Сборник задач и упражнений
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Сборник задач и упражненийTRANSCRIPT
-
- ( )
.. .. ..
2008
-
532.5(075) + 621039(075) 22.253.37 -69
.., .., .. : . .: , 2008. 124 .
, -
, . , - ( , ), , - ( -, , ).
-
. ., . .-. . . ISBN 978-5-7262-0960-9 - ( ), 2008
. .
26.09.2008. 6084 1/16. .-. . 7,75. . . 7,75. 150 .
. 4/134.
- ( ). 115409, , , 31
.
.
-
3
...................................................................................... 4 1.
.................................................................... 5 .................................................................... 8 .................................................................... 8
2. .................................................................... 12 ................................................................... 19 ..................................................................... 20
3. ..................................................................... 26 ................................................................... 50 .................................................................... 51
4. .................................................................... 41 .................................................................. 59 .................................................................... 51
5. .................................................................... 56 ................................................................... 63 .................................................................... 63
...................................................... 67 .................. 81 ................................................. 100 ....................... 108 ................................... 112
........................................................................... 124
-
4
, , . , ( , - ), , - - ( , , ).
, - , : , , , , .
, - - .
- - . -.
, , . - , , . .
. .. ( 13, 5
6), .. ( 4 6), .. ( 6 ).
-
1.
-. - , -. .
: - ()
0div =+
ur , (1.1) -
pPuuu V grad)( =+
rrrr, (1.2)
- ),( Tp= . (1.3) -
RTp = . (1.3a)
. )( T= ,
))(1( TT = . (1.3)
const = . (1.3) , , -
(). ( ) -. , -
5
-
, , .. - , (, , ) - . , , - (, - ..)
const=s . (1.4) (1.4)
const =p . (1.4)
, ..
VVP grad=r
(1.5) ( )( p= ), (1.2)
++=
)(2
grad)(
2puuu V
rrr , (1.6)
, () , urr rot
=P
P pdpp
0)(
)( . (1.7)
(1.6) ( 0
=ur )
const)(2
2=++ pu V , (1.8) l
- : , -
6
-
.
( urr rot =0) (1.8) , .. (1.8) .
( urr rot = 0 -
), ),,,( zyx ,
= gradur . (1.9) (1.6)
)()(2
2Fpu V =+++
(1.10) .
( = const) ( gzV = )
.const2
,grad,0
2
2
=++=
=
pgzu
ur (1.11)
.
((,), ((,))
),( yx , , ),( yx = = const .
yux
= , x
uy = . (1.12)
7
-
0=+
y
ux
u yx . (1.13)
),( yx , - ),,,( zyx , , ..
022
2
2=
+
yx. (1.14)
, .. - = const ))(x,y ,( yx = const -. += ),( yx + ) ,( yxi iyxz += -.
1. ? -
? 2. -
? 3. . 4.
? 5. ? ? 6. -
?
1.1. - , = 0. p0.
8
-
9
1.2. . . p0 - 0.
1.3. , u0, p0, 0.
1.4. , u0, p0, 0, T0. .
1.5. - , . 1.1.
. 1.1 1.6.
, . 1.2.
. 1.2
1.7. - - .
P
U
U
P
-
1.8. , , , - h . l. , . g.
1.9. 1.8 V.
1.10. , r0, -
, (. 1.3.). - h, p0. , m. g. . ? [1]
10
. 1.3
h
1.11. 1.10 , - - .
1.12. - . , - . . - p
)(R
0 . 1.13. p0 -
r0 p, p > p0. - , , [1].
1.14. - .
-
u0, p0, . - , .
1.15. ),,( zyx - u0 nr .
1.16. ),( yx ) u
,( yx0 n
r . 1.17. ),,( zyx
() Q [3/], . . 0r
r
1.18. ),( yx ) () V [,( yx 2/], . - . 0r
r
1.19. ),( yx ) ( - , - ). -
,( yx
0rr .
1.20. az= . , .
1.21. , )()( 20 zrzuz += , -
u , , -
r
202 ruM =
0. 1.22. , , c
n n > 1, / n.
nczz = )(
11
-
2.
j
jiVi
j
ij
i
xP
Pxuuu
+=+
, 3,2,1=i , (2.1)
,
, - 1 3.
ViP ijP
+
++=i
j
j
iijij x
uxuupP )div
32( r . (2.2)
(2.1), (2.2) - (1.1), (1.3) - (),
Dxpupq
xT
xxTucTc
jjV
jjjjpp +
+++
=
+ , (2.3)
, () . (2.3) - , , . -
D
( 2div32 u
xu
xu
xuD
i
j
i
j
j
i r
+
= ) . (2.4)
. Vq
12
-
, const = const = , -
0div =ur , (2.5)
upPuuu Vrrrrr 2grad1)( +=+
, (2.6)
0=+
+
zu
yu
xu zyx , (2.7)
+
++
=+
++
2
2
2
2
2
2
1
zu
yu
xu
xpg
zuu
yuu
xuuu xxxxxzxyxxx , (2.8)
+
++
=+
++
2
2
2
2
2
2
1
zu
yu
xu
ypg
zu
uyu
ux
uu
u yyyy
yz
yy
yx
y , (2.9)
+
++
=+
++
2
2
2
2
2
2
1
zu
yu
xu
zpg
zuu
yuu
xuuu zzzzzzzyzxz . (2.10)
(2.8) (2.10) -.
