Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://new.math.msu.su/department/hydro/papers/ak-anis_fluid.pdf.pdf Äàòà èçìåíåíèÿ: Wed Jun 9 16:37:37 2010 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:45:31 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 64

..

-


. .

Mo c 2005


. .

. , , . , , . , , , 64 c., 8 . ( 05-01-00375) -1481.2003.1. . .

©-
, 2005 . © . .





1888 . ., , , , , , , . , , . , . , , , , ( ""- ), , 340 , . , . , , , . ( ""- ), , . , , , , , . 60- . , , , . 3


. [36], . . , . . [1], . [30], [45], . [41]. , , , , . , , . [17], . [39], . [28], . . [16,26,27]. , [21], [46]. , , . . , , . . . , .

4


1.
(, ""- ) (, ""- ) 1) [17]. , , . ­ 4--4- (), ´ , 0,5 2 . 16 ­ 210 39 ­ 470 C . , . , l, , ­ d, , l , d= . || <> , (. 1). d -d . n, . , [39,40]. , Q,
1)

.

5


. 1 1 Qij = Q di dj - 3

. 2 1 < 3 cos2 - 1 > , 2


ij

,

Q=

, , di ­ d, ij ­ (i, j = 1, 2, 3), ­ . , , , Q [19], . , , , 1) 100 : , (. 2). , , , .
""­ , .
1)

6


­ . , , . , , , . , . d . [17]. : , , , , , , . . 3 .

. 3

. 4

. (. 4). C , , 7


. . , . , BA ( ) BC ( ). (N) . , , --(n-) (), : . B C A N . .

. . ( -600 C), ( 4000 C), , . . , , , , , , . - . , ( ) , , ( ), , . , 8


, ­ , . , (. 5), , (. 6) [28].

. 5

. 6

2.
, G ­ , , U . , , . G . , , , , , . , , 9


. , ­ ±S O3 , , , , . . [29]. , , . . . , .-. . ±S L3 , , ±1, [33,46]. , ­ ±S L3 , . . [10-13]. U ­ () gpq ( ^ xi xi ), gpg = xi xj ij , xi = p , xi ( p , t) ­ ^ p pq p , p xi ­ . U , , . , , - , . , , U . , . , , : Tji (xp ) = Tlk (y q ) xi y l , y k xj 10


y i = y i (xj ) ­ . T G, y i = y i (xj ) Tji (xp ) Tji (y q ) , . , , , , . 1) (): ­ ±S L3 ­ , ±1. - ij k lmn . 0 0 gij ^ ­ 1 ij k lmn gil gj m gkn , ^^ ^ 60 0 , 0 g0 = g , ^ U ( V = 0 /). , , , ±1, , , . 2) , [21]: - Hpq , . Hpq H g= ^ H i = ipq H 0
pq

. 11


S L3 , e1 H , 1ab (ai ) = 0 c d (2.1) p 0ef cf - ed = 1. U ^^^ |H |2 = H i H j gij . 3) , O3 , , . 0 ij k gij . 0 , I1 , I2 , I3 [9,25]. [14] , ±S L3 , . , ­ . , ( , ) , . , () () . , (ai ), . p , (ai ) p . U (), , 12


. , , . [22] 1-4 ±O3 . S L3 , , . ±S L3 , ­ . ±O3 , [12]. , , , . , , , , ( ), m · : m, ­ · m ( ). : Dh , Dh , Dh [19].

3.
3.1.
. . , .-. . [33,46]. , U 13


gij = xk xl kl . , U ^ ij i s, , , , . , U . , , U i . , U . . V -
t2

I=
t
1


V

v2 -U 2

d dt .

­ , v ­ . , . . [24] I + W = 0 . (3.1) (3.1) W , . W [5]. I
t2

I =
t
1


V

v2 -U 2

t

2

d dt =
t
1

v i vi -
V

U x xi k

i k

d dt =

14


t

2

=
t
1

v
V t
2

i

d U ( xi ) - ( xi ) dt xi k k U dvi + dt k k
t
2

d dt =

=
t
1

-
V

xi d dt +

(3.2)

t

2

+
V

vi x d
t
1

i

-
t
1



U nk xi d dt , xi k

nk ­ V . (3.2) , . (3.2) di x = vi , dt xi = k xi i = x , k

( i , t) [8]. , d = dm = 0 d0 = const . (3.1) W
t
2

t

2

W = -
V

pi x d
t
1

i

+
t
1

pni xi d dt ,


(3.1) (3.2) , V xi , . , (3.2) xi ( j , t) 2 xi = t2
j

U xi j

.

(3.3) 15


, U ^k ^ ^ ij ^ Ik = Tk gij Ik = Tij g ij (k ­ ), (3.3) : U U Ik = . I k xi xi j j Tk ( ), Tji ( p ) 0 t Tji ( p ) = const( p ) , (3.4)

(3.3) . (3.4) (3.3).

3.2.
(3.3) . U = U (, |B |2 ), B ­ , . : pij =
2

dv i = dt

j

pij ,

(3.5)

U ij U - B i B j . |B |2

B dBi = -Bk dt
i

vk .

