft A0

At

Ft

At (x, y) t=0 b

t=T >0

a At

0≤t≤T 1

A ⊆ IRn A

A

A

2

Ft A0

A0

x = Ft (ξ)

ξ ∈ A0

δε > 0 Ft (ξ ) − x < ε ·

ξ

ξ ∈ ∂A0

{xk }k=1,2,...

(IRn , · )

ξ ∈ A0

ε>0 x =

δ ξ √ε xi xi A0

∂A0 {ξ k ∈ A0 }k=1,2,... ∀ε > 0 ∃Mε ∈ N | ∀k, h > Mε ξ k − ξ h < ε t>0 Ft (ξ k ) = xk k = 1 2 ... {x k }k=1,2,... Ft (ξ ) ξ ∈ ∂A0 ε>0 k, h > Mε

Mε xk − xh < ε Mε 3

ξ k − ξ h < δε Ft (ξ k ) − xh < ε C IRn n ≥ 2 IRn

Ft

{xk }k=1,2,... C : [a, b] → IRn σ

→ x(σ)

k, h > Mε xk − xh = [a, b]

Ft At x(ξ, t) A0 ξ Ft−1 A0

0

t A0 Ft ξ(x, t) ∈ A0

IRn n = 2 3 Ft ξ ∈ A0 x ∈ At

At

At x(ξ, t) ∈ At

Ft−1 At

A0 P1 P2 Q1 Q2

FT

a Q6 Q7 Q2 Q3

Q5 Q6 Q3 Q4 Q5

∂AT

0 Ft

∂At ∂At Q2 Q3

Q5 Q6 ∂AT

∂AT ∂A0

a = −∞ C

σ b = +∞ C

IR a

σ1 = σ2

b x(a) = x(b)

σ x(σ1 ) = x(σ2 )

y

y P2

Q7

P1

Q6 Q2

Q1

Q5 Q3 At

A0

Q4 x

x

a

b a

b 0 a

P1 P2 Q1 Q2 . . . Q7 Q5 Q6

Q2 Q3

T b

Ft A0 Ft

Ft−1

ξ

∂AT

Ft−1

At

x Ft

A0 0 ∂At

Ft Ux ξ

Vξ

Vξ

Ux ∂A0

ξ x = FT ξ FT (Vξ ) ⊆ Ux

x = x(ξ, t) ∈ ∂At Ft (Vξ ) ⊆ Ux x(ξ, t)

0

ξ t

4

C ⊂ IRn

x ∈ C x

C

Ux n

Ft (Vξ ) A0

Vξ

ξ

At

x(ξ, t)

a

b 0 ξ ∂A0 Ft (Vξ )

a b t > 0 ∂A0

Vξ

∂At x(ξ, t) Ft (Vξ ) ⊆ Ux

ξ

Ux Vξ

t ≥ 0 x ∈ At ⊆ IRn n = 2 3 0 f ◦ Ft−1 x ∈ At x t f x f

f Ft−1

t ξ ∈ A0

0 (x) t (ξ, t)

f f

L

E

f f

ξ

E

(x)

E

k

f ◦ Ft−1 [x(ξ, t)] = f L (ξ, t) x

f

k Ft−1

Ft

ξ A0

At

ξ

u L (ξ, t) =

x(ξ, t + Δt) − x(ξ, t) = ∂t x(ξ, t) , Δt Δt → 0 lim

Ft−1

t

L

(ξ, t)

u E (x, t) = u L [ξ(x, t), t] . L

E

u

t ∈ [0, T ]

0

ξ

⎧ ⎨ dx (t) = u E [x(t), t] dt ⎩ x(0) = ξ x(ξ, t) Ft

C∞

ξ

IRn n = 2

3

ξ ∈ A0

t ∈ [0, T ]

t

x r = x/ x

x = δ u E (x) = ρ(δ)

x . x δ → 0+

δ=0 ρ(δ) ρ(0) = 0

A0 = IR3

Ft

ξ = δ0 r 0

⎧ ⎨ d (δr) = δr ˙ + δ r˙ = ρ(δ)r dt ⎩ δ(0)r(0) = δ0 r 0 . r˙

r(t) ≡ r0

r

δ˙ =

ρ(δ) δ ≡ 0

0 δ˙ = ρ(δ)

δ0 = 0

δ0 = 0

5

t f˙

f df /dt

δ

dη =t. δ0 ρ(η) ρ(δ) = U δ/R

δ0 exp(U t/R)

U

R δ(t) =

δ x(ξ, t) = ξ exp(U t/R)

U >0 U <0

R/U ρ(δ) = U [1 − exp(−δ/R)] δ(t) = R log[1 + (exp(δ0 /R) − 1) exp(U t/R)] δ(t) δ0 exp(U t/R) δ0 R δ(t) δ0 + U t δ0 R ρ(0) = 0 ρ→∞ ρ(δ) = U (R/δ)n−1 n

u(x) = U Rn−1

x x n

A0 = IRn − {0} U R ∂B0 (r)

q r q[∂Br (0)] = dS(x)

∂Br (0)

dS(x) u(x) · ν(x) x

S B0 (r)

r

ν(x) ν(x) = x/r

x=0 ∇·u ≡ 0

B0 (r) q = χn rn−1 ρ(r) ≡ χn U Rn−1 , χ2 = 2π

χ3 = 4π δ(t) = δ0 [1 + n(R/δ0 )n−1 U t/δ0 ]1/n .

δ˙ = ρ(δ)

U >0

U <0 IR − {0}

0

0

t (δ0 )

B0 (δ0 ) t = −R/(nU )(δ0 /R)n U (R/δ0 )n−1 t = 0

t = t

t x0

xc = xc (σ) ⎧ ⎨ dxc (σ) = u E [x (σ), t] c dσ ⎩ xc (0) = x0 x0 σ |u| u=0

|u| → ∞

σ=t x=0

t uE x(0) = xc (0) x(0) = xc (0) t

σ

u=0

⎧ 1 ⎪ ⎪ [(γ + χx )x0 + χy y0 ]eγσ + [ (γ − χx )x0 − χy y0 ]e−γσ ⎨ xc (σ) = 2γ 1 ⎪ ⎪ [χy x0 + (γ − χx )y0 ]eγσ + [−χy x0 + (γ + χx )y0 ]e−γσ . ⎩ yc (σ) = 2γ

u ≡ u0 , v(y; t) = Ωy sin Ωt , Ω = 2π/t (x0 , y0 ) ⎧ dx ⎪ ⎪ = u0 ⎪ ⎪ dσ ⎨ dy ⎪ = Ω sin Ωt y(σ) ⎪ ⎪dσ ⎪ ⎩ x(0) = x0 , y(0) = y0 , x = x(σ)

σ

y(x) = y0 exp

y = y(σ) Ω

sin Ωt (x − x0 ) . u0

x

u(x, y; t) = Ω = 2π/t

t

sin Ωt

Ω2 Ω2 2 xy sin Ωt , v(y; t) = − y cos Ωt , u0 2u0 u0

(x0 , y0 ) ⎧ Ω2 dx ⎪ ⎪ = sin Ωt x(σ)y(σ) ⎪ ⎪ u0 ⎨dσ Ω2 dy ⎪ = − cos Ωt y 2 (σ) ⎪ ⎪ 2u0 ⎪ ⎩dσ x(0) = x0 , y(0) = y0 , σ

y0 = 0

σ = Ωσ y0 = 0

Ω

y(σ ) ≡ 0

y(σ ) = β = Ωy0 /(2u0 )

y0 1 + β cos Ωt σ y0 = 0 t σ → −1/(β cos Ωt)

y(σ )

cos Ωt 2β sin Ωt dx = dσ , x 1 + β cos Ωt σ

1 β cos Ωt = 0 2 β cos Ωt = 0 x(σ ) = x0 exp(2β sin Ωt σ ) x(σ ) = x0 (1 + 2 tan Ωt β cos Ωt σ ) x σ x= 2 tan Ωt x(σ ) y = y(σ ) x/x0 = (y0 /y) y = 0 u(x, y; t) = Ω

x+2

x Ω 2 , + u0 sin Ωt , v(x, y) = u0 y u0 2y Ω = 2π/t

u0 y

dy x(σ) = u0 , dσ 2y(σ)

η = y2

x = η /u0

σ x η − Ωη − 2Ω 2 η = u20 sin Ωt , η(0) = y02 η (0) = u0 x0

1 (α e2Ωσ − β e−Ωσ ) 3Ωu0

1/2 1 y(σ) = ± √ , (α e2Ωσ + 2β e−Ωσ ) − 3u20 sin Ωt 6Ω

x(σ) =

Ωu0 x0 + y0 < 0

u20

sin Ωt

α(t) = 2Ω 2 y02 + 2Ωu0 x0 + u20 sin Ωt β(t) = 2Ω 2 y02 − y0 > 0

... 0 ≤ t ≤ T t=T

x0 x0 t=T

x0

⎧ ⎨ dxf (t | τ ) = u E [x (t | τ ), t] f dt ⎩ xf (τ | τ ) = x0 xf (t | τ ) τ xf (T | τ )

τ
t 0≤τ ≤T

x0 = ξ

σ =t

y τ =0 4Δτ

3Δτ

2Δτ

Δτ

5Δτ x0

x

x x0 T = 6Δτ T = 6Δτ

6

•

τ = kΔτ

k = 0 1 ... 5 x

(x0 , y0 )

t≥τ

τ

x(t | τ ) = x0 + u0 (t − τ ) , y(t | τ ) = y0 exp(cos Ωτ − cos Ωt) , t=T x(T | τ ) = x0 + u0 (T − τ ) , y(T | τ ) = y0 exp(cos Ωτ − cos ΩT ) , T = t b

8

8

7

7

6

6

5

5

y

y

τ ∈ [0, T ] T = t /2 a

4

4

3

3

2

2

1

1

0

0 0

0.25

0.5

0.75

1

0

0.25

0.5

x a x 0 = 0 y0 = 1 τ =0

T = t /2 a T = t b u0 = 1 t = 1 a

τ = 5Δτ

x0 b

(x0 , y0 ) y0 = 0 x(t) ≡ x0 y(t) ≡ y0 y0 = 0 ⎧ dy Ω2 ⎪ ⎪ − 2 = cos Ωt dt ⎪ ⎪ ⎪ 2u0 ⎨ y Ω2 dx = y sin Ωt dt ⎪ ⎪ ⎪ x u0 ⎪ ⎪ ⎩ y(τ ) = y0 , x(τ ) = x0 , Ω y0

0.75

1

x b

u0 β = Ωy0 /(2u0 )

Δτ = t /10

|β| < 1/2 y(t | τ ) =

+2

y0 . 1 + β(sin Ωt − sin Ωτ ) sin Ωt − sin Ωτ

|β| < 1/2

−2

y(t) x(t) ≡ x0

x0 = 0

x0 = 0

dx = 2Ωβ x0 x x

t

τ

dt

sin Ωt , 1 + β(sin Ωt − sin Ωτ )

ξ = tan Ωt /2

x(t | τ ) = x0 exp 4 = x0 exp

Ω 1 (t − τ ) − √ × 2 1 − δ2

4

×

δ(τ ) = β/(1 − β sin Ωτ ) < 1 tan(Ωt/2) tan(Ωt/2) dξ dξ − 2 2 tan(Ωτ /2) ξ + 1 tan(Ωτ /2) ξ + 2δξ + 1

tan(Ωt/2) + δ tan(Ωτ /2) + δ √ √ arctan − arctan 2 1−δ 1 − δ2 y = 0

x0 y0 = 0

.

η = y2

20

y

15

10

5

0 0

5

10

15

20

x

x 0 = 1 y0 = 1 t = 103

T 10 20 30

40

u0 = 2

η˙ = u0 x

x = η/u ˙ 0

η(τ ) = y02 τ

(x0 , y0 )

x˙ = η¨ η

η(τ ˙ ) = u0 x0

1 2 u0 x0 u2 y0 + + 02 (cos Ωτ + 2 sin Ωτ ) 3 Ω 5Ω

u2 x 1 u 0 0 β (τ ) = + 02 (sin Ωτ − cos Ωτ ) , 2y02 − 3 Ω 2Ω

α (τ ) =

t≥τ y(t | τ ) = ±

α e2Ω(t−τ ) + β e−Ω(t−τ ) + y0 > 0

x = η/u ˙ 0 x(t | τ ) =

1/2 u20 (cos Ωt − 3 sin Ωt) , 2 10Ω y0 < 0

Ω 2Ω(t−τ ) u2 2α e −β e−Ω(t−τ ) − 0 2 (sin Ωt+3 cos Ωt) . u0 10Ω

t = T

τ ∈

[0, T ]

T T

Ft

IR3

t Ft−1 : IR3 −→ IR3 x −→ ξ |Vt |

Vt

|Vt | =

dVt dV (ξ)

t

Vt

dVt =

V0

dV0 |J(ξ, t)| ,

dV (x) dV0

J

∂ξ x1 ∂ξ x1 ∂ξ x1 2 3 1 ∂(x1 , x2 , x3 ) = ∂ξ1 x2 ∂ξ2 x2 ∂ξ3 x2 . J= ∂(ξ1 , ξ2 , ξ3 ) ∂ξ1 x3 ∂ξ2 x3 ∂ξ3 x3

x(ξ, 0) = ξ F0 J(ξ, 0) ≡ 1 . Ft J |Vt |

d |Vt | = dt

V0

dV0 ∂t J ≡

V0

∂t J = J

J dV0

JdV0 = dVt

Vt

|Vt |

dSt u · ν = ∂Vt

Vt

dVt ∇ · u ,

dSt

∂Vt

dVt ∇ · u = Vt

dVt

d |Vt | = dt

Vt

dVt

∂t J , J

∂t J −∇·u ≡0 . J Vt

∂t J = ∇·u J ∂t J J DJ/Dt

∂t J , J

u·ν

∂Vt

Vt

dVt

Dt J

J J = εijk ∂ξi x1 ∂ξj x2 ∂ξk x3 ,

∂t J = εijk

∂ξi (∂t x1 )∂ξj x2 ∂ξk x3 + ∂ξi x1 ∂ξj (∂t x2 )∂ξk x3 +

+∂ξi x1 ∂ξj x2 ∂ξk (∂t x3 ) , ∂t xl = ul

ξl u ξ ul = ul [x(ξ, t), t] ξm ∂ξm ul = ∂xp ul ∂ξm xp . ∂t J = εijk

∂xp u1 ∂ξi xp ∂ξj x2 ∂ξk x3 + ∂ξi x1 ∂xp u2 ∂ξj xp ∂ξk x3 +

+∂ξi x1 ∂ξj x2 ∂xp u3 ∂ξk xp

= ∂xp u1 εijk ∂ξi xp ∂ξj x2 ∂ξk x3 + ∂xp u2 εijk ∂ξi x1 ∂ξj xp ∂ξk x3 + +∂xp u3 εijk ∂ξi x1 ∂ξj x2 ∂ξk xp . εijk ∂ξi xp ∂ξj x2 ∂ξk x3 , εijk ∂ξi x1 ∂ξj xp ∂ξk x3 , εijk ∂ξi x1 ∂ξj x2 ∂ξk xp , p p = 1

p=2

p=3 p = 2

p = 3

p=1 p=2

p=3 ∂t J = J

∂x1 u1 + ∂x2 u2 + ∂x3 u3

x = eαt ξ + (1 − e−αt )η y = −(1 − e α

−αt

)ξ + e

−αt

η,

,

x = ξ + η sin Ωt y = −ξ sin Ωt + η ,

Ω ∇·u J

Ft

⎧ ⎨ x = ξ + tη

⎧ −αt ⎨ x = ξ + (1 − e )ζ

⎩

⎩

y = −tξ + η + tζ z = tη + ζ ,

y = (1 − e−αt )ξ + eαt η z = −(1 − e−αt )η + e−αt ζ ,

⎧ ⎨ x = ξ + η sin Ωt

⎧ 2 ⎨x = ξ + t ζ

⎩

⎩

y = η cos Ωt + (1 − cos Ωt)ζ z = ξ sin Ωt + ζ cos Ωt ,

y = tξ + η

z = ξ + (1 + t2 )ζ .

Ft

u = u(x, t) x

x u

i = 1, 2, 3 :

ui (x) ui (x ) + ∂xk ui (x ) (xk − xk ) , ⎛

∂x1 u1 ∂x2 u1 ∂x3 u1

⎞

⎜ ⎟ ⎟ ∇u = ⎜ ⎝ ∂x1 u2 ∂x2 u2 ∂x3 u2 ⎠ . ∂x1 u3 ∂x2 u3 ∂x3 u3 (i, j)

(∇u)ij

i ∂j ui

j

S S=S Ω Ω = −Ω ∇u ≡

1 1 [∇u + (∇u) ] + [∇u − (∇u) ] = S + Ω , 2 2 S Ω

7

x ∂j ui

∂xj ui

u(x) u(x ) + S(x ) · (x − x ) + Ω(x ) · (x − x ) , S · (x − x )

Sik (xk − xk ) ei ei

i

x − x

S

∇ · u = Sii

∇ × u = εijk Ωkj ei

Ω

y y

Ω ·y ω = ωi ei = ∇ × u = εilm ∂l um ei , u Ωjk

ω ωi

εikj

i

εikj ωi = εikj εilm ∂l um = (δkl δjm − δkm δjl )∂l um = ∂k uj − ∂j uk = 2Ωjk .

Ω · y = Ωjk yk ej =

1 1 1 εikj ωi yk ej = εjik ωi yk ej = ω × y , 2 2 2

1 u(x) u(x ) + ω(x ) × (x − x ) + S(x ) · (x − x ) 2

|x − x |

u(x )

ω(x )/2

S(x )·(x− x )

S i = 1, 2 3 {ei } {εi }

{εi , i = 1, 2, 3} R {εj } Sa

χi R−1 = R S

⎞ χ1 0 0 S a = R−1 SR = ⎝ 0 χ2 0 ⎠ , 0 0 χ3 ⎛

Sa

S y a = R−1 y

ya i Sa

S

y

{εi } χi S

Sa

{εj }

{ei }

S

Sa R

S

R−1 χi > 0 S · (x − x )

x χi < 0 S ·(x−x )

i

Sa

∂i ui = ∇ · u

S

χ1 + χ2 + χ 3 = ∇ · u ,

S

Ω

ω S

u = u(x) t ϕ = ϕ(x) u = ∇ϕ ϕ u

u u · dx

ϕ

C[x0 , x]

dϕ C[x0 ,x]

x0

dx · u(x )

C

x0

x

x0

x C[x0 , x]

ϕ ϕ(x) =

C[x0 ,x]

dx · u(x ) ,

ϕ(x0 ) = 0

C C

C x0 C

dx · u(x ) ≡

x C

dx · u(x ) +

−C

−C∪C

dx · u(x ) ,

C

x

C = −C ∪ C ∂S = C

C

S

dx · u(x ) =

S

x0

C

dA(x ) ω(x ) · ν(x ) . C x0

C ∂S = C

x S

C C

C Γ u

dx · u(x ) = Γ , 0

C

x

Γ = 0

C

ϕ

C x0

x ω = ∇×u ≡ 0

xy x2 x y y2 , −y , −z , u(x) = x − y + z , z2 zy z x 2 2 y + z2 xy 2 x −z 2 2 x −z , x+y+z , . y x2 + y 2 yz 2 xyz

ϕ(x)

Ft ξ

x = x(ξ, t)

2 a L (ξ, t) = ∂t u L (ξ, t) = ∂tt x(ξ, t) .

a u L (ξ, t) = u E [x(ξ, t), t] ,

a L (ξ, t) = ∂t u E [x(ξ, t), t] + uiE [x(ξ, t), t] ∂xi u E [x(ξ, t), t] . aL

Ft−1

Du/Dt

Du = ∂t u(x, t) + u(x, t) · ∇ u(x, t) . Dt

D/Dt

Dt x

u

t

x

x

u u(x) x

u

Dt u

x(ξ, t) =

ξ cos Ωt − η sin Ωt ξ sin Ωt + η −(ξ + η) sin Ωt + ζ ξ − ζ sin Ωt η −η sin Ωt + ζ

,

ξ − t2 η + tζ t3 ξ + η ξ sin Ωt + (1 − t)ζ

,

ξ + t2 η (1 − t)η t(ξ + η) + ζ

ξ + t2 ζ η cos Ωt + ζ sin Ωt ζ(sin Ωt)/(Ωt)

,

2 ∂tt x

,

,

ξ(sin Ωt)/(Ωt) η cos Ωt + t2 ζ (ξ + η) sin Ωt + ζ

ω k ui ∂i uk ≡ ui (∂i uk − ∂k ui ) + ui ∂k ui .

εjik ωj = εjik εjlm ∂l um = (δil δkm − δim δkl ) ∂l um = ∂i uk − ∂k ui ,

ui ∂i uk ek = εjik ωj ui ek + ∂k

ui ui |u|2 ek = ω × u + ∇ . 2 2

Du |u|2 = ∂t u + ω × u + ∇ . Dt 2

Dt u

2 3

4

6 8 36 79 12 13

16 19 45 8 10

16 1.2

12 13

14 15

1932

3

2 ∇ϕ = −u 31 33

ϕ

2

∂t ϕ

2 72

79 81

2.3 84 88

u ∇×u v = ∇ϕ 2.7

uv ue

∇·u

ϕ 100

102 2.8

1.1

1.2

1.3 1.4

1.4.1 63 64 1.5.3 2.3

1.2

1.3 14 1.9

4.8

26

248

10 25

58 1.8 15 20 4.1 4.7 1.7

4 5 6 1 49 50

2

23 3

2D

2D

(x, y) ez u(x, t) =

u(x, t) v(x, t) χx

= M (t) · x =

χx (t) x + χy (t) y χy (t) x − χx (t) y

χy z ∇ × u = ez ∇ ⊥ · u = (−∂y u + ∂x v) ez

∇ · u = ∂x u + ∂y v χx

χy

M

λ(1,2) = ±γ = ±

M

v (1) =

v (2) =

v

cos α sin α cos α sin α

=

(2)

=

R=

χ2x + χ2y

1 2γ(γ − χx )

1

−χy −γ + χx

2γ(γ + χx )

−χy γ + χx

α = α ±π/2

M

,

v (1)

cos α cos α sin α sin α

,

R−1 = RT

T

R ·M ·R = R

γ 0 0 −γ

, M (ξ, η)

(x, y) ξ

η ξ

γ

α ξ

α

M

ξ

η

η −η

χx

χy

γ

α

χx = γ cos 2α , χy = γ sin 2α .

R

˙T ·R R

ξ η

= RT ·

x y

=

x cos α + y sin α

x cos α + y sin α

ξ˙ = R T · x˙ = (RT · M · R) · ξ

x˙ = M · x

˙ ξ = +γ ξ η˙ = −γ η , ξ(0) = ξ0 χx χy

η(0) = η0

˙ T · R) · R ˙T ·x ξ˙ = RT · x˙ + (R

˙T ·R R

ξ(t) = ξ0 exp(+γt) η(t) = η0 exp(−γt)

x(t) = R ·

ξ0 exp(+γt) η0 exp(−γt)

= x0 cosh γt + χx

R

x(t) x0 cos 2α + y0 sin 2α x0 sin 2α − y0 cos 2α

sinh γt .

χy χx

χy

xy˙ + y x˙ xx˙ − y y˙ = χx , = χy , x2 + y 2 x2 + y 2 m θ

x

⎧m ⎨ ˙ cos 2θ − θ˙ sin 2θ = γ cos 2α m

˙ ⎩m sin 2θ + θ˙ cos 2θ = γ sin 2α , m

χx

χy ε = θ −α

ε˙ = −(γ sin 2ε + α) ˙ , z = tan ε = tan(θ − α) z˙ = −[2γz + α(1 ˙ + z 2 )] . I α˙ = 0 γ II γ

α˙

III γ

α˙

2D α˙ = 0

z(t) = z(0) e

−2 Γ (t)

t

Γ (t) =

dt γ(t ) > 0 .

0

θ−α → 0 ξ ξ

θ−α → π Γ → Γ∞

Γ → +∞

t → +∞

t → +∞ z → z(0) exp(−2Γ∞ )

m/m ˙ = γ cos 2ε

z → 0

γ → 0

γ

1 1 2z m ˙ = − + 2 z˙ , m 2 z z +1 z(t) m m(t) = m(0)

2 cos2 ε(0) sinh 2Γ (t) + e−2Γ (t) . ε(0) = π/2 3π/2 m

η t→∞ Γ → +∞

Γ → +∞

m→0 m → +∞

Γ → Γ∞

m γ

α˙

α˙ = 0 dz = −αdt ˙ , z 2 + 2μz + 1

μ = γ/α˙ 1 |μ| > 1

γ > |α| ˙

γ < |α| ˙

2 |μ| < 1 μ = ±1

z z1,2 = −μ ± ω α˙ > 0

ω=

μ2 − 1

α˙ < 0 1 z˙

|μ| > 1

−z/ ˙ α˙ = z 2 + 2μz + 1 , z˙

α˙

z z

z1

z2

z˙

z˙ z2

z1

z2

z

α˙ < 0

z1

z

α˙ > 0 z˙ z

z z˙

+∞ α˙ > 0 z +∞

α˙ < 0 z → z2 t→∞ z(0) > z1 z −∞ z2 z(0) < z2 ε = −π/2

z(0) ε = +π/2 z → z1 t → +∞ −∞

z1

α˙ < 0 z → z2

z(t)

z2

z(0)

dz = −αt ˙ , + 2μz + 1 +∞

z(0) > z1

+∞

dz +

z(t) = α˙ > 0

z(t)

dz

−∞

z(0)

1 = −αt ˙ . z 2 + 2μz + 1

˙ [z(0) − z1 ]z2 e−2ωαt − [z(0) − z2 ]z1 . ˙ − [z(0) − z ] [z(0) − z1 ]e−2ωαt 2

z(0) > z2

z(t)

z

z(0)

2

dz = −αt ˙ , + 2μz + 1

z(0) < z2

−∞

z(t)

dz + z(0)

z(t) =

+∞

z(0) < z1

dz

1 = −αt ˙ . z 2 + 2μz + 1

˙ [z(0) − z2 ]z1 e2ωαt − [z(0) − z1 ]z2 , 2ω αt [z(0) − z2 ]e ˙ − [z(0) − z1 ]

−∞

t → +∞ |μ| < 1

z1 1 − μ2

ω = z + 2μz + 1 2 ω (ζ 2 + 1) 2

z1,2 z = ω ζ − μ z˙

α˙ z

T 1 T = |α| ˙

+∞

−∞

dz = z 2 + 2μz + 1

α˙ < 0

π α˙ 2

− γ2

.

α˙ > 0 z(0)

T0 =

π

1 α˙ 2

−

2

γ2

∓ arctan

z(0) + μ , ω

α˙ < 0 z(t) = −μ + ω tan t T

arctan

T0

α˙ > 0

z(0) + μ − ω αt ˙ . ω t − T0

t z(t) = −μ + ω tan

t < T0

− ω α˙ t ∓

π , 2 m/m ˙ = γ cos 2ε

γ

1 2z 2z + 2μ m ˙ z˙ , = γ cos 2ε = − 2 m 4 z2 + 1 z + 2μz + 1

m(t) = m(0)

z 2 (t) + 1 z 2 (0) + 1

1/4

|μ| > 1

z 2 (0) + 2μz(0) + 1 z 2 (t) + 2μz(t) + 1

IR3

Bx (r) = y ∈ IR3 | |y − x| < r r

A

, t → +∞ |μ| < 1 T

m

z1,2

x

1/4

Bx

,

⇔

A

∀x ∈ A : ∃Bx (r) | Bx (r) ⊂ A .

IR3 ∅

∪α Aα x ∈ ∪α Aα ∃

Aα x ∈ Aα

Aα

∃Bx | Bx ⊂ Aα ⊂ ∪α Aα

IR1

Ak = (0, 1 +

∩∞ k=1 Ak

1/k) C

B ⊂ IR3

!

B=B=

⊃B

C C

C , B

B B IR3 τ = {Aα }

X X

τ 1) X

∅

τ

2)

τ

τ

3)

τ

τ

(X, τ )

X

τ x ∈ X Ux

V τ

x

τ

x Bx ⊂ Ux BV ⊆ V

τ BV ∈ Bx

x V ∈ Ux (X, τX ) (Y, τY )

f f f (x) ∃ f f

f : X −→ Y , x ∈ X VX ∈ τX x∈X

UY ∈ τY f (VX ) ⊂ UY

x

⇔ ∀AY ∈ τY : f −1 (AY ) ∈ τX . y = f (x) X = Y = IR1

τ

E

K 1

K E

F

K E

F

x1,2 ∈ E

α1,2

L ∈K

L(α1 x1 + α2 x2 ) = α1 L(x1 ) + α2 L(x2 ) . L

E

K

w

E

K E

E E

{ek , k = 1, . . . , n}

x = xi e i

E

θh h = 1, . . . , n θh (ek ) = δkh , δkh

1

K E

h=k {ek } w = yi θi

n

{ei }

E

{ej }

ej = Rjp ep , (Rjp )

n×n R = {ej }

E

p

j {ei }

{ej }

{ej }

ei = R i eq , q

R ei = R i Rqp ep , ej = Rjp R p eq , q

δip − R i Rqp q

q

ep = 0 ,

δjq − Rjp R p q

E R i Rqp = δip , Rjp R p = δjq , q

R {ei }

0 h = k {θh }

q

R x

eq = 0 .

x = xk ek = xk (R−1 )pk ep = x ep , p

x = x ek = x Rkp ep = xp ep . k

n R

k

x

xi

−1

i

R R

δkh = θ

h

(ek ) = θ

h

(Rkq eq ) = Rkq θ

h

(eq ) , θh

E

θ

h

= Rhm θm , {θi }

R

{θ } j

eq θ

h

(eq ) = Rhm θm (eq ) = Rhm δqm = Rhq .

δkh = Rkq Rhq . R = R−1 δkh = θh (ek ) = θh (R k eq ) = R k θh (eq ) , q

q

θh R θh = (R−1 )hm θ eq

E m

.

E {ek }

θh (eq ) = (R−1 )hm θ

m

(eq ) = (R−1 )hm δqm = (R−1 )hq .

δkh = R k (R−1 )hq . q

R = R−1 R E

R −1

E

x∈E (R−1 )−1 = R E E

E

n

E R−1

E E

E ≡

E

E

E

K T E×F K T

K

E×F {ei , i = 1, . . . , n} T

E

F

(x, y) {f k , k = 1, . . . , p} (x, y) E ×F (ei , f k )

x ∈ E

E

n×p

i

y ∈ F F

T

i k

T (x, y) = x T (ei , y) = x y T (ei , f k ) . n×p

T

E E×F

E

F

T (ei , f k ) T

E ⊗ F

F

w ∈ E

E×F

K

v ∈ F

K w ⊗ v (x, y) = w (x) v (y) . {ei } n×p

{f j }

{θh } {γ k } F θ h ⊗ γ k

E

E ⊗ F E

RE

RF

F

E F T = Tij θi ⊗γ j

Tij

Thk = (RE )lh (RF )m k Tlm ,

E

F

T T E × F K (θh , θk ) E ⊗F E

F eh ⊗ f k E T

hk

F −1 h −1 k = (RE )l (RF )m T lm ,

E ⊗ F θh ⊗ γk

r

p

E p+q = r

q

(E, . . . , E , E , . . . , E

p

q

K

R

E T j11 ,...,jqp i ,...,i

= Rjh11 · . . . · Rjpp · (R−1 )ik11 · . . . · (R−1 )kqq Th11,...,hpq . h

i

k ,...,k

S

Ft

JdV0 Vt = F t V0

f (x, t) d dt

dV (x) f (x, t) , Vt

Vt

Ft−1

0

dV (x) f (x, t) = Vt

V0

J(ξ, t)dV0 (ξ) f [x(ξ, t), t] . V0

d dt

d dV f = JdV0 f dt V0 Vt = dV0 [∂t Jf + J (∂t f + ui ∂xi f )] V0

≡

V0

∂t J . f + ∂t f + ui ∂xi f J

JdV0

JdV0 = dV D/Dt = ∂t + u · ∇ d dt

dV f = Vt

Df +f ∇·u Dt

dV Vt

1842 1912

dV Vt

Df +f ∇·u = Dt

dV Vt

∂t f + ∇ · (f u)

=

Vt

dV ∂t f + dA f u · ν ∂V t f

f (x, t) Vt = F t V0

x

f

Vt

x(ξ, t) = (ξ + tη, −tξ + η) r exp(−|x|/r) sin Ωt

V0 f (x, t) = √ r 1 + t2 . . .

IR3

Ft

x(ξ, t) V0

t dVt = JdV0 t Vt = F t V0

0

Ft−1

0

ξ t≥0

V0

Vt

L δ

L δ V0

0

t→∞

dV (x) ρ(x, t) , Vt

ρ V0

V0

d dt ≡

Vt

dV ρ ≡ 0 , 0

V0 dV Vt

Dρ +ρ ∇·u ≡0 . Dt Vt V0

Dρ +ρ ∇·u =0 Dt

∇ · ρu = 0 , ρu

∂t ρ Dt ρ

L

=

E

J(ξ, t)ρ

L

(ξ, t) ≡ ρ

L

(ξ, 0) .

V

dV ∂t ρ V

=

∂V

dA ρu · (−ν) V ∂V

V

ρ f = ρg Dρ Dg Dg Df +f ∇·u=g +ρ ∇·u +ρ =ρ , Dt Dt Dt Dt f = ρg d dt ρ p

dV ρ g =

Vt

dV ρ Vt

Dg Dt T

∂ρ =0, ∂p T ρ

T ρ

Dt ρ =

0 ∇·u=0 ,

C S ∂S = C

x0 ∈ C

t Tf

S ∀x ∈ ∂Tf − S :

u(x) · ν(x) = 0 .

S

dA u · ν S

S

Σ ⊂ (∂Tf − S)

dA u · ν ≡

S

Tf (t)

S

∂S ⊂ Tf

dA u · ν +

dA u · ν , Σ

S

ρ(x, 0) ≡ ρ0 ρ(x, t) ≡ ρ0

P(p) = p0 (∇p)/ρ

∇P

dp , p0 ρ(p ) p

P

P(p) = (p − p0 )/ρ0

P p/ργ ≡

ρu

u(x) = u1 (x1 , x2 )e1 + u2 (x1 , x2 )e2 , (x1 , x2 ) ... e3 ω

ρu u ψ ∂2 ψ = u1 , −∂1 ψ = u2 .

u

∇·u=0 ψ≡

u(x, t) = sin(αx + βy)

−β

x(2αy + βx)

, −y(αy + 2βx) β x 1 α−β , , αx + βy −α (x + βy)2 y 2 β y 1 β αx+βy , e , 2 −α α2 x2 + β 2 y 2 −α x

α

α

,

β

x = x1 e1 +x2 e2 π/2 x⊥ = −x2 e1 + x1 e2 ∇ = e1 ∂1 + e2 ∂2 −∇⊥ ψ = u .

f f = f1 (x1 , x2 )e1 + f2 (x1 , x2 )e2 ∇ × f · e3 = ∇ ⊥ · f .

u = ∇ϕ

u = −∇⊥ ψ (x⊥ )⊥ = −x

u u⊥ = ∇ψ .

C

u⊥ · dx

∇ ∇⊥ = −e1 ∂2 + e2 ∂1

C

u⊥ · dx = 0 .

S

∂S = C

C

⊥

u · dx =

S

=

S

dA ∇ × u⊥ · ν dA ∇⊥ · u⊥

=

S

dA ∇ · u = 0 , S

ν ≡ e3

C x1

x2

C[x1 ,x2 ]

ds ν · u = −

C[x1 ,x2 ]

= C[x1 ,x2 ]

ds ν · ∇⊥ ψ

dx · ∇ψ = ψ(x2 ) − ψ(x1 ) , ν⊥

x1

x2

C ψ

3

ψ ω = ω 3 e3 ω3 = ∇⊥ · u ω3 = ∇⊥ · u = −∇⊥ · ∇⊥ ψ = −∇2 ψ .

Vt = F t V0 Vt

dV ρu , Vt

Vt •

Vt

•

∂Vt

F (x, t) ρ(x, t) ρF

g ρg Vt

dV ρF . Vt

x T (x, ν) ν

T

T (x, −ν) = −T (x, ν) Vt dA T . ∂Vt

∂Vt

x3 −e2 −e1

ν

x1

x −e3

x2

T

ν

x1 x2 x1 x3 ν x1 x3

3

1

x1 x2

A Ai xi = 0 2

x2 x3

Ai = Aνi

x2 x3 2 −e1 −e2 −e3

i=1

3

x δ

0

δ2

δ3

T i

T

−T (i)

i

δ3 x

T d ( dt

)−

= δ3

.

(i)

− T (i) (x) + O(δ)

Ai +

T (x, ν) + O(δ)

A = O(δ 3 ) . δ→0

A T (x, ν) = νi T (i) (x) . T (i) i

τ

⎞ (1) (2) (3) T1 T1 T1 ⎟ ⎜ τ = ⎝ T2(1) T2(2) T2(3) ⎠ , (1) (2) (3) T3 T3 T3 ⎛

τ T (x, ν) = τki (x)νi ek . τ

τ

2◦

(1, 1) τ τ

R

IR3

τ −1 −1 −1 Th = τhi νi = τhi Riq νq = Rhj Tj = Rhj τjq νq ,

−1 −1 τhi − Rhj τjq Riq

νq ≡ 0 , ν

Rqp

q −1 τhp = Rqp Rhj τjq

R

⇒

τ = R−1 τ R , τ

R−1

(1, 1) T

ν

d dt

Vt

dV ρu Vt

=

dV ρF Vt

+

∂Vt

dA τ · ν

d dt

dV ρu =

Vt

dV ρ Vt

Du . Dt

dA τ · ν = ei ∂Vt

dA τij νj = ei

Vt

∂Vt

∇·τ

dV ∂j τij .

ei ∂j τij

Vt ρ

Du = ∇ · τ + ρF Dt τ V

dV ∂t (ρu) = dA u · (−ν) ρu + dV ρF + dA τ · ν Vt ∂Vt Vt ∂Vt

V

ρu

Vt

d dt

dV x × ρu Vt

=

dV x × ρF + dA x × τ · ν Vt ∂Vt

x×u Dt x × u = x × ∂t u + ei uk ∂k (εipq xp uq ) = x × ∂t u + ei εipq uk (δkp uq + xp ∂k uq ) = x × ∂t u + x × u · ∇u = x × Dt u .

dA x × τ · ν = ei

∂Vt

Vt

dV εipq ∂k (xp τqk )

= ei

Vt

= Vt

dV εipq (τqp + xp ∂k τqk )

dV ei εipq τqp +

Vt

dV x × ∇ · τ .