(r, , z) -
01)(1 =+
+
zuu
rru
rrz
r , (2.11)
,21
222
2
+
=
=+
++
urr
uurpg
ru
zuuu
ru
ruuu
rrr
rz
rrr
r
(2.12)
,21 222
++
=
=++
++
r
rzr
urr
uup
rg
ruu
zu
uu
ru
ru
uu
(2.13)
13
-
zzz
zzz
rz u
zpg
zuuu
ru
ruuu 21 +
=
++
+ , (2.14)
2
2
2
2
22 11
zrrr
rr +
+
.
( ) ,,r0)sin(
sin1
sin1)(1 22 =
+
+
ur
ur
urrr r
, (2.15)
+
=+
+
++
rpg
ruuu
ruu
ru
ruuu rrrrrr
1sin
22
)sin2ctg222( 2222
2
+ ururu
rruu rr , (2.16)
=+++
+
+
ruu
ruuu
ruu
ru
ru
uu r
rctg
sin
,sincos2
sin2
sin
sin1
222222
+
++
+
=
ur
urr
uu
pr
g
r (2.17)
=++
+
+
ru
ruuu
ruu
ru
ruuu rr
ctgsin
2
,sincos2
sin2
1
222222
++
+=
urr
uur
u
pr
g
r (2.18)
2
2
2222
22
sin1sin
sin11
+
+
=
rrrr
rr.
14
-
.
. , , () , - - .
(2.6) uu rr )( , ,
0div =ur , (2.19) upPu Vrrr 2grad
1 +=
. (2.20) -
. , , , - .
, - . - - , .. - , .
: , , - , - (1.1)(1.3).
15
-
- . -, . - ( > 0) xy ,0=
0 =
+
yu
xu yx , (2.21)
yu
ydxdp
yuu
xuu xxyxx
+=
+
. (2.22)
dxduu
dxdp 0
0= . (2.23) (2.21), (2.22)
2)(Re)(
20ucx xf = , (2.24)
xxfc Re
664,0)(Re = , =xu
x0Re . (2.25)
(2.21), (2.22) . , - (x) -, ..
))(
(),(
0 xyF
uyxux = , (2.26)
0)(
ux
kx= . (2.27)
16
-
, = 5,83. 4322)( YYYYF += k k
- -
20
2(Re) SuCF D= , (2.28)
, - ( ), - , -,
0u S(Re)DC
Re 0lu= , l .
. 2.1 - . 2.2.
. 2.1
17
-
. 2.2
.
, . , .. = = 0,
(1.1)
zu
xu yu
0=
zuz , ),,( = yxuu zz . -
- :
+
+=
2
2
2
2
yu
xu
zpu zzz , (2.29)
0==
yp
xp . (2.30)
18
-
, - (.. ) .
lp
zp ==
const , p l. -
),( yxuz
lp
yu
xu zz
22
2
2 =+
(2.31)
0=zu .
:
0u l
2(Re)
20
udlp
+= . (2.32)
(2.32): (Re) , -
=0Re du , ,
-, ( , , , ..), -,
dSd 4 = , S ,
. Re = ,
A .
1.
? 2. ? .
19
-
3. , - , ()?
4. ? - ?
5. - . ?
6. ?
7. - - ?
8. ?
2.1. -
. . , , - .
0r
0u p
2.2. . 0,01, 0,03, 0,10 0,30 . 3 6 .
2.3. . - 4 . - 10500 /3, 830 /3, 2,910-7 2/.
2.4. 1250 /3 , 3 , , 8,00,2 . 300 .
2.5. . , , .
20
-
2.6. -, 16 . - 5 .
2.7. ,
53=l
,10 =u /. 610.15 = 2/ ( 1=p , ). C18 o=t
2.8. , /.
, 0,10 =u
4,0=l 0,1=b . 6100,1 = 2/, (t = 20 ).
2.9. - (. 2.3).
)( x
y
(x)
*(x)
U0
x . 2.3
2.10. 2.8 , - . , .
500 =h
2.11. , - 12=d , .
/. 3,330 =u 21
-
2.12. , 100 /, . , - 12 13 . - 20%. 44 /, 30%.
2.13. =10 =10
ddRe
4. , (2.27) , - , . )(0 xu
2.14. - , , - . - h.
0u
2.15. 2.14 - . .
2.16. , 0= . 0u
2.17. - - , - . cos0 = uu
2.18. . . . -
0r
22
-
0. r = r0 . , - .
2.19. r1, r2, , - . , .
2.20. r1 r2 - . , - 0. . ( ) .
2.21. - - 2h. .
2.22. - - r1, r2.
2.23. - - nba = . .
2.24. - - - . .
2.25. - . - Vl 2/. -
23
-
24
. .
2.26. - . . 20 , 22,6 /.
2.27. D = 100 l = 100 - d = 5 u0 = 0,4 / (. 2.4). F , -20 ( = 1,0 /, = 892,5 /3).
. 2.4
2.28. - (. 2.5). - , , - = 8 . D = 60 , - d = 2 , l = 200 . H = 0,32 . : , . .
. 2.5
F
U0
d D
H
H 2
D l
d
-
2.29. (. 2.6). , - . , - )
F(h .
r0, h0 (h0
-
3.
, -
- ().
. ( ) -
, -, .
: (); (); (); (); (); ().
, , = lu . [ ] TLu = , L [ ] =u . amF = , [ ] 2= MLTF [ ] 2 =F . ( )
- - , -
[ ] == TML . (3.1) , -
, (3.1) , .. - .
(-).
26
-
, - . :
s
xxX = , 0rrR = ;
auM = ; - LlKn = . - , . , , , - , - ,
2 .
. :
Bi
l= , 2Fo= a ,
Relu= .. -
mn
mnn aaa = 21 21 , (3.2)
, .
maaa ,...,, 21 mnnn ,...,, 21
, , (, , ). , . , . - . - , .
, , -. - , .