(3.6) 16


, A, , U = U (, |A|2 ). A : pij =
2

U ij U + Ai Aj , |A|2

(3.7)

dAi = Ak k v i . (3.8) dt , , . , , (3.5), (3.7), (3.8) [21], , ^ H i /, divH = 0.

3.3.
. . , , U = U0 () + U1 (I ) = U0 () + K I, 2

I = |A|2 , I = |B |2 ­ , K = const. : dAi =A dt
k k

vi ,

d = - dt

k

vk ,

(3.9) 17




dv i =- dt

i

p+

k

(K Ak Ai ) ,

p=

2

U0 .

(3.10)

, x1 = x t. , , , (x, t). : df f = = f x , x d x f df = = f t . t d t (3.11)

(x, t), , , d = t dt + x dx = 0 dx = - t = 0 , dt x (3.12)

0 ­ . a= dx - v1 . dt (3.13)

(3.11), (3.12), (3.13) (3.9), (3.10), : -aA1 - A1 u = 0 -aA2 - A1 v = 0 -aA3 - A1 w = 0 -a + u = 0 (3.14) -au + p - (K A1 A1 ) = 0 -av - (K A1 A2 ) = 0 -aw - (K A1 A3 ) = 0 v = (u, v , w), p= p .

18


, , . a. a (3.14), , , (x, t), . x = [a() + u()]t. (3.14) : a(a2 - (p + K A1 A1 ))(a2 - K A1 A1 )2 = 0, p + K A1 A1 0 K A1 A1 0 (3.14) , a = 0 a . 1) a = 0, , . 2) a = ± p + K A1 A1 , (3.14), , v , w, A2 , A3 ­ a > 0 : A1 = c , u= p +K c2 , 2 a= p +K c2 . 2

, A1 , . 3) a = ± K A1 , u = u0 A1 = A1 ­ , 0 v=A
2



K + v0 ,

w = A3 K + w0 ,

dx = u0 ± A1 K , 0 dt

A2 , A3 ­ (x, t). , , A v , . 19


, , x, , . [21] [19]. B , : dBi = -Bk dt dv i =- dt
i i

vk , (K B k B i )

d = - dt

k

vk ,
2

p-

k

p=

U0 .

, , : -aB1 + B1 u + B2 v + B3 w = 0 -aB2 = 0 -aB3 = 0 -a + u = 0 (3.15) 2 -au + p + (K B1 ) = 0 -av + (K B1 B2 ) = 0 -aw + (K B1 B3 ) = 0 : a3 (a4 - ba2 + c) = 0 ,
2 2 2 b = p + 3K B1 + K B2 + K B3 ,



2 2 2 c = K (p - K B1 )(B2 + B3 )) .

, , a , , a=± b± b2 - 4c , 2

(3.15) 2 . [21]. 20


, , , ( ) . , . , [21].

3.4.
, . , . , , ^ g , , A. , A ­ . U= ^ ^ U (xi , gij , Aq , s) = U (gpq , Ar , s) = U (, s, I ), = 0 / g , ^ ^ p ^p Aq . ^ I = gpq A ^ [25]: dU = pij eij , ^^ dt (3.16)

pij ­ , ^ dgij ^ 2eij = ( i vj + j vi ) = ­ dt . U , , : ^ dU U U ds U dAp = 2epq ^ + + . ^ dt gpq ^ s dt Ap dt ppq = 2 ^ U gpq ^ T= U , s 21


: ^ ^ hp dAp ds =- , dt T dt U ^ hp = . ^ Ap

[25] (3.3), (3.16), : ^ ^j dAi ^ ^ ^ dA 1 , -hi = µ1 + µ2 Ai Aj dt dt |A|2 , µ1 0 , ^^ dAp Aq ds = µ1 gpq ^ +µ T dt dt dt ds 1 = dt T hi =
2

µ1 + µ2 0 .

^p ^ dA Ap dt

2

1 0 |A|2
2

1 1 |h |2 + |h|| | µ1 µ1 + µ2
i j -

, Ai Aj j h. |A|2

Ai Aj |A|2
i j -

hj ,

hi = ||

Ai Aj (3.17) |A|2 A. 0 = const : 2 U ^k 2 xi U 2U j 2U = xpq + Ap + sp , = ^ t2 p x i xi s xi x j Ak xi q p q p p ds 1 = dt T |h|| |2 | h | 2 + µ1 µ1 + µ2 , 22


^ hp hp dAp || =- - , dt µ1 µ1 + µ2 p, q k . , , µ2 = 0, µ1 = const, A: ^ dAp 1 U = - g pq ^ . (3.18) ^ dt µ1 Aq U = U (s, J ), J = |A| = I , (3.18) : A A, , hi (Ak ) = 0. , µ1 > 0, 0 ^p ^p ^ d|A|2 ^ dA = - 1 Ap U = - 1 A Ap UJ 0 = Ap ^ dt dt µ1 µ1 J Ap (3.18) U 0. J (3.19)

U hi (Ak ) = 0 , = 0 0 ^^ Ap Ak =Ak 0 M , 2U (Mpq ) = . ^^ ^ Ap Aq Ak =Ak
0

U : 2U > 0. J 2 (3.20) 23


, (3.19), (3.20) U A [14]. [9].