Vt x×ρ

Du = x × ρF + x × ∇ · τ + ei εipq τqp . Dt

εipq τqp = 0 i=1 2

3

τ

τ

x

x

τ τ = H(u, ∇u) , x u

u(x) x IR3

x (t)

x(t) x = x0 + R−1 x , x0

IR3

{ei } x0 (t)

R(t)

{ek (t)}

R(t)

u x˙ = x˙ 0 + QR−1 x +

˙ Q=R

−1

R−1 u

R

0=

d ˙ =Q+Q ˙ −1 R + R−1 R (R−1 R) = R dt

,

R x ui

xj

−1 −1 ∂x j ui = Rkj ∂xk Qip Rpq xq + Rip up

−1 −1 = Rkj Qip Rpk + Rip ∂xk up = Qij + (R−1 ∇uR)ij , ∂xj = Rkj ∂xk ∇ u = Q + R−1 ∇uR .

H

R

R−1 H(u, ∇u)R = R−1 τ R = τ = H(u , ∇ u ) =H

x˙ 0 + QR−1 x + R−1 u , Q + R−1 ∇uR

,

∇u S

Ω

H R−1 H(u, S + Ω)R ≡ H

x˙ 0 + QR−1 x+ R−1 u, Q+ R−1 ΩR + R−1 SR

R x0

R −1

Q = −R

x˙ 0 = −QR−1 x − R−1 u(x) Ω(x)R Q H

Ω S H

S τ ∇u

u

H R−1 H(S) R ≡ H(R−1 S R) . H (S) = S k

k

H H(S) = χ0 I + χ1 S + χ2 S 2 + . . . ⎛ ⎞ 100 I = ⎝0 1 0⎠ 001 3

χk I1 =

(S) = ∇ · u , I2 = (·)

(·)

I

S

1 2 I1 − (S 2 ) , I3 = 2

(S) ,

,

S 1945 τ

H

Sk

H

τ = H(S) = (α + λ∇ · u)I + 2μS , τ 3(α + λ∇ · u) + 2μ∇ · u = 3

α+

2 λ+ μ ∇·u . 3

2 λ+ μ=0 3 3α

τii

−p

α

T (x, ν)

S=0 p(x)

ν

τ =−

p+

2 μ ∇ · u I + 2μS 3

μ

kg/(ms) 10−5 kg/(ms) ν = μ/ρ

[μ] = ML−1 T −2 /T −1 = ML−1 T −1 10−3 kg/(ms)

k τ (∇ · τ )k = −∂i

p+

2 μ ∇ · u δki + μ∂i (∂i uk + ∂k ui ) 3

2 μ ∂k ∇ · u + μ∇2 uk + μ∂k ∇ · u 3 1 = −∂k p + μ∇2 uk + μ ∂k ∇ · u 3 = −∂k p −

ρ

Du 1 = −∇p + μ ∇2 u + ∇ ∇ · u + ρF Dt 3 μ=

0

p μ=0

ρ

F F = ∇f f

F

ϕ ω=0 ∇

Φ(t)

∂t ϕ +

|u|2 + P(p) − f = 0 , 2

t ∂t ϕ +

|u|2 + P(p) − f ≡ Φ 2

x x A(x)

ρ l ri ui

ru pi

F rm = (ri + ru )/2

α = (ru −

ri )/l Fx = −2παrm l(pi + αρu2i lrm /ru2 )

h g

ui

Ai /Au 1 1 + 2gh/u2i

Ai /Au =

e

s

T

h(x, ν)

x

ν E d dt

dV ρ Vt

e+

Vt = dA (h + T · u) + dV ρ(F · u + E) ∂Vt Vt

|u|2 2

h [h] = MT −3 1) h(x, −ν) = −h(x, ν) 2) q(x) ∂Vt±

±

= (∓S) ∪ Σ d |u|2 + Vt : dV ρ e + dt Vt+ 2

Vt−

d : dt

dV ρ

Vt−

e+

|u|2 2

h(x, ν) = −q(x) · ν Vt±

∂Vt = Σ + ∪ Σ − S x Vt = dA (h + T · u) + dV ρ(F · u + E)+ Σ Vt+ + + dA [h(−ν) + T (−ν) · u] S = dA (h + T · u) + dV ρ(F · u + E)+ Σ Vt− − + dA [h(+ν) + T (+ν) · u] S

Σ− Vt−

Vt

Vt = Vt+ ∪Vt− S

Σ+

S

Vt+

Vt± ∂Vt

S

S

− S

Σ+

S

Vt+

Vt−

Σ−

dA [T (−ν) + T (+ν)] · u −

dA [h(−ν) + h(+ν)] = 0 , S

Vt h(x, −ν) = −h(x, ν) , 1) x −τk1 uk A1 − τk2 uk A2 − τk3 uk A3 + τjl uj nl A+ +h(−e1 )A1 + h(−e2 )A2 + h(−e3 )A3 + h(ν)A = O(δ 3 ) , τ

h δ3 −h(ei )Ai + h(ν)A = O(δ 3 ) h(ν) = νi h(ei ) 2)

x δ 0 q = −ei h(ei )

Vt ρ

|u|2 D e+ = −∇ · q + ∇ · (τ · u) + ρF · u + ρE , Dt 2 e

|u|2 /2 ρ

D |u|2 = u · (∇ · τ ) + ρF · u , Dt 2 u

u · (∇ · τ ) = uk ∂i τki ≡ ∂i (uk τki ) − τki ∂i uk = ∂i (τik uk ) − τki ∂i uk = ∇ · (τ · u) −

ρ

(τ · ∇u) ,

D |u|2 = ∇ · (τ · u) − (τ · ∇u) + ρF · u Dt 2

ρ

De = tr(τ · ∇u) − ∇ · q + ρE Dt

e (τ · ∇u) =

{[(−p + λ∇ · u)I + 2μS] · ∇u}

= −p ∇ · u + λ(∇ · u)2 + 2μ

2

(S · ∇u) ,

(S · ∇u) =

[S · ∇u + (∇u) · S ]

(A) =

=

[S · ∇u + S · (∇u) ]

(AB) =

=

[S · ∇u + S · (∇u) ]

=2

{S · [∇u + (∇u) ]/2}

=2

(S 2 ) .

(A ) (BA) S

S S (S 2 ) =

1

(R−1 S 2 R) =

(R−1 SR R−1 SR) =

−1 (R−1 SR) = Rip Spq Rqi = δpq Spq =

(S 2a ) = χ21 + χ22 + χ23 .

(S)

λ (∇ · u)2 + 2μ

(S · ∇u) =

= λ [ (S a )]2 + 2μ (S 2a )

= λ 3(χ21 + χ22 + χ23 ) − (χ1 − χ2 )2 + (χ1 − χ3 )2 + (χ2 − χ3 )2 + +2μ (χ21 + χ22 + χ23 ) 2 = 3 λ + μ (χ21 + χ22 + χ23 ) + 3

−λ (χ1 − χ2 )2 + (χ1 − χ3 )2 + (χ2 − χ3 )2 =μ

2 3

(χ1 − χ2 )2 + (χ1 − χ3 )2 + (χ2 − χ3 )2

=: φ ≥ 0 , φ

R3

φ φ φ i = 1 2

χi = 1/3 ∇ · u

3

λ

μ

λ+2μ/3 ≥ 0 (τ · ∇u) = −p∇ · u +

φ

q q {ei }

{ei }

ei = Rei

ν −1 −Rkl ql νk = −ql Rlk νk = −ql νl = h = h = −ql νl

q = R−1 q ν

q ∇T q = f (T, ∇T )

T

f

q f 2

(R

−1

(∇ T )i = ∂x T = ∂x xk ∂xk T = Rki ∂xk T = ∇T )i

i

i

f (T, R−1 ∇T ) = q = R−1 q = R−1 f (T, ∇T ) , R f (T, ∇T ) = −K(T )∇T

f K

q = −K∇T K(> 0)

T −

q ∇T

K

[K] = ML−1 T −3 /(L−1 Θ) = MT −3 Θ−1

3

kg/(s K) 0.6 kg/(s3 K)

2 × 10−2 kg/(s3 K) μ

K ρ

De = −p∇ · u + φ + K∇2 T + ρE . Dt φ s

ρ Dt e + p ∇ · u = ρ Dt e −

De D 1

p Dt ρ = ρ +p . ρ Dt Dt ρ

h = e + p/ρ ρ Dt e + p ∇ · u = ρ

Dt e + p

Dp D 1 1 = ρ Dt h − Dt p , + Dt p − Dt ρ ρ Dt

ρ Dt e + p ∇ · u = ρT Dt s . d˜ ˜ = T ds dQ

˜ dQ

˜ − pd(1/ρ) de = dQ

ρ

Dp Dh = + φ + K∇2 T + ρE , Dt Dt

ρT

Ds = φ + K∇2 T + ρE Dt

q 1 K K 2 q 1 ∇ T = − ∇ · q = −∇ · + q · − 2 ∇T = −∇ · + 2 |∇T |2 , T T T T T T V

T

V

dV ∂t (ρs) =

φ K h + 2 |∇T |2 dA ρsu · (−ν) + dA + dV + T T T ∂V V ∂V ≥0

+

E dV ρ T V

Dt s

s = s(T, p)

Dt T

∂s ∂s dT + dp p ∂T ∂p T 1 ∂s 1 ∂ 1 = T ρ dT − dp p T ∂T ρ ∂T ρ p

ds =

=

β cp dT − dp , T ρ

∂s ∂v =− T ∂p ∂T p v = 1/ρ

Dt p

cp :=

˜ dQ = T ∂s p dT ∂T p

β :=

1 ∂v v ∂T p

β = 1/T

ρcp

DT Dp = βT + φ + K∇2 T + ρE Dt Dt p = ρRT

ρcv

DT = −ρRT ∇ · u + φ + K∇2 T + ρE , Dt T

4.2

4.3

4.5

4.6 4.10 4.11 4.12 4.13 4.14 +

4.15 10 11 4 15 10 τij = −pδij

79 35 2 67

10

12 13 32 33 2 5

45 6 50 51 3

2.2 75

133

3.2

3.3 147 3.4 153 156 3 3.2.1 3.3 5.1 5.2 3 3.1 141 143 3.4 3.5 3.2

3.6 4

4.2 4.3

4.5

4.4

4.7

4.9 45 6 10 + 11 21 323

22

4

328 1.2

1.3 1.4 +

1.6

1.8 85 87 1.9 12 14 2 33

ρf

33 37 43 48

3.3.5

3.3.6 323 325

327 330 333 335 337

337 339 347 349 3.6 1.3 68 89

9 10 206 209 214 2.1 2.2 3 3.1 3.11 3.13

3.14 3.2 5 5.1 5.2

s v T p e

h

A

G

e s

v

e = T s−p v

⇒

de = T ds − p dv

∂e ∂e =T , = −p , ∂s v ∂v s

⇒ e

s

v

∂p ∂T =− . s v ∂v

∂s

h

s

(s, T )

(v, p)

p

h=e+p v

⇒

dh = T ds + v dp

∂h ∂h =T , =v, p ∂s ∂p s

⇒

h ∂v ∂T = . ∂p s ∂s p

s

p

A

T (s, T )

v

(v, p) A = e−T s

⇒

dA = −s dT − p dv

⇒

∂A ∂A = −s , = −p , v ∂T ∂v T A

T

v

∂p ∂s = . ∂v T ∂T v G

T (s, T )

G = e−T s+p v

⇒

dG = −s dT + v dp

⇒ A

∂v ∂s =− . T p

∂p

∂T

p

(v, p) ∂G ∂G = −s , =v, ∂T p ∂p T T

v

ρ ≡ ρ0 u ≡ 0

u=0 ∇p = ρ0 F .

p F F = g ≡ −g ez 1

g>0

ez

∂x p = 0

∂y p = 0

∂z p = −ρ0 g z

p z= p = −ρ0 gz ,

z z

ρ0 g ∼ 104

/ 1.013 · 105

z1

z2

p1 p2 p1 − p2

z1 − z2

|Ωb |

Ωb ∂Ωb ν Fb =

Fb

dA(x) p(x) [−ν(x)] = ρ0 g ∂Ωb

dA(x) z ν(x) = ρ0 g|Ωb |ez , ∂Ωb

h

A

m ρ g

z = z0 t=0 z ze = (ρAh/m − 1)m/(ρA) ω = ρAg/m z(t) = (z0 − ze ) cos ωt + ze

M BR (0) p(x) r = |x| mχ

mR = M

g χ (x) = G 2

G /

2

x ∈ Bχ (0)

Φ x

Bχ (0) ⊆ BR (0)

x ∈ Bχ (0)

mχ (−x) , r3 6.66 · 10−11 Φχ (x) = Gmχ /r

gχ

χ = R Φ(x) = GM/r

Φ(x) ∂Bχ (0) mχ Bχ (0)

dx ∇2 Φ =

Φ(x) = Φχ (x) χ ∈ (0, R)

∂Bχ (0)

dS g χ · ν = −4πGmχ = −4πG

Bχ (0)

dx ρ .

χ ∇2 Φ = −4πGρ , 6 BR (0)

Φ(x) = GM/R Φ

x ∈ ∂BR (0)

∇p = ρ∇Φ

u=0

F = ∇Φ Φ Φ

ρ

p ∇·

∇p = −4πGρ . ρ ∇

1 d r2 dp = −4πGρ . r2 dr ρ dr p(R) = p0 dp/dr = −ρ0 GM/R2

r=R ρ0 = ρ(R) ρ/ρ0 = (r/R)−α

0 < α < 3 M [R2 /(Gp0 )]1/4

(p0 /G)1/2 R2

"

R

p + (2 + α)p /r = −4πρ20 r−2α p(1) = 1 , p (1) = −4πρ20 /(3 − α) ,

α = 1 p(r) = 1 +

2πρ20 1 − r2(1−α) . 3−α 1−α r

1 + 2πρ20 /[(1 − α)(3 − α)] r→0 α=1 p(r) = 1 − 2πρ20 log r r→0

ρ/ρ0 = 1 + α(r/R)

0 < α < 1 r=0

α p(r)

ρ/ρ0 = (p/p0 )α

α ∈ (0, 1)

r→0

1<α<3

˙ −1 R Q=R ω0 ω0 l =

1 εlrs Qsr . 2

Qx = ω 0 ×x ω0

R

−1

x˙ = u

x

x˙ 0 = u0

u = u0 + ω0 × (x − x0 ) + R−1 u . ¨ a = x

¨ a = x

a = u˙ 0 + ω˙ 0 × (x − x0 ) + ω 0 × (u − u0 ) + Q(R−1 u) + R−1 u˙ . R

−1

u

u˙ 0

=

u − u0 = ω 0 × (x − x0 ) +

a0

a =

a0 + ω˙ 0 × (x − x0 ) + ω 0 × ω 0 × (x − x0 ) +

+ 2 ω 0 × (R−1 u) +

R−1 a

RC

RC ω0 ω0

u0 a0 RC u ≡ 0 a ≡ 0 ω˙ 0 = 0

ρ [a0 + ω0 × ω 0 × (x − x0 )] = −∇ p + ρF , ∇ = ei ∂xi

ρ ≡ ρ0

F =g

ω0 = 0

∇ p = ρ0 (g − a0 ) p(x )/ρ0 = (g − a0 ) · x , g − a0

5 4

−ω 0 × ω 0 × (x − x0 )

3

z

g 2

∇ p˜

1 0 0

1

2

3

4

5

1/ω0

Δp

r ρ0 g 3 /ω06

r=2 z=3 p˜ = p/ρ0 g

g/ω02

−ω 0 × ω 0 × (x − x0 )

ω0

π⊥

ω0 x0

xn = rr

1/2

F =g

ω0

ω 0 × ω 0 × (x − x0 ) x − x0 xn = (x − x0 ) − xp ∈ π ⊥ r ω0 = |ω0 |

xp ω0 ω0

r ω0 × ω0 × (x − x0 ) = −ω02 rr

∇p = ρ0 (ω02 rr + g)

∂z p = −ρ0 g

∂r p = ρ0 ω02 r p(r, z)/ρ0 =

ω02 2 r − gz , 2 ω0

2

61

2.8 2.9

1.7 1.4 18 19 3

23 27 61 65 52 57

60

64 80

3.1.4 3.4.1

3.4.2

∇2 ψ = 0 ψ

w

μ

w

ψ

w(z) = u∞ (z + r2 /z) u∞ ×

Q = 2u∞ πr2 2 ×

d = −ex

r z ρ = |z| θ = arg z

w(ρe θ ) = u∞

ρ+

r2 cos θ + ρ

ρ−

r2 sin θ . ρ

ρ = r

w r

u∞ = u∞

Γ

x α

x

χ = π/α z = re θ θ=α

sin(χθ) (r, θ)

∇2 =

r

1 1 2 ∂r (r ∂r ) + 2 ∂θθ , r r

θ=0 θ

rχ sin(χθ) ψ(r, θ) = rχ sin(χθ)

∇2 r(θ) = (cos θ, sin θ) θ(θ) = (− sin θ, cos θ) r θ

8

4

6

2

4

0

y

y

∇ = r∂r + θ/r∂θ . . .

2

-2

0

-4 -4

-2

0

2

4

-4

-2

x

α = 120◦ a 0.2

2

4

240◦ b a b

ϕ(r, θ) = r cos(χθ) χ

0

x

−∇⊥ ψ = ∇ϕ w(z) = z χ .

u = χz χ−1 χ z =0 α>π χ<1 α < π χ > 1 χ = 1

π

α = π/2

c ρ

x = c cosh ρ cos θ y = c sinh ρ sin θ , ρ ∈ [0, +∞) θ ∈ [0, 2π) (x, y)

ρ

x2 y2 + =1, (c cosh ρ)2 (c sinh ρ)2 ρ≡ρ a = c cosh ρ b = c sinh ρ ρ ≡ ρ a b c ρ c=

a2 − b 2 , ρ =

x

(x, y) y

a≥ b

x

y

1 a+b log , 2 a−b cosh ρ = a/c sinh ρ = b/c

c

4 2 x −1 c2 4 cosh 2ρ + cos 2θ − cosh 2ρ cos 2θ = 2 y 2 + 1 c cosh 2ρ + cos 2θ + cosh 2ρ cos 2θ =

cosh 2ρ cos 2θ ⎧ 2 ⎪ ⎨ cosh 2ρ + cos 2θ = 2 (x2 + y 2 ) = μ2 ≥ 0 c ⎪ ⎩ cosh 2ρ cos 2θ = 2 (x2 − y 2 ) − 1 = ν c2 cos 2θ =

ν ν/ cosh 2ρ

cosh 2ρ cosh2 2ρ − μ2 cosh 2ρ + ν = 0 ,

1 2 μ + μ4 − 4ν 2

1 ; = 2 x2 + y 2 + (x2 + y 2 )2 − 2c2 (x2 − y 2 ) + c4 c

cosh 2ρ =

cos 2θ

1 2 μ − μ4 − 4ν 2

1 . = 2 x2 + y 2 − (x2 + y 2 )2 − 2c2 (x2 − y 2 ) + c4 c

cos 2θ =

x = m cos γ y = m sin γ 2 m = m/c exp(ρ) = m +O(1/m ) exp(−ρ) = 1/m +O(1/m ) 2 cos 2θ = cos 2γ + O(1/m ) x→∞

J=

c2 ∂(x, y) = (cosh 2ρ − cos 2θ) , ∂(ρ, θ) 2 ρ=0

2 2 ∇2 = ∂xx + ∂yy =

θ=0

θ=π

1 2 2 (∂ + ∂θθ ). J ρρ

f (ρ, θ) = exp(−ρ)(A cos θ + B sin θ) ψ∞ u∞ = ψ∞ (ρ, θ) = u⊥ u∞ (cos α, sin α) ∞ · x(ρ, θ) = cu∞ (− sin α cosh ρ cos θ + cos α sinh ρ sin θ) exp(−ρ)A − cu∞ sin α cosh ρ = 0 exp(−ρ)B + cu∞ cos α sinh ρ = 0 ψ∞ +f α ψ(ρ, θ) √ = u∞ a + b

e−ρ √ (sin α cos θ − λ cos α sin θ)+ 1−λ √ + 1 − λ (− sin α cosh ρ cos θ + cos α sinh ρ sin θ) , λ = b/a < 1

∂ν ψ ∝ (λ cos θ∂x ρ + sin θ∂y ρ) ∂ρ ψ + (λ cos θ∂x θ + sin θ∂y θ) ∂θ ψ = 0

2

1

1

y

y

2

0

-1

0

-1

-2

-2 -2

-1

0

1

2

-2

-1

0

x a 60◦

1

2

x b α = 60◦

(x , y )

ρ=ρ a [(1 + λ2 ) − (1 − λ2 ) cos 2θ] > 0 2J λ cos θ∂x θ + sin θ∂y θ ≡ 0 ,

λ cos θ∂x ρ + sin θ∂y ρ =

∂ρ ψ(ρ, θ) = 0

θ=α

θ =α+π

F

ϕ −∇⊥ ψ = ∇ϕ ϕ(ρ, θ) √ = u∞ a + b

∂ρ ϕ = ∂θ ψ

∂θ ϕ = −∂ρ ψ

e−ρ √ (sin α sin θ + λ cos α cos θ)+ 1−λ √ + 1 − λ (sin α sinh ρ sin θ + cos α cosh ρ cos θ) . α = 60◦ b → 0+

h ρ U x u

x

v

y x y≤h

0≤

u(x, y)

x

y

U 1

y/h

h p1

p0 0

x

L

0.5

0 -1

-0.5

0

0.5

1

1.5

2

u/u b

a a

U p1 − p0 =1

L b

p˜ = −10 −8 . . . +10

∂x u ≡ 0

0

∂y v = v(x, 0) = v(x, h) = 0 y

v ∂y p = 0 y

x

p

dp/dx

x μ

d2 u = p . dy 2 y

x p˜ = ρh3 (p1 − p0 )/(Lμ2 )

u(0) = 0 u(h) = U h2 /ν u = ν/h

h u (˜ y) = u

y˜ −

p˜ y˜(1 − y˜) , 2

y˜ = y/h

= ρhU/μ μU/h2

ρU 2 /h

u/u =

y˜

u/u = (−˜ p )˜ y (1 − y˜)/2

y

y Q Q 1 (6 = hu 12

− p˜ ) . p˜ <

6

p˜

U

τw

τw p˜ = 1 − , 2 2 ρu p˜ < 2

= 0 p˜ = 0

p˜ = 0

R1 Ω1

>0

R2 R1 < R2

Ω2

r ∈ (R1 , R2 ) (∂θ uθ )/r + ∂z uz = 0

z uz = ur = 0 [∂r (rur )]/r + uθ = uθ (r)

θ ∈ [0, 2π) r u2θ /r = ∂r p/ρ ,

uθ

r

p θ

d 1 d (ruθ ) = 0 , dr r dr uθ (R1 ) = R1 Ω1

uθ (R2 ) = R2 Ω2 Ω2 = 0 u ˜θ =

uθ 1 1

, (1 − χ2 ω) r˜ − χ2 (1 − ω) (˜ r) = 2 R2 Ω2 1−χ r˜

χ = R1 /R2 < 1 ω = Ω1 /Ω2 r˜ = r/R2 ∈ (χ, 1) uθ (1 − χ2 ω)/(1 − χ2 ) b = χ2 (1 − ω)/(1 − χ2 ) p˜ p˜ =

a =

1 p (˜ r ) = − a2 (1 − r˜2 ) + 4ab log r˜ + b2 −1 , ρR22 Ω22 /2 r˜2 p(1) = 0 χ = 1/4 Ω1 /Ω2

0.75

0.75

r˜

1

r˜

1

0.5

0.5

0.25

0.25 -2

-1

0

u ˜ a

1

2

a χ = 1/4

-5

-4

-3

p˜ b

-2

0

b ω = −8 −6 . . . +8

kR

Ω

-1

R k < 1

Ω R kR

m mg

R Ω

τrθ

r θ

τrθ = τij rj θi

r = rj ej

θ = θ i ei τij = −pδij + μ(∂j ui + ∂i uj )

1 ∂θ u) r 1 uθ + ∂θ ur = μ ∂r uθ − r r uθ 1

+ ∂θ ur , = μ r∂r r r

τrθ = μ(θ · ∂r u + r ·

θi ∂i = − sin θ ∂x + cos θ ∂y = − sin θ(∂x r∂r + ∂x θ∂θ ) + cos θ(∂y r∂r + ∂y θ∂θ ) = M = kR

L

2πkR

dz 0

0

ds τrθ (kR, θ, z) = k 2 R2

L

dz 0

∂t u = 0

0

1 ∂θ . r

2π

dθ τrθ (kR, θ, z) .

ur = uz = 0

uθ = uθ (r, z) p = p(r, z) ΩR 1 − k2

uθ =

r R − k2 . R r

M = 4π Ω R2 L

k2 μ 1 − k2

m M μ

L

u z (x, y) ex

x

z θ

ey r

r

θ

⎛

⎞ ⎛ ⎞ cos θ − sin θ r = r(θ) = ⎝ sin θ ⎠ , θ = θ(θ) = ⎝ cos θ ⎠ 0 0

dθ dr =θ, = −r . dθ dθ ∇·u = 0

∇ x = cos θ r y sin θ ∂x θ = − 2 = − r r

∂x r =

ez

y

y = sin θ r x cos θ ∂y θ = 2 = r r

∂y r =

∇ = ex ∂x + ey ∂y + ez ∂z = (cos θ r − sin θ θ) (∂x r ∂r + ∂x θ ∂θ ) + +(sin θ r + cos θ θ) (∂y r ∂r + ∂y θ ∂θ ) + ez ∂z

sin θ ∂θ + r cos θ ∂θ + ez ∂z +(sin θ r + cos θ θ) sin θ ∂r + r θ = r ∂r + ∂θ + ez ∂z . r = (cos θ r − sin θ θ)

cos θ ∂r −

u θ ∂θ + ez ∂z · (ur r + uθ θ + uz ez ) r ∂θ uθ ur + + ∂z uz = ∂r ur + r r 1 ∂θ uθ + ∂z uz , = ∂r (rur ) + r r

∇·u =

r ∂r +

1 ∂θ uθ ∂r (rur ) + + ∂z uz = 0 r r ⎧ n ⎨ ur (r, θ, z) = r [a(z) cos nθ + b(z) sin nθ] ⎩

uθ (r, θ, z) = r n [c(z) cos nθ + d(z) sin nθ]

uz (r, θ, z) = r n−1 [e(z) cos nθ + f (z) sin nθ] , n z

a b c d e

f

∇

u · ∇u = (ur r + uθ θ + uz ez ) · = =

ur ∂r +

r ∂r +

θ ∂θ + ez ∂z (ur r + uθ θ + uz ez ) r

uθ ∂θ + uz ∂z (ur r + uθ θ + uz ez ) r

uθ u2 ∂θ ur − θ + uz ∂z ur r + r r uθ ur uθ ∂θ uθ + + uz ∂z uθ θ + + ur ∂r uθ + r r uθ ∂θ uz + uz ∂z uz ez , + ur ∂r uz + r

ur ∂r ur +

∇2 =

r ∂r +

2 + = ∂rr

θ θ ∂θ + ez ∂z · r ∂r + ∂θ + ez ∂z r r

1 1 2 2 ∂r + 2 ∂θθ + ∂zz r r

u ∇2 u = =

=

2 ∂rr +

1 1 2 2 ∂r + 2 ∂θθ (ur r + uθ θ + uz ez ) + ∂zz r r

ur ∂r ur ∂ 2 ur 2 2 − 2 + θθ2 − 2 ∂θ uθ + ∂zz er + ur r r r r 2 uθ ∂r uθ ∂ 2 uθ 2 2 − 2 + θθ2 + 2 ∂θ ur + ∂zz θ+ uθ + uθ + ∂rr r r r r 2 1 1 2 2 + ∂rr uz + ∂r uz + 2 ∂θθ uz + ∂zz u z ez r r 2 ∂rr ur +

1

∂ 2 ur 2 2 ∂r (rur ) + θθ2 − 2 ∂θ uθ + ∂zz er + ur r r r

∂ 2 uθ 1 2 2 + ∂r uθ ∂r (ruθ ) + θθ2 + 2 ∂θ ur + ∂zz θ+ r r r

1 1 2 2 ∂r (r ∂r uz ) + 2 ∂θθ + uz + ∂zz u z ez . r r

∂r

p˜ := p/ρ+ gh p/ρ gh

g

h ⎛

⎞ r cos θ h(r, θ, z) = ⎝ r sin θ ⎠ · z . z

z

(x, y, z)

g ∇

⎧ uθ u2 ⎪ ⎪ ∂t ur + ur ∂r ur + ∂θ ur − θ + uz ∂z ur = ⎪ ⎪ ⎪ r r ⎪ ⎪ 2 ⎪ 1

∂θθ ur 2 ⎪ 2 ⎪ ⎪ ∂r (rur ) + = −∂r p˜ + ν ∂r − 2 ∂θ uθ + ∂zz ur ⎪ 2 ⎪ r r r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uθ ur uθ ⎪ ⎨ ∂θ uθ + + uz ∂z uθ = ∂t uθ + ur ∂r uθ + r r 1

∂ 2 uθ ⎪ 2 ∂θ p˜ ⎪ 2 ⎪ + ν ∂r ∂r (ruθ ) + θθ2 + 2 ∂θ ur + ∂zz uθ =− ⎪ ⎪ r r r r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uθ ⎪ ⎪ ⎪ ∂t uz + ur ∂r uz + ∂θ uz + uz ∂z uz = ⎪ ⎪ r ⎪ ⎪ ⎪

1 1 2 ⎪ 2 ⎩ ∂r (r ∂r uz ) + 2 ∂θθ = −∂z p˜ + ν , uz + ∂zz uz r r (0, R) × [0, 2π) × (0, L)

(r, θ, z)

∂t u = 0 uz = uz (r, z)

ur = uθ = 0

uz = uz (r) , ur = uθ =

0 p˜ = p˜(z) .

ν d duz d˜ p = r , dz r dr dr z r d˜ p = A ⇒ p˜(z) = Az + B , dz p˜L

p˜0

p˜(z) =

p˜L − p˜0 z + p˜0 . L A

A duz = r+C , dr 2ν C uz (R) = 0 uz (r) =

r 2

R2 (˜ p0 − p˜L ) 1− 4νL R

z=0 1849

⎧ ⎨ x = r sin θ cos φ y = r sin θ sin φ ⎩ z = r cos θ

z=L

p˜0 − p˜L

θ ∈ [0, π]

r=

φ ∈ [0, 2π)

r≥0

sin θ cos φ cos θ cos φ − sin φ sin θ sin φ cos θ sin φ cos φ , θ= , φ= , cos θ − sin θ 0

∂θ r = θ

∂φ r = sin θ φ

∂θ θ = −r

∂φ θ = cos θ φ

∂θ φ = 0

∂φ φ = − sin θ r − cos θ θ

ex ey

ez

r θ

φ

ex = sin θ cos φ r + cos θ cos φ θ − sin φ φ ey = sin θ sin φ r + cos θ sin φ θ + cos φ φ ez = cos θ r − sin θ θ .

r=

x2 + y 2 + z 2 , θ = arg

&

x2 + y 2 , z

∂x r = sin θ cos φ

∂y r = sin θ sin φ

1 cos θ cos φ r 1 sin φ ∂x φ = − r sin θ

1 cos θ sin φ r 1 cos φ ∂y φ = r sin θ

∂x θ =

∂y θ =

'

, φ = arg(y, x) ,

∂z r = cos θ ∂z θ = −

1 sin θ r

∂z φ = 0 .

∇ ∇ = ex (∂x r ∂r + ∂x θ ∂θ + ∂x φ ∂φ ) + ey (∂y r ∂r + ∂y θ ∂θ + ∂y φ ∂φ ) + +ez (∂z r ∂r + ∂z θ ∂θ + ∂z φ ∂φ ) =

(sin θ cos φ r + cos θ cos φ θ − sin φ φ) × 1 1 sin φ ∂φ + × sin θ cos φ ∂r + cos θ cos φ ∂θ − r r sin θ + (sin θ sin φ r + cos θ sin φ θ + cos φ φ) × 1 1 cos φ × sin θ sin φ ∂r + cos θ sin φ ∂θ + ∂φ + r r sin θ 1 +(cos θ r − sin θ θ) cos θ ∂r − sin θ ∂θ r

= r

(sin2 θ cos2 φ + sin2 θ sin2 φ + cos2 θ) ∂r +

+(sin θ cos θ cos2 φ + sin θ cos θ sin2 φ − sin θ cos θ)

∂θ + r

∂φ

+ +(− sin φ cos φ + sin φ cos φ) r 2 +θ (sin θ cos θ cos φ + sin θ cos θ sin2 φ − sin θ cos θ) ∂r + ∂θ + +(cos2 θ cos2 φ + cos2 θ sin2 φ + sin2 θ) r sin φ cos φ sin φ cos φ ∂φ

+ ) + − + tan θ tan θ r +φ (− sin θ sin φ cos φ + sin θ sin φ cos φ) ∂r + ∂θ + +(− cos θ sin φ cos φ + cos θ sin φ cos φ) r 2 sin φ cos2 φ ∂φ

+ + , sin θ sin θ r ∇ = r ∂r +

θ φ ∂θ + ∂φ r r sin θ

θ φ θ φ ∂θ + ∂φ · r ∂r + ∂θ + ∂φ r r sin θ r r sin θ r φ θ · ∂ 1 θ · ∂ θ θ 2 2 ∂r + 2 ∂θθ ∂φ + + + 2 = ∂rr r r r sin θ φ · ∂φ r φ · ∂φ θ 1 2 ∂r + 2 ∂θ + 2 2 ∂φφ + r sin θ r sin θ r sin θ 1 1 2 1 cos θ 1 2 2 ∂θ + 2 2 ∂φφ = ∂rr + ∂r + 2 ∂θθ + ∂r + 2 , r r r r sin θ r sin θ

∇2 =

r ∂r +

∇2 =

1 1 1 2 ∂θ (sin θ∂θ ) + 2 2 ∂φφ ∂r (r2 ∂r ) + 2 r2 r sin θ r sin θ ∇·u=0 θ φ ∂θ + ∂φ · (ur r + uθ θ + uφ φ) r r sin θ uφ ur ∂θ uθ θ · ∂θ r + + θ · ∂θ φ + = ∂r ur + r r r uθ ∂φ uφ ur φ · ∂φ r + φ · ∂φ θ + + r sin θ r sin θ r sin θ

∇·u =

r ∂r +

= ∂r ur +

∂θ uθ ur cos θ ur ∂φ uφ + + + uθ + , r r r r sin θ r sin θ r

θ

1 1 1 ∂θ (sin θuθ ) + ∂φ uφ = 0 ∂r (r2 ur ) + 2 r r sin θ r sin θ

u · ∇u uθ uφ ∂θ + ∂φ (ur r + uθ θ + uφ φ) = ur ∂r + r r sin θ =

u2φ uφ ∂φ ur uθ u2 ∂θ ur − θ + − r+ r r r sin θ r u2φ uθ ur uθ uφ ∂φ uθ + ur ∂r uθ + + ∂θ uθ + − θ+ r r r sin θ r tan θ uθ uφ uφ ∂φ uφ uθ ur uφ ∂θ uφ + + + φ. + ur ∂r uφ + r r r tan θ r sin θ

ur ∂r ur +

∇2 u =

=

∇

1 ∂r r2 ∂r (ur r + uθ θ + uφ φ) + 2 r

1 ∂θ sin θ ∂θ (ur r + uθ θ + uφ φ) + + 2 r sin θ

1 + 2 2 ∂φ ∂φ (ur r + uθ θ + uφ φ) r sin θ 2 ∂φφ ur 2 ∂ 2 ur ∂θ ur ∂r ur + θθ2 + 2 + 2 + r r r tan θ r sin2 θ 2∂φ uφ 2 2uθ 2 − 2 r + − 2 ur − 2 ∂θ uθ − 2 r r r tan θ r sin θ 2 2 ∂φφ uθ ∂ 2 uθ 2∂r uθ ∂θ uθ + ∂rr + θθ2 + 2 + 2 uθ + + r r r tan θ r sin2 θ cos θ uθ 2∂θ ur −2 2 θ + ∂φ uφ + 2 − 2 r r sin θ r sin2 θ 2 ∂φφ 2 uφ ∂ 2 uφ 2∂r uφ ∂θ uφ + θθ2 + 2 + 2 + ∂rr uφ + + r r r tan θ r sin2 θ cos θ uφ 2∂φ ur +2 2 ∂φ uθ φ. + 2 − 2 2 r sin θ r sin θ r sin θ

2 ur + ∂rr

H =:

1 1 1 2 ∂θ (sin θ ∂θ ) + 2 ∂r (r2 ∂r ) + 2 , ∂φφ r2 r sin θ r sin2 θ

⎧ u2θ + u2φ uθ uφ ∂φ ur ⎪ ⎪ ρ ∂ ∂ − = u + u ∂ u + u + ⎪ t r r r r θ r ⎪ ⎪ r r sin θ r ⎪ ⎪ ⎪ 2∂φ uφ 2ur 2∂θ uθ 2uθ ⎪ ⎪ = −∂r p + μ Hur − 2 − − 2 − 2 ⎪ 2 ⎪ r r r tan θ r sin θ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u2φ ur uθ uθ uφ ∂φ uθ ⎪ ⎨ ∂θ uθ + + − = ρ ∂t uθ + ur ∂r uθ + r r sin θ r r tan θ ⎪ 1 cos θ ∂φ uφ 2 uθ ⎪ ⎪ = − ∂θ p + μ Huθ + 2 ∂θ ur − 2 −2 2 2 ⎪ ⎪ r r r sin θ ⎪ r sin θ ⎪ ⎪ ⎪ ⎪ u uθ uφ u u ∂ u ⎪ θ φ φ φ r uφ ⎪ ⎪ ∂θ uφ + + + ρ ∂t uφ + ur ∂r uφ + = ⎪ ⎪ r r sin θ r r tan θ ⎪ ⎪ ⎪ cos θ ∂φ uθ uφ 2 ∂φ ur ⎪ ⎩ = − 1 ∂φ p + μ Huφ − +2 + 2 2 2 r sin θ r2 sin θ r sin θ r sin θ

uφ ≡ 0

ur

uθ

φ

⎧ uθ u2 ⎪ ⎪ ρ ∂t ur + ur ∂r ur + ∂θ ur − θ = ⎪ ⎪ r r ⎪ ⎪ ⎪ 2ur 2∂θ uθ 2uθ ⎪ ⎪ ⎨ = −∂r p˜ + μ Hur − 2 − − 2 2 r r r tan θ ⎪ uθ u u θ r ⎪ ⎪ ρ ∂t uθ + ur ∂r uθ + ∂θ uθ + = ⎪ ⎪ r r ⎪ ⎪ ⎪ ⎪ ⎩ = − 1 ∂ p˜ + μ Hu + 2∂θ ur − uθ , θ θ 2 2 r r r sin θ H =:

1 1 ∂θ (sin θ ∂θ ) . ∂r (r2 ∂r ) + 2 r2 r sin θ p˜ = p + ρgr cos θ

z ψ ur =

∂θ ψ ∂r ψ , uθ = − r2 sin θ r sin θ

θ r

r E 2 = ∂r2 +

1 sin θ ∂θ =: Dr2 + Dθ2 , ∂θ 2 r sin θ

∂θ (∂t ur ) − ∂r (r∂t uθ ) = ∂t [∂θ ur − ∂r (ruθ )] =

1 ∂t E 2 ψ , sin θ

uθ u2 ∂θ ur − θ − ∂r (rur ∂r uθ + uθ ∂θ uθ + ur uθ ) = r r 2 sin θ 2E ψ cos θ ∂r ψ − ∂θ ψ + = 3 2 r r sin θ 1 ∂r ψ ∂θ E 2 ψ − ∂θ ψ ∂r E 2 ψ , − 2 r sin θ ∂θ

∂θ

ur ∂r ur +

∂(ψ, E 2 ψ)/∂(r, θ) ur 2 2 uθ 2 uθ Hur −2 2 − 2 ∂θ uθ − 2 − ∂r rHuθ + ∂θ ur − 2 r r r tan θ r r sin θ T1

T2

2 sin θ E2ψ E2ψ 2(∂θ ψ − r∂rθ ψ)

1 + + ∂ θ 2 2 4 sin θ r sin θ r sin θ tan θ r sin θ 2Dθ2 ψ ∂r ψ ∂r E 2 ψ − − rHuθ = − sin θ r sin θ r2 sin3 θ ∂2 ψ 2 ∂θ ψ T1 = − 4 − r rθ r sin θ sin θ ∂r ψ 2Dθ2 ψ + T2 = , r sin θ r2 sin3 θ

Hur =

E4

E4ψ , sin θ E2 · E2 ψ

∂t E 2 ψ −

r2

∂(ψ, E 2 ψ) 2 E2ψ 1 + 2 2 sin θ ∂(r, θ) r sin θ

cos θ ∂r ψ −

sin θ ∂θ ψ = νE 4 ψ r

ψ

R u∞

z r

θ

∂r ψ ∂θ ψ r− θ sin θ r sin θ ⎛ ⎛ ⎞ ⎞ ⎞ ⎛ sin θ cos φ cos θ cos φ 0 ψ ∂θ ψ ⎝ ∂ r ⎝ cos θ sin φ ⎠ ≈ ⎝ 0 ⎠ , sin θ sin φ ⎠ − = 2 r sin θ r sin θ cos θ − sin θ u∞

u=

r2

∂θ ψ ≈ u∞ r2 sin θ cos θ , ∂r ψ ≈ u∞ r sin2 θ .