-. 0),,,( 21 = naaaf (3.3)
27
-
- , - ( ), ( k < l ),
lnl k
kn , ln kl :
0),,,,,,( 2121 = kllnsssF . (3.4) ) -
. ( lnCkn
(3.3), , , (3.4).
: -. , - .
.
. p
- u0, d l , - .
),,,,( 0 ldufp = . (3.5) . 1. , -
,
[ ] 2
= , [ ]
0 =u , [ ] =d , [ ] , (3.6) =l[ ] 3
= , [ ]
2
= .
28
-
2. n, l, k -.
n , (3.6):
n = 6. d l -
. -, . l :
l = 5.
, ,
2 , 3
2 (-
l) , .. .
, , , , 3
.
, :
2 = ,
2
32
= . (3.7)
,
, : , 2 , 3
,
2 , 2
. ,
. , k :
k = 3.
29
n, l, k , -, - (n k = 3) , (n l = 1) (l k = 2).
-
3. ( - k) . : ,
3 . ,
Al, Au A
lA
, uA
,
3
3 A
. (3.8)
. -, , Al = 100, , - l d 100 100l 100d. , , Au A . -, - p , - (3.8). (3.7)
. (3.8) (3.5)
uAu0 A
lu AA2uApA
),,,,( 02
lulluu AAAlAdAAufp = . (3.9) 4.
, Al, Au A - . , ( k) (3.9) - , :
10 =uAu , 1=lAd , 1 =A . (3.10) 5. (3.10) -
01 uAu = , dAl 1= , 1=A , (3.9),
=
dudlf
up
020
,1,,1,1
, (3.11)
30
-
= Re,2
0 dlF
up . (3.12)
, - -, , 3 4 . -, 4 (3.10)
12 = up , 1=lAd , 1= lu AA . (3.13) (3.13)
dAl
1= , dAu = , 2
2dp
A = , (3.9),
= 22
0 ,,1,1dpd
lduf . (3.14)
(3.14) -,
=
Re,122
dlf
dp. (3.15)
(3.15) (3.12), , (3.15) (3.12) . , (3.15) - , (3.12)
=
= Re,Re,
Re1
1220 d
lFdlf
up . (3.16)
n k = 1. - - F() = 0
const = . (3.17) (3.2) (3.17) ,
- -
31
-
= 1321 naaaCa . (3.18) . -
) .
.
,( lgfT =
1= kn , .
= lCgT
2
= . . : +=0 ; :
. , = 21 5,0= , 5,0= glCT = . . -
k , -, , .. , . [3] - - . , . . , .
zyx ,,
.
,
( ), - - : -, , .
.
32
-
. - - 2h l.
. , - , ( ):
2
21yu
zp
zuu
yuu zzzzy
+
=+
,
2
21yu
yp
zu
uy
uu yyz
yy
+
=+
, (3.19)
0=+
z
uy
u zy ,
lzhy = 0, 0== zy uu hyhz = ,0 0uuz = .
(3.20)
. 1. ,
puuzy zy ,,,, . (3.21)
2. , . , - , . - .
0,, ulh . (3.22) . 3. , -
0,,,
uuU
uu
UlzZ
hyY zz
yy ==== . (3.23)
4. (3.23)
00 ,,, uUuuUulZzhYy zzyy ==== (3.24) 33
-
(3.24) (3.19), (3.20).
2
2
20
20
20 1
YU
hu
Zp
lZUU
lu
YUU
hu zz
zz
y +
=
+ ,
2
2
20
20
20 1
YU
hu
Yp
hZU
Ul
uY
UU
hu yy
zy
y +
=
+
, (3.25)
000 =+
Z
Ul
uY
Uhu zy ,
lZlhYh = 0, 000 == zy UuUu , hYhhZl = ,0 00 uUu z = .
(3.26)
(3.25) (3.26) , , - : , , , , - . 5.
. (3.25) , ,
hu20 hu0 :
2
2
Re1
YU
ZP
lh
ZUU
lh
YUU zzzzy
+=
+ ,
2
2
Re1
YU
YP
ZU
Ulh
YU
U yyzy
y +
=+
, (3.27)
0=+
Z
Ulh
YU zy ,
10,1 = ZY 0== zy UU , . 11,0 = YZ 1=zU
(3.28)
(3.27) -
20u
pP = ,
34
-
, -
: =hu0Re
lh .
- . , .
(3.27) (3.28) , -
zy UU ,
)Re,,,(1 lhZYfU y = , )Re,,,(2 lhZYfU z = , (3.29) )Re,,,(3 lhZYfP = .
, )Re,,(4 lhZfP = , (3.30)
)(Re,)Re,,1()Re,,0( 4420
lhFlhflhfupP ==
. (3.31)
,
:
1) ; 1)
. -
, , - .
,
35
-
= . (3.32) -
, - - .
, - :
= XX xrr
. (3.33) , , -
, , .. - .
:
= . (3.34) ,
, , , - . = 1, .
CCC x ,,
i , -
, .. l. k .
. kl ,
: ,
. -
. , , -
() . ,
.
36
-
, - .
- , :
-, , - .
, (3.32)(3.34). , -
1=CCC xu .
1. ? 2. u , , r . 3. -. 4. ,
? ? 0= kn5. ,
? 1= kn6. , ?
? 7. . 8. . 9. ?
3.1.
lp 0, d, - . -
37
-
.
3.2. - , F, , 0, d, - . - - .
3.3. 3.2 -, . . .
3.4. -
0r
, - . , , , - .
3.5. -
-
) . .
,,,( = yfu3.6.
. 0= . , - , - ?
3.7. - - . -
0=0. -
38
-
, - , ?