4.
4.1.
() [24]. - [17,26,35] , : xi ( j , t) ­ ; di = k di , di ­ k d ­ , , ; T ­ ; , - d , |d| = 1 . (4.1)

, =
0





g0 = const . g ^ g0 = 1. 24


, [39]. . . , V - (n ­ ), : I + W + W = 0 . -: t
2

(4.2)

d +


I=
t1


V

d dt ,

v2 = +p 2 - F1 (T , d,

1 1 - 0
i

1 - F0 (T ) + I1 |d|2 - 2

(4.3)

d) + (|d|2 - 1) + Ug (r ) , (4.4)

= -F (d, n) + (|d|2 - 1) .

, :
t
2

t

2

W = -
V

(vi x + I1 di d )d
t t
2 1

i

i

+
t
1

j (pj xi + qi di )nj d dt + i

+
t
1

F - 2 di di - x di


i



i d dt ,

, F l i i , = F xi - x n. dl :
t
2

W =
t
1



-s T -
V

ij

(i

x

j)

- Li J di d dt , 1

(4.5)

25




J di = di -

[k

x i] dk .

: , ­ , d (4.1); s ­ ; I1 ­ , d ; dd d= ; dt F = F1 ­ [36]; F0 ­ ; F ­ ; Ug = (g , r ) ­ ; J ­ , d , W = 0 [7]. : DJ di = dJ di d di = - [ , d]i , dt dt

d , 2 = rot v . N [39]. ­ ; L1 ­ . B
(ij )

=

B ij + B 2

ji

,

B

[ij ]

=

B ij - B 2

ji

.

(4.3), (4.4) k
t2

I =
t
1


V



d dt +


1 ( a ) d dt , ^ a ^

26


t

2


t
1



t

2

d dt =
t
1

v i vi + p
V

V

1 1 - 0

+ p

1 -

-

F0 F1 F1 i F1 i T - T + I1 di di - d - i dk + T T di dk + Ug i x + (|d|2 - 1) + 2di d xi =
V t
2

i

d dt =
2

t

v xi + I1 di di d
i t
1

-

-
t t2
1




F1 F1 i d - p xk - i di x i dk dk l dvi i x + dt 1 1 - 0

l

nk d dt +

+
t
1

-
V

p + (|d|2 - 1) -

- +

i

p xi - F1 di k

F1 (F1 + F0 ) T - i di + 2di di + T d di -
k

(4.6) d dt ,
l

k



F1 d di k

i l

xl - I di =
k i

1

d2 di d dt2
k

i

( a ) = - a ^ ^ +

k

di -

xl ·

di .

F i F d + n di ni

- F a + ^

a( (|d|2 - 1) + 2 di di ) = ^
i

F i =- a ^ d + x di -


F l x ni - F x nl
i

i

-

(4.7)
i

F l x ni - F xi x nl

- (|d|2 - 1) - 2 di d

.

27


(4.6), (4.7) : a = a xi xi = a[ (xi xi ) - b ni xi ] , ^ ^ ^ ni = -xi n
k

xk ,

b = ni xi ­ . (4.6), (4.2) , : dv i = j pij - g i , (4.8) dt d2 di F1 k I1 2 + - k qi - 2di + L1i = 0 , (4.9) dt di |d| = 1 , = 0 , (4.10) (F0 + F1 ) = -s T pij = -p ij - q
i k j

dk + F1 . i jd

ij

- L1 dj ] ,

[i

(4.11) (4.12)

j qi =



: pij nj -
j qi nj +



i

= pij ) ni , (e

(4.13)

F j - 2 di = q(e)i nj , (4.14) di dF i = F xi - dl xl ni , (4.15) d dn e . : i i ( m )+ + ( m )- = 0 , (4.16) 28


m± ­ . , . (4.8)-(4.15) V , , ij Li , 1 , .

4.2.
- [45], F1 d: 2F1 = C
ij kl

(

i

dj ) (

k

dl ) ,

(4.17)

C ij kl = C ij kl (T ). , m · : m , C ij kl [22]: C
ij kl

= k1 g ij g

kl

+ k2 g ik g j l + k3 g il g

jk

+ k4 g ij dk dl + k5 g ik dj dl +

+k6 g il dk dj +k7 g kl di dj +k8 g j l dk di +k9 g j k di dl +k10 di dj dk dl . (4.18) (4.17), (4.18), , (4.1) di j di 0, [30,36]: 2F1 = k1 ( +k3 (
ij i j

di )2 + k2 (
k

ij

d )(

i l

dj )+
l

d )(

di ) + k8 (d

k

di ) (d

di ) .

(4.19)

k2 , k3 , , k2 , k3 , : (
i

di )2 - (

ik

d )( di

k

di )

i

(d

i

k

dk - dk

k

di ) .

(4.20) (4.21) 29

i

dk = -[d, rot d]k ,


i

d

k

i

dk = ( +

i

di )2 + (d, rot d)2 + |[d, rot d]|2 +
i

(4.22)

(d

k

ki

d )-

k

(dk

i

di ) .