ψ≈

1 u∞ r2 sin2 θ . 2 ψ

⎧ 1 2 2 sin θ ⎪ ⎪ E 4 ψ = ∂rr ∂θ + 2 ∂θ ψ=0 ⎪ ⎪ r sin θ ⎪ ⎨ ∂θ ψ = 0 , ∂r ψ = 0 r=R ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ ≈ 1 u∞ r2 sin2 θ 2

ψ(r, θ) = f (r) sin2 θ ,

f (R) = 0 , f (R) = 0 ,

f (r) ≈ E4ψ

1 u∞ r 2 2

r→∞.

ψ f

⎧ 4 IV r f − 4r2 f + 8rf − 8f = 0 ⎪ ⎪ ⎪ ⎨ f (R) = 0 , f (R) = 0 ⎪ ⎪ ⎪ ⎩ f (r) ≈ 1 u∞ r2 r→∞. 2 f (r) = rn r4 n(n − 1)(n − 2)(n − 3)rn−4 − 4r2 n(n − 1)rn−2 + 8r nrn−1 − 8rn = = rn (n − 1)(n − 2)(n − 4)(n + 1) = 0 , f f (r) = A, B, C

A + Br + Cr2 + Dr4 , r

D D=0

C = u∞ /2 f (r) =

r 2

r u∞ 2 R R −3 +2 , 4 r R R ur = ∂θ ψ/(r2 sin θ)

ur 3 R 1 + = 1− u∞ 2 r 2

R 3

cos θ r

uθ = −∂r ψ/(r sin θ) uθ 3 R 1 + = −1+ u∞ 4 r 4

R 3

sin θ r

⎧ 2 2 2 uθ ⎪ ⎨ ∂r p = μ Hur − 2 ur − 2 ∂θ uθ − 2 r r r tan θ ∂ p 2 uθ ⎪ ⎩ θ =μ Huθ + 2 ∂θ ur − 2 r r r sin θ

p = −ρgr cos θ −

3 cos θ μu∞ R 2 2 r

z τrr r

r

τθr

θ τrr = ri τij rj = ri (−pδij + 2μSij ) rj = −p + μ ri (∂i uj + ∂j ui ) rj r=R

τθr τrr = −p + 2μ∂r ur τθr = μ

r∂r

3 μu∞ cos θ 2 r

uθ 1

3 μu∞ + ∂θ ur = − sin θ r r 2 r

F = ez 2πR = ez

= ρgR cos θ +

2

0

π

dθ (τrr cos θ − τθr sin θ) sin θ

4 3 πR ρg + 6πμRu∞ , 3

vlim

R z m¨ z = −m g − az˙ , m = m−V ρ g

m a = 6πRμ z(0) ˙ =0

z(t) ˙ = vlim (−1 + e−at/m ) vlim > 0 z(t) < 0 0 vlim < 0 z(t) > 0

t>0

vlim = m g/a . m > m <0

4 180

230 2 410 411 6.6

6 6.4

6.14

9.5 9.12 2D

9.4

B 6

1

48 49 17

4 18 20

58 60 63 71

68 72

69 330

332 333 338

158 159

4.11 261 262 6.4 6.6

518 519 D

5.2 6.1

6.2 6.7 10.3 211 214 226 6.3.1 6.3.2 9 9.3.2 5

9.3.3 5.1

5.2 5.1

474 478 42 43 43 45 125 130 134 174 175

223 226

226

1

γ

y

2 μ∂yy u

u →∞

2

3

60

d d → +∞ uv∞

=

uv1 e1

m m um ∞ = u 1 e1 + u 2 e2 v + u 2 e2

d

um ∞

Fb Am

F

+

F−

R

uv∞

d Av

R+ R−

d R

R+ d Av

2 Am um ∞

A

v

R−

R 2 Am

uv∞ v um 1 = u1 =: u1 ,

R 1

m v v ρ (−um 1 d · u1 + u1 d · u1 ) = 0

2

m v v v m ρ (−um 1 d · u2 + u1 d · u2 ) = ρu1 · (u2 − u2 )d ,

ρ Am

R± (uv2 − um 2 )d

Av Γ R

F b = F1 e1 + F2 e2 F

R+

R±

+

R

−F + R−

R

F − = −F + R

Am

Av 1

pm d − pv d − F1

2

−F2 ,

ν Am

∂R

ν = −e1

Av

(pm − pv ) d =

ν = +e1

1

1 v 2 um + uv2 2 ρ |u∞ | − |um Γ . d=ρ 2 ∞| 2 2

F1 = ρ

v um 2 + u2 Γ , F2 = −ρ u1 Γ . 2

1 um 1

D 2

L

D = ρΓ u1

2 v 2 v m m |um uv2 − um 2 ∞ | + |u∞ | − (u2 − u2 )u2 , L = −ρΓ . m m 2|u∞ | 2|u∞ |

d → +∞ Γ uv2 → um 2 D = 0 , L = −ρ|u∞ |Γ . L > 0 Γ

2

66

Ωb A

B

u∞

∂Tf

A B A

∂Ωb

Tf u∞

u ≡ u∞

T M

B

u∞

∂Ωb

T

Tf u∞ M

∂Ωb

B

A

B

M

T

Ωb A u∞

0=

d dt

dV ρu T

= T

dV ρ∂t u

=− =−

dV T

ρ∂i (ui u) + ∇p

+

A

+ B

+ M

dA

−∂Ωb

−∂Ωb A

ρuu · ν + pν

T u·ν

B

Fb = −

+

A

Fb

dA pν .

M

p + ρ|u|2 /2 A

B

T ν F b · u∞ = 0

u∞

M

u∞

∂Ωb

Ωb 2 x=0

,

M

+ B

Ωb Γb ϕ

∇2 ϕ = 0

∇·u = 0 u → u∞

uν = 0 x→∞

∂Ωb

⎧ 2 ⎪ ⎨∇ ϕ = 0 ∂ν ϕ ≡ 0 ⎪ ⎩ ∇ϕ → u∞

∂Ωb Γb ∇2 ψ =

ψ −ω

Γ = Γb +

dx ω Ωb

u(x) ∼ u∞ + Γ K(x) ψ≡

=: ψb

⎧ 2 ⎪ ⎨ ∇ ψ = −ω ψ ≡ ψb ⎪ ⎩ ∇ψ → u⊥ ∞

∂Ωb u

u u = u∞ + u 2 ∂BR (x0 )

BR (x0 ) ϕ∞ (x) = u∞ · x

uτ ϕ

∂Ωb

u∞ Γ = 0 u BR (x0 ) ⊃ Ωb

Γb = 0

ds u · ν . uν ∼ 1/R2

ϕ = ϕ∞ +ϕ

∇ϕ

⎧ 2 ⎪ ⎨∇ ϕ = 0 ∂ν ϕ = −u∞ · ν ⎪ ⎩ ∇ϕ → 0

R → +∞

2

∂Ωb

Γb = 0 u

1

ψ = ψ∞ + ψ

ψ∞ (x) = u⊥ ∞ · x − Γ G(x)

G(x) = (log |x|)/(2π) ψ

1 ⎧ 2 ⎪ ⎨ ∇ ψ = −ω ψ = ψb − ψ∞ ⎪ ⎩ ∇ψ → 0

∂Ωb

ψb Ωb ∇2 G = 0

∂Ωb Ωb

x=0 2 Ωb

∇ψ ϕ

ψ

ω

∂Ωb f = ϕ

ψ

g=G

ϕ (x) = −

BR (0)

ds

ϕ

R

ψ

ϕ (x )∂ν G(x − x ) − G(x − x )∂ν ϕ (x )

∂Ωb

ψ (x) = −G ∗ ω(x) +

− ds ψ (x )∂ν G(x − x ) − G(x − x )∂ν ψ (x ) , ∂Ωb

ds = ds(x ) Ωb ϕ ψ Φ

Ψ

ν G∗ω

∂ν ϕ (x )

ψ (x ) ∂Ωb

x

∂Ωb

Ψ

4

G(x − x0 )

x0 ∈ Ωb

−G ∗ ω

ϕ (x) = − ψ (x) =

ds ϕ (x )∂ν G(x − x ) + Φ (x)

∂Ωb

ds G(x − x )∂ν ψ (x ) + Ψ (x) .

∂Ωb

ds μ(x )∂ν G(x − x )

∂Ωb

μ

ds χ(x )G(x − x )

∂Ωb

χ 3 2 ∂Ωb

3 x ∈ ∂Ωb

ϕ 1 ϕ (x) + 2

ds ϕ (x ) ∂ν G(x − x ) = Φ (x) .

∂Ωb

ν = ν(x)

∂Ωb

x

ν · ∇x ψ (x ) = ν · ∇x

ds G(x − x )∂ν ψ (x ) + ν · ∇x Ψ (x )

∂Ωb

x → x ψ 5

x 3 O(s ) τ

x ∈ ∂Ωb ∩ Bε (x ) K

∂ν G(x − x ) = −

ε s

ν · (x − x ) K = + O(s ) , 2 2π |x − x | 2

∂ν G(x − x ) ∂ν G(x − x )

x = x

2

x = x +τ s +K ν s /2+ x ∈ ∂Ωb

1 ∂ν ψ (x) − 2

ds ∂ν G(x − x )∂ν ψ (x ) = ∂ν Ψ (x) .

∂Ωb

∂ν ψ = −uτ

u

∂Ωb

ϕ

∂ν ψ

∂Ωb

ϕ ψ

ψb − ψ∞

∂ν ψ

−uτ − ∂ν ψ∞

ψ ψ

uτ

∂Ωb

ψ (x) = −G ∗ ω(x) − ds uτ (x )G(x − x )+ ∂Ωb

+ ds ψ∞ (x )∂ν G(x − x ) − G(x − x )∂ν ψ∞ (x ) .

∂Ωb

Ωb Γ G(x)

ψ Ωb ψ(x) =

u⊥ ∞

· x − G ∗ ω(x) − u(x) = u∞ + K ∗ ω(x) +

u

ds uτ (x )G(x − x ) ∂Ωb

ds uτ (x )K(x − x ) .

∂Ωb

Ωb

∂Ωb

uτ

w w w(x)+w(x) x=x=x x0

(x) > 0

(x0 ) > 0

⎧ Γ ⎪ [log(x − x0 ) − log(x − x0 )] ⎪ ⎪ ⎪ 2π ⎪ ⎨ q [log(x − x0 ) + log(x − x0 )] w(x) = ⎪ 2π ⎪ ⎪ ⎪ d ⎪ Q d ⎩ − + 2π x − x0 x − x0

d

d

d

x

6

w(x)

w(x) w(x)

w(x) x

x1 (0)

q x2 (0) = −x1 (0) δ(t) = |x1 (t)|2 δ(t) = δ0 + αt

δ0 = δ(0) α = 3q/(2π) x1 > 0 y1 > 0 y1 (0) = y1 0

x1 (0) = x10 x1 x1 (t) =

x2 (0)

{(x1 20 − y1 20 ) [δ(t)/δ0 ]1/3 + δ(t)}/2 , −

y1 (t)

w w r

xc

w(x) = w (x) + w

r2 . x − xc x = xc + re

θ ∈ [0, 2π)

θ

θ u∞ r

w (x) = u∞ x . . .

1

x = x0 |x0 | >

⎧

Γ 1 ⎪ ⎪ log(x − x0 ) − log x − + log x ⎪ ⎪ 2π x0 ⎪ ⎪ ⎨ q

1 − log x log(x − x0 ) + log x − w(x) = ⎪ 2π x 0 ⎪ ⎪ ⎪ d/x20 d Q ⎪ ⎪ ⎩ − + 2π x − x0 x − 1/x0 x

x0 1/x0 x0

1/x0

z ζ z = z(ζ) , z(ζ) z(ζ) =1. lim ζ →∞ ζ ζ

ζ

z ζ

ζ 1,2

σ1,2

3

2

2

1

1

0

0

y

η

3

-1

-1

-2

-2

-3

-3 -3

-2

-1

0

1

2

3

-3

-2

ξ a

-1

0

1

2

3

x b α = 60◦ z =x+ y b

ζ =ξ+ η a

=

dζ 1 dζ 2 , dσ1 dσ2

dζ 1 /dσ1

dζ 2 /dσ2

e

sin αc z 1,2

αc

cos αc z = z(ζ)

σ1,2 i=1 2 dζ i dz i = z (ζ i ) dσi dσi

z = dz/dζ ζ

e

αf

=

m

z

dz i /dσi dz/dζ

dζ 1 dζ 2 1 dz 1 dz 2 1 dζ dζ 2 = 2 z z = 1 =e 2 m dσ1 dσ2 m dσ1 dσ2 dσ1 dσ2

αc

.

z = z(ζ) m wc (ζ)

w f (z) wf (z) = wc [ζ(z)]

uf uf [z(ζ)] =

dζ dwc uc (ζ) dwf [z(ζ)] = [z(ζ)] (ζ) = . dz dz dζ z (ζ)

4

2

2

0

0

y

y

4

-2

-2

-4

-4 -4

-2

0

2

4

-4

-2

x a

0

2

4

x b a

b α = 60◦

z(ζ) = ζ +

1 ζ 4

ζ = e θ z = e θ + e−

θ

= 2 cos θ

−2 2 α

ζ → ζ α ζ = ζ e

α

ζ → z

α

z = ζ +

1 ζ

z → z α

α z = z e−

α

.

4

2

2

0

0

y

y

4

-2

-2

-4

-4 -4

-2

0

2

4

-4

-2

x a u∞ = 1 α = 60◦ z 1 = −0.2 + 1.7 z 2 = 1.8 − 1.7 Γb = 0 a Γ1 +22.23233 Γ2 +24.67786

χ2 ζ

α

uc (ζ) = u∞

uf (ζ) = u∞

ζ = ±χ z = ∓(1 + cos 2α − sin 2α)

∂Ωb

4

Γ1 −6.84080 Γ2 +8.91820

a

z=ζ+

α = 60◦

2

Γb = 4 b

b

χ = e−

0

x b

1−

1 ζ2

ζ2 − 1 . ζ 2 − χ2 ζ = ±1

z 1,2 = z(ζ 1,2 ) z Γ1,2 uc (+χ) = 0 uc (−χ) = 0

4

uc "

uf

uc

a11 Γ1 + a12 Γ2 = b1 a21 Γ1 + a22 Γ2 = b2 .

α = 60◦ b

Γb

a

w

Ωb Ωb

Fb =

ds p (−ν) = ρ ∂Ωb

ds ν ∂t ϕ + ρ ∂Ωb F ns

ds ν ∂Ωb Fs

|u|2 . 2

ν = − τ 7

a11 = 1/(χ − ζ 1 ) − 1/(χ − 1/ζ 1 ) + 1/χ a12 = 1/(χ − ζ 2 ) − 1/(χ − 1/ζ 2 ) + 1/χ a21 = 1/(χ + ζ 1 ) − 1/(χ + 1/ζ 1 ) + 1/χ a22 = 1/(χ + ζ 2 ) − 1/(χ + 1/ζ 2 ) + 1/χ b1 = −2π (1 − 1/χ 2 ) − Γb /χ

b2 = +2π (1 − 1/χ 2 ) − Γb /χ

∂t ϕ = ∂t w +

F ns = ρ

ds ν ∂t ϕ = − ρ

dz ∂t w ,

∂Ωb

∂Ωb

dz = τ ds |u|2 dz = ∂z w ∂z w dz = ∂z w dw = ∂z w dw = (∂z w)2 dz , dw = dw Fs = − ρ dz (∂z w)2 . 2 ∂Ωb

dw

Fb = − ρ

dz ∂t w − ∂Ωb

2

ρ

dz (∂z w)2

∂Ωb

z0

|z 0 | > 1

q w(z, t) =

q 1 log[z − z 0 (t)] + log z − − log z . 2π z 0 (t) t

z

z˙ 0 z˙ 0 /z 20

q + − 2π z − z0 z − 1/z 0 1 1 q 1 + − ∂z w(z) = 2π z − z 0 z − 1/z 0 z ∂t w(z) =

∂t w

F ns = ρq

z0 z

z˙ 0 . z 20

z˙ 0

Fs =

z0 ρq 2 . 2π |z 0 |2 (|z 0 |2 − 1)

q

α Γb

0

uτ τ ∂Ωb

M (t) M (R, t)

Ωb R

BR (0)

M (t) :=

dx xω(x, t) , M (R, t) := Ωb

dx xω(x, t) ,

Ωb ∩BR (0)

Ωb ⊂ BR (0) ω

R

M (R, t) ∂t M (R, t) =

x∇ · (ωu)

dx x∂t ω =

Ωb ∩BR (0)

dx x[−∇ · (ωu) + ν∇2 ω] ,

Ωb ∩BR (0)

x∇2 ω ωu = (−∂k ∂k ψ)(−ei ∂i⊥ ψ) = ei [∂k (∂k ψ∂i⊥ ψ) − ∂k ψ∂i⊥ (∂k ψ)] = ei [∂k (∂k ψ∂i⊥ ψ) − ⊥ = −∂k (u⊥ k u) − ∇

1 ⊥ ∂ (∂k ψ∂k ψ)] 2 i

|u|2 , 2

x∇ · (ωu) = ei xi ∂k (ωuk ) = ei [∂k (xi ωuk ) − ωuk ∂k xi ] ⊥ = ∂k (xωuk ) + ∂k (u⊥ k u) + ∇

x∇2 ω = ei xi ∂k ∂k ω = ei [∂k (xi ∂k ω) − ∂k ω∂k xi ] = ∂k (x∂k ω) − ∇ω .

|u|2 . 2

Ωb ∩ BR (0) ∂BR (0) ∪ (−∂Ωb )

1 ∂t M (R, t) = − ds −xωuν +νx∂ν ω −νων +uuτ − |u|2 τ . 2 ∂BR (0) ∂Ωb

ω × u = ωu⊥ ∇2 u =

2 2 ∂11 u1 + ∂22 u1 2 2 ∂11 u2 + ∂22 u2

∂t u + ∇

∇·u = 0

=

−∂2 (∂1 u2 − ∂2 u1 ) ∂1 (∂1 u2 − ∂2 u1 )

= ∇⊥ ω .

|u|2 + ωu⊥ = −∇˜ p + ν∇⊥ ω , 2

p˜ = p/ρ τ ∂t uτ + ∂s

|u|2 + ωuν = −∂s p˜ + ν∂ν ω , 2

s

∂Ωb x(s)

∂Ωb

−xωuν + νx∂ν ω = ∂t (xuτ ) + ∂s

1 1 x|u|2 − |u|2 τ + ∂s (x˜ p) − p˜τ , 2 2

s

∂Ωb

uν ≡ 0 ∂t M 1 (R, t) = −

1 ds − xωuν + νx∂ν ω − νων + uuτ − |u|2 τ + 2 ∂BR (0) ds [∂t (xuτ ) − |u|2 τ + uuτ − p˜τ − νων] .

∂Ωb

ω uuτ −

R → +∞

τ τ |u|2 = (u∞ + u )(u∞ · τ + u · τ ) − (|u∞ |2 + 2u∞ · u + |u |2 ) 2 2 1 Γ u∞ |u∞ |2 = (u∞ · τ )u∞ − τ+ +O , 2 2πR R2

Γ ∂BR (0)

∂t M (R, t) +

uuτ −

ds xuτ ∂Ωb

=

ds

1 τ |u|2 = Γ u∞ + O 2 R

=

ds (˜ pτ + νων) + ∂Ωb

ds (|u|2 τ − uuτ ) +Γ u∞ + O ∂Ωb

1 . R

≡0

Mc = M +

ds xuτ , ∂Ωb

u = uτ τ ρ

u ≡ 0

∂Ωb

R → +∞ ˙ ⊥= ρM c

ds (−pν + μωτ ) + ρΓ u⊥ ∞ .

∂Ωb

∂Ωb ei τij νj = −pν + 2μ∂s u⊥ + μωτ , u=0

μ=0 Fb ⊥ ˙ ⊥ F b = ρM c − ρΓ u∞

∂Ωb

u∞ −Γb

u∞

Γb

u∞ Γ ≡ 0

Γb

−Γb u∞

0 /|u∞ |

0−

Γ

⊥

0+

˙ F b = ρM c

Γb < 0 −Γb u∞

Fb =

−Γb u∞

ρ(−Γb )u⊥ ∞

˙ ⊥ M

u∞

u∞

y ˙ /2 M x ˙ /2 M

y=0

y y

I(t) = Ωb

dx |x|2 ω(x, t) .

R ∂t I(R, t) =

Ωb ∩BR (

=

Ωb ∩BR (

0) 0)

dx |x|2 ∂t ω dx |x|2 [−∇ · (ωu) + ν∇2 ω] ,

Ωb ∩ BR (0)

∇ · (|x|2 uω) − 2x · uω

ωx · u

|x|2 ∇ · (ωu) ≡

ωx · u = (∇⊥ · u) x · u = ∇⊥ · [u(x · u)] − u · ∇⊥ (x · u) x = ∇⊥ · [u(x · u)] − · ∇⊥ |u|2 2

x ⊥ . = ∇ · u(x · u) − |u|2 2 |x|2 ∇ · (uω) = ∇ · (|x|2 uω) − ∇⊥ · [2u(x · u) − x|u|2 ] ,

|x|2 ∇2 ω ≡ 4ω + ∇2 (|x|2 ω) − 4∇ · (xω) . BR (0) ∂t I(R, t) = − = ∂BR (0)

ds

∂Ωb

− |x|2 ωuν + 2(x · u)uτ − |u|2 xτ +

1 . +ν ∂ν (|x|2 ω) − 4ωxν + 4νΓ + O R ∂BR (0)

R → +∞

xτ = x·τ ≡

0 2

u = u∞ +

uν u∞ · ν +

∂BR (0)

Γ 2π

ds x · uuτ .

1 1 + x 2π

1 Mc +O 2 x R3 ∂BR (0)

ν · Mc τ · Mc Γ + , uτ u∞ · τ + , 2 2πR 2πR 2πR2

1/R3 ∂BR (0) x · u uτ = Ruν uτ 1 Γ u∞ · M c u∞ · ν + +O , = R u∞ · ν u∞ · τ + 2πR 2πR2 R3 ∂BR (0) 2 u∞ · M c

R → +∞ ∂t I ∂Ωb

∂Ωb

ds (−xτ u2τ + 2νωxν − ν|x|2 ∂ν ω) . uν ≡ 0

∂Ωb

ν|x|2 ∂ν ω

−ν|x|2 ∂ν ω = −∂t (|x|2 uτ ) − ∂s

u2

u2 |x|2 p˜ + τ + 2xτ p˜ + τ , 2 2

2

d dt

ds (xτ p˜ + νxν ω) − ∂Ωb

ds |x|2 uτ .

∂Ωb

2/ρ M

M = e3 ·

ds x × (−pν + νωτ ) =

ds x · (pτ + νων) .

∂Ωb

∂Ωb

Ic Ic :=

2

Ωb

dx |x| ω +

ds |x|2 uτ

∂Ωb

2u∞ · M c

M= u∞ = 0 4νΓ

1 ˙ ρ I − 2μΓ − ρu∞ · M c 2 M=0 I

u∞ = 0 u∞

p˜ → p˜∞

∇

x→∞

p˜∞

p˜ − p˜∞ + (|u|2 − |u∞ |2 )/2

= −∂t u − ωu⊥ + ν∇⊥ ω ,

P

∇2 P = −∇ · (ωu⊥ ) Ωb

∇·u = 0 f =P P

Ωb ∩ BR (0)

g=G x

P =−

+ +

−

∂Ωb

∂BR (0)

BR (0)∩Ωb

∂BR (0)

∂BR (0)

−

ds P τ · K + ds G (−∂t uν + ν∂s ω) +

∂Ωb

dx ω u · K , ∂ν P

R → +∞

P

2

ds (P τ · K − ν G ∂s ω) +

∂Ωb

BR (0)∩Ωb

dx ωu · K .

8

x G

ds = ds(x ) K

x − x

Ωb

−F b R

−F b

+F b

Ωb

x p∞

R ∂BR (0) x

(−F b )/(2πR)

G(x − x )∂s ω(x ) ≡ ∂s [G(x − x )ω(x )] − ω(x )ν(x ) · K(x − x ) , p˜

∂Ωb 2

2

p˜ + |u| /2 p˜∞ + |u∞ | /2 + +

1 2

∂Ωb

ds (pτ − νων) · K +

∂Ωb

ds u2τ τ · K +

BR (0

)∩Ωb

dx ωu · K ,

R → +∞ p˜ + |u|2 /2

1/|x| x ∈ ∂Ωb

K(x − x ) = K(x) + O(1/|x|2 ) K(x) p˜ + |u|2 /2 p˜∞ + |u∞ |2 /2 + K ⊥ (x) · ds (−pν + νωτ ) + ∂Ωb 1 2 + K(x) · ds |u| τ + 2 ∂Ωb + dx ωu · [K(x − x ) − K(x)] + BR (0)∩Ωb

+K(x) ·

BR (0)∩Ωb

dx ωu + O

1 . |x|2 1/ρ

K(x) ·

∂Ωb

ds (u2τ τ − uuτ ) + Γ u∞

Fb

, u = uτ τ

u≡0

1/|x|2

R

∂Ωb

+∞

p + ρ|u|2 /2 = p∞ + ρ|u∞ |2 /2 + K ⊥ (x) · [F b + ρΓ u⊥ ∞] + O

1 . |x|2

u(x) = u∞ + Γ K(x) + O(1/|x|2 )

p(x) = p∞ +

1 x −F b · +O |x| 2π|x| |x|2

x x/|x| −F b

|x|

|u|2

0 ∞

|u(x)| ∼ |Γ |/(2π|x|) ψω/2

Γ

u∞ = x→

BR (0) ⊃ Ωb R

Ωb

0

ψ ≡ ψb

ω BR (0) ψ

ω

∂Ωb ∂Ωb

E(R, t) =

1 2

BR (0)

1 =− 2

dx ∇ψ · ∇ψ

1 ds ψ uτ + 2 ∂BR (0)

BR (0)

dx ψ ω , ψ

∂BR (0) O(1/|x|)

u(x) = u∞ + Γ K(x) + O(1/|x|2 ) ψ(x) = −u∞ · x⊥ − Γ G(x) +

x→∞ 1 1 π Γ2 log R + O − ds ψ uτ = R2 |u∞ |2 + , 2 ∂BR (0) 2 4π R BR (0)

E(R, t) =

1 π 2 Γ2 R |u∞ |2 + log R + 2 4π 2

BR (0)

dx ψ ω + O

1 . R R

u∞ = 0

Γ = 0

1 Ef = 2

dx ψω . IR2

ψ

ψb

u∞

BR (0)

∂t Ef (R, t) = ∂t E(R, t) =

BR (0)∩Ωb

dx u · ∂t u ,

u · ∂t u = −∇ · [(˜ p + |u|2 /2) u] + ν [∇⊥ · (ωu) − ω 2 ] . uν ≡ 0

∂Ωb

dx u · ∂t u = −

BR (0

)∩Ωb

ds [˜ p − p˜∞ + (|u|2 − |u∞ |2 )/2] uν − ν

∂BR (0)

dx ω 2 .

BR (0)∩Ωb

R → +∞ ρE˙f = −μ

dx ω 2 +

Ωb

1 u∞ · F b 2

ω2

6.4

404 406

6.5 422 424 427 444 449 6 11 65 13 51 53 ∂z w 166 167 168 13 8

15

62

VI 72.b

370.b 9.4.1 12.6

140 144 20 21 134 136 136 137 137 138

138 140 140 141 147 149

141 143 3.12.3

3.12.3 3.12.5 3.3 3.31

3.4 3.43

3.44 3.51 3.53 3.6 3.7 3.71 5 5.2 5.31 5.33 5.4

5.5 6.1 6.8 7.1

S

2D I(x )

x ∈ S

S ∩I

S

3D

3D

S

2D 2 ν(x )

x

S 3D 2D

x

τ 1,2 (x ) τ (x )

S S

ν

S+

S

I(x )

I

x ∈ S + x → x ∈ S −

S ∩I I+

I− x ∈S

S−

x ∈ S

x x ∈ I+

x→

μ x → x ∈ S ±

dσ(x ) μ(x ) ∂ν G(x − x ) ,

S

x

dσ(x ) ν(x ) · ∇x

∂ν

μ x x ∈ S

S

μ(x )

μ≡1 x ε>0 x

S ∩ Bε (x ) Bε (x )

ε

lim x → x ∈ S ± ≡

lim x → x ∈ S ±

dσ(x ) ∂ν G(x − x ) ≡

S

S∩

Bε (x )

+

S ∩ Bε (x )

dσ(x ) ∂ν G(x − x )

ε → 0+

lim x → x ∈ S ± ≡

lim

dσ(x ) ∂ν G(x − x ) ≡

S

lim x → x ∈ S ±

ε→0+

+ lim

ε→0+

S ∩ Bε (x )

dσ(x ) ∂ν G(x − x ) +

dσ(x ) ∂ν G(x − x ) . lim x → x ∈ S ± S ∩ Bε (x ) x ∈ S ∩ Bε (x ) x

x

ε

S ∩ Bε (x )

dσ(x ) ∂ν G(x − x )

x

lim

ε→0+

lim x → x ∈ S ±

= lim

ε→0+

S ∩ Bε (x )

S∩

Bε (x )

dσ(x ) ∂ν G(x − x ) =

dσ(x ) ∂ν G(x − x ) .

∂ν G(x − x )

2D

x = x σ

1 2 k ν σ + O(σ 3 ) 2 ν = ν − k τ σ + O(σ 2 )

x = x + τ σ +

k

x

∂ν G(x − x )

∂ν G(x − x ) = ∂ν G(x − x ) S lim

ε→0+

=

S∩

1 [k + O(σ)] . 4π 3D

∂ν G(x − x )

x ε

lim x → x ∈ S ± Bε (x )

S∩

Bε (x )

dσ(x ) ∂ν G(x − x ) =

dσ(x ) ∂ν G(x − x ) ,

lim

lim x → x ∈ S ±

ε→0+

+

∂ Bε (x )

S ∩ Bε (x )

∂Bε (x )

S x → x ∈ S ± [S ∩ Bε (x )] ∪ ∂ ∓ Bε (x ) G

−

∂ Bε (x ) x

[S ∩ Bε (x )] ∪ ∂ ∓ Bε (x ) ∓

S ∩ Bε (x )

dσ(x ) ∂ν G(x − x ) =

Bε∓ (x )

dx ∇2x G(x − x ) ≡ 0 . S ∩ Bε (x )

∂ Bε (x )

dσ(x ) ∂ν G(x − x ) .

dσ(x ) ∂ν G(x − x ) =

∂ ∓ Bε (x )

dσ(x ) ∂ν G(x − x ) ,

∂ ∓ Bε (x ) S x

∂ ∓ Bε (x ) x

x

lim

ε→0+

lim x → x ∈ S ±

= lim

ε→0+

∓

∓

∂ Bε (x )

∂ Bε (x )

dσ(x ) ∂ν G(x − x ) =

dσ(x ) ∂ν G(x − x ) ,

S+

S ∩ Bε (x )

S−

x x ∈ ∂ + Bε

∂ν G(x − x ) = ν(x ) · ∇x G(x − x ) =

⎧ 1 ⎪ ⎨−

2πε ⎪ ⎩− 1 4πε2

x ∈ ∂ − Bε

∂ν G(x − x ) = S ∓

|∂ Bε (x )| =

⎧ 1 ⎪ ⎨+

2πε ⎪ ⎩+ 1 4πε2

2D 3D

x

"

πε + O(ε2 ) 2πε2 + O(ε3 )

2D 3D

2D 3D ,

x → x ∈ S −

−1/2

lim x → x ∈ S ±

S

+1/2

dσ(x ) ∂ν G(x − x ) ,

S

S

μ = 1

S

1 + 2

dσ(x ) ∂ν G(x − x ) = ∓

x → x ∈ S +

μ

μ(x ) ≡ μ(x ) + μ(x ) − μ(x) + [μ(x) − μ(x )]

lim x → x ∈ S ±

dσ(x ) μ(x ) ∂ν G(x − x ) =

S

= lim

ε→0+

lim x → x ∈ S ±

S ∩ Bε (x )

dσ(x ) μ(x ) ∂ν G(x − x ) +

L1

+ μ(x ) lim

ε→0+

lim x → x ∈ S ±

S ∩ Bε (x )

dσ(x ) ∂ν G(x − x ) +

L2

+ lim

ε→0+

lim x → x ∈ S ±

S ∩ Bε (x )

dσ(x ) μ(x ) − μ(x) ∂ν G(x − x ) +

L3

+ lim

ε→0+

lim [μ(x) − μ(x )] x → x ∈ S ±

S ∩ Bε (x )

dσ(x ) ∂ν G(x − x )

L4

3D

L1

L1 =

dσ(x ) μ(x ) ∂ν G(x − x ) .