3.8. (. 3.1), - , , , - T = f(m, s, , g). - T .
39
. 3.1 . 3.2
3.9. (. 3.2) - , , - . -
.
)
)
,,,( = AgfE
3.10. , - , , , - . -
, , .
,,( = Efr
3.11. (. 3.3) - - , g h, ..
. . ),,( hgfM =
A
-
40
. 3.3
h
3.12.
D - ( ), g , -
. -
D . 3.13. , -
, -, V, g . -.
3.14. 3.8 . 3.15. 3.9 . 3.16.
. . u
3.17. 3.16 -, - . .
-
4.
-
, - - .
, - .
- ( , , - .), - . -, : uuu rrr += , ur ; ur ; ur -.
, . - . -
+
=+
''
jij
i
jij
jii uuxu
xxP
xuuu , (4.1)
41
'' jiuu ( ). ( i = j) -
-
, ( i = j) - .
(4.1) - - .
- :
0
=
j
j
xu
. (4.2)
- '' jiuu , , - (4.1)(4.2) .
(4.1) - (4.2) , .
- - x :
TxyP
dyuduluuP xyji
Txy
'''' == . (4.3) . (1877 .)
Vl= , ( , , V - , l -) (4.3) :
'' luyt = . (4.4) -
dyudluu xxy
''' -
: 42
-
dyud
dyudl
dyud
dyudlP xxxxTxy = 22' , (4.5)
2'll = - ( ); - .
l
, - :
yl = , (4.6) .
= 0,4.
:
dyudl xt
2= . (4.7) , -
x, (4.1) :
0)( '' =+
jix uudyud
yxP , (4.8)
.
( 0=
xP ) -
y - :
const'' == jix uudyud . (4.9)
(4.9) , - ),,( = x yfu
43
-
=
*
*
yufuux , (4.10)
= u* .
: *uux= =
*yu , ,
const = , , -
: )(= . (4.11)
)( .
- -, . , , , -
C+= ln1 . (4.13)
: 30 ,5 t , = 5,5, - :
- : = ; 7,11
-
- : ; = 5 .
0
*3,0uux= . (4.16)
0,2 - , 75% 25% - 0,8 - .
-
+
=yfHyuf
uux
1*
*, (4.17)
)(* =
f
yuf ,
)12(2
sin11 += yf ;
= 1,375 .
:
2)(Re
20
uC xf
= , (4.18) ,
)(RexfC
=xu
x0Re .
-:
45
-
755
2,0 10Re105 ,Re0576,0
-
- :
25,0 Re316,0Re)( = (4.25) 510Re
:
237,0 Re221,00032,0(Re) += . (4.26) 85 10Re10
-
. u
Re(Re)8
=d
u . (4.30)
, - ( - ..). -, -, . ( k )- -.
(k )- -
( k )- - (4.31), (4.32), - k (4.33) (4.34) :
0
=
j
j
xu
, (4.31)
+
=+
''
jij
i
jij
jii uuxu
xxP
xuuu
, (4.32)
,
32
+
+
+
+
=+
j
iij
i
j
j
it
jk
t
jjj
xuk
xu
xu
xk
xxkuk
(4.33)
48
-
.
32
22
3
2
2
1
+
+
+
+
+
=+
jk
it
j
iij
i
j
j
it
j
t
jjj
xxuC
kC
xuk
xu
xu
kC
xxxu
, (4.34)
,,,, 321 kCCC , ij .
() ji uu (4.32) - t -
= ji uu kxu
xu
iji
j
j
it
+
32
. (4.35)
, t , - :
= 2kCt , (4.36)
. C
,,,,, 321 kCCCC , (4.28)(4.33) , , - , , - - :
])50/Re1/(4,3exp[09,0 2tC += , 44,11 =C ,
49
-
))Reexp(3,01(92,1 22 tC = , 0,23 =C , 0,1=k , 3,1= ,
. = /Re 2kt
1. . -
? 2. -
? 3. . -
? 4. ? u5. -
? 6.
. 7. -
? 8. -
-?
9. l , ? ) , ) .
10. .
11. ? 12. ? -
: , -?
13. - ?
50
-
4.1. . -
. Re = 50000. - . :
) -, ;
) , P .
4.2. , . P -. .
4.3. : . - .
4.4. Nu = = 0,023Re0,8Pr0,4 ( ). , . .
8,0u
4.5. . Re 104, 105. (: , . -
0,8)2,0lg(Re
1
= .)
4.6. ( P )
. -. ( ).
51
-
4.7. Re, - - 2,1=ds . (: .. - )
4.8. - - . d, s.
4.9. N . , - , ,
iF
id il i . .
4.10. - N
- . l,
iF
id
25,0 )(316,0
= iii du , . 4.11.
. , - p . p , - -
+= 2
0
2
0 21
248,31)(
rrurux ,
.
0u
4.12. 16,0 4 /c 300 1 - 10 = 10 , :
52
-
) 12 ; ) 2 ; ) -
? 4.13. , -
, , - ( ), :
) 16,0 12,0 ; ) 300 330 ? 4.14. -
,
lp , , 150 , T = 293 50 /.
4.15. , - = 1,3; - . - 0,5 , - 0,125 , 15 / .
Dc
4.16. 150 60 /. l = 24 - - , 89 .
, , .
4.17. d1 =150 d2 =100 1 3/. :
= 2
1
22
15,0 dd .
53
-
4.18. , - d1 = 200 d2 = 300 , - 0,2 3/ T = 293 . , d1 d2 6? : - , .. hhh += ,
gu
ddl
gdpdh
2
2
= , g
uuh2
)( 221
= . - =sin , =1/(1,8 lg Re 1,5)2.