(4.21), (4.22) d . 2F1 = K1 (div d)2 + K2 (d, rot d)2 + K3 |[d, rot d]|2 + +K
24 i

(4.23)

(dk

k

di - di div d) ,

[17,36]. Ki . K24 , K2 - K4 . "" , [26]: K1 ­ , d; K2 ­ , d; K3 ­ , d; Ki [17]. K1 = 6 , K2 = 4 , K3 = 7, 5 ( T = 250 [34,38]). (4.23) , . , . , , - ­ [17], K1 = K2 = K3 = K . (4.23) F1 (4.20)-(4.21) : 30


1 K ( i dj ) ( i dj ) . (4.24) 2 - , . F1 =

4.3.
m · , , . : F1 = C
ij kl

(

i

dj )(
ij kl

k

dl ) + L

ij k

di

j

dk ,

C L [22]: L
ij k

(4.18),

= k1 g ij dk + k2 g ik dj + k3 g kj di + k4 di dj dk .

, (4.23) : 2F1 = K1 (div d)2 + K2 ((d, rot d) + q )2 + K3 |[d, rotd]|2 , (4.25) |q | 107 1/. q . . , . d d = (sin cos , sin sin , cos) , 31


, (4.25) , , K 0, x, y , z = /2 = q z (. . 2). , d -d, /q .

4.4.
, , .
t
2

W =
t
1



(-s T - q ) d dt ,
V

(4.5) : dq = ij eij + Li DJ di . 1 dt : ds + dt
i

qi 1 = - 2q T T

i

i

T+

1 ij 1 eij + Li DJ di 0 . T T1

(4.26)

eij , i T , DJ di , ij , q i , Li ­ , , 1 gij , T , di , DJ di , eij , i T , di Li . 1 : qi = -(Ô
i

T + Ôd di dj

j

T).

Ô = 0, 126, Ôd = 0, 084 (/(·)). Ô = 0, 58, Ôd = 0. , Ô 0, Ô + Ôa 0. 32


, Li : 1 ij = Qij k DJ dk + Rij kl ekl , L1i = Qij DJ dj + Rij k e
jk

ij



.

[22,25], , , : Qij = 1 gij + 2 di dj , , Li di = 0, , 1 Qij di = 0 , R
ij k

= l1 gij dk + l2 gik dj + l3 gj k di + l4 di dj dk . Qj
ki

, = Rij k . Rij k = Rikj , l1 = l2 . Li di = 0, 1 Rij k di = 0 , l3 = 0, 2l1 + l4 = 0. Rij k = 2l1 (gi(j d
k)

- di dj dk ) .

Rij kl Cij kl (4.18). , , L1 : ij = 2l1 DJ d(i dj ) + 22 eij + 23 gij dk dl ekl + + 24 (d dj eik + di d ej k ) + 5 di dj ekl d d , L1i = 1 DJ di + 2l1 (eik dk - di dj dk ej k ) . (4.28) (4.27) 3 , gij dk dl ekl p [17]. 5 [39]: l1 , 2 , 4 , 5 , 1 , , . 10­100 ·. , , , , . 33
k k kl

(4.27)


4.5.
(4.9). I1 10-19 2 , , , , K 102 /, . (4.9) I1 : d di +d dt
k k

v

i

1 l1 -+ 2 1

+ dk

i

v

k

1 l1 + 2 1

-

(4.29)

-2 i =

l1 i di i d edd - 2 + = 0, 1 1 1

F1 k - k qi , edd = eij di dj . di l1 /1 = -1/2 - [7], (3.8). , 1 = -2l1 i = 0 (4.29) gpq dp dq = 1, ^ ^^ 1 ^p ^q dgpq ^ edd = d d 2 dt ^ ddp ^ + edd dp = 0 , dt ^ dp ^ dp = 0 , |d0 | ^ dp ­ 0 d. , , d d0 . , , , 34


(3.8) 3, A, . , v 10-4 / l = 10-3 (4.29) i /1 = K/v l 10-2 . , d . 2l1 /1 = -1.03, . , , , , l1 /1 1/2 [19], (3.6).

4.6.
(4.8)­(4.15) F dn . - [16] - [23]. , d . , d||n dn. d||n n , . dn, : d , , . ­ b, . b , n , , 35


Ki . , b n . d -d, , , , F d, F d2 . b F , : 1 F = + (1 - d2 ) , (4.30) b 2 , ­ , 0.04 /, 10-8 Â 10-3 / [34,38] - . , (4.30) [4], . b , . (4.2) b bn = cos , , [0, /2]. b , d = n, b = n F , d2 , d n . b b : b = b1 n + b2 d + b3 k , k ­ , n d. (d, b)2 = (b1 dn + b2 )2 , 1 = |b|2 = b2 + b2 + b2 + 2b1 b2 dn . 3 1 2 (4.31) (4.32)

2b1 b2 dn (4.31)-(4.32) , (d, b)2 d, n , 36


­ , b d n b3 = 0, n. dn = 1 : b = b1 n + b2 d ,
2 b2 = b2 + b2 d2 + 2b1 b2 dn , n 1 n

(4.33) (4.34)

b2 + b2 + 2b1 b2 dn = 1 . 1 2 (4.33), (4.34) , b2 = ± b2 2 +b -b
2 n 22 2 dn

= 1, (4.35)

1 - b2 n . 1 - d2 n

(4.33) (4.35) b1 : b1 = bn dn 1 - b2 n . 1 - d2 n (4.36)

(4.31) (4.36) (d, b) = (b1 dn + b2 ) = =(
2 2

bn

dn

1 - b2 n 1 - d2 n

dn ±

1 - b2 n 1 - d2 n

2

=

(1 - b2 )(1 - d2 ) ± dn bn )2 ( n n

(1 - b2 )(1 - d2 ) + |dn bn |)2 . n n

, F = F (d, b, n) F = F (dn ) : 1 F = + (1 - ( (1 - b2 )(1 - d2 ) + |dn bn |)2 ) . n n 2 (4.37)

(4.23), : 1 K24 ni (dk 2
k

di - di div d) .

