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.. () . - NEW USING OF ASYMPTOTICAL METHODS FOR HYDRODYNAMICAL STABILITY PROBLEMS Shatrov A.V. Kirov (This article deals with two points Pade-approximants (TPPAs) and their applications to hydrodynamical stability problems. The TPPA method presented in this work can be applied to calculation the oscillatory boundary layer growth over the top and battom plates of rotating channel) [1-8] , . , , , . , .
184


.. -- -10, 2002, .184-185

, [5-8]. [5], , . -, , [9-12]. , , - [9-11] . () , . [13], . [3,4] . [13]. 1. . Oxyz Oz. , , Ox. , , , () . : u, v, w ­ Ox, , Oz; v ­ ;
P* = P - 1 2 x 2 + y 2

(

2

)

- ;

- ;
185


­ ; W0 ­ . w v + =0 (1) z y
w

w w 1 P * 2w +v = 2u - + 2 z z z y u u 1 P * 2u +v = -2w - + 2 z y x y

(2)

w

(3)

, . (2) (3) 1 P * (4) - =0 z (5) = 2W0 x (4) (5), (2) (3)
-
w

1 P *

w w 2w +v = 2u + 2 z y y u u 2u +v = 2 (W 0 - w) + 2 z y y

(6) (7)

w

(6) (7) . y = 0 : u = 0, v = 0, w = 0 (8)
y : u 0, w W
0

(9)
186


.. -- -10, 2002, .184-187

z = z 0 : w = w0 ( y ), u = u 0 ( y ) z0 ­ , (6)
w 0 = W 00


(10)

. ­ (10) .

w 1 - W 0

dy

2

(11)

- w)2 , (1) (6) , ,

(W0

w(W0 - w z

[

)

2

]

+

v(W0 - w y

[

)

2

]

= -4u (W0 - w) - 2 (W0 - w

)

2w y 2

(12) (12) 0 , , (11)
d ( 0W dz
3 0

)

w w = -4 u (W 0 - w)dy - 2 y dy + 2W 0 y 0 0


2

y =0

(13)

: = w, = -v y z y z w = , = , ( , ) = , W0 W0 0 (z )
u , ( , ) = = d W0 W0 0 0 (6), (7), (13) (8), (9) (10)

( , ) =

187


2 F + 2 f = f + 2 - 2

(14)

2
2

+

F + 2 f (1 - ) = f - 2

(15)

f=

02 ,
2 0

df = F = -8 f (1 - )d - 4 d + 4 d 0 0

(16) (17) (18)

= 0 : = = 0, = 0 : 1, 0 = 0 : = 0 ( ), = 0 ( ),
f 0 = f (
0

)

(19)

2. . . Oz. 2.1. - . ( ). (14) ­ (16) , , . f , . u t = u xx + f (u ) , [11]. , . f. ,
188


.. -- -10, 2002, .184-189

. , « », f . , () = 0, f const , , df 0 F 0. d (14), (15)
2 + 2 f = 0 2 2 + 2 f (1 - ) = 0 2

(20) (21)

~ ~ = 1 - exp(- ) cos ~ ~ = exp(- ) sin ,

(22)

~ = 0 , = - . (22) [1]. , , () . . (14), (15) , , , , , , () 0. f 0, f ~ (14) (15) (17), (18) ( () , () = 0). 189


. 2.2. ( ). . , , f = 0.

+ +

= 0 2 = 0 2


(23) (24) (25)

= d
0

(17), (18). (23) ­ (25) , 1. (), , , (19). ( (17))

= 2a 2 + 3a 3 + 4a 4 = 2a 2 + 6a 3 + 12a 4 2 = 6a 3 + 24a 4 + 60a 3
2 3

= a 2 2 + a 3 3 + a 4 4 + a 5 5 + O(
4 5 3 5

) + 5a + O( ) + 20a + O( ) + O ( )
6 5 4 2 3

(26)

... (26) (23)
6a3 + 24a4 + 60a52 + 1 (a22 + a33 + a44 + a55 ) (2a2 + 6a3 +12a42 + 20a53 ) = 0 2

a 3 = 0, a 4 = 0, a 5 = -

a 22 , 60

190


.. -- -10, 2002, .184-191



~ 2 a 2 -

a 22 4 12 a2 ~ 2 a 2 2 - 2 60

(27)
5



~ 1-


A exp(- 2 + c 2 - c

)

(28)

c = (1 - )d
0



(29) (30)

a2 =

1 (1 - )d 2 0

, (23) exp 2 - c , , 0 ,

(

)