4.19. 100 100 /, 6105 , 293 . , M .
4.20. , ])(1[ 0max
nrruu = , , n = 2, 4, 6, 8.
maxu
4.21. 5 /, 760 . . - 288 . , 3 , ?
4.22. , - Rex = 3,2105, , , , - , - 20 / T0 = 288 p0 = 760 . .
4.23. (T0 = 288 p0 = 760 . .)
54
-
30 /. - ( y = 0 y = ) x = 50 . x = 200 . - Re 5105.
4.24. , . : 3 , - 3 . 4 /, 800 /3, 14 , 0,03, 4,5.
55
-
5.
( -
, ) - -.
=
1
V
dVV
, =
1
V
dVV
, (5.1)
V , V V, . . 5.1 - .
. 5.1 V -
, - . - 56
-
, -.
, s ( 5.2, ). ( - , ). s 60 (. 5.2, ).
) )
. 5.2 (5.1) -
- , - VV :
= . (5.2) -
, , - , () .
, V.
(5.1) , , - , - . -, - , - (5.1).
57
-
-
( const = ) 0=
j
j
xu
, (5.3)
jij
ijij
i Tx
guux
u+=
+ . (5.4)
(5.4) Tji - , , -,
j
i
j
j
iijijijij uux
uxupTTT )()(
++=+= . (5.5)
(5.3) (5.1), - V V:
01
=+ dSuVx
u
Sn
j
j . (5.6)
(5.6) , - ,
dSnV S
+=
1 r , (5.7)
S V, nr
- S. (5.7) - , -, .
(5.7) , . -
(5.2) (5.6)
0=
j
j
x
. (5.8)
(5.4) 58
-
.1
1
dSTnV
Tx
g
dSuuV
uux
u
jiS
jjij
i
iS
njij
i
++=
=++
(5.9)
(5.9) - , i- - .
dSTnV
f jiS
ji =
1 . (5.10)
-- , ..
+= iii uuu , (5.11) . u
(5.11) - -
+= jijiji uuuuuu , (5.12) jiuu -, .
ji uu
(5.8) (5.10) -
ij
jiiji
j
i Tx
fguux
u++=
+ , (5.13)
, - , .
ijT
+
+
+= jiji
j
j
iijij uuuux
uxupT )( . (5.12)
59
-
(5.8) (5.13), - , - fi ijT , .
,
, -- ( - )
uKf rr = (5.13)
fi = kijuj. (5.13)
, -. - () () . -
jiijij nnkkkk )( += , (5.14) , . nr
[4] - - (z) (r)
2d
uk z= ,
2d
uk r= . (5.15)
z, r [5].
[6] - 1)
60
1 , , , -.
-
+
+
+
++=
i
j
j
iij
nij x
uxu
nuuPT 121
34div
32 r
+
+
+
i
k
k
ijk
j
k
k
jki x
uxunn
xu
xu
nn2 , (5.16)
inn,r ,
; ; - :
222 )cossin( uccpP ++= , (5.17) , - . - ( ) c c -; 1, 2 , , :
+= 21 cos)( , (5.18) 2 = , (5.19)
, - - . , -, . 5.3. - -
j
iijij x
uT = , ij -
.
,
61
-
Red= , (5.20)
= Re ud , , d - , ,
4
38
.
. 5.3
-. ,
Re)( dsii = , . (5.21) =,i
i - 0,01.
, - , - (5.13) ,
2d
uk = , (5.22) ,
+
+
+=i
j
j
iijij x
uxuudivPT
32 r . (5.23)
(5.23) = .
62
-
1. -
? 2. -
? 3. ? 4. -
?
5. - ?
6. - ?
7. - .
8. ?
5.1.
, : 1) - - = = ; 2) - = = = .
5.2. 0 - 0 -
const=
xp . ,
)( yux
00c=
=yxx
lu
yu , lc
.
63
-
5.3. . . 5.4.
h2
const=
xp
. ( ) - ,
: )
; )
)( yuxhy =
0)( == hyux 0=
= hyx
yu .
. 5.4
5.4. 5.3 - 1u 2u .
5.5. - (5.15) - , [5] .
k
5.6. - (5.15) - , -
k
64
-
[5] - .
5.7. ,
. ,
h
nr nr .
0
y
x
U nr
h
. 5.5 -
. . 5.5. , .. , - . (5.16), (5.17). , , - , . -
==
0=y hy = : ) 0
,0
=
= hy
x
yu
, ) -
0)()0( ==== hyuyu xx . 5.8. 5.7
h
65
-
.
5.9. 5.7 - - .
5.10. () l, 0r , - , - r
))1(2( )( 20
2
rrkkqrq rrvv = .
t. - .
- -
z
0
2
= dg
lPPmz ,
lz
rrkkttzrt rr
= 2
0
2
00 )1(2),( .
, -, - = [1 (t t)] = . . - z r . - .
66
-
1.1. , . zgpzp 00)( = z1.2.
pg grad0 = r gzp
yp
xp =
==
,0 , , . - RTp = const=T
00 p
p= )
(zp
dzP
gp
dp0
0= . ,
=
0
00 exp)( p
zgpzp ,
=
0
00 exp)( p
zgz .
1.3. 2
20
0upp += .
1.4.
RTp = , (1.4.1) ( )
0
0
const
pp == , (1.4.2)
const)(2
2=++ gzpPu . (1.4.3)
(1.4.2)
1
00 )()( ppp = 67
-
Cppp
ppdp
pdppP +
=
== 1
0
0
1
0
0 1)()(
CppP +=
1)( . (1.4.4)
- , , (1.4.4)
0
020
112 =
+ ppu . (1.4.5) RTp = (1.4.1), (1.4.5)
RuTT
+=20
0 21 , 2
0
211 M
TT += , (1.4.6)
00 auM , 00 RTa . (1.4.1) (1.4.2) - :
0T
11
2
0
211
+= M , 12
0
211
+= Mpp .