37


4.7.
, [39]: ij = 1 eij + 2 di ej k dk + 3 dj eik dk + 4 di dj ekl dk dl + +5 di DJ dj + 6 dj DJ di Li : Li = 1 DJ di + 2 eij dj , (4.39) i . , 1 = 5 -
6 i

(4.38)

2 = 2 - 3 ,

(4.40) (4.41)

2 - 3 = 5 + 6 .

1 = 83, 2 = -35, 3 = 46, 4 = 6, 5, 5 = -1, 6 = -77, 5 (·). 1 = 2, ­ . , (4.41) . , , (4.26) : 1 0, 1 /2 + 2 + 3 + 4 0, 5 - 6 0,

4(21 + 2 + 3 )(5 - 6 ) - (5 + 6 + 2 - 3 )2 0 . . : i (4.42) iv = 0 , I
1

dv i + dt
j

i

(p + F1 ) = F i + F
1

j

ij ,

(4.43) (4.44) 38

d2 di = dt2



j

di

-

F1 + Li + di , di


, (4.38)-(4.44) (4.8)-(4.12). , F1 (4.43) j k dk i F1 . d,j ,i L L1 (4.8)-(4.12) , (4.9). L , (4.8) , ( 1 , 4.1 4.4):
ij

=

ij 1

- L1 dj ] ,

[i

, L1 . L L1 : Li = -Li - 2l1 di ekl dk dl . 1 : dv i + dt
i

(p + F1 ) = F i +

j

(Lj di ) +

ij j 2

,

(4.12) : 1 1 [i ij ij - L1 dj ] = 1 - Li dj + Lj di = 2 + Lj di . 21 21 , ij Li , , . , v d, . , (4.43), ( )
ij 1

39


. , , 2 3 , 5 6 (4.38), (4.40), (4.41). , , [41], . : 1 = 1 (1 + 3 - 6 ), 2 2 = 1 (5 + 1 + 2 ), 2 3 = 1 , 12 = 4 .

, [3]. , I1 0, , , , , . (4.44), d (3.17):
j i - dj di



F
j

1



j

di

-

F1 +L di

i

= 0,

(4.45)

(4.45) , d, 0 0. (4.44) d [7], (4.45) d L ij :
j

ikl d

k

F1 j dl

= ij

k

-

F1 kd

l

j

dl +

jk

.

(4.46)

, (4.46) , , 40


:
ij

= ji .

(4.47)

(4.46) DJ d d v : DJ di =
i (j - di dj ) (5 - 6 )

-

F1 + dj

F
k

1



k dj

+

(4.48)

i (j - di dj ) (3 - 2 )ej dk , k (5 - 6 ) , . , K/ 2 ( K/ 2 = 3, 7 · 10-6 ), , , . , .

+

5.
5.1.
­ . , , , , , . 30- . [41] . . [3], . [39], , [31,32,47]. : h 10-4 Â 10-3 (, 41


), V 10- , Ki ,

 10-3 /. i (K K ) = < 10-4 << 1, V h . (4.43)-(4.44) : ij - j i = 0 ,
5

dv i + i (p - Ug ) = j ij . (5.1) dt : v = (u(y ), 0, 0) , d = (cos (y ) sin (y ), sin (y ) sin (y ), cos (y )), x1 = x , x2 = y x3 = z ­ . 12 - 21 = 0 : (d1 )2 (2 - 3 + 5 - 6 ) + (d2 )2 (3 - 2 + 5 - 6 ) = 0 . (5.2)

(5.2), : 6 - 5 1 A) cos 2 = = , = ± arccos , = /2; 2 - 3 2 B) = 0. , . (4.41) , A = 0, 0, = ±9, 60 . , > 1 [17], , B. . 42


: . . 1) ( ). H , ( y = 0) , (y = H ) U , . (5.1) d2 u = 0, , dy 2 , u = U y /H , . (5.1) p0 : U 2 + 3 + 5 + 6 p = p0 + d2 + 4 (d2 )3 d1 . H 2 2) H . y = 0, g = (sin , - cos , 0). (5.1) :
2 2



12

= -g sin ,

(5.3) (5.4)

(

22

- p) = g cos .

, , , y = H du/dy = 0. , (5.3), (5.4) : g y (2h - y ) ) u = sin , 2b g (h - y ) (2D cos + (5 + 6 ) cos sin ), p= 2D 43


+

D = 1 + 2 cos2 + 3 sin2 +

4 sin2 2 + 4

6 5 sin2 - cos2 , D>0 . 2 2 g y B) u = (2h - y ) sin , p = g (h - y ) cos 1 , (1 = 0, i = 0) . , , , . , , , , [39], [40,42].