S

L2 = ∓

μ(x ) , 2 S

μ 3D

S ∩ Bε (x )

2D ε → 0+

L4 = 0 S

μ

1

μ

M (x , x )

S

μ(x ) = μ(x ) + M (x , x ) · (x − x ) . x − x ≡ (x − x) + (x − x )

lim dσ(x ) μ(x ) ∂ν G(x − x ) = x → x ∈ S ± S μ(x ) + dσ(x ) μ(x ) ∂ν G(x − x ) =∓ 2 S S

χ x → x ∈ S ±

ν(x )·∇x

dσ(x ) χ(x ) G(x−x ) = S

dσ(x ) χ(x ) ν(x )·∇x G(x−x ) ,

S

∫

x

χ x

x ∈ S

−χ(x )

lim x → x ∈ S ±

S χ≡1

dσ(x ) ν(x ) · ∇x G(x − x ) ≡

S

dσ(x ) ν(x ) · ∇ x G(x − x ) +

≡ lim

lim x → x ∈ S ±

+ lim

dσ(x ) ν(x ) · ∇ x G(x − x ) . lim ± x → x ∈ S S ∩ Bε (x )

ε→0+

ε→0+

S ∩ Bε (x )

ε

S∩ x lim

x ∈ S ∩ Bε (x ) x

x Bε (x )

lim x → x ∈ S ±

ε→0+

= lim

ε→0+

S∩

lim

ε→0+

x =x

dσ(x ) ν(x ) · ∇x G(x − x )

S ∩ Bε (x )

dσ(x ) ν(x ) · ∇ x G(x − x ) =

dσ(x ) ∂ν G(x − x ) .

Bε (x )

∂ν G(x − x )

ε

dσ(x ) ∂ν G(x − x ) = dσ(x ) ∂ν G(x − x ) , S ∩ Bε (x ) S ∩ Bε (x )

lim

ε→0+

lim x → x ∈ S ±

S ∩ Bε (x )

dσ(x ) ν(x ) · ∇x G(x − x ) .

ν(x ) ≡ ν(x ) + [ν(x ) − ν(x )] ∇x G(x − x ) ≡ −∇ x G(x − x )

−

dσ(x ) ∂ν G(x − x ) + dσ(x ) ν(x ) − ν(x ) · ∇ x G(x − x ) . S ∩ Bε (x ) S ∩ Bε (x )

− lim

ε→0+

lim x → x ∈ S ± x→x ∈S

S ∩ Bε (x )

+

x=x

|ν(x ) − ν(x )| ≤ N |x − x | 2D d = 2

dσ(x ) ν(x ) − ν(x ) · ∇ x G(x − x ) ≤

S ∩ Bε (x )

1 , 2

x → x ∈ S −

N > 0 d=3

dσ(x ) ∂ν G(x − x ) = ±

|x − x | N . dσ(x ) ≤ 2(d − 1)π S ∩ Bε (x ) |x − x|d−1

3D

ε → 0+

χ = 1

lim

lim x → x ∈ S ±

= lim

lim x → x ∈ S ±

+ lim

lim x → x ∈ S ±

ε→0+

S

dσ(x ) χ(x ) ν(x ) · ∇ x G(x − x ) =

ε→0+

S ∩ Bε (x )

ε→0+

S ∩ Bε (x )

dσ(x ) χ(x ) ν(x ) · ∇ x G(x − x ) +

dσ(x ) χ(x ) ν(x ) · ∇ x G(x − x ) . Bε (x )

lim

ε→0+

lim x → x ∈ S ±

S ∩ Bε (x )

dσ(x ) χ(x ) ν(x ) · ∇x G(x − x ) ≡

≡ χ(x ) lim

ε→0+

lim x → x ∈ S ±

S ∩ Bε (x )

+ lim

ε→0+

lim x → x ∈ S ±

dσ(x ) ν(x ) · ∇ x G(x − x ) +

S ∩ Bε (x )

dσ(x ) χ(x ) − χ(x ) ν(x ) · ∇x G(x − x ) , χ

lim dσ(x ) χ(x )ν(x ) · ∇x G(x − x ) = x → x ∈ S ± S χ(x ) + dσ(x ) χ(x ) ∂ν G(x − x ) =± 2 S

ζ

w wc w c (ζ) = w(ζ) + w

1 ζ

,

(Γv , ζ v ) w(ζ) = Γv /(2π ) log(ζ − ζ v ) w c (ζ) =

Γv 2π

log(ζ − ζ v ) − log

1 − ζv ζ

=

Γv [log(ζ − ζ v ) + log ζ − log(1 − ζ v ζ)] 2π

=

Γv 2π

Γv = 2π

log(ζ − ζ v ) + log ζ − log

log(ζ − ζ v ) + log ζ − log

1 −ζ ζv 1 ζ− ζv

− log ζ v

− π − log ζ v

,

ζ w c (ζ) =

Γv log(ζ − ζ v ) 2π

−

Γv log 2π

ζ−

1 ζv

+

Γv log ζ 2π

1/ζ v

ζv

{(Γk , ζ k )}k=1,...,n

n ˜ w(ζ) ˜ w(ζ) = −u∞

ζ+

1 1 + ζ 2π

n # k=1

Γk

log(ζ −ζ k )−log

ζ−

1 Γb0 + log ζ , 2π ζk

0

Γb0

−u∞ (ζ + 1/ζ)

1 2π

−u∞ /ζ

n #

Γk

log(ζ − ζ k ) − log

ζ−

k=1

1

ζk

Γk log(ζ − ζ k )

−Γk log(ζ −1/ζ k )

−u∞ ζ

k Γb0 /(2π ) log ζ

Γb0

z(ζ ) = ζ +

p2 ζ

ζ c p

ζ c = ξc + ηc

ξc |ζ c | 1

−p

|ζ c | < 1

ξc = 0

p p= ζc

1 − ηc2 − ξc . ξc = 0

3 2

2

ζ c

−p

0

1

+p

y

η

1

-1

0 -1

u∞

u∞

-2

-2

-3 -2

-1

0

1

2

3

-3

-2

-1

0

ξ

1

2

3

x ζ

z ζ c =

0.1+ 0.3

ζ = −p

u∞

ηc = 0

ξc ηc ηc b z = −2p

ζ (θ) = ζ c + e

(θ+θ )

∂ζ z(ζ )

ζ (0) = ζ c + e

,

ξc a

ζ = −p

θ

= −p . θ

dz p + ζ c τ (θ) = dθ = − e dz |p + ζ c | dθ

θ

ζ (θ) − p ζ (θ) ζ (θ) + p , |ζ (θ) − p| |ζ (θ)| |ζ (θ) + p| θ =0

τ (θ)

θ = 0

θ >0

τ (θ)

τ (0± ) = ±

p + ζ c |p + ζ c |

2 , τ (0− ) + τ (0+ ) = 0 x

θ <0

y

y

0.8 0.4 0 -0.4 1

2

-1

0

1

0.8 0.4 0 -0.4 -2

-1

0

1

y -2

-1

0

1

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

0.8 0.4 0 -0.4

2

0.8 0.4 0 -0.4

-2 0.8 0.4 0 -0.4

2

y

y

0

0.8 0.4 0 -0.4 -2

y

-1

y

y

-2

0.8 0.4 0 -0.4

0.8 0.4 0 -0.4

2

x a

x b a

ηc = 0.125 ξc ηc 0.2

ζ =

ζ c

ξ

ξc ηc

ξc ξc = 0.05 0.1 0.15 b ξc = 0.125

p + ζ c

ηc = 0.05 0.1 0.15

ζ = −p |ζ c − p| < 1

|ζ c − p|2 =

1 − 4pξc

u∞

x

χ(t) = e

α(t)

, ζ

ζ =

ηc

0.2

ζ ζ p2 + ζ c , z(ζ) = + ζc + , ζ χ χ + ζc χ

η

η ζ c

η

ξ

ξ

ξ O

O

O u∞

α

u∞

ζ =ζ −

ζ c

ζ = ζe

α

= (ζ − ζ c ) e

ζ

−ζ c

u∞

α α

ζ O

ζ

−α

ξ

ζ c = ζ c χ ζ

α

z(ζ) G(ζ) = ∂ζ z(ζ) =

1 [ζ + χ(ζ c + p)][ζ + χ(ζ c − p)] , χ (ζ + χζ c )2

ζ(z) ζ(z) =

χ (z − 2ζ c ) ± 2

z 2 − 4p2

ζ 1 1 = . F (ζ) = ∂z ζ(z) |z = z(ζ) = ∂ζ z(ζ) G(ζ) |ζ| > 1

F F (ζ) = −2 χ3 p2

ζ + χζ c [ζ + χ(ζ c + p)]2 [ζ + χ(ζ c − p)]2

F (ζ) = +2 χ3 p2

3ζ 2 + 6χζ c ζ + χ2 (3ζ 2c + p2 ) [ζ + χ(ζ c + p)]3 [ζ + χ(ζ c − p)]3

F (ζ) = −24 χ3 p2

(ζ + χζ c )[ζ 2 + 2χζ c ζ + χ2 (ζ 2c + p2 )] . [ζ + χ(ζ c + p)]4 [ζ + χ(ζ c − p)]4

˜ w ζ ˜ ˜ |ζ(z) ∂z ζ |ζ(z) = ∂ζ w ˜ |ζ(z) F [ζ(z)] . u(z) = ∂z w[ζ(z)] = ∂ζ w

ζ

−p

−χ(ζ c + p)

ζ

F u

˜ lim ∂ζ w(ζ) =0, ζ → −χ(ζ c + p) ˜ (ζ) = ∂ζ w(ζ) ˜ u

−χ(ζ c + p)

z(ζ g )

ζ

ζg ζ

z χ(ζ g − ζ c )

ζ χ(ζ g − ζ c )

tn+1

˜n w

tn+1

n

t ∈ [tn , tn+1 ) ˜ n (ζ) + ˜ n+1 (ζ) = w w

χ(ζ g − ζ c )

Γn+1 1 log[ζ − χ(ζ g − ζ c )] − log ζ − , 2π χ (ζ g − ζ c )

Γn+1 ˜ n+1 = ∂ζ w ˜ n (ζ) + ˜ n+1 (ζ) = u u 2π

1 − ζ − χ(ζ g − ζ c )

ζ−

1 χ (ζ g − ζ c )

˜ n [−χ(ζ c + p)]+ u +

Γn+1 2π

1 − −χ(ζ c + p) − χ(ζ g − ζ c )

1 −χ(ζ c + p) −

1 χ (ζ g − ζ c )

Γn+1 Γn+1 =

9

˜ n [−χ(ζ c + p)] 2π χ u . 1 1 − 1 p + ζg p + ζc + ζg − ζc χ = 1/χ

1

=0,

˜n u

Γn+1 −χ(ζ c + p)

(Γv , z v ) ˜ w z

ζ z(ζ) ˜v w

ζ v

z˙ v =

v

lim z → zv

=

lim z → zv

=

lim z → zv

Γv 1 u(z) − 2π z − z v

1 Γv ˜ ∂z ζ − ∂ζ w 2π z − z v ˜ v ∂z ζ + ∂ζ w

lim z → zv

Γv ∂z ζ − ζ − ζv 2π

= =

lim z → zv

Γv Γv ∂z ζ 1 − 2π ζ − ζ v 2π z − z v

Γv ˜ v ∂z ζ |ζ + = ∂ζ w lim v 2π z → z v

Γv ∂z ζ − ζ − ζv 2π

.

(z − z v )∂z ζ − (ζ − ζ v ) (ζ − ζ v )(z − z v )

2 3 2 ζ + (z − z v )∂zzz ζ ∂zz ζ 1 ∂zz = , lim 2 z → z v (z − z v )∂zz ζ + 2∂z ζ 2 ∂z ζ ζ v

Γv ˜ v ∂z ζ |ζ + z˙ v = ∂ζ w ∂ζ (∂z ζ) |ζ v v 4π Γv ˜ v (ζ v ) F (ζ v ) + = ∂ζ w F (ζ v ) . 4π z(ζ)

F

(Γv , z v ) z → zv

Δt

˜ v (ζ) = u

Γv 1 . 2π ζ − ζ v |ζ − ζ v |2

ζ − ζv

ε

˜ v (ζ) = u

ε2 ζ − ζv Γv , 2π |ζ − ζ v |2 + ε2 z˙ j

ζ → ζv Γj z˙ j = F (ζ j ) + 4π

− u∞

1− N #

1 + 2π

1 ζ 2j

Γk

k=1

+

1 2π

Γb0 + ζj

ζj − ζk − |ζ j − ζ k |2 + ε2

1 ζj −

1 ζk

F (ζ j ) ,

Uj

j

ζj

ζ ζv =

ζ v (z v , t) ζ˙ v (t) = F (ζ v ) z˙ v (t) + ∂t ζ(z v , t) , F zv

Rns

Rs

∂t ζ

Rns = ∂Ωb

1 dz ∂t w , Rs = 2

dz (∂z w)2 .

∂Ωb

F ns Fs

w − ρ

R

R = − ρ (Rns + Rs ) . ζ

˜b Ω

z

Ωb

t z = z(ζ, t) , α = α(t) ˜ w ˜ w(z, t) = w[ζ(z, t), t] , ˜ w u∞ ζ

z dz dz = ∂ζ z dζ = G dζ w

z

˜ ∂z ζ = ∂ζ w ˜ F . ∂z w = ∂ζ w ˜ + ∂ζ w ˜ ∂t ζ ∂t w = ∂t w

z

R ns =

˜ ∂Ω b

˜ + ∂ζ w∂ ˜ t ζ)G , Rs = dζ (∂t w

1 2

˜ ∂Ω b

˜ 2F . dζ (∂ζ w) z

α

ζc

p

z ∂t ζ = α˙ ζ , ˜ w

t ∈ [tn , tn+1 ) tn = (n − 1) Δt n

˜ = −u˙ ∞ ∂t w

˜ = −u∞ ∂ζ w

ζ˙ 1 ζ˙ k + k2 ζ − ζk ζk ζ − 1 k=1 ζk

# n 1 1 Γb0 1 + Γk − + . 1 2π ζ − ζk ζ ζ− k=1 ζk

ζ+

n 1 1 # − Γk ζ 2π

1−

1 ζ2

Rns Rns

R ns =

˜ ∂Ω b

˜ + ∂ζ w∂ ˜ t ζ)G dζ (∂t w

=

− u˙ ∞

dζ ˜ ∂Ω b

+ α˙ ζ

+

Γb0 1 2π ζ

ζ˙ k ζ˙ k 1 + 2 + ζ − ζk ζk ζ − 1 k=1 ζk n 1 # 1 1 + Γk − + 1 2π ζ − ζk ζ− k=1 ζk

n 1 1 # ζ+ − Γk ζ 2π

− u∞

1−

1 ζ2

G(ζ) , ζ = −χζ c

G

Rns =

Rns

n 1 # 1 Γb0 α˙ ˙ ∞ )ζ + Γk (−ζ˙ k + αζ) ˙ + − (u˙ ∞ + αu 2π 2π ζ − ζk

dζ ˜ ∂Ω b

n 1 # 1 ˙ ∞) − Γk +(−u˙ ∞ + αu ζ 2π k=1

˙ ζ

k=1

k 2 ζk

+ αζ ˙

1 ζ−

1 ζk

G(ζ) .

ζ=0 (−u˙ ∞ + αu ˙ ∞ ) G(0) , 1/ζ k −

k = 1 2 ... n

1 1 Γk ˙ (ζ k + α˙ ζ k ) G . 2π ζ 2 ζk k G(ζ) ζ = −χζ c

(ζ + χζ c )2

H (ζ) =

αζ ˙ k ˙ζ /ζ 2 + αζ ˙ k k 1 χ

−ζ˙ k + αζ ˙ αζ ˙ k

n ζ˙ − α˙ ζ k 1 # Γb0 α˙ Γk k + − (u˙ ∞ + α˙ u∞ )ζ − 2π 2π ζ − ζk k=1

n 1 1 # Γk ζ˙ k + α˙ ζ k +(−u˙ ∞ + α˙ u∞ ) − × 2 1 ζ 2π ζk ζ− k=1 ζk ×[ζ + χ(ζ c + p)][ζ + χ(ζ c − p)] .

ζ

[ζ + χ(ζ c + p)][ζ + χ(ζ c − p)] ζ = −χζ c

−χp2

− (u˙ ∞ + α˙ u∞ ) + +

n 1 # Γk 2π

+

n #

Γk −

u˙ ∞ − α˙ u∞ + χ2 ζ 2c

ζ˙ k + α˙ ζ k ζ˙ k − α˙ ζ k + . (ζ k + χζ c )2 (1 + χζ c ζ k )2

k=1

Rns = 2π

ζ = −χζ c

H

(−u˙ ∞ + α˙ u∞ ) G(0) + χp2 (u˙ ∞ + α˙ u∞ ) − ζ˙ k + α˙ ζ k 2

G

1

ζk

k=1

ζk

+ χp2

p2 (u˙ ∞ − α˙ u∞ ) + 2 χζ c

˙ α˙ ζ k − ζ˙ k ˙ ζk 2 ζk + α − χp . (ζ k + χζ c )2 (1 + χζ c ζ k )2 ζ˙ k

2 ... n

Rs ˜ ∂ζ w

˜ 2 = u2∞ − (∂ζ w)

n n u∞ # Γk 1 # Γk Γj − + 4π 2 (ζ − ζ k )(ζ − ζ j ) π ζ − ζk k,j=1

k=1

−

1 Γb0 # Γk u∞ Γb0 1 − 2 + π ζ 2π ζ ζ − ζk

+

1 # 2π 2

n

k=1

n

k,j=1

−

2u2∞ +

Γk Γj (ζ − ζ k )

1 ζ− ζj

n u∞ # + π k=1

−

1 # 4π 2 k=1

+

ζ−

n 2 1 Γb0 u∞ 1 # Γk + + 4π 2 π ζ2 ζ − ζk ζ2 k=1

n

Γk

Γk2 + 1 2 ζ− ζk

u∞ Γb0 1 + π ζ3

1 ζk

+

k=1

+

u2∞ + ζ4

−

1 4π 2

−

n #

k, j = 1 k = j

Γk Γj

1 1 ζ− ζ− ζk ζj

n 1 Γb0 # + 2π 2 ζ k=1

Γk ζ−

+

n u∞ 1 # Γk . 1 π ζ2 k=1 ζ − ζk

ζ = 0

ζ = 0

F (ζ)

˜b ∂Ω ζ =0

ζ = 0 ζ = 0

ζ = 0

χ(p − ζ c ) −χ(ζ c + p) R ˜ ∂ζ wF

G ˜ u

−χ(ζ c + p)

˜ 2F (∂ζ w) ζ = −χ(ζ c + p) ζ = χ(p − ζ c ) F ˜ 2 (∂ζ w)

˜ [−χ(ζ c + p)] = 0 u F

2

1 2 2 χ p˜ u [[χ(p − ζ c )] . 2 ζ=0

−

n u∞ Γb0 Γb0 # Γk + F (0) , π 2π 2 ζk k=1

ζ = 0

1 1 1 # Γk Γj u∞ # Γk F − F . 2 1 π 2π ζk ζj k=1 k,j=1 ζ k − ζj n

n

ζ=0 −

2 F (0) + ζ k F (0) Γb0 u∞ # F (0) − Γk , 2 4π π ζ 2k n

2u2∞ +

k=1

10

1 ζk

ζ = χ(p − ζ c ) ˜ 2 (∂ζ w)

˜ u

2

ζ = 0 −

n 1 # 2 1 Γk F . 4π 2 ζk k=1

ζ=0 u∞ Γb0 F (0) 2π u2∞ F (0) . 6 n #

1 1

Γk Γj F −F + 1 1 ζ ζj k − k, j = 1 ζj k = j ζ k

−

1 4π 2

+

n 1

Γb0 # Γk ζ k F − F (0) , 2π 2 ζk k=1

ζ=0

ζ = 1/ζ k

n 1

u∞ # 2 2 Γk ζ k F (0) + ζ k F (0) − ζ k F . π ζk k=1

1/2 Rs = π

p ˜ 2 [χ(p − ζ c )] + u 2χ2 Γ2 u∞ Γb0 u2 u∞ Γb0 F (0) + 2u2∞ + b02 F (0) + F (0) − ∞ F (0) + − π 4π 2π 6 +

−

n # k=1

Γk

Γb0 2π 2

ζk −

1

1 F (0) − ζ k F + ζk ζk

Γk 1 F + 4π 2 ζk u∞ 2 1 1 + ζk − 2 F (0) + ζ k − F (0) + π ζk ζk

+

+(1 − ζ 2k )F +

n 1 # 2π 2

k,j=1

+

1 ζk

+

1 Γk Γj F + 1 ζj ζk − ζj

n 1 1 1 # Γk Γj F −F . 1 1 4π 2 ζ ζj k − k, j = 1 ζj k = j ζ k

y

/χ π/2 1/χ

x α

u∞ /χ

Γk = 0 n

α

k = 1 2 ...

u∞

v 1 = χ(ζ c + p) , v 2 = χ(ζ c − p) , Γb ˜ ∂ζ w(−v 1) = 0 . Γb = 2π

1 v1

− v1

|v 1 | = 1

u∞ , F ζ=0

χ (v 1 + v 2 )2 4 v1v2 χ v1 + v2 F (0) = − (v 1 − v 2 )2 4 v 21 v 22 2 χ v + v 1 v 2 + v 22 F (0) = (v 1 − v 2 )2 1 2 v 31 v 32 (v 1 + v 2 )(v 21 + v 22 ) 3 F (0) = − χ (v 1 − v 2 )2 . 2 v 41 v 42 F (0)

=

ζ = −v 2 ˜ (−v 2 ) = u∞ (1 + v 1 v 2 ) u

v1 − v2 . v 1 v 22

Rs = −u∞ Γb /χ

R = ρu∞ Γb

3

3

,

0.3

L

D

4

Γb

4

χ

2

2

1

1

0.2

0.1

0

0 0

50

100

150

200

0 0

50

100

t

150

200

0

50

100

t

L u∞

D =1

150

200

t

α = 5◦ × 10◦

Γb ζ c = 0.1 Γb0 = 0 15◦

20◦

Γb t→∞ Γb Rns = 0 Rs Rs = π −

−

p ˜ 2s [χ(p − ζ c )] + u 2χ2

u∞ Γb Γ2 u∞ Γb u2 F (0) + 2u2∞ + b2 F (0) + F (0) − ∞ F (0) , π 4π 2π 6 ˜ s (ζ) = −u∞ u

1−

1 Γb 1 + 2π ζ ζ2

n {Γk }k=1,...,n

n #

Γk = −Γb ,

k=1

Γb

1.6

y

0.8 0 -0.8 -4

-2

0

2

x t = 0.4 0.8 1.2 1.6 α = 20◦

2

n u∞

Γb L

D α t→∞ α = 20◦

Rs = π

− +

p u2 ˜ 2 [χ(p − ζ c )] + 2u2∞ F (0) − ∞ F (0) + u 2 2χ 6

n #

Γk

k=1

+

Γk 1 F + 4π 2 ζk u∞ π

ζ 2k −

1 2

ζk

1

+(1 − ζ 2k )F +

n 1 # 2π 2

k,j=1

+

˜ (ζ) = −u∞ u

ζk −

F

1 ζk

n 1 1 # 1− 2 + Γk 2π ζ k=1

−F

1 ζk

1

,

ζj

1 − ζ − ζk

1 ζ−

1 ζk

F (0) +

+

ζk

1 Γk Γj F + 1 ζj ζk − ζj

1 # Γk Γj 1 1 4π 2 − k, j = 1 ζj k = j ζ k

F (0) +

.

t→∞

ζk → ∞

k =1 2

... n

˜ (ζ) → −u∞ u

1−

1 1 + 2 2π ζ

−

n #

Γk

1

k=1

ζ

,

t→∞ ˜ (ζ) → u ˜ s (ζ) , u ˜s u

F (ζ) ζ=0

1 Γk Γk 1 F = F (0) + O , 2 2 4π 4π |ζ k | ζk 1

1 1 u∞ 2 ζk − 2 F (0) + ζ k − F (0) + (1 − ζ 2k ) F = π ζk ζk ζk

1 u∞ 2F (0) − F (0) + O , = 2π |ζ k | 1 1

1 1 F −F = F (0) + O , 1 1 |ζ k | ζ ζ k j − ζk ζj O(1/|ζ k |) t→∞ Rs → π

− +

p u2 ˜ 2s [χ(p − ζ c )] + 2u2∞ F (0) − ∞ F (0)+ u 2 2χ 6

n # k=1

+

ζk

Γk

u∞ u∞ Γk F (0) + F (0) − F (0) 4π 2 π 2π

F (0) # Γ k Γj 4π 2 k, j = 1 k = j

#

.

n #

Γk2 , Γk Γj = Γb2 − k=1 k, j = 1 k = j

+

y

2

2

0

0

-2

-2

y

-6

-4

-2

0

2

2

2

0

0

-2

y

-4

-2

0

2

-6

-4

-2

0

2

-6

-4

-2

0

2

-6

-4

-2

0

2

-2 -6

-4

-2

0

2

2

2

0

0

-2

-2 -6

y

-6

-4

-2

0

2

2

2

0

0

-2

-2 -6

-4

-2

0

2

x

ζ c = 0.1

z j (t)

U (j) (t)

x

ζ c = 0.1 + 0.05 u∞ = 1 α(t) = 10◦ sin(2πt/T ) 1.25 2.5 3.75 5

t

j j

T =1

II Δt2 = Δt/2

tk+1

u∞ (tk )

I)

U j (tk )

tk α(tk ) {z m (tk )} (a)

U j = U j (tk ) z j (tk + Δt2 ) = z j (tk ) + Δt U j (tk )

u∞ (tk + Δt)

α(tk + Δt) {z m (tk + Δt2 )} U j (tk + Δt2 ) (a) (a) U j = U j + U j (tk + Δt2 ) (a) z j (tk+1 ) = z j (tk ) + Δt2 U j ,

II)

(Δt)3 Δt4 = Δt/4 I)

u∞ (tk ) U j (tk )

II)

α(tk ) {z m (tk )} (a)

U j = U j (tk ) z j (tk + Δt4 ) = z j (tk ) + Δt2 U j (tk )

u∞ (tk + Δt2 ) α(tk + Δt2 ) {z m (tk + Δt4 )} U j (tk + Δt4 ) (a) (a) U j = U j (tk ) + 2U j (tk + Δt4 ) z j (tk + 2Δt4 ) = z j (tk ) + Δt2 U j (tk + Δt4 ) {z m (tk + 2Δt4 )} U j (tk + 2Δt4 ) (a) (a) U j = U j (tk ) + 2U j (tk + 2Δt4 ) z j (tk + 3Δt4 ) = z j (tk ) + Δt U j (tk + 2Δt4 )

III)

IV )

IV

Δt6 = Δt/6

u∞ (tk + Δt)

α(tk + Δt) {z m (tk + 3Δt4 )} U j (tk + 3Δt4 ) (a) (a) U j = U j (tk ) + U j (tk + 3Δt4 ) (a)

z j (tk+1 ) = z j (tk ) + Δt6 U j

,

(Δt)5

Γb a

y

1

0

b

-1

c

a -7

-6

-5

-4

-3

-2

x 0.6 0.3

0.6

0.6

0.3

0.3

0

0

-0.3

-0.3

y

0 -0.3 -0.6 -0.9

-0.6 -7.2

-6.9

-6.6

-6.3

-6

-0.6 -6

x a

-5.7

-5.4

-5.1

-5.1

-4.8

x b t = 5.5

a b

c

b α˙ c

N {xk = Rk e

θk

}k=1,...,N

11

Γb

Rk > 1

-4.5

x c

-4.2

0.8 0.6

Γb

0.4 0.2 0 -0.2 -0.4 0

1

2

3

4

5

3

4

5

3

4

5

t 15 10

L

5 0 -5 -10 -15 0

1

2

t 1.2 0.8

D

0.4 0 -0.4 -0.8 -1.2 0

1

2

t Γb D

[δ] = L−2 ω(x, t) =

N #

Γk δ[x − xk (t)] ,

k=1

Γk

w(z) = u∞

k

z+

N 1 # Γk + z 2π

log(z − xk ) − log

k=1

Γ + log z 2π

Γ = Γb +

N # k=1

Γk ,

z−

1

+ xk

L

j w (j) (z) := w(z) − Γj /(2π ) log(z − xj ) u(z) =

dw (z) dz

= u∞

1−

N 1 # Γk + z2 2π

k=1

x˙ j = u∞

1−

N ∗ N # 1 # Γk 1 Γk + − 2π xj − xk 2π x2j k=1

1

1 − z − xk

1 z− xk

1 1 xj − xk

k=1

+

+

Γ 2π

Γ 2π

1 z

1 xj

j = 1 ... N k

j

1 # Γ R2 − 1 − Γk k2 2π 2π Rk + 1 N

uτ (θ) = −2u∞ sin θ +

k=1

M c = −2π u∞ +

N #

Γk x k −

k=1

1 xk

1 , 2Rk 1− 2 cos(θ − θk ) Rk + 1

˙ c= M

N #

Γk x˙ k +

k=1

x˙ k , x2k

u∞

pk (z) =

12

1 , q k (z) = z − xk

1 z−

1 xk

,

θ ∈ [0, 2π) θ

∂t w(z) = −

N

1 # x˙ k Γk x˙ k pk (z) + 2 q k (z) . 2π xk k=1

F ns = ρ

N #

Γk

k=1

x˙ k . x2k

z ∂z w(z) = u∞

1−

N 1 1 # Γ + Γk [pk (z) − q k (z)] + z2 2π 2π k=1

N #

Γk Γj [q k (xj ) − q j (xk )] ≡ 0 ,

k,j=1

∂Ωb

N #

Γk

k=1

N #

dz (∂z w)2 = −2

Γk

k=1

u∞

1−

N ∗ #

Γj pj (xk ) ≡ 0 ,

k=1

N ∗ 1 1 # + Γj pj (xk )+ x2k 2π

N 1 # Γ − Γj q j (xk ) + 2π 2π j=1

Fs = ρ

N #

Γk x˙ k − u∞ Γ

k=1

1

+ 2u∞ Γ . xk

,

k=1

F ns Fb = ρ

N # k=1

Γk

x˙ k +

1 , z

x˙ k − Γ u∞ , 2 xk

2u∞ · M c I˙c =

N #

Γk (xk x˙ k + x˙ k xk ) .

k=1

k

Γk

u∞ −

xk −

N # Γj j=1

N ∗ 1 1 # Γj + xk − + xk xk 2π

j=1

2π

xk xk − + x k − xj x k − xj

xk xk − . xk − 1/xj xk − 1/x j k

I˙c = u∞

N #

Γk

xk −

k=1

= u∞

N #

Γk

1 1 + xk − xk xk

xk −

k=1

= 2u∞

N #

Γk

1 1

+ xk − xk xk

xk −

k=1

M=0

1 − 2π u∞ = 2u∞ · M c , xk

u∞ μ=0

N

Ef =

N 1 # 1 Γk ψ(k) (xk ) + Γb ψb , 2 2 k=1

ψb

ψb (t) =

[w(e θ , t)] =

[w b (t)]

w b (t) =

N 1 # Γk log xk (t) . 2π k=1

μ=0

A = −u∞ · M ⊥ c + 2Ef A A=

N #

u∞ M c +

Γk (w (k) − w b ) + Γ w b

k=1

=

u∞

N #

Γk

2xk +

k=1

+

+

1 2π

N ∗ #

1 1 − + xk xk

Γk Γj log(xk − xj ) −

k,j=1

1 2π

N #

Γk Γj log(1 − xk xj ) +

k,j=1

N

Γ # Γk log xk . π k=1

A

A˙ =

u∞

N #

Γk

k=1

+

N # k=1

+

N # j=1

+

1 Γk x˙ k 2π 1 Γj x˙ j 2π

N x˙ k Γ # Γk π xk

2x˙ k −

# N ∗ j=1

# N ∗ k=1

x˙ k x˙ k + 2 + 2 xk xk

# Γj − xk − x j N

j=1

# Γk − xj − xk N

k=1

Γj xk −

1 xj

Γk xj −

1 xk

+

+

.

k=1

A˙

N xk

Γk

xk = xk (t) xk k = 1 2 ... N

k N

y p = y p (t) p = 1 2 . . . M

M

⎧ M ⎪ Γq 1 # ⎨˙ yp = 2π q=1 y p − xq ⎪ ⎩ y p (0)

20

20

15

15

y

y

N ≥4

10

5

10

5

0

0 -3

-2

-1

0

1

-3

x a

-2

-1

x b

a (∓1/4, 0)

±2π

b (∓1/2, 0)

π/50

−π/100

0

1

20

15

15

y

y

20

10

5

10

5

0

0 -3

-2

-1

0

1

-3

-2

x a

-1

0

1

x b a

b

a b (−1/50, 0) (−1/100, 0)

n=2 ±2π

(∓1/4, 0)

2

y

a b 10−10

(−1/50, 0) (−1/100, 0) y b (−1/50, 0) b

0.8

0.8

0.6

0.6

d

1

d

1

0.4

0.4

0.2

0.2

0

0 0

2

4

6

8

10

0

2

t a a a

4

6

8

t b b

b a

b

a b

a

b ... u

x ρ

t

10

"

∇·u=0 Dt u = −∇˜ p + ν∇2 u ,

p˜ = p/ρ u x t x

t

Dt = ∂t + u · ∇

L L U =

U 2 /L LU = = ν νU/L2 u · ∇u

ν∇2 u

≥ 104 1 2

∂t u u · ∇u

x ω =∇×u

3D 2D

3

4

E D(t)

D(t)

dx ω 2 (x, t) =: E(t) .

L η

L η

τ

T T τ

2D L T η τ

η τ

ρui ui /2 u

∂t

1 ui ui ui ui + ∂k uk ui ui = −∂i ui p˜ + ν ∇2 − ν ∂k ui ∂k ui , 2 2 2 uk ui ui ui p˜ ui ui d ui ui = −ν ∂k ui ∂k ui =: −ε . dt 2 −ε f

fi u i − ε

5

k

k k k

x ∈ IR

P IR → IR+ ∪{0} IR+ x

P (x ) dx

x x +dx α β β >α

x x ∈ (α, β) =

β

dx P (x ) .

α

P P

+∞

dx P (x ) = 1 ,

−∞

(−∞, +∞) β = +∞

1.4

1.4

1.2

1.2

1

1

0.8

0.8

P

P

x α = −∞

0.6

0.6

0.4

0.4

0.2

0.2

0 -1.5

-1

-0.5

0

0.5

1

x a

0 -1.5

-1

μ

-0.5

0.5

0

1

1.5

x b

a = 3/4 x = −0.15 σ 0.82057 100 5 · 103 a x −0.16684 σ : 0.81989 x −0.15089 σ : 0.82051

b = 1/4 δ = 2/5 5 · 105 b

1.5

a b δ

μ = 4/5

⎧ ⎪ ⎨ 1/(2δ) −a − δ < x < −a P (x) = 1/(2μ) b < x < b + μ ⎪ ⎩ 0 x

x

x=

σ

1 b−a μ−δ 1 1 + , σ 2 = (a+ b)2 + (a+ b)(δ + μ)+ (5μ2 + 6μδ + 5δ 2 ) . 2 4 4 4 48

α

β

x

0 x ≤ −a − δ (x + a + δ)/(2δ) −a < x ≤ b (x − b + μ)/(2μ) b < x < b+μ 0

1

(·) 0

xi

−a − δ < x ≤ −a 1/2 1 x ≥ b+μ yi

1

yi

xi

−∞

dx P (x ) = yi , yi

xi

σ2

x

(x − x)2

1 exp − . P (x) = √ 2σ 2 σ 2π

+∞

2

dx e−x =

√ π,

−∞

x

(·)

σ

1

x

+∞

x= −∞

σ2

dx x P (x) , σ 2 =

σ

+∞

−∞

dx (x − x)2 P (x) .

0.6

0.6

P

0.8

P

0.8

0.4

0.4

0.2

0.2

0 -0.5

0

0.5

1

1.5

2

0 -0.5

2.5

0

x a

1 σ = 0.5 σ 0.49990

a

x 0.99251 σ 0.50295

x

0

x)/(σ 2)]}/2 0

y ∈ (0, 1) 0

1

2

2.5

dx P (x ) =

1

x= b x 0.99979

2

{

−∞

1.5

dx e−x ,

√

x

1

x b

2 (x) = √ π

0.5

√ [(β − x)/(σ 2)] −

[(α −

(·)

1 1+ 2

x−x

√ =y, σ 2

xx,σ (y) yi

x+ √dx x)/(σ 2)]}/2

{

xi = xx,σ (yi ) x √ [(x + dx − x)/(σ 2)] − [(x −

P (x) = 6(−x2 + 3x − 2) 1 2

x

-1.5 0.1 -2

log10 E

(F)

0.05 0

-2.5

-0.05 -3 -0.1 -3.5 -0.1 -0.05

0

0.05

0.1

0.15

3

(F) a

4

5

6

log10 N b

a −20 ≤ k ≤ +20 Q = 100 N = 64·103 E Q = 100

b L2 N = 103 2 · 103 4 · 103 . . . 512 · 103 N

−3/2 −1/2 P (x) = 4x + 6 −3/2 ≤ x ≤ −1 P (x) = −4x − 2 −1 ≤ x ≤ −1/2 x

(x1 , x2 , . . . , xm ) = x ∈ IRm

m m>1 P (x) dx

x (x2 , x2 + dx2 ) · · · × (xm , xm + dxm ) P

(x1 , x1 + dx1 ) ×

IRm

P 2

F[P ](k) = e−

kx

< (x − x)m > x

1 2π

+∞

−∞

dx P (x) e−

k(x−x)

1 F [P ](k) = 2π

m

+∞

dx P (x) e−

kx

−∞

=

∞ 1 # (− )q k q +∞ dx P (x) xq 2π q=0 q! −∞

Q 1 # (− )q k q < xq > , 2π q=0 q!

xm

x

< xq > Q < xq >

xq

F [P ] < xq >

P F [P ]

-1.5 0.1 -2

log10 E

(F)

0.05 0 -0.05

-2.5

-3

-0.1 -3.5 -0.05

0

0.05

0.1

0.15

3

(F) a

4

5

log10 N b

a −10 ≤ k ≤ +10 Q = 100 N = 64·103 E Q = 100

b L2 N = 103 2 · 103 4 · 103 . . . 512 · 103 N

6

F [P ](k) =

1 sin kδ/2 e 4π kδ/2

F [P ](k) =

k(a+δ/2)

+

sin kμ/2 − e kμ/2

k(b+μ/2)

,

k2 σ2

1 exp − + kx . 2π 2 k k a

a

P

m j = (j1 , j2 , . . . , jm ) k = 1 2 ... m |j| j! j1 !j2 ! · . . . · jm ! x = (x1 , x2 , . . . , xm ) ∈ IRm xj11 xj22 ·. . .·xjmm F [P ](k) =

F [P ](k) =

1 (2π)m

IRm

jk

j1 + j2 + . . . + jm xj

P (u)

dx P (x) e− k·x

∞ 1 # (− )q dxP (x) (k · x)h m (2π)m q=0 q! IR Q # 1 # (− )q m (2π) q=0

|j| = q

kj < xj > . j!