1.5. = pu 2 . 1.6.
= 1
2
2
min0
2
SSup ,
)1)((2
2min0 =
SSpu .
1.7. 2
min
02
2
=
SSup ,
= pS
Su 20
min .
1.8. - - glFV
3= gr , .. .
-
68
-
, . - , , , . , . - . , , , , . , .. - .
2))()(( lhplhpFS +=SF
glhplhp )()( =+ , , . , , .
3glFS =
3 )( glF =
1.9. - VgFV = gr , .. .
, , , , [2]. , , d (. 1), -
= dnzpFd S rr
)( , (1.9.1) , - .
nr )(zp
. 1 (1.9.1)
.
69
-
= dnzpF xSx )( , ,
= dnzpF ySy )(
= dnzpF zSz )( . (1.9.2)
(1.9.2) zyx nnn ,, nr
. SF
r
: x0 d - (. .2). x0
. 2
- d nr .
, , . , (1.9.2) - . , , ..
dnx dnxx0
SxF
0=SxF . -, 0 . , , , .
=SyF
.
SzFd
. -z0
70
-
- d nr . , - d
z0d , = dnzpdF zSz )( 1 , = dnzpFd zSz )( 2 , (1.9.3)
. 1z 2z , zZ ddn = , zz ddn = , zd -
, , - , ,
zgzpzp )0()( +==
dVgdzzgFddF ZSzSz 12 )( =+ , (1.9.4) zdzzdV )( 12 .
(1.9.4) gVFSz = . (1.9.5)
(1.9.1) (1.9.5) , , VgFz )( = . (1.9.6)
, , , , , . .
1.10. - (. 3)
. 3
)( zh)( 0 gpzp + = . , -
dr0 = cos0rz ,
,)cos(,)cos(
0
0
zz
xx
dSrpdFdSrpdF=
= (1.10.1)
-
xdS zdS= drrdS 00 sin -
x z:
71
-
= drdSx 220 sin2 , = drdSz cossin20 . (1.10.2)
. ) , , , , , .
(zp
0p
hgrdrrhgFx 2
022
00
0 sin2)cos( ==
, (1.10.3)
==
drrhggmF z cossin)cos(2
00
0
= 3032 + grgm . (1.10.4)
, , . gm
, , - , . - .
1.11. - - , .. (1.10.4)
30 3
2 rm = . 1.12. -
(1.1) (1.2) :
+
++
)(sin
1)(1 22 urur
rr r
0)sin(sin1 =
+ ur , (1.12.1)
72
-
+
++
rrr
r ur
ur
uuusin
+rpP
ruuu
ru
Vrr
=+
122 , (1.12.2)
++
+
+ u
ruu
ru
ru
uu
r sin
_
=+ prPr
ctguur
uuV
r 1 , (1.12.3)
++
+
+ u
ruu
ru
ruuu r sin
=+ prPr
ctguruu
Vr 1
2
. (1.12.4)
: const= , ,0=VP
r0== uu ),( = ruu rr .
0)( 2 =
rurr, (1.12.5)
rp
ruuu rrr
=
+ 1 . (1.12.6)
-
ru
Rd
dRRrur &== )()( , (1.12.7)
0)( prp == . (1.12.8) ,
(1.12.5), 2)(),(
rCrur
= . ( ) (1.12.7), -
73
-
22 )()(),(r
RRrur= & . (1.12.9)
(1.12.9) (1.12.6) - , r (1.12.8)
4
42220
22),(
rRR
rRRRRprp &&&& +=
. (1.12.10) (1.12.10) , -
)5,1()( 20 RRRpRrp &&& ++== . (1.12.11)
1.13. , - , 1.12. - , 1.12, - (.. ) - ) (. (1.12.11)). (R )( p - )(u ( ), (1.12.11) )()( = Ru &
))((123
02 ppu
dduR =+ . (1.13.1)
( ) -
const =pV)(R
= 30 )()( Rrpp . (1.13.2)
)( p (1.13.1) - )(R
= dR
dudd ,
74
-
=
0
30
223 2)( p
RrpRuR
dRd , (1.13.3)
pRrpRuR +
=
0
30
323
11
32 . (1.13.4)
, , , 0rR = 0=u .
+
=
13
21)1(3
2 300)1(3
03
02
Rrp
Rr
Rrpu . (1.13.5)
- . , -. (1.13.5)
0p)(R
, - ) .
( ), ) . , -
, .. .
(1.13.5)
(R02 u
(RmaxR
2u maxR
02 =u
+=
)1(3
max
0
0
3
0
max 1)1(
1R
rp
pr
R . (1.13.6)
(1.13.5) . )(u -, . , . , ,
maxR
0p
75
-
0 . (1.13.5) - , , 0 . . - - .
)( u
0r )( 02 == rRu
, , -
0)1( p
p , ,
(1.13.6), 3
1
0
0max )1(
pprR . (1.13.7)
1.14. ),( r - , ,
011 22
2 =+
rrr
rr. (1.14.1)
-
00
=
=rrr (1.14.2)
0u
r
=
cos0ur r
,
=
sin1 0ur r
. (1.14.3)
(1.14.1) (1.13.3)
+= cos)(),(2
00 r
rrur . (1.14.4)
76
-
= cos)1(),( 22
00 r
rurur , += sin)1(),( 22
00 r
ruru . (1.14.5)
22),(),(),( 200
22 uprururp r +=++
. (1.14.6) , , . 1.15. , -
,
xnux=
0 , ynuy 0
= , znuz 0=
, (1.15.1) -
,
=== cos,cos,cos zyx nnnnr ,, nr . (1.15.1) ),(10 zyCxnu x += , -
, )(),( 201 zCynuzyC y += , . 302 )( CznuzC z +=
)(),,( 0 Cznynxnuzyx zyx +++= . (1.15.2) 1.16. )(),( 10 Cynxnuyx yx ++= , )(),( 20 Cxnynuyx yx += . 1.17. -
, . 4.