5.2.
5.2.1. [15,37]. . [30], d, . , ( ij = 2µeij ), F1 = 1 K( 2
i dj

)(

ij

1 d )+ K 2

24

(

i dj

)(

ji

d )-(

k

dk )2 . (5.5) 44


, g , :
i

v i = 0,
j

(5.6) (5.7) (5.8)



dv i = dt

pij + g i ,
i

j k - dj dk

F1 = 0, i dj

pij = -

ik

d



F1 - p g ij + 2µeij . k jd

, vn = D, (4.13)-(4.15), , , : pj nj = i
j k - dj d k i

- pa ni ,
j

(5.9) = 0, (5.10)

F1 dF n+ n ji id d dn

1 - (sin 1 - d2 + cos |dn |)2 , n 2 ­ , D ­ , pa ­ . , = 1, 2, a = gij xi xj , xi = xi / u ­ , u ­ . (5.7) (5.8) F = + dv i + dt
i

(p + F1 ) = µv i + g i .

, p . 45


C (5.5) K24 , (5.7) (5.8), (5.9) (5.10). (5.8) (5.9) [18]: e 2µ e
nn n

= 0, +
i

(5.11) di dF = d dn (5.12)

+ d




dF d dn dF d dn

=b

F - dn

+ p - pa ,

b ­ , d ­ . 5.2.2. . x, y , z , z g , x ­ k. d d = (sin cos , sin sin , cos) . 0 = 0 ­ p0 = pa - g z . , > 0, (5.6)-(5.8) :
i

v i = 0,

i vt +

i

p = µv i , ~

~ = 0 ,

= 0. (5.13) ~

t . = 0 ~ 46


. , 0. (5.13) v i = Re(ui (z ) E )) , ~ = Re(r(z ) E ) , p = Re(q (z ) E ), ~ = Re(s(z ) E ) ~

z = Re(Q E ), k ­ , Q ­ E = exp(i( t - k x)) . (5.11), (5.12) z -, : v1 = -i(A exp(k z ) + B l/k exp(lz )) , v3 = A exp(k z ) + B exp(lz ) , r = C exp(k z ) , A = -B 1 + Q=- B , 2µk 2 i 2µk 2 , l=k 1+ i , µk 2 v2 = 0 ,

q = -i A /k exp(k z ), s = i sin 0 r(z ) ,

C=

i k cos 0 Q , k K (1 - ( sin sin 0 )2 ) +

= K24 /K , B ­ . ( 2 - k g ) - (k ) k 3 = 4µ k ~ =+ ~
2

i + (1 - l/k ) µ k 2 / ,

(5.14)

cos 2 k K 0 , k K + /(1 - ( sin sin 0 )2 ) . (5.14) , | | > 1/ sin , 0 , , k , Im < 0, . 47


= 0, s 0, .. . , cos 0 = 0, .. d0 , . : i g + k2 ~ , 2µ k | | 0; µ k2 | | . µ k2

2iµ k 2 / +

g k + k 3 /, ~

5.3.
5.3.1. , , , , [43,44]. , [18,37]. , . , , . (4.42)-(4.45) , , :
i j k - dj d

(p + F1 - Ug ) = 0 , F1 - dj
i

(5.15) = 0. (5.16)

k

F1 i dj

(4.13)-(4.14) pj nj = i - pe ni , (5.17) i 48


j k - dj dk





F1 dF n+ n ji d dn id

j

= 0,

(5.18)

i = F xi -

dF dl xl ni . d dn

(5.15) p = (Ug - F1 ) + const , (5.19)

. ( ) , [20], (4.16), . , (4.15), , . , (5.18) (5.17) d2 F d d d2 n
dn

+

dF d dn

i

di = b F - d

n

dF d dn

+ p - pe .

u , , , . 5.3.2. , E=
V

F1 d +


F d ,

(5.20)

V , , 49


(5.19). , , , R0 10-5 . K 6 · 10-12 , , , : 1 = 2 = 10
-1

 10

-6

,

R0 103 Â 10-2 K . 2 ­ - . 1 . = 0 ( 1 = 2 = 0) E V : d = const, . > 0 1 . , 2 1 ( ), , ±dn 0 (4.37), (5.20). , 2 1 , ( ) (5.16), |dn | = cos [16]. i (4.11) -p ij - qk j dk . , . 50


, . E . r, , r = R() d= cos , sin ,0 , r n= R, -R /R, 0 R2 + R 2 .

u = + (r, ) ­ d. R() ­ , u(r, ) ­ x = /2 - . E + V ( ­ ) (4.37)
R

E + V = K
0 0

r2 u2 + u2 + r

sin2 u sin2

sin drd+

(5.21)

+ 2
0

R

R2 + R

2

+


sin2 ( - h( )) sin d+ 2

2 + 3

R3 sin d ,
0

= - u - arctg(R /R) , ur , u ­ r h = arccos |dn |: h( ) = | | [- /2; /2] . h( ) = , 0, /2, , . 51


0 r R0 ; 0 /2. u = 0 R = R0 . r1 = r/R0 , y (x) = (R - R0 )/(1 R0 ), w(r, x) = u/2 .