Q

P f : IR → IR

f ∈ L1 (IR)

{xi }

N

f P (x)

f (xi ) α β

P (x) [α, β) [xk , xk+1 ) δ k = 1 2 ... n k

α<β

n k xi

xk+1

xk

x<α

x>β x1 = α

dxP (x) = (xk+1 − xk )P (ξk )

ξk

[xk , xk+1 ) f

[xk , xk+1 ) f

[xk , xk+1 ) = =

xn+1 = xk+1 − xk

β N

nk , N (xk , xk+1 )

nk

N →∞

f (xi ) nk x ˜i i = 1 2 . . . nk

nk 1 # f (˜ xi ) N i=1

nk [f (ξk ) + O(xk+1 − xk )] N

(xk+1 − xk )P (ξk )f (ξk ) + O(xk+1 − xk )2 , N → ∞ n

f

f=

n #

(xk+1 − xk )P (ξk )f (ξk ) + O(xk+1 − xk )2

k=1

δ → 0

→

β

dx P (x )f (x ) ,

α

n → ∞

+∞

f=

dx P (x) f (x) . −∞

P {x(i) }i=1,2,...,N

x N x

0.708

1.26

0.704

1.256

0.7

1.252 1.248 1.244 1.24

0.696

1.236 3

4

5

6

3

4

log 10 N a

5

6

log10 N b f (x) = x2

x

a P (x)

N b

f 0.69583

1.25

< f >=

N 1 # f (xi ) , N i=1

N N

N

f (x) = sin(πx) P (x)

x

u : IR → IR x

ux u(x) x

IR u(x) N

N (q) {ux }q=1,2,...,N u(x)

{u(q) (x)}q=1,2,...,N ux

1

3

2

0.6

u1 (x)

P (ux )

0.8

0.4

0

0.2 0 -0.5

1

-1 0

0.5

1

1.5

2

2.5

0

0.25

0.5

ux a

0.75

1

x b

a u1 : [0, 1] → IR b

x (q) {u1 (x)}

8 u1

N = 4000 0.99012

P (ux )

x

0.50241

ux

P (ux )

x ux

u(x) u(x) P (ux ) (q)

x u1

(q)

{u1 (x)} {u2 (x)} ω ∈ IR u1 (x) ≡ ω

u2 P

(q)

u1 (x) a P

1 N = 4000

1/2 b

8 u2 (x)

(q)

(q)

{ux } x

P

u2 (x) x u1 (x)

u2 (x)

x a

P (ux ) N = 4000 c d

2

b

x = x1

x = x2

[0, 1] e

1

0.8

0.8

P (ux2 )

P (ux1 )

1

0.6 0.4 0.2 0 -0.5

0.6 0.4 0.2

0

0.5

1

1.5

2

0 -0.5

2.5

0

0.5

1

1.5

2

2.5

ux2 b

3

3

2

2

u(2) (x)

u(1) (x)

ux1 a

1

0

1

0

-1

-1 0

0.25

0.5

0.75

x c

1

0

0.25

0.5

0.75

1

x d

1.1 1

x, σ

0.9 0.8 0.7 0.6 0.5 0.4 0

0.25

0.5

0.75

1

x e 1 P (ux )

x = x1 = 3/8 (q)

u2 : [0, 1] → IR 1/2 a b x = x2 = 6/8 N = 4000

{u2 (x)}

c

d

e u2

x x

x P (ux )

x u(x) u(x)

. . . ux(n) ) x(n)

P (ux(1) ux(2) x(1) x(2) . . .

n

Q(j) (x(1) , x(2) , . . . , x(m) ) =< ujx1(1) ujx2(2) · . . . · ujxm(m) > m m j m (x(1) , x(2) , . . . , x(m) ) = x m j = |j| = j1 +j2 +. . .+jm ≥ m jq j u(x) x = xq

q

j

{y (1) , y (2) , . . . , y (j) } u Q (y (j)

(1)

{x(1) , x(2) , . . . , x(m) } , y (2) , . . . , y (j) )

x(q)

jq

u(x) Q(j)

(m)

(0)

x(1) x(2) . . .

x j x Q(j) (x(0) + x(1) , x(0) + x(2) , . . . , x(0) + x(m) ) = Q(j) (x(1) , x(2) , . . . , x(m) ) m x(1) x(2) . . . x(m) Q(j) u(x) (a, b) δ = (b − a)/N

N (0) xi

i

< ujx1(1) ujx2(2) · . . . · ujxm(m) > =

N 1 # j1 (0) (0) (0) u [xi + x(1) ]uj2 [xi + x(2) ] · . . . · ujm [xi + x(m) ] δ , N δ i=1

N

δ

Q(j) (x(1) , x(2) , . . . , x(m) ) = b 1 = dξ uj1 [ξ + x(1) ]uj2 [ξ + x(2) ] · . . . · ujm [ξ + x(m) ] , b−a a (a, b) x(0) = −x(m) m ∈ IRm−1

Q(j) (ξ ) m−1 IRm−1

u(x) Q(j) (1) (2) (m−1) ξ = (x , x , . . . , x ) x → ∞

Q(j) (ξ ) → 0 u(x) x (i) = x(i) − x(m) i = 1 2 . . . m − 1 u(x) u(0) x

> (m − 1)

Q(j) m−1 xi = xi − xm (k1 , k2 , . . . , km−1 ) = k 1 χj (k ) = dξ Q(j) (ξ ) exp(− k · ξ ) , (2π)m−1 IRm−1 dξ = dx (1) dx (2) · . . . · dx (m−1) Q(j)

u(x) u : IRn → IRn

IRm−1 u(x) n

n=3

x {x(1) , x(2) . . . , x(m) } m

m−1

u(x)

u(x) j

m j ≥m {y (1) , y (2) . . . , y (j) }

u i1 i2 . . . ij m

j

(1) Q(j) , y (2) , . . . , y (j) ) =< ui1 y (1)ui2 y (2) · . . . · uij y (j) > . i1 ,i2 ,...,ij (y u ∇·u = 0 {y (1) , y (2) , . . . , y (j) }

u(x)

l y (l)

x(q) = x(q)

∂x(q) Q(j) i1 ,i2 ,...,il−1 ,a,il+1 ,...,ij a

u(x) j−1

u(x) u(x)

(0) y (0) ∈ IR3 Q(j) + y(1) , y (0) + y(2) , . . . , y (0) + i1 ,i2 ,...,ij (y (j) (j) (1) (2) (j) y ) ≡ Qi1 ,i2 ,...,ij (y , y , . . . , y ) D ⊂ IRn

y

(0)

= −x

x(q)

(m)

x

(q)

(q)

= x

−x

m m−1

(m)

q = 1 2 ... m − 1 u(0) u(x) (1) (2) (m−1) {x , x , . . . , x }

x→∞

χj (k(1) , k(2) , . . . , k(m−1) ) x(q) (1) (2) (j) (l) {y , y , . . . , y } l y = x(q) j−1 j ka(q) χi1 ,i2 ,...,il−1 ,a,il+1 ,...,ij

j = 2

IR3 j=2

IR3

m = 2

u(x, t) t m=2

(i, j)

Rij (r) =< ui (x)uj (x + r) > , r=x

(2)

−x

(1)

x(1) = x x(2) = x+r

r =0

(i, j)

∂rj Rij = 0

i

Rij (0) =< ui (x)uj (x) >

∂ri Rij = 0

j y =

x+r

Rij (r) =< ui (y − r)uj (y) > u(x)

u(x, t) Rij (−r) =< ui (x)uj (x − r) > y = x−r Rij (−r) =< uj (y)ui (y + r) >= Rji (+r) i<j i>j −r Rij (r) 2 Rij (r)

=

1 |D|

2

dx ui (x)uj (x+r)

≤

D

1 |D|

D

dx u2i (x)

1 |D|

dy u2j (y) .

D+r

Rii (0)

i

Rjj (0)

u(x) |Rij (r)| ≤ [Rii (0)Rjj (0)]1/2 ,

i=j

r |Rii (r)| ≤ Rii (0) , Rii (r) r=0

r = 0

r ui (x)

r=0

ui (x+r) Φ

(i, j) Rij Rij (r) =

IR3

dk Φij (k) e k·r

dk = dk1 dk2 dk3

IR3

Φij z1 z 2

z i z j Φij (k)

z3

z

z k Φ < ui (x)uj (x) >

IR3

Φij < ui (x)uj (x) >= Rij (0) =

dk Φji (k) e k·r r = 0 = 3

IR

Φij (k) dk (k1 , k1 + dk1 ) × (k2 , k2 + dk2 ) × (k3 , k3 + dk3 )

IR3

dk Φji (k) .

z i z j Φij (k)

kj Φij = 0 , ki Φij = 0 ,

r

k r

Sij (r) =

1 4πr2

k

dS(r)Rij (r) , Ψij (k) =

dS(k)Φij (k) .

∂Br (0)

∂Bk (0)

(i, j) Bk (0) ∩ Bk+dk (0)

Ψij (k) dk

< ui (x)uj (x) > = Rij (0) dk Φij (k) =

IR3

+∞

=

dk 0

3

Rij Rij

Φij

dS(k)Φij (k)

∂Bk (0)

Φij

Φij (k) = Rij

1 dr Rij (r) e− k ·r . (2π)3 IR3

1 Φij (k) = dr Rij (r) e k·r (2π)3 IR3

=

1 dr Rij (−r)e− k·r (2π)3 IR3

1 = dr Rji (r)e− k ·r = Φji (k) , (2π)3 IR3 Φij (k) = Φij (−k) z l z h Φlh

z1 z2 z l z h Φlh = z l z h Φlh = z h z l Φhl .

z3

+∞

= 0

Sij (r) k

dk Ψij (k) .

Ψij (k) r

sin kr . dS(r) e k·r = 4πr2 kr ∂Br (0)

Sij (r) = = = = = =

Sij (r) Ψij (k) 1 dS(r) Rij (r) 4πr2 ∂Br (0) 1 dS(r) dk Φij (k) e k·r 4πr2 ∂Br (0) IR3 1 dkΦ (k) dS(r) e k·r ij 4πr2 IR3 ∂Br (0) sinkr Φij (k) dk kr IR3 +∞ sinkr dk dS(k) Φij (k) kr ∂Bk (0) 0 +∞ sinkr Ψij (k) , dk kr 0

rSij (r) =

+∞

dk 0

1 Ψij (k) sin kr . k r

Ψij (k) = = = = = =

−k

Ψij (k) Sij (r)

dS(k) Φij (k) 1 dS(k) dr Rij (r) e− k·r 3 3 (2π) ∂Bk (0) IR 1 dr R (r) dS(k) e− k·r ij (2π)3 IR3 ∂Bk (0) sin kr k2 Rij (r) dr 2π 2 IR3 kr k 2 +∞ sin kr dr dS(r) Rij (r) 2π 2 0 kr ∂Br (0) +∞ 2 k dr sin kr rSij (r) , π 0 ∂Bk (0)

2 1 Ψij (k) = k π

0

+∞

dr rSij (r) sin kr .

r Sij (r)

1 1 Ψii (k) = 2 2

E(k) =

Ψij (k)/k

Ψij (k)

dS(k) Φii (k) ,

∂Bk (0)

1 1 1 dk E(k) = dkΦii (k) = Rii (0) = < ui (x)ui (x) > , 3 2 IR 2 2

+∞

0

E(k) dk

Bk (0) ∩ Bk+dk (0) k

kL = 2π/L L (i, j)

Φij (k)

k Φij (k) = C ij + kl C ijl + kl km C ijlm + O(k 3 ) , C ij C ijl

C ijlm k

Φij (k) k3

O(k 3 )

k=0

0 = kj Φij (k) = kj C ij + kj kl C ijl + kj kl km C ijlm + O(k 4 ) 0 = ki Φij (k) = ki C ij + ki kl C ijl + ki kl km C ijlm + O(k 4 ) , k C ij = 0 ;

# π(i,l)

C ijl = 0 ,

# π(j,l)

C ijl = 0 ;

# π(i,l,m)

C ijlm = 0 ,

# π(j,l,m)

C ijlm = 0

π(q1 , q2 , q3 , . . . , qr ) q1 q2 q3 . . . qr

z i z j Φij z1 z 2 z3 z i z j kl C ijl + z i z j kl km C ijlm + O(k 3 ) k z i z j kl C ijl

−k

k

C ijl = 0 .

Φij (k) = kl km C ijlm + O(k 3 ) , Ψij (k) Ψij (k) =

dS(k) (kl km C ijlm + kl km kq C ijlmq + . . .) .

∂Bk (0)

dS(k) = k 2 sin φ dφdθ

k ⎛

⎞ sin φ cos θ k = k ⎝ sin φ sin θ ⎠ . cos φ ki

dS(k) ki1 · ki2 · . . . · kim = 0 .

∂Bk (0)

1 q

2

m 3 p+q+s=m

s

p m

k

dS(k) ki1 · ki2 · . . . · kim =

∂Bk (0)

= k m+2

π

dφ sin φ sinp+q φ coss φ

0

2π

dθ cosp θ sinq θ , 0

p+q 2π dθ cosp θ sinq θ = 0 , 0

p+q

s

(p + q) + s = m

π

dφ sinp+q+1 φ coss φ = 0 .

0

k k 4πk 4 /3

Ψij (k) k kL E(k) =

1 2 Ψii (k) = πC iill k 4 + O(k 6 ) , 2 3

k ≥ kη exp(−lk)

k

l

η

L η

ε = −3/2 du2 /dt L T

ν

[ε] = L2 T −3 [ν] = L2 T −1

η = ν α εβ

L1 T 0 = L2(α+β) T −(α+3β) α = 3/4 β = −1/4

η

η = (ν 3 /ε)1/4 . L

η kL k kη

ε E(k) 1941

k

log(E/E0 )

−5/3 +4

a

b

c

d log(Lk)

0

log(Lkη )

Lk E0

k = 2π/L

E

a

E(k) ∝ k4 b k kL

k kL c

d k ≥ kη

E k E k [E] = L3 T −2

E E

ε α

k

β

E(k) ∝ εα k β .

L3 T −2 = L2α−β T −3α , α = 2/3

β = −5/3

ε2/3 k −5/3

εω = −∂t < ω 2 > E

[εω ] = T −3

εω

k εω

ε

L3 T −2 = L−β T −3α , α = 2/3

β = −3

2/3

εω k −3

ω wij (r) =< ωi (x)ωj (x + r) > , ωi (x)

ωj (x + r)

ω wij

Rij

ω

∇×u

D ⊂ IR3

1 < ωi (x)ωj (x + r) > = dx ωi (x)ωj (x + r) |D| D 1 dx eipq ∂p uq (x) ejlm ∂rl um (x + r) = |D| D eipq ejlm ∂rl dx um (x + r)∂p uq (x) . = |D| D y = x+r r

−∂rp uq (y − r)

eipq ejlm ∂rl dy um (y)∂rp uq (y − r) |D| D+r eipq ejlm 2 ∂rl rp dy uq (y − r) um (y) . =− |D| D+r

∂p uq (x) = r+D r

wij (r) = −

x=y−r wij (r) = −eipq ejlm ∂r2l rp

1 dx uq (x)um (x + r) |D| D

= −eipq ejlm ∂r2l rp Rqm (r) .

eipq ejlm = δij δpl δqm + δil δpm δqj + δim δpj δql − δij δpm δql − δil δpj δqm − δim δpl δqj ,

2 2 wij (r) = −δij ∇2 Rmm (r) − ∂im Rjm (r) − ∂lj Rli (r) + ≡0

≡0

2 +δij ∂lm Rlm (r) +∂ 2 Rmm (r) + ∇2 Rji (r) , ij ≡0

wij Rij 2 wij (r) = (∂ij − δij ∇2 )Rmm (r) + ∇2 Rji (r) .

Rji

Rij

Rlm wij

wii (r) = −∇2 Rmm (r) .

Ωij (k) =

1 dr wij (r) e− k·r , (2π)3 IR3

Ωij (k) = (δij k 2 − ki kj )Φmm (k) − k 2 Φji (k) , Ωij Ωii (k) = k 2 Φmm (k) ,

L=

< ui (x)ui (x) > < ωj (x)ωj (x) >

1/2

= IR

3

IR3

dk Φii (k) 1/2

dk Ωjj (k)

+∞

dk E(k)

= 0

0 +∞

dk k 2 E(k)

1/2

E

L

E 0

dk E(k) 1 =

+∞

0

dk E(k) 2 ,

+∞

0

dk k 2 E(k) 1

+∞

0

dk k 2 E(k) 2 .

+∞

k k

L1 L2 L

f : IR → IR

2π f ∈ L1 ([−π, π))

1 2π

ck =

+π

f (x) e−

kx

dx ,

−π

k

1 |ck | = 2π g : IR → IR

+π

f (x) e

− kx

−π

1 dx ≤ 2π

+π

|f (x)| dx .

−π

|k| → +∞ (a, b) g ∈ L1 (a, b)

g

b

lim y→±∞

y

dx g(x) e

yx

=0,

a

402−403 1

|k| → +∞ +∞ #

ck e

kx

k=−∞

f f n S n (x) S n (x) =

+n +π 1 # dt f (t) e− 2π −π

kt

e

k=−n

=

1 2π

+π

dt f (t) −π

+n # k=−n

e

k(x−t)

kx

=

=

= =

1 2π 1 2π 1 2π 1 2π

1 = 2π

+π

dt f (t) e−

−π

+π

dt f (t) e−

n(x−t)

2n #

+π

dt f (t) e−

+π

dt f (t)

n(x−t)

e

e

h(x−t)

(2n+1)(x−t)

e

e

+π

dt f (t)

−1 −1

(x−t)

(n+1/2)(x−t)

− e−

(n+1/2)(x−t)

(x−t)/2

− e−

(x−t)/2

e

−π

(k+n)(x−t)

h=0

−π

e

k=−n

−π

+n #

n(x−t)

sin

−π

1 (x − t) 2 . x−t sin 2

n+

n

1 sin Dn (x) = 2π

1

n+ x 2 , x sin 2 2π

+π

S n (x) =

+π

dt Dn (x − t) f (t) = −π

−π

f

dt Dn (t) f (x + t) . −π

n f f (x+ )

f x

+π

dt Dn (t) f (x − t) =

S n (x) f (x− )

f x

f +∞ #

ck e

k=−∞

kx

=

f (x− ) + f (x+ ) . 2

x f : IR → IR 1 F (k) = F[f ](k) := 2π k f (x) f : IRn → IR

n

+∞

dx e−

−∞

kx

f (x) ,

f ∈ L1 (IR)

f (x) ∼ |x|−n−ε

f ∈ L1 (IRn ) n

ε>0

x→∞ x

xi i = 1 2 . . . n 1 (2π)n =

1 (2π)n

+∞

dx1 e

− k1 x 1

dx2 e

−∞

IRn

+∞

− k2 x 2

ki

+∞

· ... ·

−∞

dxn e−

kn x n

f (x) =

−∞

dx f (x) e− k ·x = F[f ](k) .

k ∈ IRn F (y)

k → ±∞ f ∈ L1 (IR) k ∈ IR

k

ε >0 η ∈ (k − δε , k + δε ) |F (η) − F (y)| < ε f : IRn → IR x ∂s f

δε > 0 f xs

F[∂s f ](k) = ks F[f ](k) . ∂BR (0)

ν R

dS

F[∂s f ](k) =

1 lim dx ∂s f (x) e− k ·x (2π)n R→∞ BR (0)

=

1 dS(x) νs (x)f (x) e− k·x + lim n (2π) R→∞ ∂BR (0)

−

=

f (x) ∼ |x|−n−ε

dx f (x) (− ks ) e− k·x BR (0)

ks dx f (x) e− k ·x , (2π)n IR3

∂BR (0) ε>0

R

f

x→∞ F[f1 ] F[f2 ] =

f1,2 : IR → IR

1 F[f1 ∗ f2 ] , 2π

f1 ∗ f2 x

+∞

f1 ∗ f2 (x) =

+∞

dy f1 (y)f2 (x − y) = −∞

dy f1 (x − y)f2 (y) . −∞

n 1/(2π)n 1/(2π) x = x1 + x2 1 F[f1 ](k) F[f2 ](k) = 2π

+∞

dx1 f1 (x1 )e −∞

1 = (2π)2 1 = (2π)2 =

+∞

+∞

dx1 −∞

−∞

+∞

dx e

− kx

1 2π

− kx1

+∞

dx2 f2 (x2 )e−

dx2 f1 (x1 )f2 (x2 ) e−

dx1 f1 (x1 )f2 (x − x1 )

−∞

−∞

1 F[f1 ∗ f2 ](k) . 2π f ∈ L1 (IRn )

+∞ kx

− dk e

F (k) =:

x ∈ IR

+K

lim

dk e

K→+∞

−∞

ξ ∈ [−π, +π)

k(x1 +x2 )

+∞

F

kx

F (k) ,

−K

2π m

Φx (ξ + 2mπ) := f (ξ + x)

F = F[f ]

f ∈ L1 (IR) x

x

kx2

−∞

x

+∞

− dk e

kx

F (k) =

−∞

−

f (x ) + f (x+ ) . 2 F[f1,2 ]

F1,2 f1 f2 = F

−1

[F1 ] F

−1

[F2 ] = F −1 [F1 ∗ F2 ] .

k = k 1 + k2 f1 (x) f2 (x) = F −1 [F1 ](x) F −1 [F2 ](x)

+∞

= − dk1 F1 (k1 ) e −∞

+∞

= − dk e −∞

k1 x

+∞

− dk2 F2 (k2 ) e −∞

+∞

kx

dk1 F1 (k1 )F2 (k − k1 ) −∞

= F −1 [F1 ∗ F2 ](x) ,

k2 x

F1,2

⎧ 2 ⎨ ∂t u + u0 ∂x u = ν∂xx u ⎩

u(x, 0) = a0 (x) = exp(−x2 /σ 2 ) x → ±∞

u→0

u t

t,

x

x

u0

ν

σ

u(x, 0) 0

x

|x| > 2σ x u ˆ(k, t) u(x, t)

x

∂t u ˆ = −(νk2 + u0 k) u ˆ,

u ˆ0 (k) = F[a0 ](k) =

1 2π

+∞

dx e−

kx

e−x

2

/σ 2

2 2 /4

e−σ k 2π

=

−∞

+∞

dx e−(x+

σ 2 k/2)2 /σ 2

.

−∞

h(z) = exp(−z 2 /σ 2 )

z = x+

y

+R

dx e−(x+

σ 2 k/2)2 /σ 2

−R

+e

−R2 /σ 2

+R

=

σ 2 k/2

dx e−x

2

/σ 2

+

−R

dη e

(−2 Rη+η 2 )/σ 2

−e

−R2 /σ 2

0

σ

√

σ 2 k/2

dη e(2 Rη+η

0

R → +∞ π R

0

σ 2 k/2

dη e(±2 Rη+η

2

)/σ 2

≤

σ 2 k/2

eη 0

2

/σ 2

dη

2

)/σ 2

dη ,

y

σ 2 k/2

x −R

0

+R k>0

exp(−R2 /σ 2 ) R → +∞ u ˆ0 (k) =

u ˆ(k, t) =

2

σ √

u(x, t) = √

π

exp

2

σ √

−

π

σ2 4

2 2

k /4

+ νt

.

k2 − u0 tk

x − u0 t 2

σ √ exp − . σ 2 + 4νt σ 2 + 4νt u0 exp(−x2 /σ 2 )

ξ = x − u0 t

e−σ

t=0

(νt/σ)−1/2 exp[−ξ 2 /(σ 2 + 4νt)]

{y(1) , y(2) , . . . , y (j) } j

1 p(x)ui (x + r) 3

2

u : IR3 → IR3 (j)

Qi1 ,i2 ,...,ij =< ui1 (y (0) + y (1) )ui2 (y (0) + y (2) ) · . . . · uij (y (0) + y (j) ) > y (0) (j)

x

(2)

(m)

−x

x

m−1

Qi1 ,i2 ,...,ij

... x

1

(m−1)

=x

(m−1)

(1)

= x(1) − x(m) x

(m)

−x

y

(0)

(2)

=

(m)

= −x

G ◦ : G×G → G (a ◦ b) ◦ c = a ◦ (b ◦ c) e a◦e=e◦a =a a◦a a b∈G

−1

= a−1 ◦ a = e

a◦b = b◦a

(G, ◦) a b∈G

I) : II) : III) : IV ) : V):

I

π

−

π

+

π+

r−

π

π+

π−

r−

π−

I

r+ r+

r+

r−

II z

RC(x, y, z) (x, y) α 2π − α

2π α (x , y ) (x, y)

x y

=

cos α − sin α x sin α

cos α

y

= R(α)

x y

.

R(α1 )R(α2 ) = R(α1 + α2 ) . . .

III z

ez RC(x, y, z)

x

x

(x, y) z

z

x

z = z0

2z0 − z

z = z2

z = z1

2(z2 − z1 ) (x, y) (x, y)

d1 = cos β x

d2 = sin β x − 2(x · d)d 1 − 2d21 −2d1 d2 − cos 2β − sin 2β S(d) = = , − sin 2β cos 2β −2d2 d1 1 − 2d22

z d = (d1 , d2 )

−1

α

− cos 2β − sin 2β − sin 2β cos 2β

·

cos α − sin α sin α cos α

z =

− cos(2β − α) − sin(2β − α) − sin(2β − α) cos(2β − α)

β = β − α/2 z

β1

β2

,

x

z

− cos 2β1 − sin 2β1 · − sin 2β1 cos 2β1 cos 2(β1 − β2 ) = sin 2(β1 − β2 )

− cos 2β2 − sin 2β2 − sin 2β2 cos 2β2 − sin 2(β1 − β2 ) cos 2(β1 − β2 )

=

2(β1 − β2 )

IV

(x, y, z)

RC RC (x , y , z )

RC

RC

RC

RC

RC

SO(3)

V

d = (d1 , d2 , d3 ) x

x = x − 2(x · d)d ⎛

⎞ 1 − 2d21 −2d1 d2 −2d1 d3 ⎟ ⎜ S(d) = ⎝ −2d2 d1 1 − 2d22 −2d2 d3 ⎠ , −2d3 d1 −2d3 d2 1 − 2d23 −1 d

e

d

e

x y

(2)

(1)

Q(j) i1 ,i2 ,...,ij

. . . x y

(2)

x

m−1

j Q(j)

(1)

(m−1)

. . . y

Q(j) i1 ,i2 ,...,ij

(j)

0 x

(1)

(2)

. . . x

(m−1)

(1)

x

(2)

. . . x

x

m−1

Q(j) i1 ,i2 ,...,ij

x

Q(j)

(m−1)

V

IV

Q(j)

III II

Q(j) i1 ,i2 ,...,ij

I V m−1 1940

x (1) x (2) . . . x (m−1) V I

m (j)

... y a(1) a(2) . . . a(j)

x

(1)

j

x

(2)

. . . x

(m−1)

|a(i) | = 1

j

y j

(1)

y

(2)

i = 1 2 ... j

˜ (j) (y (1) , . . . , y (j) | a(1) , . . . , a(j) ) =< u(y (1) ) · a(1) · . . . · u(y (j) ) · a(j) > . Q

a(1) a(2) . . . a(j)

˜ (j) (y (1) , . . . , y (j) | a(1) , . . . , a(j) ) = a(1) · . . . · a(j) Q(j) (y (1) , . . . , y (j) ) , Q i1 ···ij i1 ij a

˜ (j) Q

(i)

V

˜ (j) Q

S=

x

(1)

, x

(2)

, . . . , x

(m−1)

| a(1) , a(2) , . . . , a(j)

V

w (1) , w(2) , w(3) ∈ S

V (w (1) , w(2) , w(3) ) = w (1) ·w(2) ×w(3)

w(1) w(1) w(1) 1 2 3 (1) (2) (3) = w1(2) w2(2) w3(2) = εijk wi wj wk . (3) (3) (3) w1 w2 w3

2

u(x)

3

S V (w V (w

(1)

,w

V (w = =

(2)

(1)

,w

, w

(3)

(2)

(1)

, w

(2)

, w

(3)

)

)

, w

(3)

(1) (2) (3) εijk w i w j w k

) V (w

(1)

, w

(2)

, w

(3)

) =

(1) (2) (3) εpqr w p w q w r

(δip δjq δkr + δiq δjr δkp + δir δjp δkq − δip δjr δkq − δiq δjp δkr − δir δjq δkp ) × (1)

(1)

(2)

(2)

× w i w p w j w q

(3)

(3)

w k w r

= a11 a22 a33 + a12 a23 a31 + a13 a21 a32 − a11 a23 a32 − a12 a21 a33 − a13 a22 a31 ,

˜ (j) (y (1) , . . . , y (j) | a(1) , . . . , a(j) ) Q S a(i) i = 1 2 ... j F (y

(1)

· y

(1)

, y

(1)

· y

(2)

, . . . , y

(j)

· y

(j)

) y

˜ (j) (y , . . . , y Q i1 . . . ij (1)

F (y

(1)

· y

(1)

, y

(1)

· y

(2)

, . . . , y

(j)

(1)

(j)

· y

· a(1) a(2) · a(3) · . . . · y

(j)

· a(j)

a(j) | a(1) , . . . , a(j) ) (1) (j) Qj (y , . . . , y ) (j)

1

(j)

) y i1 δi2 i3 · . . . · y ij .

(1)

m=2 Q(2) i1 i2 Q(3) i1 i2 i3

x = x(1) − x(2) = r (r) = A ri1 ri2 + B δi1 i2 V (r) = A ri1 ri2 ri3 + B ri1 δi2 i3 + C ri2 δi1 i3 + D ri3 δi1 i2 V r · r = r2

r = |r|

A, B, C, D, . . . V V

u(x) x(1) = x(1) − x(2) = r

s = s(x) Q(1) (r) =< u(x(1) ) s(x(2) ) > (r) = A ri , Q(1) i V A

r < u(x + r) p(x) >

4

x

x(3) = r(2)

(1)

(1) Q(3) , r(2) ) = i1 i2 i3 V (r (1) Q(4) , r(2) ) = i1 i2 i3 i4 V (r w

(i)

·w

(j)

= aij

i, j = 1 2 3

5

= x(1) − x(3) = r(1) (α) (β) (γ)

ri1 ri2 ri3 + (α) (β) (γ) (δ)

ri1 ri2 ri3 ri4 +

m=3 (2) x = x(2) − (α)

riq δir is

(α) (β)

+

rip riq δir is +

+

δip iq δir is

{1, 2} r (1) · r(1) r (1) · r(2) Q(j)

{1, . . . , j} r(2) · r (2)

j

V

IV

S S

m=2 Q(2) i1 i2

(1)

x = x(1) − x(2) = r (r) = Q(2) (r) + A rk εki1 i2 i1 i2 V IV

(3) Q(3) i1 i2 i3 IV (r) = Qi1 i2 i3 V (r) +

p q

s

rip rk εkiq is + A εi1 i2 i3

r·r = r2

{1, 2, 3}

x

m=3 x (2) = x(2) − x(3) = r (2) (1) Q(3) , r(2) ) = i1 i2 i3 IV (r

(1)

r IV = x(1) −x(3) = r(1)

(1) Q(3) , r (2) ) + i1 i2 i3 V (r (α) (β)

+

rip rk εkiq is +

+

rip riq rk rh εkhis + Aεi1 i2 i3

(1) Q(4) , r(2) ) = i1 i2 i3 i4 IV (r

(α) (β) (γ) (δ)

(1) Q(4) , r(2) ) + i1 i2 i3 i4 V (r (α)

+

rip εil iq is +

+

rip ril rk εkiq is +

(α) (β) (γ)

(α) (β) (γ) (δ) (μ)

+

α β γ

rip ril riq rk rh εkhis

{1, 2}

δ p l q

s

{1, . . . , j}

4 5

III λ = (λ1 , λ2 , λ3 ) j

a(1) a(2) . . . a(j) λ

S =

x

(1)

, x

(2)

, . . . , x

(m−1)

, λ | a(1) , a(2) , . . . , a(j)

a(1) a(2) . . . a(j)

a

(2)

... a

(j)

r (k) k = 1 2 . . . m − 1 λ

a(1) m=2

1 (r) = Q(1) (r) + B λi , Q(1) i i III V

x(1) − x(2) = r

r · r = r2 r · λ = rλ (1) m = 2 x =

r 1

(2) Q(2) i1 i2 III (r) = Qi1 i2 V (r) + (3) (3) Qi1 i2 i3 III (r) = Qi1 i2 i3 V (r) +

λip riq + C λi1 λi2 λip riq ris +

+E λi1 λi2 λi3 + p q

λip δiq is

s

r2

V 3 (1)

Qi

(r) = Qi

(1)

II

1 III

(r) + C λk rl εkli ,

r2

rλ

(2) Q(2) i1 i2 II (r) = Qi1 i2 III (r) + +

Q(3) i1 i2 i3

II

(r) = Q(3) i1 i2 i3

λk rl rip εkliq +

λk λip rl εkliq +

+

{1, . . . , j} rλ

IV

1 2

1

λip λiq ris +

λk εki1 i2 +

rk εki1 i2 III

(r) + F εi1 i2 i3 +

λk rip εkiq is +

+

λk λip εkiq is +

rk λip εkiq is +

+

rk rip εkiq is +

λk rl rip riq εklis +

+

λk rl λip riq εklis +

λk rl λip λiq εklis

I λ = (λ1 , λ2 , λ3 )

μ = (μ1 , μ2 , μ3 ) λ

μ u(x)

˜ (j) (y (1) , . . . , y (j) | a(1) , . . . , a(j) ) Q

m≤j S =

x

(1)

, x

(2)

, . . . , x

(m−1)

j

, λ, μ | a(1) , a(2) , . . . , a(j) a(i) 1

a(1)

(r) = A ri + B λi + C μi , Q(1) i I A B

r·r r·λ

C r

u(x) a(1) a(2) . . . a(j) Q(2) i1 i2 I (r) =

A ri1 ri2 +

j

S rip λiq +

rip μiq +

λip μiq + B λi1 λi2 + C μi1 μi2 + D δi1 i2

+ Q(3) i1 i2 i3 I (r) =

r·μ m=2

A ri1 ri2 ri3 +

λip riq ris +

+

μip riq ris +

λip μiq ris +

+

λip λiq μis +

λip μiq μis +

+B λi1 λi2 λi3 + C μi1 μi2 μi3 +

p q

+

λip δiq is +

+

rip δiq is

μip δiq is +

{1, 2, 3}

s r

r (1) r(2) . . . r(m−1)

u(x)

IV λ |λ| = 1

|μ| = 1

μ

I j=2

m=2

r Q(2) i1 i2 IV

λ=μ=0 λ μ

i1 = i2 Q(2) i1 i2 (r) =

p

A ri1 ri2 + B δi1 i2 + C rk εki1 i2 + + rip λiq + rip μiq + λip μiq + D λi1 λi2 + E μi1 μi2 + + +F λk εki1 i2 + G μk εki1 i2 + H rk λl εklip riq + +I rk μl εklip riq + L rk λl εklip λiq + M rk λl εklip μiq + +N rk μl εklip λiq + O rk μl εklip μiq + P λk μl εklip riq + +Q λk μl εklip λiq + R λk μl εklip μiq

(3) (4) (4) (4) (6) (6) (4)

{1, 2}

q r

31

u(x)

3 j=3 2 (i1 , i2 , i3 ))

m=2

Si1 i2 i3 (r) =< ui1 (x)ui2 (x)ui3 (x + r) > , r

2

(i1 , i2 , i3 )

(i1 , i2 ) (i1 , i2 , i3 )

Si1 i2 i3 (r) = A ri1 ri2 ri3 + B ri1 δi2 i3 + C ri2 δi1 i3 + D ri3 δi1 i2 , A B C i1

D

r

i2 C

B

Si1 i2 i3

Si2 i1 i3

0 ≡ Si1 i2 i3 (r) − Si2 i1 i3 (r) = (B − C ) (ri1 δi2 i3 − ri2 δi1 i3 ) , r

B ≡ C

(i1 , i2 , i3 ) S

Si1 i2 i3 (r) = A ri1 ri2 ri3 + B (ri1 δi2 i3 + ri2 δi1 i3 ) + D ri3 δi1 i2 . S r

A B

D

u(x) ∇·u = 0

1 Q(1) (r) r =0 A

A

r

A

r

0 = ∂ri Q(1) i = (A ∂ri r)ri + A∂ri ri = rA + 3A .

r→0

< p(x)u(x + r) >

A≡0

A ∝ r−3

A r=0

(i, j)

Rij (r) =< ui (x)uj (xv+r) > r

Rij (r) = F (r) ri rj + G(r) δij , F

G

r F

r

G G

F

u(x) 0 ≡ ∂rj Rij (r) = F

rj rj G F ri rj + F δij rj + F ri 3 + G δij = rF + 4F + ri , r r r

G

F

G

r r

rF + 4F +

rˆ

G =0. r

r rˆ = r/r rˆ⊥ u · rˆ = up

u · rˆ⊥ = un < up (x)up (x + r) > < un (x)un (x + r) > .