- -
Q
. 4
24 rQur = .
77r
ur = ,
-
24 rQ
r = , -
rQzyx = 4),,( . (1.17.1)
(1.17.1) -. (1.17.1) r ()
, 222 zyxr ++= 20
20
20 )()()( zzyyxxr ++= -
( ). 000 ,, zyx1.18. , -
rVur = 2 , . (1.18.1) 0=u
=
=rr
ur1 ,
rru
== 1 . (1.18.2)
(1.18.1) (1.18.2)
rVr ln2
),( = , = 2),(Vr . (1.18.3)
2y2ln
2),( xVyx += , x
y . Vyx arctg2
),( = (1.18.4)
0rr
20
20 )()(ln2
),( yyxxVyx += ,
02 xx 1.19.
0arctg),( yyVyx = . (1.18.5) -
78
-
0=ru , ru = 2 . (1.19.1)
=
=rr
ur1 ,
rru
== 1 . (1.19.2)
(1.19.1) (1.19.2)
=2
),(r , rVr ln2
),( = . (1.19.3)
xyyx arctg
2),(
= , 22ln2
),( yxyx += . (1.19.4)
0rr
0
0arctg2
),(xxyyyx
= ,
20
20 )()(ln2
),( yyxxyx += . (1.19.5)
1.20. , ax= ay= , aux = , 0=yu . 1.21.
)( 20 zrzu += )sin(cos)exp( += irirz , , += i ,
+= cos)( 20 rrru , = sin)( 20 rrru . (1.21.1)
= cos)1(),( 22
0
rrurur , += sin)1(),( 2
20
rruru . (1.21.2)
(1.21.1) (1.21.2) , (0rr = ,0)( 0 == rr
), 0)( 0 == rrur== sin2)( 0 urru . ( r )
. , - .
u
0r u 79
-
1.22. -
nczz = )(iyxz += )exp( = irz , 22 yxr += ,
xyarctg= , , += i , -
)cos( = ncrn , . (1.22.1) )sin( = ncrn
)cos(1 = ncnru nr , . (1.22.2) )sin(1 = ncnru n (1.22.1) , 0= ,
0= n= (. 5).
. 5 ,
u
1= nr cnru0= 1= nr cnru n=
(1.22.2). . 5. - , 0
, . 0= 0 ,
-, .. -
nczz = )()11(2 n = ,
(0n 0=r ) . 80
-
. 6
1 . 2= nr cnru =n (. 6).
2.1.
. , = 0. , ur(r, ) u(r, ). - (2.15) (2.18) -
012 =+++
rctguu
rru
ru rr , (2.1.1)
++
+=
2
2
222
2 1221 rrrr urr
ur
urr
urp
+
u
rctgu
ru
rctg r
22222 , (2.1.2)
++
+=
2
2
22
2 121 urr
urr
upr
+ +
2222 sin
2r
uur
ur
ctg r , (2.1.3)
-
0),( 0 == rrur , 0),( 0 == rru , (2.1.4) , .. - ,
81
-
= cos),( 0urur , , = sin),( 0uru= prp ),( . (2.1.5)
(2.1.5)
= cos)(0
Rfuur , = sin)(
0Rg
uu ,
= cos)()( 00 Rhu
rpp , (2.1.6)
R , 0rrR = . (2.1.6) (2.1.1) (2.1.3)
hgf ,,
0)(2 =+ gfR
f , (2.1.7)
)(42 2 gfRf
Rfh += , (2.1.8)
)(22 2 gfRg
Rg
Rh ++= . (2.1.9)
(2.1.7) (2.1.9) , )
) )(Rg (Rh(Rf
0888 23 =++ ffRfRfR , (2. 1.10)
nRf = . (2.1.11) (2.1.11) (2.1.10), -
, n ,01 =n ,22 =n , (2.1.10)
,13 =n 34 =n
34
13
221)(
+++= RCRCRCCRf . (2.1.12) ) ) . -
(2.1.4) (2.1.5).
)(Rg (Rh (Rf
82
-
+= cos)21
231(),( 3
300
0 rr
rrurur , (2.1.13)
= sin)41
431(),( 3
300
0 rr
rruru , (2.1.14)
= cos23),( 2
00 r
ruprp . (2.1.15)
, ,
==
= drr
urrpFrr
sin2)sincos),(( 200
00
. (2.1.16)
00 6 ruF = , (2.1.17)
(2.28)
Re24=D , =
du0Re . (2.1.18)
2.2.
2
81)( gdu =
(Re)341
DCgdu
= .
. 2.2. 3 /, 27 /, 0,3 /, 1,64 /. 8,66 /, 12,2 /.
0,4Re (Re)DC
2.3. 1,23 /. 2.4. 0,86 . 2.5. . 2.6. 0,325 / 2.7. =32 . 2.8. 0,84 . =F
2.9. dyu
yxux x
=0 0
)),(1()( . 430
22),( YYYu
yxux += ,
83
-
)(xyY = , 0
83,5)(u
xx
= 0
75,1)(u
x
= . 2.10. - -
- ( ) , - .
, - , , - , - )(x .