, (5.21) 1 2 , 1 , 2 K E 2 2
R
0

r2 u2 + u2 + r
0 0

u2 sin2

sin drd+

+

2 R0 0

+

sin2 ( - ) + y 2 + y sin2 ( - ) - 2

(5.22)

(y )2 ) |(y + u) sin 2( - )| + + sin d , 2 2 - E K R0 (2 + 2 2 (1 + 1 + 2 + 2 + 1 2 )) , 1 1




y sin d = 0 .
0

, (5.22), 0 < r1 < 1, 0 < x < /2 (wx cos x)x w 2 (r1 wr1 )r1 + - =0 (5.23) cos x cos2 x wr1 (1, x) = 1 sin 2(x + ) , 2

w(r1 , 0) = w(r1 , /2) = 0 . 52


, , (5.22) (y cos x) = (2y + 1 ) cos x - 1 cos(3x + 2) . 2 (5.24)

/2 0 y cos x dx = 0 (5.24) 1 . 1 1 = - (sin 2 + cos 2) . 6 y+ 1 sin 2(x + ) cos x , 2

. , , , . (5.24) x /2 y (x) y = 1 sin 2. 2 5.3.3. (5.23), (5.24). = 0, . w = w1 12 r1 sin 2x, 4 y = y1 1 (3 cos 2x - 1) . 24 (5.25)

, . w y (x) (5.22). = /2. w y (5.25) . ­ 53


. . 7 , 1 = 1 , . , (5.25) .

. 7

. 8

d = (r, , ) = 0, rc = r sin , z = r cos : drc = tgu u = 2 w . dz (5.26)

(5.25), = 0 = /2 2 z 2 rc = C exp ± 2 , 4R0 C [0, 1]. z , ­ .

54


5.3.4. : w = cos 2 w1 + sin 2 w2 , y = cos 2 y1 + sin 2 y2 , , w2 , y2 (5.23), (5.24) = /4. = /4 (5.24), (y cos x) = 2y - 1 6 cos x + 1 sin 3x 2 (5.27)

: y (0) = 0 y ( /2) = 1/2, , . . = /4 , R0 = 1, . 8 , . 1 = 1. = /4 (5.23) wr1 (1, x) = 1 cos 2x , 2

w(r1 , 0) = w(r1 , /2) = 0 . W , w = -Wx (r1 , x) . , , W
2 2 r1 · W = (r1 Wr1 )r1 +

(Wx cos x)x = 0, cos x

(5.28) 55


w (5.23). , r1 < 1, |x| /2 W 11 Wr1 (1, x) f (x) = - | sin 2x| . (5.29) 64 (5.28), (5.29) [6] W (r1 , x) = 1 4


2 - ln(1 - r1 cos + Z f (x0 ) sin 0 d0 d0 , Z

r1 ­ , W (r1 , x), 0 = /2 - x0 , = /2 - x, ­ , (r1 = 1), Z ­ .
2 Z 2 = 1 + r1 - 2r1 cos ,



cos = sin sin 0 cos 0 + cos cos 0 ,

(r1 , , ) ­ , W , (0 , 0 ) ­ . , w(r1 , x) : w(r, x) = 1 4


Z r1 + r 1 2r1 +3 Z - Z r1 cos + Z 2 Z

·

(5.30)

· [sin 0 cos cos 0 - cos 0 sin ]f (x0 ) sin 0 d0 d0 . (5.30) , , r1 , x . , (5.25) (5.30), (5.27), . 56


5.3.5. , . , (5.21), u(r, ) r = R() 0, u, . - R . R(). , u, (r2 ur )r + (u sin ) sin 2 - =0 sin 2 sin2 (5.31) (5.32)

u(r, 0) = u(r, /2) = 0 K (R2 ur - R u )
r =R

R R2 + R 2 sin 2( - h) 2 , 0 < < /2, = - u - arctg(R /R), . u 0 u a(r), a(r) ­ . (5.32) 0: = K a(R)R - R R2 + R
2



sin 2( - h) , 2

(5.33)

h = -arctg(R /R) 0 . (5.33) , a(r) R 0 sin 2 = 0 = 0; /2. , R /R , . (5.33) a(R(0)). (5.31)-(5.32) R sin R2 + R R + R sin2 ( - h) + sin 2( - h) 2 2 = 57

2


= sin 2R2 + R + R2 + R

K 2
2 2

u2 - r2 u2 + 2R ur u + r

sin2 u sin2

+ R2 +
r =R

(5.34) RR + sin2 ( - h) - sin 2( - h) . 2 2 R2 + R 2

, sin = 0 , (5.34) . (5.34) 0. 0, , R > 0, : R + R sin2 ( - h) + sin 2( - h) 0 . 2 2

z = R /R 1 = /al, : 2z + 1 [(z 3 + z ) cos2 + sin 2] = 0 , . , 1 , : z=- 1 sin 2 + O(4 ) . 1 2 + 1 cos2

, = 0; /2 . [2], , , .