u2 =

1 1 < ui (x) ui (x) >=< up (x) up (x) >=< un (x) un (x) >= Rii (0) , 3 3 r

f (r) =

< up (x) up (x + r) > < un (x) un (x + r) > , g(r) = , 2 u u2

u(x + r)

z un (x + r) u(x) x+r

un (x)

up (x + r)

up (x)

x

ˆ⊥ r

y

ˆ r x

up un ˆ x+r r r ˆ⊥ r r up (x) = u(x) · rˆ ˆ⊥ up (x + r) = u(x + r) · rˆ un (x + r) = u(x + r) · r

x ˆ⊥ un (x) = u(x) · r

f f

g

g

u2 f (r) = rˆi rˆj Rij (r) = r2 F + G u2 g(r) = rˆi⊥ rˆj⊥ Rij (r) = G , F

G

r

f

g

r F G = u2 g

u2 (f − g)/r2

F =

G

Rij (r) = u

2

ri rj [f (r) − g(r)] 2 + g(r) δij r f

g=f+ f

g

r 1 d 2 f = (r f ) . 2 2r dr

g m

g

f

+∞

dr rm g(r) = 0

+∞

rm f (r) + 0

1 2

0

+∞

dr rm+1 f (r)

m−1 =− 2

+∞

dr rm f (r) . 0

g g(r) f (r)

g(r)

A g = g(r)

GA > 0

r

B

GB < 0

g rA GA + rB GB

Lp =

0

rA

rB

r

+∞

dr f (r)

Ln =

0

A

+∞

dr g(r) = 0

B

Lp 2

f

g

f (0) = g(0) = 1 g

f

r

f g 1

A

λ 0

√

2λ

r B

f = f (r) g = g(r) r

r=

√

r = 0

f = f (r) g = g(r)

2λ

r = λ 2 g = g(r)

A

B

r

f (r) = 1 +

r2 f (0) + O(r4 ) , g(r) = 1 + r2 f (0) + O(r4 ) . 2 |f (r)| ≤ 1

u(x) |g(r)| ≤ 1

r = 0 g (0) = 2f (0) < 0

f (0) < 0

f (0) = g(0) = 1

λ f (0) = −

f (r) = 1 −

1 , λ2

1 r2 r2 + O(r4 ) , g(r) = 1 − 2 + O(r4 ) . 2 2 λ λ λ ε λ η = ν 3/4 /ε1/4

wii (r) = −∇2 Rmm (r) = −u2 ∇2 (f + 2g) = ∇2 r2 = ∂ri (2ri ) = 6 ωi (x)ωi (x) = wii (0) = 15 u2 /λ2 λ

L

5 u2 2 2 ∇ r + O(r2 ) . 2 λ2 √ λ= 5L

r = 0 L

Φij Φij (k) = A(k)ki kj + B(k)δij , A(k) B(k) kj Φij = Ak 2 ki + Bki = (Ak 2 + B)ki = 0 B(k) = −k 2 A(k)

A(k) E(k)

1 Ψii (k) 2 1 = dS(k)Φii (k) 2 ∂Bk (0) 1 = · 4πk 2 · Ak 2 + 3B = −4πk 4 A , 2

E(k) =

A(k) = −E(k)/(4πk 4 ) Φij (k)

(i, j)

E(k) ki kj δij − 2 . 2 4πk k

Φij (k) =

r Sij (r) Ψij (k)/k E(k) Rii 1 u2 Rii (r) = [rf (r) + 3f (r)] := R(r) , 2 2 r Ψii (k)/k = 2 E(k)/k

rSii (r) = 2 rR(r) rR(r)

r E(k)/k

1/2

R(r) f (r)

E(k) 2 1 E(k) = k π

+∞

0

u2 dr rR(r) sin kr = π

u2 E(k) = π

+∞

dr k 2 r2

0

E(k)/k

0

dk ds 0

dr (r sin kr − kr2 cos kr)f (r) ,

sin kr − cos kr f (r) . kr

+∞

rR(r) =

+∞

E(k) sin kr , k

r u2 u2 d 3 [3r2 f (r) + r3 f (r)] = [r f (r)] = r 2 2 dr

+∞

dk E(k) 0

sin kr k

r u2 f (r) = 2

dk 0

E(k)

+∞

1 k 2 r2

f = f (r) k=0

sin kr − cos kr E(k) . kr E = E

r=0

f

Ωij (k)

Φij (k)

Ωii (k) = k 2 Φii (k)

Ωij (k) = (δij k 2 − ki kj )

E(k) E(k) ki kj δij − 2 = k 2 Φij (k) . 2 − k2 2 2 4πk 4πk k (i, j)

wij (r)

f (r)

wij 8 f + 7f + rf r 1 ri rj

4 2 2 (4f + rf )δij + rf + 4f − f ∂ri rj Rmm = u r r r2 r ri rj 2 ∇2r Rji = u2 + − f − 2f + f 2 r r2 r

2 + f + 3f + f δij , 2 r ∇2r Rmm = u2

wij (r) = u2

r ri rj

r 2 2 f + 2f − f f + 4f + f δij . − 2 r r2 2 r

(3)

Qijl (r)

Sijl (r) = ui (x)uj (x)ul (x + r)

Sijl (r) = Ari rj rl + B(ri δjl + rj δil ) + Drl δij , A B

D

r

3

2

rl A ri rj rl + A(δil rj rl + ri δjl rl + ri rj δll ) + r rl + B (ri δjl + rj δil ) + B(δil δjl + δjl δil ) + r rl + D rl δij + Dδll δij r B + δij (2B + rD + 3D) ≡ 0 , = ri rj rA + 5A + 2 r

∂rl Sijl (r) =

r

rA + 5A + 2B /r = 0 2B + rD + 3D = 0 . Siil

Siil (r) = (Ar2 + 2B + 3D)rl ≡ 0

Ar2 + 2B + 3D = 0 . r4

r5 A + 5r4 A + 2r3 B =

d 5 (r A) + 2r3 B = 0 , dr

r2 3r2 D + r3 D + 2r2 B =

d 3 (r D) + 2r2 B = 0 dr r5 A + 3r3 D + 2r3 B ≡ r3

3 = 0

r = 0

r

B=−

r 3 D − D 2 2

B

A A=

D

1 D . r

D(r) (i, j, l)

Sijl (r)

rˆ

⊥

ˆ r up (x) u(x + r) · rˆ = up (x + r) ˆ ⊥ = un (x + r) u(x + r) · r

x

∞1 u(x) · rˆ = u(x) · rˆ ⊥ = un (x) x+r

k h q u3 k(r) = up (x)up (x)up (x + r) u3 h(r) = un (x)un (x)up (x + r) u3 q(r) = un (x)up (x)un (x + r) k = k(r) h = h(r) q = q(r) A B D ri rj rl Sijl = Ar3 + 2Br + Dr r3 ri⊥ rj⊥ rl Sijl = rD u3 h = r3 r⊥ rj r⊥ u3 q = i 3 l Sijl = rB r u3 k =

u3 k = −2rD u3 h = rD r2 3 u3 q = − rD − D , 2 2

k = k(r) (i, j, l) D = u3

−

Sijl (r) = u3

k − rk 2k + rk k , A = u3 − , , B = u3 − 3 2r 2r 4r

1 1 rl

ri ri rj rl 1 rj (k − rk ) 3 + (2k + rk ) δjl + δil − k δij 2 r 4 r r 2 r k = k(r)

D h = h(r) r

q = q(r)

D = D(r) k h Υ Υijl (k) =

(i, j, l)

1 (2π)3

q IR3

(i, j, l) dr Sijl (r) e− k·r

Υ Υijl (k) = Aki kj kl + B(ki δjl + kj δil ) + Dkl δij ,

A B

D

k

k

kl Υijl = (Ak 2 + 2B) ki kj + Dk 2 δij ≡ 0 , B = −k 2 A/2 D ≡ 0 (i, j, l)

Υ

A A(k) = Υ (k)

Υ = Υ (k) Υijl (k) = Υ (k)

ki kj kl −

k2 (ki δjl + kj δil ) 2

Υ (k) k(r) S Siji (r) =

u3 rk + 4k u3 rj =: K(r) rj , 2 r 2 r K = K(r)

Υiji (k) = − Υ (k) k 2 kj u3 1 = dr K(r)rj e− k·r (2π)3 2 IR3 i u3 ∂k dr K(r) e− k·r = 2 (2π)3 j IR3 +∞ u3 ∂k dr K(r) dS(r) e− k·r . = 2 (2π)3 j 0 ∂Br (0) r

dS(r) = r2 sin ϕdϕdθ

∂Br (0) r = r(sin ϕ cos θ, sin ϕ sin θ, cos ϕ)

sin kr dS(r) e− k·r = 4πr2 kr ∂Br (0)

u3 Υiji (k) = ∂k 2 (2π)3 j = u3 = u3

(2π)2 (2π)2

0

+∞

dr K(r) 4πr2

sin kr kr

1 +∞ dr K(r)r sin kr k 0 kj +∞ 1 +∞ kj

− 3 dr K(r)r sin kr + dr K(r)r cos kr r k 0 k 0 k

∂kj

d kj +∞ kj +∞ sin kr =u dr K(r)r sin kr + 3 dr K(r)r2 − 3 2 (2π) k 0 k 0 dr +∞ kj = −u3 dr [3rK(r) + r2 K (r)] sin kr (2π)2 k 3 0 kj +∞ sin kr 3 = −u , dr ∂r [r3 K(r)] (2π)2 k 2 0 kr

3

Υ =

Υiji

Υ (k) Υ

u3 1 Υ (k) = (2π)2 k 4

8πk 5 Υ (k) =

2 π

+∞

0

+∞

0

dr ∂r [r3 K(r)]

sin kr . kr

dr [u3 r(r∂r + 3)K(r)] sin kr ,

u3 r(r∂r + 3)K(r) =

+∞

dk [8πk 5 Υ (k)] sin kr .

0

4

u3

1 K(r)rj = 2

dk Υiji (k) e k ·r

IR3

+∞

=− 0

= −∂rj

dk Υ (k)k2

dS(k)kj e k·r ∂Bk (0)

+∞

dk Υ (k)k

2

0

dS(k) e k ·r , ∂Bk (0)

∂Br (0)

Siji (r) = −∂rj

+∞

0

1 rj ∂r = −4π r r

u3 K(r) = 8π

1 r3

0

+∞

dk Υ (k)k3 sin kr

,

0

rj +∞

sin kr kr

dk Υ (k)k2 4πk2

dk Υ (k)k3 sin kr −

1 r2

0

j +∞

dk Υ (k)k4 cos kr

.

z

z

(x, y)

(x , y )

ϕ ∈ [0, π]

N z

z

ez ×ez N⊥ N⊥

a N

N (x, y) (x , y ) x θ ∈ [0, 2π) N x χ ∈ [0, 2π) N = (cos θ, sin θ, 0) RC N⊥ = (− sin θ, cos θ, 0) N = (cos χ, sin χ, 0) RC N⊥ = (− sin χ, cos χ, 0) (ϕ, θ, χ) a ez (x, y, z) z cos ϕ ez N e x ex × N = sin χ ez ey e y = e z × e x

e x =

ey =

ez =

sin θ sin χ cos ϕ + cos θ cos χ − cos θ sin χ cos ϕ + sin θ cos χ − sin χ sin ϕ − sin θ cos χ cos ϕ + cos θ sin χ cos θ cos χ cos ϕ + sin θ sin χ cos χ sin ϕ sin θ sin ϕ − cos θ sin ϕ cos ϕ

RC ez RC R(ϕ, θ, χ)

R=

b RC RC e x e y RC

sin θ sin χ cos ϕ + cos θ cos χ − cos θ sin χ cos ϕ + sin θ cos χ − sin χ sin ϕ − sin θ cos χ cos ϕ + cos θ sin χ cos θ cos χ cos ϕ + sin θ sin χ cos χ sin ϕ sin θ sin ϕ − cos θ sin ϕ cos ϕ

R

R(ϕ1 , θ1 , χ1 )R(ϕ2 , θ2 , χ2 ) ϕ θ u3 r(r∂r + 3)K(r)

χ

z

z z

ϕ

v z

z

ez

y

ey

y O

y

N

x

χ

x

y

v y

e x v x w x

θ x

w y

N x

a

b

a b RC RC ex (x , y ) cos χ N − sin χ N⊥ = v x + w x v x = − sin χ cos ϕ N⊥ w x = ey − sin χ sin ϕ ez (x , y ) sin χ N cos χ N⊥ = v y + w y v y = cos χ cos ϕ N⊥ w y = cos χ sin ϕ ez ez cos ϕ ez − sin ϕ N⊥

sin

θ1 − χ 2 θ1 − χ 2 = u , cos =v 2 2

(3, 3) cos ϕ = v 2 cos(ϕ1 + ϕ2 ) + u2 cos(ϕ1 − ϕ2 ) ,

sin2 ϕ = =

ϕ ∈ [0, π] v sin(ϕ1 + ϕ2 ) − u2 sin(ϕ1 − ϕ2 )

2

A

v 2 sin(ϕ1 + ϕ2 ) + u2 sin(ϕ1 − ϕ2 )

2

2

+ +

2uv sin ϕ1

(3, 1)

B

2uv sin ϕ2

2 2

= A2 + B 2 = C 2 + D2 ,

D

C

sin ϕ ≥ 0

sin ϕ =

√

A2 + B 2 =

√

C 2 + D2

sin θ sin ϕ = A sin θ2 + B cos θ2 sin θ

1 sin θ =

|A sin θ2 + B cos θ2 |/ sin ϕ ≤

A sin θ2 + B cos θ2 √ . A2 + B 2 (3, 2)

cos θ

cos θ =

A cos θ2 − B sin θ2 √ . A2 + B 2 θ

(1, 3)

(2, 3) C sin χ1 − D cos χ1 C cos χ1 + D sin χ1 √ √ , cos χ = C 2 + D2 C 2 + D2

sin χ = χ

(ϕ1 , θ1 , χ1 )

ϕ=0 θ

ϕ1

χ

ϕ=0

⎧ 2 v cos(ϕ1 + ϕ2 ) + u2 cos(ϕ1 − ϕ2 ) = 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v 2 sin(ϕ1 + ϕ2 ) = 0 ⎪ ⎨ u2 sin(ϕ1 − ϕ2 ) = 0

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

uv sin ϕ1 = 0 uv sin ϕ2 = 0 .

uv = 0 u=0 u=0 v2 = 1 cos(ϕ1 + ϕ2 ) = 1 sin(ϕ1 + ϕ2 ) = 0 ϕ1 + ϕ2 = 0 ϕ2 [0, π) v =0 χ 2 = θ1 + π θ1 ∈ [0, π) χ2 = θ1 ∈ [π, 2π) (ϕ2 , θ2 , χ2 ) θ1 − π (ϕ1 , θ1 , χ1 ) ez × ez χ2 u2 = 1 ϕ1 = ϕ2 θ2 = χ1 + π χ1 ∈ [0, π) θ2 = χ1 − π χ1 ∈ [π, 2π) v=0

ϕ1

d = (sin γ cos μ, sin γ sin μ, cos γ)

e = (sin δ cos ν sin δ sin ν

cos δ) x S(d) · S(e)

(ϕ, θ, χ)

α cos α = d · e = sin γ sin δ cos(μ − ν) + cos γ cos δ .

(3, 3)

S(d) · S(e)

ϕ cos ϕ = u2 cos 2(δ − γ) + v 2 cos 2(δ + γ) ,

u = cos

sin2 ϕ = =

μ−ν μ−ν , v = sin . 2 2

2

u2 sin 2(δ − γ) + v 2 sin 2(δ + γ)

2

E

u2 sin 2(δ − γ) − v 2 sin 2(δ + γ)

+ +

2uv sin 2γ

(3, 1) sin θ =

E2 + F 2 =

F

2uv sin 2δ

2

H

G

sin ϕ =

2

G2 + H 2 .

(3, 2)

θ

+E cos ν + F sin ν −E sin ν + F cos ν √ √ , cos θ = , E2 + F 2 E2 + F 2 sin ϕ

(1, 3)

(2, 3)

χ sin χ =

RC

+G cos μ + H sin μ −G sin μ + H cos μ √ , cos χ = √ , G2 + H 2 G2 + H 2

RC

f (ϕ, θ, χ) RC

d = (d1 , d2 , d3 ) RC

⎛

RC

RC

RC

RC

RC

e x ey ez

1 − 2d21 −2d1 d2 −2d1 d3

⎞⎛

RC

RC

RC ex1 ey 1 ez 1

⎞

⎝ −2d2 d1 1 − 2d22 −2d2 d3 ⎠ ⎝ ex2 ey 2 ez 2 ⎠ . −2d3 d1 −2d3 d2 1 − 2d23

αx αy

αz

RC cos αx = d · ex = sin γ cos αy = d · ey = sin γ

ex3 ey 3 ez 3

d = (sin γ cos β, sin γ sin β, cos β)

+ u2 cos(χ − θ − β) + v 2 cos(χ + θ − β) + 2uv cos γ sin θ

2 2 − u sin(χ − θ − β) + v sin(χ + θ − β) 2

2

cos αz = d · ez = cos γ (u − v ) − 2uv sin γ sin(χ − β) ,

− 2uv cos γ cos θ

R u = cos(ϕ/2)

⎛

E

v = sin(ϕ/2)

ex1 − 2d1 cos αx ey 1 − 2d1 cos αy ez 1 − 2d1 cos αz

⎞

⎝ ex2 − 2d2 cos αx ey 2 − 2d2 cos αy ez 2 − 2d2 cos αz ⎠ . ex3 − 2d3 cos αx ey 3 − 2d3 cos αy ez 3 − 2d3 cos αz RC −1

(1, 1) d1 = cos ω

RC

+1 β ω ex1 = cos β d2 e y + d3 e z ζ d · ex = (1, 1)

ex2 ey + ex3 ez | sin β| | sin ω| cos αx = cos β cos ω + sin β sin ω cos ζ

ex1 − 2d1 cos αx = cos β − 2 cos ω cos αx = −(cos β cos 2ω + sin β sin 2ω cos ζ) , −1

R

+1

E rR(r)

k =0

E(k) f

R q : IR+

E(k)/k

f → IR

0 E(k) r →∞

M(m) [q]

R(r)

m M(m) [q] =

+∞

dr rm q(r) . 0

k=0 sin kr E(k) =

2 π

+∞

dr kr R(r) sin kr = 0

E(k)

∞ 2 # (−1)p k2p+2 M(2p+2) [R] . π (2p + 1)! p=1

k k 5

M(2) [R] M

(2)

[R] = 0

+∞

u2 dr r R(r) = 2

f (r)

2

0

+∞

dr ∂r [r 3 f (r)] = 0 ,

E(k)

E(k) E(k) =

∞ # k2q q=2

∂k2q E(k) k=0 ,

(2q)! E

E = E(k)

k → 0+

4 ∂k2m E(k) k=0 = (−1)m+1 mM(2m) [R] . π R(r) f

R

M(m) [R] = −

u2 (m − 2) M(m) [f ] . 2

4 ∂k2m E(k) k=0 = (−1)m m(m − 1) u2 M(2m) [f ] . π m = 0 E(0) = 0

∂k2 E(k) k=0 = 0 E = E(k) k kL r =0

m = 1

f R

E(k)

sin kr

dk 0

# (−1)p 2p+1 E(k) sin kr = r k (2p + 1)! +∞

+∞

rR(r) =

p=0

+∞

dk k2p E(k) ,

0

R(r) R(r) =

+∞ # (−1)p p=0

(2p + 1)!

R = R(r)

r 2p M(2p) [E] .

r

R(r) =

∞ # r 2p p=0

(2p)!

R(r)

∂r2p R(r) |r=0 .

r=0 R(r)

(−1)m ∂r2m R(r) r=0 = M(2m) [E] , 2m + 1

R r =0 R f = f (r)

r 2 R = u2 ∂r (r 3 f )/2

r2

E

R(r)

∞ # r 2m m=0

∞ #

u2 r 2l 2l ∂r2m R(r) r=0 = ∂r r 3 ∂r f (r) r=0 , (2m)! 2 (2l)! l=0

r=0

f (r)

R(r)

2m + 3 2m ∂r2m R(r) r=0 = u2 ∂r f (r) r=0 . 2 R(r)

u2 ∂r2m f (r) r=0 =

f = f (r)

2(−1)m M(2m) [E] , (2m + 1)(2m + 3)

E = E(k) r = 0

k=0 f

E m=0

1

0

2

f = f (r) E = E(k)

+∞

drR(r) = u2

0

+∞

dr f (r) 0

rR(r) u2

+∞

+∞

dr f (r) ≡ 0

dr 0

+∞

= 0

+∞

= 0

=

6

sin x/x z = x + y e−z /z

π/2

1 rR(r) r

+∞

dk 0

E(k) dk k

+∞

dk 0

sin x/x

dr r

E(k)/k

E(k) k

E(k) sin kr k

+∞

dr 0

+∞

dx 0

sin kr r sinx . x

2

+∞

u

0

π dr f (r) = 2

+∞

E(k) , k

dk 0

k=0

E(k)

k kL

2 f r 2 R(r) = u2 ∂r [r 3 f (r)]/2

f = f (r) u2

+∞

E = E(k) dr r2 f (r) =

0

+∞

0

+∞

= 0

+∞

dr r2 2

E(k) dk k2

dk 0

+∞

dr 0

E(k) sin kr − cos kr 2 2 k r kr

sin kr −2 kr

+∞

+∞

dr

dk

0

0

k

+∞

dr

+∞

r

E(k) cos kr = k2

dk

0

E(k) cos kr , k2

0

y

x −δ +δ

−Δ

−δ

Δ

+ −Δ

e −z + z

dz

δ

I1

Δ→∞ I1 =

−δ

+Δ

+ −Δ +π/2

I2 =

+δ

dθ e−δe

θ

dz

Cδ

e −z + z

−π/2

CΔ

e −z =0, z

δ → 0

cos y − sin y dy = −2 y

Δ

δ

→ π

dθ e−Δ(cos θ+

dz

I3

−π/2

I3 =

I3

I2

I1 I2

+Δ

sin θ)

→0.

+π/2

+∞

dx 0

π sin x = . x 2

sin y dy → −2 y

+∞

dy ds 0

sin y y

R

+∞

= 0

+∞

= 0

=−

dr r

+∞

dk 0

E(k) d sin kr k2 dk

k=+∞ dr E(k) sin kr k=0 − 2 r k

+∞

dk

∂k

0

u2

E(k)

k2

E

+∞

+∞

dx 0

E(k)

+∞

dk 0

∂k

E(k)

sin kr 2 k

π E k=+∞ sin x =− =0, x 2 k2 k=0

dr r2 f (r) = π

0

+∞

dk 0

k=0

E(k) , k3

f

n+1

n

u = u(x, t) p˜ = p(x, t)/ρ

x

i

∇2x = ∂x2k xk

∂t ui + uk ∂xk ui = −∂xi p˜ + ν∇2x ui j x 2 2 p˜ = p(x , t)/ρ ∇ = ∂x x

x

k

∂t uj

u = u(x , t)

x

k

+ ul ∂xl uj = −∂xj p˜ + ν∇2x uj , uj

ui

uj ∂t ui + ui ∂t uj + uk uj ∂xk ui + ui ul ∂xl uj = = −uj ∂xi p˜ − ui ∂xj p˜ + ν (uj ∇2x ui + ui ∇2x uj ) , x

u(x, t)

u (x, t)

x ∂t ui uj + ∂xk ui uk uj + ∂xl ui ul uj = = −∂xi uj p˜ − ∂xj ui p˜ + ν (∇2x ui uj + ∇2 ui uj ) . x

x

x

r = x −x

∂xp =

r ∂xp |x x

= ∂rp

∂xp = ∂xp |x

= −∂rp

x

ui uj

∂t ui uj = ∂rk (ui uk uj − ui uk uj ) + ∂ri uj p˜ − ∂rj ui p˜ + 2ν∇2r ui uj , u(x, t) ui uj

∇2r = ∂r2k rk u (x , t)

∂t < ui uj > =

∂rk (< ui uk uj > − < ui uk uj >)+ +∂ri < uj p˜ > −∂rj < ui p˜ > +2ν∇2r < ui uj > . N

∂t ui uj =

∂rk (ui uk uj − ui uk uj )+ +∂ri uj p˜ − ∂rj ui p˜ + 2ν∇2r ui uj .

(i, j) Tij (r) = ∂rk [ui (x)uk (x)uj (x + r) − ui (x)uk (x + r)uj (x + r)] Pik (r) = ∂ri uj (x + r)˜ p(x) − ∂rj ui (x)˜ p(x + r) ,

∂t Rij = Tij + Pij + 2ν∇2r Rij u(x, t) 2 ∂t u + u∂x u = ν∂xx u+f

[−π, +π) × (0, +∞) f

f

T

P (i, j)

Γij (k) =

1 (2π)3

IR3

dr Tij (r) e− k·r , Πij (k) =

1 (2π)3

IR3

dr Pij (r) e− k·r ,

∂t Φij = Γij + Πij − 2νk 2 Φij

Γ

Π

exp(−2νk 2 t) k kL

k kL

x

x

u(x, t) u(x , t) Tij Tij (r) = ui (x)uk (x)∂rk uj (x + r) − ui (x)uk (x + r)∂rk uj (x + r) , T (0) = 0 IR3

dk Γ (k) ≡ 0 .

Γ kL k kη

E = E(k) k Π P

Pii (r) = ∂ri ui (x + r)˜ p(x) + ∂ri ui (x − r)˜ p(x) ≡ 0 , Πii (k) ≡

r 0

Π

k

k k

k

r=0 Pij (r) ≡ 0 Tij (r) = ∂rk [Sikj (r) − Sjki (−r)] f = f (r, t) ∂t Rii = Tii + 2ν∇2r Rii Rii Rii = u2 (rf + 3f ) = (r∂r + 3) u2 f Siki (r) = u3 K(r)rk /2 Tii (r) =

Tii

1 ∂r [u3 K(r)rj − u3 K(r)(−rj )] = (r∂r + 3) u3 K(r) , 2 j

K(r) = (∂r + 4/r)k(r) r

Rii (r)

∇2r Rii = u2

(r∂r + 3)

r=0

rf + 7f +

∂t u2 f − u3

8f 4 = (r∂r + 3) u2 f + f . r r

∂r +

4 4 k − 2νu2 ∂r2 + ∂r f = 0 . r r (r∂r + 3)F = 0

F ≡0 ∂t u2 f = u3

∂r +

F

4 4 k + 2νu2 ∂r2 + ∂r f r r 1938 r

u2 u2 (t)

t

r = 0 r

f (r) = 1 − r2 /(2λ2 ) + O(r4 ) k = k(r) k(r) = O(r3 )

f = f (r)

−

d 3u2 15νu2 =ε= , dt 2 λ2 λ

λ η

u L

= uL/ν

L

η

1

L λ

ε λ

λ 1/2 η ∝ L L

−1/2 L

,

η λ L

E(k) . 2πk 2

Φii (k) =

Γij Tij 1 Γij (k) = (2π)3

IR3

dr ∂rp [Sipj (r) − Sjpi (−r)] e− k·r

Υ Γij (k) = kp [Υipj (k) − Υjpi (−k)] . Υijk Γij

k2 ki δpj + kp δij + 2 k2 − Υ (k) − kj kp ki − − kj δpi − kp δji 2 ki kj = k 4 Υ (k) δji − 2 , k

Γij (k) = kp

Υ (k)

ki kp kj −

Γii (k) = 2k 4 Υ (k) , k Π ∂t E = 4πk 6 Υ − 2νk 2 E 1947

4πk 6 Υ

−2νk 2 E t

k E(k, t) k E(k, t) ν

k

4πk 6 Υ Γii

0=

1 2

IR3

dk Γii (k) =

+∞

dk 0

dS(k)

∂Bk (0)

1 Γii (k) = 2

+∞

dk 4πk 6 Υ (k) .

0

4πk 6 Υ (k)

Φij (k, t) = C ijlm (t) kl km + O(k 3 ) . C ijlm Γ Π

Γ Γij 1 Γij (k) = (2π)3 =

IR3

kp (2π)3 −

dr Tij (r) e− k·r

IR3

IR3

dr ui (x)uj (x + r)up (x) e− k·r +

dr ui (x)uj (x + r)up (x + r) e− k·r

Π

kp − k ·r − dr S (r) e dr Sjpi (−r) e− k·r ipj 3 (2π) IR3 IR3

kp = dr Sipj (r) e− k·r − dr Sjpi (r ) e k·r 3 (2π) IR3 IR3

=

= kp [Υipj (k) − Υjpi (−k)] .

...

Υijl (k) Υijl (k) = Gijl + kp Gijlp + . . . Gijl = Gjil Gijlp = Gjilp kl Υijl (k) ≡ 0 k 0 = kl Gijl + kl kp Gijlp + . . . # Gijl = 0 , Gijlp = 0 , . . . π(l,p)

(i, j, l) Υijl (k) = km Gijlm + O(k 2 ) = −km Gijml + O(k 2 ) . Γij

(i, j)

Γij (k) = kl km (Giljm + Gjlim ) + O(k 3 ) # = kl km Giljm + O(k 3 ) π(i,j)

#

= kl km

2

#

Giljm +O(k 3 )

π(l,m) π(i,j)

Γijlm

= kl km Γijlm + O(k 3 ) . (i, j) p˜ = p/ρ

Π

Θi (k) =

1 (2π)3

IR3

dr ui (x + r)˜ p(x) e− k·r .

r = −r ui (x)˜ p(x + r) = ui (x − r)˜ p(x)

1 (2π)3

=

1 = (2π)3 Πij 1 Πij (k) = (2π)3

IR3

IR3

dr ui (x − r)˜ p(x) e− k·r

IR3

dr ui (x + r )˜ p(x) e− (−k)·r = Θi (−k) .

dr [∂ri uj (x + r)˜ p(x) − ∂rj ui (x)˜ p(x + r)] e− k·r

= [ki Θj (k) − kj Θi (−k)] . Θi (k) Θi (k) = T i +kj T ij +. . . 0 Ti = 0 ,

ki Θi (k) ≡ 0 #

ki T i +ki kj T ij +. . . ≡

T ij = 0 . . .

π(i,j)

Θi (k) Θi (k) = kj T ij + O(k 2 ) , (i, j)

Π

Πij (k) = (ki km T jm + kj kh T ih ) + O(k 3 ) = kp kq (T jq δip + T ip δjq ) + O(k 3 ) 1 # = kp kq (T ip δjq + T jp δiq ) + O(k 3 ) 2 π(p,q)

= kp kq

# # 2

T ip δjq +O(k 3 )

π(i,j) π(p,q)

Πijpq

= kp kq Πijpq + O(k 3 ) . (i, j) Φ

Γ

Π

Cijlm Φ Γijlm + Πijlm Γijlm + Πijlm ≡ 0 k

t

p/ρ =: p˜ x

∇2x p˜ = −∂x2i xl ui ul , p˜ = p˜(x, t) x

x uj (x , t) = uj

u(x, t)

u =

∇2x uj p˜ = −∂x2i xl ui ul uj . r = x − x

∇2r uj (x + r)˜ p(x) = −∂r2i rl ui (x)ul (x)uj (x + r) . Θj

k 2 Θj (k) = −ki kl Υilj (k) . Θj Υilj 0 = k 2 km T jm + ki kl km Giljm + O(k 4 ) = ki kl km (δil T jm + Giljm ) + O(k 4 ) , k #

(δil T jm + Giljm ) = 0 .

π(i,l,m)

(i, j) Γ

Π Πijlm + Γijlm = =

# 2

π(i,j) π(l,m)

# 2

# #

π(i,j) π(l,m)

δjm T il +

# 2

#

π(l,m) π(i,j)

(δil T jm + Giljm ) ,

Giljm

#

0=

(δil T jm + Giljm ) +

π(l,m)

+

#

#

(δli T jm + Glijm ) +

π(l,m)

(δml T ji + Gmlji )

π(l,m)

=2

#

(δil T jm + Giljm ) +

π(l,m)

#

(δlm T ji + Glmji ) ,

π(l,m)

Giljm = Glijm #

(δil T jm + Giljm ) = −

π(l,m)

1 # (δlm T ji + Glmji ) = −(δlm T ji + Glmji ) , 2 π(l,m)

Γijlm + Πijlm = − =−

# 2 2

(δlm T ji + Glmji )

π(i,j)

[δlm (T ji + T ij ) + (Glmji + Glmij )] ,

Φij

Γij

˙ ijlm kl km = O(k 3 ) , C C ijlm ≡

k kL C ijlm

1 2 ∂kl km Φij (k, t) k = 0 2 1 2 = ∂ dr Rij (r, t) e− k·r k = 0 kl km 3 2(2π) IR3 1 =− dr rl rm Rij (r, t) 2(2π)3 IR3

C ijlm (t) =

C ijlm ≡ l m i j

Πij

f 1

t2 t1 0

r

f (r, t1 )

f (r, t2 )

t2 > t1

IR3

dr rl rm Rij (r, t) ≡

f f

IR3

IR3

dr ri rj Rij (r, t) = 4πu2 (t)

+∞

g l=i

m=j

dr r4 f (r, t) ,

0

u2 (t) f (r, t) u2 (t) 0

+∞

dr r4 f (r, t) ≡ 1939

u2 (t) f (r, t2 )

f (r, t1 ) x

x+r

t t1 < t2

2.4

u(x, t)

t t0

t0 u(x, t) = u(x, t0 )+

1 1 2 ∂t u(x, t0 ) (t−t0 )+ ∂tt u(x, t0 ) (t−t0 )2 +. . . 1! 2!

t0

p˜ = (p − p∞ )/ρ

−∂x2k xi uk ui

BR (0)

1/|x|2

x→∞

u u→0 p → p∞

∂t u = −u · ∇u − ∇˜ p + ν∇2 u , ∇2 p˜ = 0

R

dx [G(x − x )∇2x p˜(x ) − p˜(x )∇2x G(x − x )] = BR (0) = dS(x ) G(x − x ) ∂ν p˜(x ) − dS(x ) p˜(x ) ∂ν G(x − x ) , ∂BR (0)

∂BR (0)

G(x) = −1/(4π|x|) ∂ν p˜(x ) ∂ν G(x − x ) R→∞

∂BR (0)

x ∈ ∂BR (0)

1 4π

R→∞ G(x − x ) R → ∞ 1/R2

x ∈ ∂BR (0)

p˜(x) =

x

dx

∇2

1/R2 1/R p˜

∂x2 x uk (x )ui (x )

IR3

k

ν = ν(x )

i

|x − x |

,

∂t u u

∂t u = F1 (u, uu) .

2 ∂tt u = −∂t u · ∇u − u · ∇∂t u − ∇∂t p˜ + ν∇2 ∂t u = F2 (u, uu, uuu)

n

u

u

n+1 m (t − t0 )n m+ 1 m + (n + 1)

m+2 t

t0 t t0 u(x, t)

k 1/k p˜ 2 p˜ = −G ∗ ∂ki uk ui

δ(x) = δ(x1 )δ(x2 )δ(x3 ) F [∇2 G](k) = −k 2 F [G](k) = F [δ](k) ≡

1 , (2π)3 F [G](k) =

−1/[(2π)3 k 2 ]

p] U = F [u] P˜ = F [˜

kp ki 2 2 P˜ = −F[G ∗ ∂ki uk ui ] = −(2π)3 F [G]F [∂pi up ui ] = − 2 F [up ui ] . k F [up ui ] = F {F −1 [Up ]F −1 [Ui ]} = F {F −1 [Up ∗ Ui ]} = Up ∗ Ui

1 P˜ (k) = − 2 k

Ul = Ulr + Uli

IR3

dk k · U (k ) k · U (k − k ) ,

l=1 2

3

k·U

kl Ul

u ∇·u=0 i k · U (k, t) = kl Ulr (k, t) + km Um (k, t) ≡ 0 ,

k U r = (U1r , U2r , U3r ) k

t

U i = (U1i , U2i , U3i ) k

∂t u = −∂l (ul u) − ∇˜ p + ν∇2 u

k3 U i (k)

k

U r (k) k2

π k1 k U Ur

Ui

ˆ = − km ∂t u k + 2 k

IR3

IR3

=

IR3

≡

IR3

=

IR3

dk Um (k )U (k − k ) dk k · U (k )k · U (k − k ) − νk 2 U .

dk k · U (k )U (k − k ) ≡ dk [(k − k ) · U (k ) + k · U (k )] U (k − k ) dk k · U (k − k ) U (k ) dk [k · U (k − k ) − (k − k ) · U (k − k )] U (k ) dk k · U (k − k ) U (k ) ,

∂t U (k, t) =

IR3

k

k = k − k

≡

IR3

π

IR3

U U i (k )Uj (k )

dk k · U (k − k )

− U (k ) +

Ui

k k · U (k ) − νk 2 U 2 k

Ui

Ui

Ui U i (k )Uj (k )

< U i (k )Uj (k ) >

(i, j) Φij (k) < U i (k )Uj (k ) >= Φij (k )δ(k − k ) k dk

δ(k) < U i (k)Uj (k) >

k

k + dk

1/k j Ui (k, t) i

Uj (k, t)

kj 2 U i × ∂t Uj = dk k · U (k − k ) − Uj (k ) + 2 k · U (k ) − νk Uj + 3 k IR

ki = dk k · U (k − k ) − U i (k ) + 2 k · U (k ) − νk 2 U i Uj × ∂t U i − k IR3

∂t U i Uj =

kl dk U l (k − k )U i (k )Uj (k) − Ul (k − k )U i (k)Uj (k ) + = IR3

kl km + k2

IR3

dk

kj Ul (k − k )Um (k )U i (k) +

−ki U l (k − k )U m (k )Uj (k) 2

−2νk U i (k)Uj (k) ,

+

k

IR3

dk kl

=

dk

IR3

dk

IR3

dk

IR3

U l (k − k )U i (k )Uj (k) − Ul (k − k )U i (k)Uj (k )

kl U l (k − k )U i (k )Uj (k) − kl U l (k − k)U i (k)Uj (k )

k · U (k − k ) k · U (k − k ) U (k − k ) ≡ U (k − k) k k

U i (k)Ui (k)

k

3 27 28

3.3 48 E(k) k 0

53 +∞

E 5 90

Lp

Ln

92

,

f (r)

g(r)

λ

94 95 6 240 143

241 E(k)

6.2 k 6.4 f

g 265 Lp

270 271

Ln

6

205 207 250

14

15 29 30 35 35 39 λ

f Lp

Ln

2

g

3 181 189 f

g

189 194

195 202

202 209 211 215 13 3

4 6

7

426 431 431 433

440 450 450 456

kε k

ε

DN S

(ν/ε)1/2 η

1/2 L

9/4 L

104

DN S

DN S

...

