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
U 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 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 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˜
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 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 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=
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 a 4
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
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 )
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 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 1
M > 1
M 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