58



1. . ., . . . . . . 1973 .7, .106-213. 2. . ., . ., . . . , 1980, 3, . 3-10. 3. . . . .: , 2002. 224 . 4. . ., . ., . . . , . 152, . 3, . 449-477. 5. . . . .: , 1983, 448 . 6. B. C. . .4. .: , 1981, 512 . 7. . ., . . . . .: 1996, . 1, 395 ., . 2, 394 . 8. . ., . . . .: , 1961. 9. . . . .: , 1978, 303 . 10. . . . , 1978. . 240, N 2. 298-301. 11. . . . . . -. . 1. . , 1997, N 0 6. 56-59. 12. . . . 2003, . 2, N 0 1. 126-183. 13. . . . . - - , 2001, 94 . 14. . ., . . . . . -. . 1, . . 1996, N 0 2. 59-62. 59


15. . ., . . . . . -. . 1. . . 2001, N 0 1. 42-43. 16. . ., . . - . : , 1994, 214 . 17. . . . .: . 1977, 400 . 18. . ., . . . , 1998, N. 3. 171-177. 19. . ., . . . .: , 1988, 144 . 20. . ., . ., . . . .: , 1963, . 1,2. 21. . ., . . .: , 1962, 246 . 22. . ., . . . , 1963, .27, N 0 3. 393-417. 23. . ., . . . .: - , 1991, 271 . 24. . . . . ., 1965, . 20, 5, 121-180. 25. . . . . 5, . .: , 1994, T. 1, 2. 26. . . . .: , 1983, 320 . 27. C . . . , 1987. . 153, . 2. 273 - 310. 28. . . .: , 1980, 344 . 29. . . . .: - , 1951, 172 . 30. . . .: , 1977, 247 . 31. Atkin R. J. Poiseuille flow of liquid crystals of the nematic type. Arch. Rat. Mech. Anal., 1970, V. 38, N. 3, pp. 224-240. 60


32. Atkin R. J., Leslie F. M. Couette flow of nematic liquid crystals Quart. J. Mech. Appl. Math., 1970, V. 23, N. 2, pp. 3-24. 33. Coleman B. D. Simple liquid crystals. Arch. Rat. Mech. Anal., 1965, V. 20, N. 1. pp. 41-58. 34. Demus D., Demus H., Zaschke H. Flussige Kristalle in Tabellen. Leipzig: Deutscher Verl. Grundstoffindustrie, 1974, 356 p. 35. Ericksen J. L. Continuum theory of liquid crystals of nematic type. Mol. Cryst. Liq. Cryst., 1969, V. 7, pp. 153-164. 36. Franc F. C. On the liquid crystals. Disc. Faradey. Soc., 1958, V. 25, pp. 19-28. 37. Golubiatnikov A. N., Kalugin A. G. On short surface waves in nematic liquid crystals. Mol. Cryst. Liq. Cryst., 2001, V. 366, pp. 2731-2736. 38. Kelker H., Hatz R. Handbook of Liquid crystals. Weinheim: Verl. Chemie, 1980, 917 p. 39. Leslie F. M. Some contitutive equations for liquid crystals. Arch. Rat. Mech. Anal., 1968, V. 28, N. 4, pp. 265-283. 40. McIntosh J. G., Leslie F. M., Sloan D. M. Stability for shearing flow of nematic liquid crystals. Continuum Mech. Thermodyn., 1997, V. 9, pp. 293-308. 41. Miesowicz M. Nature. 1935, V.136, p. 261. 42. Pieranski P., Guyon E. Instability of certain shear flows in nematic liquids. Phys. Rev. A, 1974, V. 9, N. 1, pp. 404-417. 43. Press M. J., Arrot A. S. Theory and experimental on configurations with cylindrical symmetry in liquid crystal droplets. Phys. Rev. Lett., 1974, V. 33, N. 7, pp. 403-406. 44. Press M. J., Arrot A. S. Elastic Energies and director fields in liquid crystal droplets. I. Cylindrical symmetry. J. de Phys. Coll. C. 1, 1975, V. 36, N. 3, pp. 177-184. 45. Ossen C. W. Neue Grundlegung der Theorie der anisotropen Flussigkeiten. Arkiv. Math., Astron., Fys., 1925, V. 19a, N. 9, ¨ pp. 16-19. 46. Wang C.-C. A general theory of subfluids. Arch. Rational Mech. Anal., 1965, V. 20, N. 1, pp. 1-40. 47. White A. E., Cladis P. E., Torza S. Study of liquid crystals in flow . Mol. Cryst. Liq. Cryst., 1977, V. 43, pp. 13-31. 61



1 2 3 3.1 . . . . . . . . . . . . . . . . . . . 3.2 . . . . . . . . . . . . . . . . . . . . . . . . 3.3 3.4 . . . . . . . . . . . . 4 4.1 . . . . . . . . . . 4.2 . . . . . . . . . . . 4.3 4.4 4.5 . . . . . . . . . . . . . . 4.6 . . . . . . . . . . . . . 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 9 14 14 17 18 21 25 25 30 32 33 35 36 39 42 42 45 45 47 49 49 51 55

... ... ...

5 5.1 . . . . . . . . . . . 5.2 . . . . . . . . . . . . . . . . . 5.2.1 . . . . . . . . . . . . . . 5.2.2 . . . 5.3 . . . . . . . . . . . . . . 5.3.1 . . . . . . . . . . . . . . 5.3.2 . . . . . . . . 5.3.3 . . . . . . . . . .

62


5.3.4 5.3.5

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

56 58 61

63


M., - , 64 .

01.09.2005 . 60â90 1/16. 4 .. 16 100 . ­ . , . 04059 20.02.2001 . - 64