DN S

L

L

L

L η

u

u

u u = u − u .

1

u = [2k/3]1/2

L

L

= uL/ν

106

109

∇ · u = 0 , ∇ · u = 0 , u

u ui uj

ui uj = ui uj + (ui uj + ui uj ) + ui uj . ui uj = ui uj + ui uj

K = ui ui /2 K = ui ui /2 i

k = ui ui /2

ρ(∂t ui + uj ∂j ui ) = −∂i p + μ∇2 ui , ui ρ(∂t ui + uj ∂j ui ) = ∂j

− pδij + μ(∂j ui + ∂i uj ) − ρui uj

.

τij

τ ij

i (i, j)

τ

−pδij

2μS ij τ ij

τ −2ρk τ = τ (u, S, p)

2

k

... τ S τij −

2 2 ρkδij + 2μ S ij = − ρkδij + μ (∂j ui + ∂i uj ) , 3 3

τ

μ

−2ρk ν = μ /ρ

μ

μ

ν

ν

τ

μ

τ −2ρk/3

p˜ = p/ρ ∂t ui + uj ∂j ui = ∂j [ν (∂j ui + ∂i uj )] − ∂i p˜ + ν∇2 ui . ν ν

ν k

α β

ν =k ε

ν + ν

ε [k] = L2 T −2 L2 T −1 = L2(α+β) T −(2α+3β) ,

[ε] = L2 T −3

α=2

β = −1 2

ν = Cν

k , ε

Cν k ε kε k

ui ∂t ui + uj ∂j ui = −uj ∂j ui − uj ∂j ui + ∂j ui uj − ∂i p˜ + ν∇2 ui , ui

i ui ui ui ui u u + uj ∂j = −ui uj ∂j ui − uj ∂j i i + ui ∂j ui uj − ∂i ui p˜ + νui ∇2 ui , ∂t 2 2 2

νui ∇2 ui ≡ ν

∇2 k − ∂k ui ∂k ui

,

∂k ui ∂k ui ≡ (∂k ui −∂i uk )(∂k ui −∂i uk )/2+∂k ui ∂i uk ω = ∇ × u ∂k ui −∂i uk = εpki ωp 2 . νui ∇2 ui = ν ∇2 k − ∂pq up uq − ωi ωi

νui ∇2 ui −νωi ωi = −ε , ε 3

ε

u

k

up uq −∂i ui p˜

∂j uj k

∂t k + uj ∂j k = −ui uj ∂j ui −ε , Π

Π (i, j)

τ −2 kδij /3

Π

∂m um

ε k Π kε ⎧ ∂j uj = 0 ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ∂t ui + uj ∂j ui = ∂j − p˜δij + (ν + ν ) (∂j ui + ∂i uj ) ⎪ ⎨ ∂t k + uj ∂j k = Π − ε ⎪ ⎪ ⎪ ⎪ ∂t ε + uj ∂j ε = ε Cε1 Π − Cε2 ε /k ⎪ ⎪ ⎪ ⎪ ⎩ 2 ν = Cν k /ε , Π = ν (∂k um + ∂m uk ) ∂k um . Cν Cε1

Cε2

f : IR3 × IR → IR x ∈ IR t ∈ IR f 3

C∞

1

2 3 δs

IR3 × IR δt

Hf [g](x, t) =

1 3 δs δt

g : IR3 × IR → IR

IR3

δs

+∞

dy

dτ f −∞

x−y t−τ , δs δt

g(y, τ ) .

δt

IR3 × IR

|d| = 1 d g

+∞ x−y t−τ 1 (y, τ ) Hf [∂d g] ≡ 3 dy dτ ∂d , f g(y, τ ) + δs δt IR3 δs δt −∞ x−y t−τ (y, τ ) f , . −g(y, τ )∂d δs δt d

f (y, τ ) ∂d f

x−y t−τ , δs δt

(x, t) f ≡ −∂d

x−y t−τ , δs δt

,

4

y 5

∂ (y, τ ) ∂ d

g

(x, t) d

d y

τ

x

t

y

τ

x

t

Hf [∂d g] = ∂d Hf [g] . Hf

Hf2 = Hf

δs

δt

δt → 0 + F

f

F (k , ω ) =

1 (2π)4

dx e− k ·x

IR3

+∞

k = 0

f (x , t ) , k g

F

k = kδs

ω t

−∞

G

dt e−

Hf [g]

ω

(k, ω) = (2π)4 F (kδs , ωδt ) G(k, ω) ,

F (k , ω )

ω = ωδt ω = 0

f

F

u = Hf [u] u = u − u

u

∇·u=0.

ui uj

ui uj + τij

ui uj ≡ ui uj + (ui uj − ui uj ) = τ

(i, j)

τij ≡ (ui uj − ui uj ) + (ui uj + ui uj ) + ui uj , ui uj + ui uj

ui uj

τ

∂t ui + uj ∂j ui = ∂j

− p˜δij + ν(∂j ui + ∂i uj ) + τij

,

τij τ

τ −

1963

(τ )I/3 (i, j)

S ij = (∂j ui + ∂i uj )/2 ν = Δ2 |S| , |S| = (S lm S ml )1/2

Δ Δ

"

∂j uj = 0 ∂t ui + uj ∂j ui = ∂j

− p˜δij + (ν + Δ2 |S|)(∂j ui + ∂i uj ) τ

p˜

,

g(x) g(x)

g(x)

log(E/E0 )

log(LkN )

x a

log(Lk)

b g : IR →

a IR λ1 < 2Δx1

λ1 > 2Δx1 b

λ1 = 2Δx1 kN

E(k)

x1 Δx1

λ1 = 2π/k1

a π/Δx1 = kN η b

λ1

k1 kη = 2π/η kN

2Δx1

1 0.75

1/(2Δx) x −Δx

0

F (k1 )

f (x)

0.5 0.25

+Δx

0 -0.25 -15 -10 -5

0

5

10

15

k1 a

b a

2π b

f : IR → IR f˜ (x) = [f (x + Δx) − f (x − Δx)]/(2Δx) f˜ f

x

F [f˜ ](k1 ) =

e+

− e− 2 Δx

k1 Δx

k1 Δx

F [f ](k1 ) =

sin k1 Δx F [f ](k1 ) . k1 Δx

2π a b f

U (k)

f˜

f

k k

k = |k|

k

kN kN

Δ

2.3 16 2.4 22 4 4.2

4.3.2 kε

8.2

9

9.3

9.3.1

9.4 kε kω 21

21.3.1

kε

kω

21.4 21.4.3

21.5

ue Ωb ∂Ωb μ∇2 u

ρDt u δ = U L/ν U

δ L Ωb

Ωb δ L L"δ

U

L

U

ue

u

v O(v) = δU/L O(ρDt u) = ρU U/L δ/L = −1/2 −1/2

=∞

O(μ∇2 u) = μU/δ 2 δ →∞ v

0

U

u·ν = 0 ν

∂Ωb =∞

L U ⎧ ∇·u = 0 ⎪ ⎪ ⎨ Dt u = −∇p ⎪ ⎪ ⎩ u| ·ν =0 , ∂Ωb

⎧ ∇·u =0 ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dt u = −∇p +

1

∇2 u

u|∂Ωb = 0 , →∞

−1

u ∇2 u

→ ∞

Ωb

1

L

1D

1D

→∞

→∞

∂Ωb

1955 x ∈ (0, 1) x "

εy + y = a y(0) = 0 , y(1) = 1 , ε

ε

y ∈ C 2 ([0, 1]) I ⊂ [0, 1]

2 y = y(x)

a ∈ (0, 1)

y > 0 y (x0 ) = y0

x0

y(x0 ) = y0

x

y(x) = a(x − x0 ) + y0 − ε[y (x) − y0 ] y

ε → 0+ y(x) = ax + y0 − ax0 , a ∈ (0, 1)

x x0

Iε = [ξ0 (ε), ξ1 (ε)] ⊂ I ε → 0+ y (x) > Mε

Mε → +∞

y(ξ1 ) − y(ξ0 ) =

ξ1

Mε x ∈ Iε

dη y (η) > (ξ1 − ξ0 )Mε ,

ξ0

ξ1

ε → 0+

ξ0 ε → 0+

y = y(x)

Iε ⊂ I

x0 < 1 y > a y > 0

a Iε x0 = 1, y0 = 1 ξ0 = 0 b

0.8

0.8

0.6

0.6

y

1

y

1

0.4

0.4

0.2

0.2

Iε

Iε

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

x a

0.4

0.6

x b x 0 = y0 = 1 x = 0.4 a

a = 0.7 Iε x=0 b

0.8

1

1D

f0 (x) = ax + (1 − a) . Iε x ˜ = x/ε [0, 1]

α

ε → 0+ α Iε = [0, εα ]

gε (˜ x) = y(x) ε1−α gε + gε = εα a ,

2

C ([0, +∞]) 1−a α>1 α=1

g0

g0

=

2

εα−1 g0 ≡ 0

ε → 0+

0 0 < α < 1 g0 ≡ 0

⎧ g + g0 = 0 ⎪ ⎨ 0 g0 (0) = 0 ⎪ ⎩ lim g0 (˜ x) = lim+ f0 (x) = 1 − a , x ˜→+∞

f0

x→0

g0 g0 (˜ x) = (1 − a)(1 − e−˜x ) . ε

yε (x) ax + (1 − a)

f0 (x)

+(1 − a)(1 − e−x/ε ) − g0 [˜ x(x)]

(1 − a)

f0 (0+ ) = g0 (+∞)

= ax + (1 − a)(1 − e−x/ε ) =: Yε (x) . Yε (x)

x=1

Yε yε (x) = ax + (1 − a)

yε (x) − Yε (x) = (1 − a)

1 − e−x/ε , 1 − e−1/ε

1 − e−x/ε −1/ε e < e−1/ε . 1 − e−1/ε yε = yε (x)

a ε

Yε = Yε (x)

ε = 1/4

0.006

0.8

0.005 0.004

ye − Y e

ye Y e

1

0.6 0.4

0.003 0.002

0.2

0.001

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

x a

0.4

0.6

0.8

1

x b

a = 0.7

ε = 1/4

a

yε = yε (x)

Yε = Yε (x) (yε − e−1/ε 0.018316 x=1

b Yε )(x) Yε

b yε = yε (x) x=1

Yε = Yε (x)

m k μ f (t) = F t(T − t) T x(T ) = x1

x(t)

m¨ x + μx˙ + kx = f x(0) = x0 , x(T ) = x1 .

m kT 2

F/k

2kT

1908

ue u

x x>0 y=0

x

v

y (x, y)

p

z y≥0

y y

ue

δ(x) x 0

δ x

⎧ ∂x u + ∂y v = 0 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ u∂ u + v∂y u = −∂x p/ρ + ν(∂xx u + ∂yy u) ⎪ ⎪ ⎨ x 2 2 u∂x v + v∂y v = −∂y p/ρ + ν(∂xx v + ∂yy v) ⎪ ⎪ ⎪ ⎪ lim (u, v, p) ≡ lim (u, v, p) ≡ (ue , 0, pe ) ⎪ ⎪ y→+∞ ⎪ ⎩ x→−∞ u(x, 0) ≡ v(x, 0) ≡ 0

(u, v, p) ≡ (ue , 0, pe ) ,

(ue , 0, pe ) y ζ=

y , α(ν)

lim α(ν) = 0

ν→0+

α(ν) u(x, ζ) v(x, ζ) p(x, ζ)

u v

p ν ν→0

+

ν → 0+

y>0

y=0 ue

u = ∂y ψ = ∂ζ 2

u ψ˜ = ψ/α

(x, ζ)

ψ , v = −∂x ψ ; α ν → 0+ y > 0 3 ν → 0+

ψ˜ u = ∂ζ ψ˜ , v = −α∂x ψ˜ = α˜ v

v˜ = −∂x ψ˜ ,

ν → 0+ ˜ v˜) (ζ, ψ,

v˜

⎧ u = ∂ζ ψ˜ , v˜ = −∂x ψ˜ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ∂x p 1 2 ⎪ ⎪ ⎪ + ν ∂xx u∂x u + v˜∂ζ u = − u + 2 ∂ζζ u ⎪ ⎪ ρ α ⎪ ⎨ 2 1 ∂ζ p 1 2 + ν α∂xx v˜ + ∂ζζ α (u∂x v˜ + v˜∂ζ v˜) = − v˜ ⎪ α ρ α ⎪ ⎪ ⎪ ⎪ ˜ 0) ≡ 0 ⎪ u(x, 0) ≡ 0 , ψ(x, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim lim (u, α˜ v , p) = (ue , 0, pe ) + ζ→+∞ ν→0

α

α2 (u∂x v˜ + v˜∂ζ v˜) = − ν → 0+

∂ζ p 2 2 + ν α2 ∂xx v˜ + ∂ζζ v˜ , ρ

∂ζ p = 0

u∂x u + v˜∂ζ u = ν → 0+ ν/α2 → +∞ a(x)ζ + b(x)

p

x p ≡ pe

ν 2 2 2 α ∂xx u + ∂ζζ u , 2 α α = α(ν) 2 ∂ζζ u=0

ν → 0+ √ α= ν.

ν/α2 → 0 u(x, ζ) = ν/α2

ν → 0+ α ⎧ ⎪ u = ∂ζ ψ˜ , v˜ = −∂x ψ˜ ⎪ ⎪ ⎪ ⎪ 2 ⎪ u ⎨ u∂x u + v˜∂ζ u = ∂ζζ ˜ 0) ≡ 0 , u(x, 0) ≡ 0 ⎪ ψ(x, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim u(x, ζ) ≡ ue , ζ→+∞

x1

u

x2 > x1

u = ∂ζ ψ˜ ˜ ζ) = ψ(x,

˜ 0) ≡ 0 ψ(x,

ψ˜

ζ

ds u(x, s) . 0

u

ζ

u = ∂ζ ψ˜ , v˜ = −∂x ds u(x, s) 0 ζ 2 u, u∂x u − ∂ζ u∂x ds u(x, s) = ∂ζζ 0

v˜ ˜ (x = ax, ζ = bζ, ψ˜ = cψ)

˜ (x, ζ, ψ)

u=

a c

b ∂ ψ˜ b = u c ∂ζ c ∂u u ∂x

∂u ∂ − ∂ζ ∂x

ζ

ds u (x , s)

=b

0

b

∂ 2 u , ∂ζ 2

b/c = 1 a/c = b b

a

b=c=

√

a

c

a √ a ζ/ x

u a

10

0.35 1

0.3

8 0.8

4

f

f

f

6

0.25

0.6

0.2 0.15

0.4

0.1 2

0.2

0

0.05

0 0

2

4

6

8

10

12

0 0

2

4

6

η

8

10

12

0

2

4

η

6

8

10

12

η

f η =b

η

a

f (η)

η

η

x ζ √ y η=√ ue = $ = νx x ue y = δ(x) = x u = ue f (η) ∂ζ =

ue /x ∂η

−1/2 x

f = f (η) η=0 ψ˜ ∂x η = −η/(2x)

x

= ue x/ν

√ x

x

y,

ψ˜

x 3

⎧$ ⎨ ue ∂η ψ˜ = ue f x ⎩˜ ψ(x, 0) ≡ 0 , ψ˜

v˜

˜ η) = √ue x f (η) , v˜(x, η) = √ue [ηf (η) − f (η)] . ψ(x, 2 ue x u/ue ⎧ ⎨ f + 1 f f = 0 2 ⎩ f (0) = f (0) = 0 ,

lim f (η) = 1 ,

η→+∞

1908 f (η) ∼ η

f

η 1

d η log f ∼ − , dη 2 f (η) ∼ exp(−η 2 /4)

η 1 x

g(ξ) "

g + gg = 0 g(0) = g (0) = 0 , g (0) = 1 ,

g (ξ) → 2χ2 ξ → +∞ f (η) = g(ξ)/χ

η

2χξ

⎧ g = g1 ⎪ ⎪ ⎪ ⎨ g = g 2 1 ⎪ g2 = −gg2 ⎪ ⎪ ⎩ g(0) = g1 (0) = 0 , g2 (0) = 1 . ue

f (η)

f (0) = 0

1, 2

x

f (0) =: a 0.33

u/ue

[η − f (η)] → b 1.72

η → +∞

⎧ u ⎪ = f (η) ⎪ ⎪ ⎨ ue v 1 −1/2 = [ηf (η) − f (η)] ⎪ ⎪ u 2 x ⎪ ⎩ e p ≡ pe . lim (ηf − f ) = lim [η(f − 1) + η − f ]

η→+∞

η→+∞

f − 1 +b η→+∞ 1/η

= lim

f +b=b, η→+∞ −1/η 2

= lim

lim

η→+∞

b v = ue 2

−1/2 x

. 0

v Yε u/ue

ω = −∂y u + ∂x v

v/ue

xω 1 =− (ηf − f + η 2 f ) + ue 4 1/2 x

η → +∞ x=1

1/2 x f

. −

−bue /(4x

1/2 x )

x

ω y → +∞ ue ≡

1/2 x aue /x

x

ue =

x η=y x

ω →

ue (x)/(νx)

2

1.5

1.5

1.5

1

0.5

y/x

2

y/x

y/x

2

1

0.5

0

0.5

0 0

0.25

0.5

0.75

1

1

0 0

u/ue

0.1

0.2

0.3

-10

x

-4

-2

y x

10

v/ue

x → −∞

y → +∞

x x ue = ue (x) x pe /ρ = −ue ue dpe /dx = pe

x

pe

x ue ≡

(u, v, p) = (ue (x), 0, pe (x)) , ⎧ ⎪ u = ∂ζ ψ˜ , v˜ = −∂x ψ˜ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨ u∂x u + v˜∂ζ u = −pe /ρ + ∂ζζ u ⎪ v˜(x, 0) ≡ 0 , u(x, 0) ≡ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim u(x, ζ) = ue (x) , ζ→+∞

−pe /ρ √ ζ/ x

u/ue = f (η) $

˜ η) = ψ(x,

0

ω y/x 103 −1/2 bRex /2

102

-6

xω/ue

ue ue /x

-8

v/ue

ue (x)x F (x, η)

η=y

ue (x) =ζ νx

$

ue (x) , x

u/ue

F

˜ 0) = √ue x F (x, 0) ≡ 0 ψ(x,

ψ˜

x F (x, 0) ≡ 0

pe /ρ β(x) = −

xue (x) xpe (x) = , 2 ρue (x) ue (x)

β≡0 ∂ζ η = ue /x

∂x η = −η(1 − β)/(2x)

u = ∂ζ ψ˜ x

∂ζ u x 2 u x ∂ζζ

3 ∂ηηη F+

x

= ue ∂η F $ 1 ue [(1 − β)η∂η F − 2x∂x F − (1 + β)F ] = 2 x

v˜ = −∂x ψ˜ ζ $ ue 2 ∂ F = ue x ηη u2 3 = e ∂ηηη F . x

1+β 2 2 2 F ∂ηη F + β[1 − (∂η F )2 ] = x(∂η F ∂xη F − ∂x F ∂ηη F) , 2 F u(x, ζ) u(x, ζ)/ue (x)

∂η F

η

G

u(x, ζ) = ∂η F (x, η) = G(η) ue (x) F (x, 0) ≡ 0

η

F (x, η) =

η

ds G(s) + F (x, 0) ,

0

˜ 0) = F (x, 0)√ue x ≡ 0 ψ(x, η 0 η

ψ˜ F

⎧ ⎨ F + 1 + β F F + β(1 − F 2 ) = 0 2 ⎩ F (0) = F (0) = 0 , lim F (η) = 1 . η→+∞

β 1931

F (η) ∼ η

η 1

β˜ = (1 + β)/4 η 1

F

d ˜ ∼ −2β 1 − F 0 . log F + 2βη dη F

10

0.8

F

0.8

6

F

F

1

1

8

0.6

0.6

4

0.4

0.4

2

0.2

0.2

0

0 0

2

4

6

8

10

0 0

2

η

4

6

8

10

0

2

η

6

8

10

η F

F

F η β −0.09043 0.05 0.10 . . . 0.50

−0.09 −0.08 . . . −0.01 0

η

4

η ˜

2

˜

2

F (η) ∼ F (η)eβη e−βη , F2

˜ 2) F (η) exp(βη η

η≥η F2

F

η

F → 1

η → +∞

η F (η) ∼ 1 − F2

+∞ η

η

2

+∞

˜

2

dξe−βξ = 1 −

$

π F2 4β˜

1 − F η − F (η) → b(β)

( ˜ . ( βη)

η → ∞ (x)

+∞

+∞

b − [η − F (η)] ∼ F2

dξ

F (η) ∼ η − b − F2

+∞

η

˜

2

dξ e−βξ +

η

( π β˜ η

F2 = η−b− 2β˜

˜

dξ e−βξ

2

,

ξ

η

+∞

1 −βη ˜ 2

e ˜ 2β

( ˜ 2

˜ + e−βη . ( βη) (1 − F )/F

1 − F (η) ∼ F (η)

+∞

˜

dξ e−βξ

η

2

+∞

˜ 2 e−βη

˜

dξ e−2βη(ξ−η) =

≤ η

1 1, ˜ 2βη

η

1−F

2

β H =η−F H +

1+β 2 (η − H)H + β (H − 2H ) = 0 . 2

η 1 F (η) < 1 η =0 w = F (η) w = η−b β b = b(β) β H = H(η) η=0 H0 H η H

F (η) ∼ η

0 η→∞

w

2 (x) = √ π

x

+∞

dξ e−ξ

2

(x)

2

1.5 1 0.5 0 -4

-2

0

2

4

x (0) = 1 x → +∞

(x) → 0

1

3.5 3

0.8

b(β)

a(β)

2.5 0.6 0.4

2 1.5

0.2

1

0 -0.1

0

0.1

0.2

0.3

0.4

0.5 -0.1

0.5

0

β F (0) = a(β)

0.1

0.2

0.3

0.4

0.5

β lim [η − F (η)] = b(β)

η→+∞

F β

η → +∞

H

η=0

η − F (η)

η ⎧ 1 ⎨ dη = dH H (H) ⎩ η(0) = 0 , M

0

H

H

H=0 ⎧ ⎨ dM = H (H) dH ⎩ M (0) = 0 . M M (H) =

H

˜ H (H) ˜ = dH

0

η(H)

2

d˜ η H (˜ η) .

0

H ⎧ 2 ⎪ ⎨ dH + (1 + β)[η(H) − H]H + (1 + 3β)M (H) = (1 + 5β)H − 2a dH ⎪ ⎩ H (0) = 1 ,

η

M

(1 + 3β) M (H0 ) − (1 + 5β) H0 + 2a(β) = 0 , H → H0 β a H = H0

H

H

2

H →0

H

2

0

η → +∞

a a

β

β = 0 pe < 0 β > 0

pe > 0 β < 0 F (0) = −β =

x p , ρu2e e

β −0.09 ∂y u = 0 ⎧ u ⎪ = F (η) ⎪ ⎪ u ⎪ e ⎨

v 1 −1/2 (1 − β)ηF (η) − (1 + β)F (η) = x ⎪ ⎪ u 2 e ⎪ ⎪ ⎩ p = pe (x) , ue (x)/x

1 xω =− (1−4β+β 2 )ηF −(1−β 2 )F +(1−β)η 2 F + 1/2 ue 4 x β = x u =β ue e

1/2 x F

ue = ue (x)

⇒

due dx =β ue x

⇒

ue (x) = ue (x0 )

x x0

β .

.

1 0.8

2

2

1.5

1.5

1

1

0.5

0.5

y/x

0.6 0.4 0.2 0 0

0.5

0 -0.9

1

1 0.8

0 -0.6

-0.3

0

0.3

0.6

-3

2

1

1.5

0.75

1

0.5

0.5

0.25

-2

-1

0

1

y/x

0.6 0.4 0.2 0 0

0.5

0 -0.9

1

1

-0.3

0

0.3

0.6

2

0.8

y/x

0 -0.6

-9

-6

-3

0

0.5 0.4

1.5

0.6

0.3 1

0.4

0.2 0.5

0.2 0 0

0.5

0 -0.9

1

0.1 0 -0.6

u/ue

x

-0.3

0

0.3

0.6

-25

-20

v/ue

= 102

x

-15

x

= 103

x

= 10

y β

1921 u = f (ξ) = c1 ξ + c2 ξ 2 + c3 ξ 3 + c4 ξ 4 , ue ξ = y/δ δ 1

ξ

f (1) = f (1) = 0 i = 1 ... 4 λ = ue δ 2 /ν

-10

xω/ue

ue = due /dx

f (1) = ci

-5

0

∃b = 0 | xpe ≤ −b2 , ∀x > 0 ⇒ ue

x,

lim u(x, ζ) = ue (x)

ζ→+∞

ζ

x

a>0 u1 (x, ζ) u2 (x, ζ) a x→+∞ ue (x) − ue (x) ≤ ue (x) −→ 0 , u2

u1 x0

u2 (x0 , ζ) ≥ 0

˜ ζ) ξ = x − x0 , ψ = ψ(x,

u2 (x, ζ) ≥ 0 , ∀x > x0

˜ 0) ≡ 0 ψ(x, j = ∂(ξ, ψ)/∂(x, ζ) = u = 0 ξ≥0, ψ≥0

x ≥ x0 , ζ ≥ 0

∂x = ∂ξ − v˜∂ψ , ∂ζ = u∂ψ 1 2 2 1 2 2 2 2 2 ∂ u ≡ (∂ψ u) + u∂ψψ u ⇒ ∂ζζ = u∂ψψ u , 2 ψψ 2 ⎧ 2 2 ∂ξ u − u2e = u∂ψψ u2 ⎪ ⎪ ⎪ ⎨ u(ξ, 0) ≡ 0 ⎪ ⎪ ⎪ ⎩ lim u(ξ, ψ) = ue (ξ) ψ→+∞

3

∀I ⊂ R+ ∀ε > 0 , ∃MεI > 0 | ∀ζ > MεI : |u − ue | < ε , ∀x ∈ I MεI

x

.

u2

u1

ϕ = u22 − u21 2 2 ∂ξ ϕ = u2 ∂ψψ u22 − u1 ∂ψψ u21 2 2 ≡ u2 ∂ψψ (u22 − u21 ) + (u2 − u1 ) ∂ψψ u21

2 2 2 ∂ u1 ϕ , ≡ u2 ∂ψψ ϕ + u1 (u2 + u1 ) ζζ 2 α = 2/u1 (u1 + u2 )∂ζζ u1 < 0

u1 ϕ

ϕ

⎧ 2 ∂ ϕ − u2 ∂ψψ ϕ = αϕ ⎪ ⎪ ξ ⎪ ⎨ ϕ(ξ, 0) ≡ 0 ⎪ ⎪ ⎪ ⎩ lim ϕ(ξ, ψ) ≡ 0 ψ→+∞

ϕ ξ = 0 ψ ≥ 0 ϕ(0, 0) = 0 ϕ(0, 0) = 0 ϕm ≤ ϕ(0, ψ) ≤ ϕM

ϕM

ϕ(0, ψ) ϕm

∀ξ > 0 : ϕm ≤ ϕ(ξ, ψ) ≤ ϕM

∃(ξ0 , ψ0 ) ϕ > ϕM ϕ ξ1 > 0 0 , ∃Mε1 > 0 | ∀ψ > Mε1 : −ε1 ≤ ϕ ≤ ε1 F1

∃(ξ1 , ψ1 ) ∈ S ε1 >

4

I × [0, +∞) ε > 0 ∃MεI > 0 ζ > MεI ue (x) + ε ≤ Ue + ε I × [0, MεI ] u [0, MεI ] I × [0, +∞) U ψ > MεI

I ⊂ R1

x ue (x) − ε ≤ u(x, ζ) ≤ Ue = max {ue (x) , x ∈ I} Ue + ε I× u(x, ζ) ε>0 MεI > 0

|ϕ| = |u21 − u22 | = (u1 + u2 ) |u1 − u2 | ≤ (U1 + U2 ) (|u1 − ue | + |u2 − ue |) < 2 (U1 + U2 ) ε , x

u

ψ

(ξ0 , ψ0 ) , ϕ > ϕM

S ϕM

0

ϕ=0

ξ0

ξ

(ξ, ψ)

[0, ξ0 ] × [0, Mε1 ] F1 > ε F1 S = [0, ξ0 ] × [0, +∞) F1 ≤ ε ε2 = F1 /2 ∃Mε2 > 0 | ∀ψ > Mε2 : −ε2 ≤ ϕ ≤ ε2 = F1 /2 ϕ [0, ξ0 ] × [0, Mε2 ] F2 S ξ0 > 0 ξ1 > 0

ϕ

F2 ξ1 = ξ0

(ξ1 , ψ1 ) 2 (ξ1 , ψ1 ) ∂ξ ϕ = 0 , ∂ψψ ϕ≤0 ξ1 < ξ0 2 ∂ξ ϕ ≥ 0 , ∂ψψ ϕ ≤ 0 ϕm ≤ ϕ ≤ ϕM ϕM > 0 ϕm < 0

ξ0 a2 =

a>0 |ϕ| ≤ a2 ϕM = 0 a = ϕM 2

ξ≥0

1 0< (ξ1 , ψ1 ) S

ξ1 = ξ0

(|ϕm |, ϕM )

ψ≥0 a2 = |ϕm |

ϕm = 0 ϕm

ϕM

2

0 ≤ |u1 − u2 | ≤ |u1 − u2 | (u1 + u2 ) = |ϕ| ≤ a2 ⇒ |u1 − u2 | ≤ a

x pe < 0 x0 pe > 0 pe < 0

pe (x)

τw = μ ∂y u|y=0 , u/ue τw = a(β)ρu2e Rex−1/2 =: T (β)ρu2e Re−1/2 , x a(β) = F (0)

F β

δ1 y H δ1 (H) H − δ1 (H)

u/ue 0

H y 1

ue y=H H − δ1 (H) δ1

δ1 (H)

δ1 (x) = =

lim δ1 (H)

H→+∞

lim

+∞

=

dy 0

$ δ1 =

νx ue

+∞

0

v

u(x, y) ue (x)

u(x, y)

, ue (x)

dη [1 − F (η)] − F (η)]

=: Δ(β) x

−1/2 x

,

η → +∞ δ1 β = 0

b(β)

lim

1−

−1/2 x

= b(β) x

η→+∞ u

−1/2 lim [η x η→+∞

=x

1−

dy

H→+∞ 0

H

η−F (η)

1 lim v(η) ue η→+∞ 1 −1/2 = lim {η[f (η) − 1] + [η − f (η)]} 2 x η→+∞ dδ1 b(0) −1/2 , = = x 2 dx

(η) =

y = δ1 (x)

θ

+∞

θ(x) =

dy 0

u(x, y) ue (x)

1−

u(x, y) ue (x)

δ1

β = −1/3 5

β > −0.09

β = −1/3 θ

6

$ θ=

νx ue

+∞

dη F (1 − F ) = x 0

F

−1/2 x

+∞ F (1 − F ) 0 +

+∞ 0

η → +∞

dη F F

,

θ(x) = 2

a − bβ x 1 + 3β

−1/2 x

= Θ(β) x

−1/2 x

θ

.

β > a/b β=0

a/b 0.19 x

−1/2 x

β

ξ

τw δ1 θ ⎧ ∂x u + ∂y v = 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨ u∂x u + v∂y u = −pe /ρ + ν∂yy u u(x, 0) ≡ v(x, 0) ≡ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim (u, v) = (ue , 0) y→+∞

1 − F F 2 F = lim =0, η→+∞ 1/F η→+∞ F

lim F (1 − F ) = lim

η→+∞

F 1+β 2

+∞

dη F F 0

+∞

= F (0) − β

d(η − F ) (1 + F )

0

+∞ = a − β(η − F ) (1 + F ) 0 + β +∞

0

+∞

+∞

d(η − F ) − β 0

= a − 2bβ + bβ − β

0

θ

+∞

0

dη F F = 2

dη F F

0 +∞

dη F F ,

0

+∞

d[η(F − 1)] +

= a − 2bβ + β +β

a − bβ . 1 + 3β

dη (η − F ) F

v∂y u ≡ v∂y (u − ue ) = ∂y [v(u − ue )] + (u − ue )∂x u = ∂y [v(u − ue )] + ∂x [u(u − ue )] − u∂x u + uue u u 1− − u∂x u + = ∂y [v(u − ue )] − ∂x u2e ue ue u ue + ue ue −ue 1 − ue v∂y u ∂y [v(u − ue )] − ∂x

u2e

u u u 2 1− − 1− ue ue = ν∂yy u ue ue ue 0

y

+∞

d 2 τw (u θ) + δ1 ue ue = dx e ρ

d 2 (u Θx dx e

d x dx

−1/2 ) x

−1/2 x

−1/2 ue ue x

+ Δx

d dx

=

$

=

−1/2 x

T ρu2e

νx 1−β = ue 2

ρ

,

−1/2 x

u2e 2

ue Θx ue

−1/2 x

+Θ

1−β 2

−1/2 x

+ Δx

−1/2 ue x ue

=T

β = xue /ue 1 + 3β Θ + βΔ − T 2

−1/2 x

=0. x T Δ

−1/2 x

Θ

−1/2 x

1 + 3β a − bβ 2 + bβ − a ≡ 0 , 2 1 + 3β β 1949 pe

τw , δ1 , θ

H=

β

θτw δ1 , l= θ μue

−ρue ue ,

j=

θ2 θ2 pe = − ue μue ν

x Hs =

ΔxRe−1/2 x ΘxRex−1/2

≡

Δ = Hs (β) Θ

ls =

T ρu2e Re−1/2 ΘxRe−1/2 xue −1 x x Rex ≡ ΘT = ls (β) = ΘT μue ν

js =

xue x pe −1 x Θ2 x2 Re−1 x Rex = −Θ2 2 ue ue ≡ −Θ2 β = js (β) . pe = Θ2 μue ν u2e ρ ue β = β(js ) Hs = Hs (js ) , ls = ls (js ) . H = Hs (j)

l = ls (j)

θ ue dθ2 = 2{[Hs (j) + 2]j + ls (j)} =: L(j) . ν dx L(j) 0.45 + 6j u dθ2 ν + 6 e θ2 = 0.45 dx ue ue x = x0

u6e

L(j)

u6e (x)θ2 (x) = u6e (x0 )θ2 (x0 ) + 0.45ν ue ν

=

x

x0

ds u5e (s) .

−pe /(ρue )

x ue (x) u (s) λ6 , ds u5e (s) = −μ ds u5e (s) e dλ = −μ pe (s) pe (λ) x0 x0 ue (x0 ) ue (s) x

u6e (x)θ2 (x) =

= u6e (x0 )θ2 (x0 ) + 0.45μ = u6e (x0 )θ2 (x0 ) θ (x0 )

ue (x)

dλ

ue (x0 )

1+

= ue (x0 )θ(x0 )/ν

0.45 θ (x0 )

λ6 −pe (λ) ue (x)/ue (x0 )

dχ 1

χ6 −˜ pe (χ)

,

p˜e (χ) = θ(x0 )pe [χue (x0 )]/[ρu2e (x0 )]

x

θ

p˜e ≡

= σe

σe < 0 ue (x)/ue (x0 )

χ7 1 ue (x) > ue (x0 ) u6e (x)θ2 (x) x0 σe > 0 ue (x)/ue (x0 ) 1

ue (x)/ue (x0 )

ue (x) < ue (x0 ) u6e (x)θ2 (x) pe < 0

pe > 0

x

1

u(x, y) = ue ex y

y>0 y → +∞

x y u = u(x, y) v = v (x, y; t) p = p (x, y; t) z

u

v

v = v(x, y) p

p + p

u = u (x, y; t) p = p(x, y)

z u = 0 y → +∞ x

u=u

y ⎧ ∂x (u + u ) + ∂y (v + v ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t (u + u ) + (u + u )∂x (u + u ) + (v + v )∂y (u + u ) = ⎪ ⎪ 2

⎪ ⎪ 2 ⎨ = −∂x (p + p )/ρ + ν ∂xx (u + u ) + ∂yy (u + u ) ⎪ ⎪ ∂t (v + v ) + (u + u )∂x (v + v ) + (v + v )∂y (v + v ) = ⎪ ⎪ 2

⎪ ⎪ 2 ⎪ (v + v ) + ∂yy (v + v ) = −∂y (p + p )/ρ + ν ∂xx ⎪ ⎪ ⎪ ⎪ ⎩ (u + u )(x, 0) = (v + v )(x, 0) ≡ 0 ∂t u = ∂t v = 0 ⎧ ∂x u + ∂y v = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u∂x u + v∂y u = −∂x p/ρ − ∂x u u − ∂y u v + ν ∂ 2 u + ∂ 2 u xx yy 2 2 ⎪ u∂x v + v∂y v = −∂y p/ρ − ∂x u v − ∂y v v + ν ∂xx v + ∂yy v ⎪ ⎪ ⎪ ⎪ ⎩ u(x, 0) = v(x, 0) ≡ 0 .