A = 2a 2 - - 2 + c exp( 2 - c )d 2 0

(31)

. ... Gi 2 3 4 5 = 1 + b2 + b3 + b4 + O( ) (24) 1 b2 = 0, b3 = 0, b4 = - a2 1 . 24 ,
1 a2 1 24

~ 1 -

(32)

191


B exp(- 2 + c 2 - c ~ - B exp(- 2 + c )

~

)

(33)

(24) ) , exp( 2 - c ) ,
B = - 1 + - 2 + c exp( 2 - c )d 0 2

(34)

2.3. - - , , , , Gi Ge , 2, , , , . . ,

a = 1-

(1

+ A 3 ) exp(- 2 + c 1 + + + 2 2 + 4 4

)

(35)

Gi a2 2a 2 - 2 4 (1 + 1 + 2 2 + 4 4 ) = 1 + 1 + 2 2 + 4 4 - 12 ( 2 - c )2 ... - (1 + A 3 )1 - 2 + c + 2 : 1 = c + 2a 2 c2 . 2 = 2a 2 (2a 2 + c ) - 1 + 2 Ge

4

+ A 3 )(2 - c ) = A(1 + 1 + 2 2 + 4 4 ) 4 = 2. , - 192

(1


.. -- -10, 2002, .184-193



a = 1-

c2 1 + (2a 2 + c ) + 2a 2 (2a 2 + c ) - 1 + 2 + 4 2

(1

+ A 3 ) exp(- 2 + c

)

(36)
4

, (32), (33) . (23) ­ (25) , () m. - (37) a = m exp(- 2 + c ) 0 + 1 Gi c - 1 0 = m , 1 = m . 1 1 m ( m ) = 0 , (15)





( )
m

= f (


[

m

)(2

- (

m

)) - 2]

(38)

,
a ( ) = a d
0

(39) (40)

a ( ) = a d
0



(6.40),


0



a d = 0

(41)

- (36) (37) (29), (30), (31), (38), (41)
193


2.4. . (14), (15). (42) ­ (46) : 1). (0), (0), (0) , , (0 ) (0 ) 2 (0 ) 2 (0 ) , , , ; 2 2 2). f(1), F(0) (45), (46), ; 3). (42) ­ (44), (1), (1), (1); 4). - (1) (1) a , a , , 1.-3., . «» Maple-6.1, , FORTRAN-. q ~ ~ * = = (1 - a )d , q* = = a d , = 0 , 0 0 =


a

- . -

* q* , . ( ) * q* (22) . (0 < 20) * q* (42) (43). , = 20, 1%. ,
194


.. -- -10, 2002, .184-195

, , Oz . , . , . |-2u| , . (22), . 3. . , ­ , - . = 0 f = f0, = 0(), = 0(), * ( ) - 0 ( ) * ( ) - 0 ( ) *, *, ( ). 0() 0() : , -, (42) ­ (46) f = 0 f0 f. f (16) (22) 0,1083. f0 () () . 0 = 0, f = f0. , f0 = 0,1050; 0,1100 . , 195


. ( f0 = 0,1290, 0,1390) 0(), 0() «» ( = -0,1, -0,15). f0 = 0,1290 * q ** 0 ,, , , . f0 = 0,1390 f . , , 0() ( 0( ) < 0 ) , . , 0() ( > 0 =

s < 0 > s, s ­ ), , f0 > 0,1390 .
-00-4.0-125.

. 1. . . ., 1975, 304 . 2. Ludwig H. Die ausgebildete KanalstrЖmungen in einem rotierenden System // Ing. Arch., 1951, b.19, s. 296-308. 3. .. . ­ : 1981, . 190-235. 4. .. : . . . . .-. , , ., 1987, 351 . 5. .. // . : - , 2000, . 180-205. 6. .. , : . . . . .-. , , ., 1983, 170. 7. .., .. // , 1978,
196


.. -- -10, 2002, .184-197

.35, 1, . 87-92. 8. .., .. . // . , , 3, 1982, . 154-157. 9. .., .. . ­ .: . , 1988, 240. 10. .., .., .. , // . . -, 1937, .1, 6. 11. .. // . .-. . , ., . II. ­ , 2000, .26-27. 12. .. -- // . .-. . , ., . II. ­ , 2000, .31-32. 13. Kurosaka . The oscillatory boundary layer growth over the top and battom plates J. Trans of. ASME, Ser D, 1973, v. 95, No 1, p. 139-146.

197