−u u −u v

−v v (1, 1) (1, 2) = (2, 1)

ρ (2, 2) ν → 0+ u (x, y; t) = βχ(x, y; t)

ν ν → 0+

χ

γ

β = β(ν) v (x, y; t) = βγ(x, y; t)

ζ = y/α v = α˜ v 2 β2 1 2 ∂ζ χγ + ν ∂xx u + 2 ∂ζζ u α α 2 1 β 1 2 2 ∂ζ γγ + ν α∂xx v˜ + ∂ζζ v˜ . α(u∂x v˜ + v˜∂ζ v˜) = − ∂ζ p/ρ − β 2 ∂x χγ − α α α u∂x u + v˜∂ζ u = −∂x p/ρ − β 2 ∂x χχ −

α

ν → 0+

ν/α2 √ α = ν ν

p ≡ pe

ν → 0+

2

β /α −β 2 ∂x χχ

−∂x p/ρ

2 ν∂xx u

⎧ ∂x u + ∂y v = 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨ u∂x u + v∂y u = −∂y u v + ν∂yy u u(x, 0) ≡ 0 , v(x, 0) ≡ 0 , u v (x, 0) ≡ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim (u, v, u v ) = (ue , 0, 0) . y→+∞

−∂y u v

u τw (x) τw (x)

x τw > 0 τ w (x)/ρ =: u (x) , x u

x ue

u

δ(x) x

δ

:= ue δ/ν δ

a

b

δ

x

τ w (x) δ1

θ

u

x

+∞

θ(x) = 0

u(x, y) dy ue

u(x, y) 1− ue

.

θ(0) = 0 θ(x) =

u (x)

1 u2e

0

x

dξ u2 (ξ) .

x θ(x)

x x

u(x, 0) = 0 u

u

b ν/u y+ (x, y) =

yu (x) . ν y+ < 10 u(x, y)

x

u(x, y) = u (x) f (y+ ) , f

y y=0 ∂y u(x, 0) =

τ w (x) = μ ∂y u(x, 0)

f (0) = 1

ρ u2 (x) f (0) , μ u y+

y

y δ

III δ

ux

0

II I

0

a

ux

b a b

u(x, y) ≡ ue

y = δ

δ

b I

II

III

f (y+ ) = y+ . y+ y

y+

η(x, y) =

y δ(x) (x)

u (x)δ(x)/ν η

y+ η=

y+ ν yu = . u δ ν y+

y x > x x

=

δ

y

η y+

η y /δ(x ) = y/δ(x) y+

103 < y+ < 105 u

η u (x) y+

ue − u

u b ue − u(x, y) = u (x) F (η) . θ(x) = O(δu /ue ) δ(x)/x

u /ue

u /ue

0.25

u /ue

35 30 25 20 15 10 5 1

0.3

u/u

0.2 0.15 0.1

log10 y+ 0

1

2

3

0.05

5

4

0 0

1

2

a

3

4

5

log10

δ

6

b a y+

b

u /ue δ

u /ue 1

u /ue δ

b

y+

f (y+ ) =

ue u(x, y) = − F (η) , u (x) u (x)

η

7

8

η

η

y+

η y+ f (y+ ) ≡ −ηF (η) , y+ 1/K

η

K 0.39

f (y+ ) 1 u(x, y) = log y+ + a , u (x) K a4

a u/u

y+ F (η) = −

1 log η − b , K a−b 6 η x

b u ue ue 1 1 log log + = u K u K δ (x)

u 100 <

δ

b

δ δ

+a−b.

= ue δ(x)/ν

x log10

δ

< 108

x u u /ue u /ue

x

= Tw

Te

Tw Tw

Te

Dt T K/(ρcp ) ∇2 T δ K/(ρcp )

δT ν = μcp /K

L δ O[K/(ρcp )] = O(ν)/

O(ν) = ue L(δ/L)2

DT

O

O

Dt

= δ2 /

−1/2 L l · ue /L

ue

δ Te − Tw ue (Te − Tw ) + ue L L δT Te − Tw δ = ue 1+ L δT

=

K K Te − Tw Te − Tw ∇2 T = + ρcp ρcp L2 δT2 =

1

ue

δ 2

Te − Tw δ 2 . + L L δT

δT 1

O

1 δ 2 = O(1) δT

⇒

√ δ = O( δT

) 1,

1 1

O

δ 2 δ =O δT δT

⇒

δ = O( δT

) 1

⇒

δ = O( δT

)1.

1 1

O

δ 2 δ =O δT δT

1

1

1 δ/δT

δT

T Dp DT = βT + φ + K ∇2 T ρcp Dt Dt E

δ ue L

∂ 1 β=ρ , K ∂T ρ p

μ

f (ρ, p, T ) = 0

ρ − ρ0 = 0 , ρ0 β p, u

p−R ρ T =0 β = 1/T

∂ρ/∂p = 0 f (ρ, T ) = 0 , β = 0

ρ cp

DT = φ + K ∇2 T , Dt K=0

cp Tw = Te

μ=0 T ≡ Te

T ≡ Te

μ=

K → 0

μ K

μ/K = /cp cp

K/cp

⎧ ∂x (ρu) + ∂y (ρv) = 0 ⎪ ⎪ ⎪ ⎪ μ ⎪ 2 2 ⎪ ⎪ ρ(u∂x u + v∂y u) = −∂x p + ∂x (∂x u + ∂y v) + μ(∂xx u + ∂yy u) ⎪ ⎪ 3 ⎪ ⎪ μ ⎪ 2 2 ⎪ ⎪ ρ(u∂x v + v∂y v) = −∂y p + ∂y (∂x u + ∂y v) + μ(∂xx v + ∂yy v) ⎪ ⎨ 3 2 2 ρcp (u∂x T + v∂y T ) = (u∂x p + v∂y p)βT + μφ + K(∂xx T + ∂yy T) ⎪ ⎪ ⎪ ⎪ f (ρ, p, T ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y = 0 , x > 0 : u = v = 0 T = Tw ∂y T = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim (ρ, u, v, T, p) = (ρe , ue , 0, Te , pe ) , y→+∞

φ φ=

4 (∂x u)2 + (∂y v)2 − ∂x u ∂y v + (∂y u)2 + (∂x v)2 + 2 ∂y u ∂x v , 3 μ = λ + 2μ/3 μ

α = α(μ)

μ α→0

μ→0

y→ζ=

K

y . α(μ)

ψ ρu = ρe ∂y ψ , ρv = −ρe ∂x ψ .

ρu = ρe ∂ζ 3

ψ/α μ→0

ψ , α

μ→0

ψ˜ = ψ/α v˜ = v/α

ρu = ρe ∂ζ ψ˜ , ρ˜ v = −ρe ∂x ψ˜ .

⎧ ∂x (ρu) + ∂ζ (ρ˜ v) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ μ 1 2 ⎪ 2 ⎪ ρ(u∂x u + v˜∂ζ u) = −∂x p + ∂x (∂x u + ∂ζ v˜) + μ(∂xx u + 2 ∂ζζ u) ⎪ ⎪ ⎪ 3 α ⎪ ⎪ μ ⎪ α2 ρ(u∂ v˜ + v˜∂ v˜) = −∂ p + ∂ (∂ u + ∂ v˜) + μ(α2 ∂ 2 v˜ + ∂ 2 v˜) ⎪ ⎪ x ζ ζ ζ x ζ xx ζζ ⎪ 3 ⎨ 1 2 2 ρcp (u∂x T + v˜∂ζ T ) = (u∂x p + v˜∂ζ p)βT + μφ˜ + K(∂xx T + 2 ∂ζζ T) ⎪ ⎪ α ⎪ ⎪ ⎪ f (ρ, p, T ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ζ T = 0 y = 0 , x > 0 : u = 0 v˜ = 0 T = Tw ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim (ρ, u, v˜, T, p) = (ρe , ue , 0, Te , pe ) , y→+∞

1 4 (∂x u)2 + (∂ζ v˜)2 − ∂x u ∂ζ v˜ + 2 (∂ζ u)2 + α2 (∂x v˜)2 + 2 ∂ζ u ∂x v˜ . φ˜ = 3 α y μ → 0 ∂ζ p = 0 p ≡ pe α2 /μ → 0 α2 /μ → +∞

2 μ → 0 ∂ζζ u = 0

μ→0

α=

p = p(x)

u(x, ζ) = A(x)ζ + B(x)

√ μ

x

⎧ ∂x (ρu) + ∂y (ρv) = 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ρ(u∂x u + v∂y u) = μ∂yy u ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎨ ρcp (u∂x T + v∂y T ) = μ(∂y u)2 + K∂yy T f (ρ, pe , T ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y = 0 , x > 0 : u = v = 0 T = Tw ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lim (ρ, u, v, T ) = (ρe , ue , 0, Te ) .

∂y T = 0

y→+∞

=1

η

u v

p η

Tw

T (x, y) − Te = T˜ (η) . Tw − Te Tw

=

=

2 2 ∂yy = (η/y)2 ∂ηη

η ∂x = −η/(2x)∂η ∂y = η/y∂η T˜ +

f T˜ = −

2

0< "

μu2e , K(Tw − Te )

<2 "

T˜ (0) = 1 lim T˜ = 0

η→+∞

f 2 .

T˜ (0) = 0 lim T˜ = 0

η→+∞

f /2 = −f /f f

f d T˜ dη f

f

=−

2−

, T˜

T˜ = −

T˜ = − A

η

dξ f 0

B

f

η

dξ f

2−

(ξ) + Af

0

(ξ)

ξ

dτ f 0

2−

η (τ ) + A dξ f 0

(ξ) + B ,

+∞

(η) =

Γ

dξ f

(ξ) , Λ

(η) =

η

T˜ = −

dξ f

A=0, B=

(η) +

T˜ (η) =

Λ

(η)

(τ ) ,

(η)

+B .

Λ (0) − 1 , B=1 Γ (0)

A=

Λ

2−

T˜

Γ2−

T˜ (η) =

dτ f

0

(0) − Γ2− (η) + Af

(0) − Λ (η) + A Γ (0) − Γ

f Λ

ξ

(ξ)

η

T˜ T˜ = −

+∞

1−

(0) ,

Λ

Λ (0) Γ Γ (0)

(η)

Λ =1

γ

(ξ) = 22

γ

−1 3

= −g

χ

−1

, ε

g (ξ) → 0

Γ

[η(ξ)] , λ

(ξ) = 4χ4 Λ

2−

= g

= −g

, λ

ξ → +∞

γ γ

7

f ∈ (0, 2)

ε

λ

[η(ξ)]

,

ε

λ

(0)

5

1.2

4

1 0.8

Γ

Λ

3 2

0.6 0.4

1

0.2

0

0 0

5

10

15

0

5

10

η a Γ

Γ

(η)

(ξ) ∝

ξ

λ

(η) b = 0.2 0.4 . . . 1.8

dτ g

2−

(τ ) ,

0

⎧ ⎪ ⎪ γ = −g ⎪ ⎪ ⎪ ⎪ ⎨ ε = −g 2− ⎪ ⎪ λ = g ε ⎪ ⎪ ⎪ ⎪ ⎩ γ (0) = 0 , ε (ξ)

Λ

ε

ε

γ

(η) a

(η)

Λ

ε

ξ → +∞ γ ∞ λ∞ γ ∞ λ (ξ) − λ∞

15

η b

(0) = 0

(0) = 0 , λ

(0) = 0 ,

(ξ) (ξ) −

γ

γ∞

γ

λ∞

(ξ)

λ

(ξ)

χ γ (ξ) Γ (η) Λ

λ (ξ) Λ (η)

Γ

(η) h = −qy (0) qw (0) = −K∂y T (0)

qy (0)

y

(η)

1.6 6

1.4 1.2

(0)

4 3

Λ

Γ

(0)

5

2

1 0.8 0.6

1

0.4

0

0.2 0

0.5

1

1.5

2

0

0.5

1

a

1.5

2

b (0) a

Γ

Λ

(0) b

b

√ 1/(2

)

Λ

Γ

Λ

= −f

= −f

η

dξ f

2−

(ξ) = f

0

(0)

(η) − Γ2−

Γ2−

f (0) = a 0.33 h = K ∂y T (0)

$

= K (Tw − Te )

a Γ (0)

= ρu3e

√

Λ

(0) 1/(2

Λ

h ρu3e 2

ue ∂η T˜ (0) νx

a Γ

(0)

(

√

−2

(0) − 1

Tw √ > 2 h > 0 √ <2 h<0

−1/2 x

) √

Tw < Te h > Te

. b

)

−1/2 x

,

(0)

=1 T0 u2 , 2cp

T0 = T +

ρcp (u∂x T0 + v∂y T0 ) =

K ( 2cp

− 1)

2 u∂yy u + (∂y u)2

2 + K ∂yy T0 ,

K/cp = μ ρ (u∂x T0 + v∂y T0 ) = μ ∂yy T0 . T0 = Au + B A=0

u ∂y T0 (0) = 0 T0

Tw = Te +

A T0 ≡

B

u2e . 2cp A =

−(Tw − Te )/ue + ue /(2cp ) T

B = Tw

T = Tw − (Tw − Te )

u uue + ue 2cp

1−

u . ue

qw qw = K ∂y T (0) = K ∂y T0 (0) K (Tw − T0e ) ∂y u(0) ue cp =− (Tw − T0e )τw , ue

=−

τw = μ∂y u(0) > 0 Tw < T0e

Tw > T0e

x x

=

·

×

x x

cf = τw /(ρe u2e /2)

=

1 2

x=

x cf

qw x K(T0e − Tw )

,

= (ρe ue x)/μ

=1 =1 Γ1 (η) = 1 − f (η) , Λ1 (η) =

1 2 (1 − f ) , 2 f

T˜ =

u2 1 − /2 u 1 1− 2 + 1− , 2 ue 1 ue T˜

7

x

y

y δ << L

δ

L

e 8 b

e 10

b 12 c

e

f g

16

296

17 18

b

c

a

21 5.5 205

206

21 22 8

K

j

9

= u2e /[cp (Tw − Te )]

L(K) 0.47 − 6K

η 104

22 24

25

26 2.2 →∞ 1

68

73 74 80 81 5.7 305 308

5.8 5.9

10.2 10.5 η = y/δ(x) 10.6 4 41

158 162 7.3 7.5

7.3 437 448 8

8.1 8.2 η 8.3 8.3.1

ξ u/ue δ 8.4

δ

5.5

μ→0

243 245

δ1

250 252

θ

493 497 6.5 6.5.2

505 508 9.2 9.2.2 9.2.4 9.2.6 9 6.2 6.4 6.5.1 6.6 26 30

8 260 265 269 271 271 274

μ→0

u u(x, ζ) → ue (x) ζ → +∞ I = [x0 , x1 ] ε>0

F → 1

η > mε 0 < β ≤ 1 η=ζ

Mε = mε x ∈ I

ue (x)/x

pe < 0

η → +∞ mε |F (η) − 1| < ε/ue (x1 ) xi /ue (xi ) i=0 β>1 i=1 ζ > Mε η > mε ζ > Mε

x ue (x) ε≤ε. ue (x1 )

|u(x, ζ) − ue (x)| = ue (x)|F (η) − 1| ≤ ζ > Mε

$ η=ζ

ue (x) > Mε x

x/ue (x)

η > mε

$

$

xi ue (xi )

ue (x) = mε $ x

x ue (x)

0<β≤1 d dx

$

x 1−β . = √ ue (x) 2 xue

> mε ,

β>1

l T

p

ρ p→0

T → 0

ρ1 u 1 p 1 h 1

s1

ρ2 u 2 p 2 h 2

ρ 1 < ρ2

p1 < p2

s2 u1 > u2 h1 < h2

s 1 < s2

l

|˜ ρ|/ρ0 1

u(x, t) = u0 + u ˜(x, t)

p ρ

p(x, t) = p0 + p˜(x, t) |˜ p|/p0 1 ρ(x, t) = ρ0 + ρ˜(x, t) |˜ u|/u0 1

s p −s/cv e ≡ ργ s ≡ s0

p˜

ρ˜

p = p(ρ, T ) dp =

Rρ0

∂p ∂p dρ + dT . T =T 0 ∂ρ ∂T ρ=ρ0 p0 /ρ0 dρ

s ≡ s0 dT dT =

dp = γ

γ − 1 p0 dρ R ρ20

p0 ∂p dρ = dρ . ρ0 ∂ρ s=s0

∂p/∂ρ

p˜ = a20 ρ˜

a20 = a0

∂p . ∂ρ s=s0

ρ˜

u ˜

u0

x ρ˜ u˜

"

∂t ρ˜ + (ρ0 + ρ˜)∂x u˜ + (u0 + u ˜)∂x ρ˜ = 0 (ρ0 + ρ˜)[∂t u ˜ + (u0 + u ˜)∂x u˜] + a20 ∂x ρ˜ = 0 .

⎧ ˜ + u0 ∂x ρ˜ = 0 ⎪ ⎨ ∂t ρ˜ + ρ0 ∂x u a2 ⎪ ˜ + u0 ∂x u ˜ + 0 ∂x ρ˜ = 0 , ⎩ ∂t u ρ0 ρ0

L0 ρ˜ = 0 , L0 u˜ = 0 ,

2 2 2 L0 = ∂tt + 2u0 ∂tx + (u20 − a20 )∂xx ,

t

2 2 2 2 2 ∂tt ρ˜ ∂tx ρ˜ ∂xx ρ˜ ∂tt u ˜ ∂tx u ˜ 2 ∂xx ρ˜ 2 ∂tx u ˜

x

2 ∂xx u ˜ 2 2 ∂tt ρ˜ ∂tx ρ˜ 2 ∂tt u ˜

2 ∂xx u ˜ ρ˜

L0 L/a0

(ξ, τ )

L u0 /a0 = M0

(x, t)

L0 L0 =

a20 2 2 ∂τ τ + 2M0 ∂τ2ξ + (M02 − 1) ∂ξξ . 2 L L0 M0

η1 = ξ − (M0 + 1)τ C2 g2

η1 f1

η2 = ξ − (M0 − 1)τ ρ˜

g1

L0 C2

u ˜

η2 f2

ρ˜(ξ, τ ) = f1 (η1 ) + f2 (η2 ) , u˜(ξ, τ ) = g1 (η1 ) + g2 (η2 ) . ρ˜(x, 0) = ρ˜0 (x) u ˜(x, 0) = u ˜0 (x) "

f1 (ξ) + f2 (ξ) = ρ˜0 (ξ) ˜0 (ξ) . g1 (ξ) + g2 (ξ) = u ρ˜

u ˜

t=0

C2

4

4

3

3

τ

5

τ

5

2

2

1

1

0

0

-1

-1 -5

0

5

10

15

-5

0

5

ξ a

10

15

ξ b

ρ˜0 (ξ)/ρ0 u ˜0 (ξ)/u0 τ = 0 1 ... 5 u ˜(ξ, τ )/a0

M0 = 0.5 a

(ξ, τ ) η1,2 = ξ − (M0 ± 1)τ = ξρ

M0 = 1.5 b ξρ = +1 ξu = −1 ρ˜(ξ, τ )/ρ0 ξ η1,2 = ξ − (M0 ± 1)τ = ξu

⎧ ρ0 ⎪ u˜ (ξ) + M0 ρ˜0 (ξ) ⎨ −∂τ ρ˜(ξ, 0) = (M0 + 1)f1 (ξ) + (M0 − 1)f2 (ξ) = a0 0 a0 ⎪ ˜(ξ, 0) = (M0 + 1)g1 (ξ) + (M0 − 1)g2 (ξ) = M0 u˜0 (ξ) + ρ˜ (ξ) , ⎩ −∂τ u ρ0 0 ρ˜0

u˜0 ξ

ξ f1,2

f1,2 =

g1,2

1 ρ0 1 a0 ρ˜0 ± u˜0 ± , g1,2 , u ˜0 = ρ˜ 2 a0 2 ρ0 0

q=

1 ρ˜0 1 ρ˜0 u ˜0 u ˜0 , r= + − 2 ρ0 a0 2 ρ0 a0

ρ˜(ξ, τ )/ρ0 = q(η1 ) + r(η2 ) , u ˜(ξ, τ )/a0 = q(η1 ) − r(η2 ) . ρ˜(ξ, τ )/ρ0

u ˜(ξ, τ )/a0

τ

a M0 < 1 b M0 > 1 q(ξ) r(ξ) q η1 = ξ −(M0 +1)τ ≡ η2 = ξ − (M0 − 1)τ ≡ (ξ, τ ) a M0 < 1 b M0 > 1 M0 < 1 M0 > 1

r

t=0 s x

t p

ρ p/ργ ≡

x

t

⎧ ⎨ ∂t ρ + u∂x ρ + ρ∂x u = 0 ⎩ ∂t u + u∂x u + 1 ∂x p = 0 . ρ (x, t) ∂D

(x, t)

D

∂t q + v ∂x q = 0 , v

q

v

r a

x(t)

u x

r w

w

t (x, t)

⎧ ⎨ dx = v[a(x, t), u(x, t)] dt ⎩ x(0) = x0 ,

q x= D

Cx0

x0 w

v

Cx0

II

Cx 0 Cx 0

I

Cx0

q

q˜(t) = q[x(t), t] q˜

Cx0

x0 r

Cx 0

Cx 0

Cx0

q

II

I u

a

p/ργ ≡

a2 = γp/ρ

⎧ dp ⎪ ⎪ −γ ⎨ p dp ⎪ ⎪ ⎩ − p

dρ =0 ρ dρ da =2 , ρ a

δ = (γ − 1)/2 γ da dρ 1 da dp = , = . p δ a ρ δ a

⎧ a a ⎪ ⎨ ∂t + u∂x + a∂x u = 0 δ δ a ⎪ ⎩ ∂t u + a∂x + u∂x u = 0 . δ

⎧ ∂ (a/δ + u) + (u + a)∂x (a/δ + u) = 0 ⎪ ⎪ t ⎨ q

v

r

w

⎪ ∂ (a/δ − u) + (u − a)∂x (a/δ − u) = 0 , ⎪ ⎩ t q = a/δ + u r = a/δ − u ∂D

v = u+a w = u−a D

r

8 0

D

6

Fp C

4

t

¨ ξ, ˙ ξ ξ,

-0.5

-1 2

Fa

-1.5 0 0

1

2

3

-2

0

2

t a ξ¨

a

ξ˙

4

x b

ξ t

b

(x, t)

D

t

γ = 7/5

C

(ξ(t), t) Fa

a0 = 1

Cx0

Fp

(x, t) ∈ D II a

Cx 0 I

(x, t) I

∂D Cx0

q

˙ ξ(0) =0

II

Cx 0

r

a = δ(q + r)/2

(x, t)

u(x, 0) ≡ 0 t = 0

x x = ξ(t)

u = (q − r)/2

a(x, 0) ≡ a0 x>0 x = 0 t > 0

x = ξ(t)

˙ ξ(t)

t

t = 0 t>0

ξ¨

T T

" ¨ = ξ(t)

4t(t − 1) t ∈ [0, 1] 0 t ∈ (1, +∞) ,

ξ¨ = −1

t = 1/2 a

˙ ξ(t)

t>1

b

D x = ξ(t)

ξ˙0 (x, t) C

x = ξ(t) ∂t u

˙ u(ξ(t), t) = ξ(t) ∂D u(x, 0) ≡ 0

C

a(x, 0) ≡ a0

x q = r = a0 /δ 1/(u ± a)

u a = a0 u = 0 ±1/a0

dt/dx

t = x/a0 Fa

D b

D t = (x, t)

Fa

a0 t = x/a0

I r = a0 /δ C ˙ ξ(t)

1

t ˙ = ξ(t)

⎧2 ⎨ t2 (2t − 3) 3

⎩−2

3

ξ(t) =

⎧1 ⎨ t3 (t − 2)

t ∈ [0, 1]

⎩ −2t + 1

t ∈ (1, +∞)

3

3

3

ξ(t)

8

t

6

4

2

0 -2

0

2

4

x

II I

˙ ) dx/dt = −a0 +(1−δ)ξ(t

t = t (x, t)

t

q = a /δ+u

˙ ) ξ(t

C t u = (a − a0 + δu )/(2δ) a = a a a ∝ ρδ δ = (γ + 1)/2 t

II u = u[ξ(t ), t ] =

a = (a + a0 + δu )/2 ˙ ) a = a0 + δ ξ(t a0

t

t = t +

D

a = a[ξ(t ), t ]

˙ )] 1/[a0 + δ ξ(t

˙ ) u = ξ(t

(x, t)

ξ(t ) − x . ˙ ) a0 + δ ξ(t

t = x/a0

2

˙ ) > −a0 /δ ξ(t

(x, t) t = T + [x − ξ(T )]/[a0 + δ ξ˙0 ] Fp a > 0

1.05 0 1 0.95

u

a

-0.2

t = 0.5

-0.4

0.9

-0.6

0.85

-0.8 -2

0

2

4

-2

0

2

4

-2

0

2

4

-2

0

2

4

1.05 0 1

u

a

-0.2

t=2

0.95

-0.4

0.9

-0.6

0.85

-0.8 -2

0

2

4

1.05 0 -0.2

t=4

0.95

u

a

1

0.9

-0.4 -0.6

0.85

-0.8 -2

0

2

4

x

x

a

b a t = 0.5 2

b

4

b Fa

Fp

(x, t)

b

II Fp q = a /δ + ξ˙0 II

D

II a = a[ξ(t ), t ] a

u a = a0 + δ ξ˙0

t C u = ξ˙0

t

t>T

∂x a

x=

∂x u

ξ(t) t = 0.5

4

t

3

2

1

0 0

1

2

3

4

x II

x ¨ = 4t(1 − t) ξ(t)

0 ≤ t ≤ 1

0

t > 1

II

t

t = x/a0

a = a0 u = 0 ˙ ) a = a = a0 + δ ξ(t ˙ ) > 0 ξ(t

a > a0 C

t

t = x/a0 ˙ ) u = ξ(t C

1/(u + a) 0

1

ξ˙ a a

u a

ρu · ∇h = u · ∇p u · ∇p = −ρu · ∇|u|2 /2 u u = |u| u·∇

h+

u2 =0, 2

H = h + u2 /2 H H = H0 H =h+

h = cp T = a2 = ∂p/∂ρ = γp/ρ p = ρRT

u2 ≡ H0 . 2

a2 γRT = . γ−1 γ−1

cp = γR/(γ − 1) δ = (γ − 1)/2

a2 + δu2 ≡ a20 . a0 T0 = a20 /(γR)

√ a0 / δ u u = a √ a = a0 / δ

u δ

(γ + 1)/2 M = u/a

a

M2 =

δM 2 M2 2 = . , M δM 2 + 1 δ − δM2 1

M M

M

M δ /δ

ρ u

p ⎞

⎛ ⎞ ⎛ ρ ρ2,1 ⎜ u ⎟ ⎜ u2,1 ⎟ ⎟ lim ⎜ ⎟ = ⎜ ⎝ p2,1 ⎠ x→±∞ ⎝ p ⎠ h2,1 h

h

ρ 1 < ρ2 , u 1 > u 2 > 0 , p 1 < p 2 , h 1 < h 2

P r = 3/4 1

cp,v 2

l 1922

3

+∞

2 1

−∞

x d ρu = 0 , dx ρu = m

x

ρ1 u 1 = ρ2 u 2 . ρDu/Dt = ∇ · τ λ μ

1−D T ρuu =

x d d [(−p + λu ) + 2μu ] = −p + [(λ + 2μ) u ] , dx dx x

x

mu + p − (λ + 2μ) u ≡ mU p = −mu +

d [(λ + 2μ)u ] , dx

Dp/Dt + tr(S · ∇u) − ∇ · q (λ + 2μ) u 2 2

mh = up +(λ+2μ)u +

1−D

ρDh/Dt = tr(S · ∇u) =

d d u2 d d (kT ) = −m + [(λ+2μ)uu ]+ (kT ) . dx dx 2 dx dx H = h+ u2 /2

mH − (λ + 2μ)uu − kT ≡ mH .

U

H x → −∞

x → +∞

ρ1 u 1 = ρ2 u 2 = m mu1 + p1 = mu2 + p2 = mU H 1 = H2 = H

1 2 1

2

M1 H = (γ + 1)a∗ 2 /[2(γ − 1)]

H Mk2 =

k=1

ρk =

M2

Mk∗ 2 , γ+1 − δMk∗ 2 2

2

m 1 1 γ+1 −Mk∗ 2 , uk = a∗ Mk∗ , pk = m(U −a∗ Mk∗ ) , hk = , ∗ ∗ a Mk 2 γ−1 k = 1 ρ1 u 1

u1

2

m a2 = γp/ρ

1+

ρ2 u 2

1 1 = u2 1 + , 2 γM1 γM22

a∗ M1∗ +

1 1 = M2∗ + , M1 M2 M1∗ = M2∗

M1∗ M2∗ = 1 , u1 > u2

M1∗ = 1/M2∗

M 2 ρ2 u 2 p 2 ρ1 u 1 p 1

h2 h1

M1

δM12 + 1 γM12 − δ u1 δ M12 ρ2 = = ρ1 u2 δM12 + 1 1 p2 = (γM12 − δ) p1 δ h2 − h1 δ γM12 + 1 = 2 (M12 − 1) , h1 M12 δ M22

=

δ = (γ − 1)/2 δ = (γ + 1)/2 a∗ 2 2 (1 + δM12 ) . = a21 γ+1

de = cv dT = T ds − p d

1 , ρ cv

cp d

s dρ dp − =0. +γ cv ρ p

p −s/cv e ≡ ργ

exp

.

s2 − s1 p2 ρ2 −γ = cv p 1 ρ1 =

(γM12 − δ)(δM12 + 1)γ . δ 1+γ M12γ

m = M12 − 1

m 1

M22 1 − m ρ2 /ρ1 1 + m/δ u2 /u1 1 − m/δ p2 /p1 = 1 + γm/δ (h2 − h1 )/h1 2 δm/δ 2

exp[(s2 − s1 )/cv ] 1 + γδm3 /(3δ ) .

u2 /u1 1/M12 = (γ + 1)/2 u2 /u1 − δ 1 γ + 1 u2 p1 γ−1 = = u1 1 + = u1 1 + − 2 ρ1 u 1 γm1 2γ u1 2γ u2 γ+1 γ+1 = u1 (u1 + u2 ) , +1 = 2γ u1 2γ

U = u1 +

1

a21 u2 1 u21 = + 1 = u21 + = 2 2 γ−1 2 2 (γ − 1)M1 1 γ + 1 u2 γ+1 1

+ u1 u2 . = − 2 2(γ − 1) u1 2 2(γ − 1)

H = h1 + = u21

x → ξ(x) := P r m

λ + 2/3 μ = 0

0

x

d dξ d m d ds , = = Pr μ(s) dx dx dξ μ dξ

du 4 Pr m = mU 3 dξ 4 du dT H − Pr u − cp =H, 3 dξ dξ mu + p −

P r = μ/(k/cp ) mu + p − m H−

du = mU dξ

dH =H, dξ

P r = 3/4

H = Aeξ + H , ξ → +∞

H

H

A=0

H H u γ pu γ p = h = cp T = γ −1 ρ γ−1 m

u

γ−1 du = u(u − U ) + dξ γ =u

u−

H−

⇒

u2 H− γ−1 2 , p= m γ u

u2 2

γ−1 γ+1 (u1 + u2 ) + 2γ γ

=

γ+1 2 [u − (u1 + u2 )u + u1 u2 ] 2γ

=

γ+1 (u − u1 )(u − u2 ) < 0 , 2γ

γ+1 u2

u1 u2 − 2(γ − 1) 2

U

u

H

γ+1 u du = dξ (u − u1 )(u − u2 ) 2γ

u1 u γ+1 u u2 (u1 − u2 ) dξ . du ≡ du = − − u − u1 u − u2 u − u1 u − u2 2γ r=

u2 /u1 < 1

u γ+1 1 1 1 1 u − r dξ , −r u = d − d r 1− u u1 u 2γ r 2 −1 u1 u2 u γ+1 1 u 1 log 1 − − r ξ + − r log −1 = r u1 u2 2γ r

= log A

u 1/r δ 1 1− −r ξ u1 r . u r = A e γ −1 u2

A

ξ=0

u(0)2 = a(0)2 = u1 u2

u 1/r

x 1− (1 − r)1/r 3 δ 1 ds u1 −r m u r = 1 r exp 4 γ r 0 μ(s) −1 − 1 u2 r

μ = o(ml) u1 x=0

μ/m l x → +∞

u2

x → −∞

d2 u u1 u2 du δ √ u1 + u2 δ , 1− 2 . =2 u1 u2 − = 2 dξ x=0 γ 2 dξ γ u μ Δ u = u2

u = u1

√ δ M1 + 4 γ 1+r 4 γ mΔ √ = = ReΔ = μ 3 δ 1−r 3 δ δ M1 −

1 + δM12 1 + δM12

,

mΔ/μ

U

U2

mU x μ x

=m 0

x

ds . μ(s)

m/U

1

10 9

0.8

mΔ/μ

8

u

0.6 0.4

7 6 5

0.2

4

0 -12

3 -8

-4

0

4

8

1

2

3

4

Rex a

5

6

7

8

9

10

M1 b

a

u Rex = mx/μ u1

u2

u Δ

Δ ReΔ

M1

γ 2 − 1)/(3δ ) M1 2.331

γ = 7/5

b ReΔ = 4γ(γ +

H = 3/4

u = u(x)

μ

a Rex = mx/μ

ReΔ b ReΔ

M1 → +∞ Rex a

b a H h = a2 /(γ − 1) = H − u2 /2 a = −

M =

γ−1 uu 2a

u u u − 2 a = 1 + δM 2 M <0, a a u

T ds = dh −

dp ρ

⇒

ds = cp

dp dh −R , h p

a

4

0.8

0.6

ρ

p

3 0.4

2 0.2

1 -12

-8

-4

0

4

0 -12

8

Rex a

-4

0

4

8

Rex b H p

γ ρ Rex p1

-8

b

a ρ1

ρ2 p2

u2 p 2 u , h = − uu =− 2 p h u u u2 H− H− 2 2

H+

u2 H+ −uu 2 u s = cp + R u2 u2 u H− H− 2 2 R u γ = u2 + h + u2 − 2 u γ−1 u H− 2 u2 R u a2 − = γ−1 γ−1 u2 u H− 2 R 1 u 2 = a (1 − M 2 ) 2 γ−1 u a γ−1 = R (M 2 − 1)

−

u u

0.8

0.4

0.7

0.3

h

s/cv

0.6 0.5

0.2

0.1

0.4 0.3 -12

-8

-4

0

4

0 -12

8

-8

-4

0

Rex a

4

8

Rex b H

γ b

h

a s/cv

Rex h1 s1

h2 s2 /cv

x<0

x=0

a

x>0 x=0 2

s2

u=

√ u1 u2 =

u(x) H U

m U μ γ>1

H < U 2 /(2ν)

u1,2 =

m(U − u1,2 ) p2

h2

H

U u1

H

u2

ν = 1 − 1/γ 2 ∈ (0, 1) .

γ U± γ+1

U 2 − 2νH

h1,2 = H − u21,2 /2 p1 h1 p1 H > U 2 /2 ,

, ρ1,2 = m/u1,2 p1,2 =

h1 H > U 2 /(4ν) . √ 2

γ

1

√ 2

γ

β=

m (1 − r)1/r βx 3 γ+1 1 −r , χ(x) = r e , 1 8 γ r μ −1 r

f (u; x) =

1−

u r u 1/r − χ(x) −1 =0. u1 u2 x

u(x) u x=0

f (u; x) = 0 u x=0

(u1 + u2 )/2 u

x

x 1 − u(x)/u1 s

10−3

u(x)/u2 − 1

x x

u = −βχ

1 ru1

u

x

r+1 u −1 u2 . u u 1/r u 1/r−1 r 1− −1 1− + u2 u1 u2 u1

A x

x u ρuA ≡ m , m uu +

p ρ 2 = uu + a =0, ρ ρ x

ρ /ρ = −u /u − A /A

A u = (M 2 − 1) . A u A < 0 u > 0

u <0

p = −ρuu

A At

M < 1

M = 1 M > 1

M > 1

M <1 m/A T δ /(2δ) p m = ρu = Ma ∝ M = M (1 + δM 2 )−δ /(2δ) , A RT T0 1/2

R/γp0 /T0 m/A → 0

M →0

M → +∞ m/A M

m 4

p/ρ ≡ p/p0 = (T /T0 )γ/(2δ) γ

= ρ a At

(a/a0 )2 = T /T0 = (1 + δM 2 )−1

2.5

2

M2

1.5

1

0.5

0 0

0.5

1

1.5

2

2.5

M1 A2 = A1

A2 /A1 = 1.2 1.4 . . . 2 A2 /A1 = 1.2

M1 A = A1

A2 /A1 = 2

x1 A = A2

x2

A2 M1 = A1 M2

1 + δM22 1 + δM12

δ /(2δ) .

A2 /A1

M1 A1

M2

A2 1

A2 /A1 M1 m A2 = A1 2 M1 > 1

A1 A2

M1 m A2 = A2

m/A2 M2 = 1 M1 < 1

M1 A2

A2 = A2

34

36 163

36.2

36.3

37 38 3.1 3.2

2 2.8

3 4

5.1 5.3 9 83

84 103 104 4.3

85 4.4 92 94 5

5.6

118 5.8 5.9 6 125 128 128 130 134 138 140

143

13

1D 643 650 16

9

1990 1998 2000 1993 1994 1993 2002 2002 II 1993 1995 1986 1993 1975 1996 1980 1980 2002 1969 1945

1987

VI 1987 1996 1994

2000 1971 1958 1999 2000 2003

1999 1992 1979 III

II

1982 1983 1972 1993 2001 1994 2002 1

1976

f

g

2D 2D 3D 3D

2D 3D 2D 3D

f

g

h

2D 3D ¨

2D 3D

kε

2D 3D