-
)(2)(
0
00 xh
huxu
= , (2.10.1)
2)(
2)(
20
2 xuupxp += . (2.10.2)
. - . p
dxu
xubudxbxppFll
==
0
2
20
2
0
)1)((2
))(( . (2.10.3)
2.9
075,1)(
ux
= . (2.10.4)
- 1,1)( = l . 84
-
5,0
00
2
02
20 )()(4)(41))(21(1)(
lx
hl
hx
hx
uxu
== ,
HdXXh
llbuF 7,11)(42
1
0
5,0
0
2== . (2.10.5)
2.11. 9,56 .
2.12. 3,336,3
2,11000 ==u /.
460 1066,2
1015012,03,33Re =
== du . . 2.2
2,1=DC 6,12413012,0
23,332,12,1
2
220
=== lduCF D .
.
41480 == uFM4
6 1014,310443,04148 === r
MG /, -
100 ( 100 /) 1,13 1,55 .
2.13. )(
83,5)(0 xu
xx
= , , - , , - - .
)(0 xu
)sin(2sin2)( 00 rxuuxu = . 0=x = 0. - ,
)(0 xu
== urx 283,5)0( 0 dr Re83,50 = . 102 0 = rd 410Re =d 29,0= .
85
-
2.14. hyuuz 0= , -
huPyz 0= . 2.15. ),(
21)( 0 yhyz
phyuyuz
= )2(2
10 yhzp
huPyz
= . 2.16. , y 0.
),( yuz . : ),( yuz 2
2
yuu zz
=
. 0)0,( ==yuz , (2.16.2)
0)0,0( uyuz == > . (2.16.3) (2.16.1) (2.16.2)
(2.16.3) , -
zu
),,,( 0uyfuz = . (2.16.4) (2.16.4) (. 3 -
) , )( 210 = yfuuz , (2.16.1) .
0uuU z= , = 2
y . (2.16.5)
(2.16.1) 02 =+ UU (2.16.6)
(2.16.2) (2.16.3) 1)0( ==U , 0)( =U . (2.16.7)
(2.16.6)
+=
0
212
)( CdeCU
86
-
(2.16.7)
)2
(erf1),(
0 = y
uyuz , (2.16.8)
) (erf x =x
x dxex0
22)(erf .
2.17. , .
0y),( yuz .
: ),( yuz2
2
yuu zz
=
. (2.17.1)
== cos),0( 0uyuz . (2.17.2) -
. - 0 . =y
- = ieyfyU )(),( .
= yyuyuz 2cos2exp),( 0 . (2.17.3)
2.18.
, (0=zu z 0=z
) ,
( 0= ). -
, (2.11) , 0=ru .
87
-
(2.12), (2.13)
rp
ru
= 12
, (2.18.1)
01 2 =
ru
ru
rrr
. (2.18.2)
00 )( rrru == , 0)( prp = .
:
00)( r
ruru = 0rr , (2.18.3)
rruru 00)( = , (2.18.4) 0rr
20
2202
00 2)(
rruuprp += 0rr , (2.18.5)
2
20
20
0 2)(
rruprp = . (2.18.6) 0rr
(2.18.3) (2.18.6) 00 ru = . ,
(2.18.4) - ( ) z
=== rdr
rrurdr
ruQ
rr
22
22
)(2
20
20
2
00
= 0
ln2020 rrru . (2.18.7)
, (2.18.4) . - (2.18.4), -.
0rr
88
-
2.19.
= rrr
rrr
ru2
22
12
2
21)( , -
21
22
222
1rr
rr
udrdrP
rrr
=
=
=
.
21
22
21
222
11 42)( rrrr
llrrPM r == .
2.20. , 0),( == uuru rz
89
0=
zp .
:
)ln()ln()(
12
10 rr
rruruz = ,
)ln()ln(
)(12
20 rr
rruruz = .
, , .
2.21.
= 2
2
0 15,1)( hyuyuz , l
phu =3
2
0 , Re96
= . 2.22. .7.
. 7
(2.31)
:
lp
drdur
drd
rz
=)(1 . (2.22.1)
-
,0)( 1 == rruz 0)( 2 == rruz
+= 212
12
121
22 )ln(
)ln()(4
)( rrrrrrrr
lpruz . (2.22.2)
rdrrurr
ur
rz 2)(
1 2
12
12
20 = . (2.22.3)
,
+
=ln
118
22
21
0 lrp
u , (2.22.4)
12 rr= . (2.22.4) -
. (2.32) -
p 0u
Re)(
= A ,
+
=ln
11
)1(64)(2
2
2A . (2.22.5)
2.23. . 8.
),(),( yxByxuz = , (2.23.1)
22
2
21),(
by
axyx =
, (2.23.1) - .
a b
. 8
90
-
(2.23.1) (2.31), - - , -, B, -
22
22
2 baba
lpB +
= . (2.23.2) , (2.23.1) B, (2.23.2),
-, .
dyby
axBdx
abu
a a
xb
= 0
1
02
2
2
2
0
2
2
)1(4 dXXB 1
0
232 )1(
38 . (2.23.3)
22
22
0 42 baba
lpBu +
= . (2.23.4) (2.23.1) (2.23.4), ,
0max 2uu = nba = .
(2.23.4) - . (2.32) .
p 0u
Re)(
nA= , 2
2
2
22
)1()1(264
)()(264)(
nn
babanA +
+++= . (2.23.5)
(2.23.5) ,
ba
abd +=4
,
=0Re du .
91
-
2.24. . 9.
x
y
a
. 9 -
(2.31)
lp
yu
xu zz
=
+
2
2
2
2, 0
=zu . (2.24.1)
, 23ay = , xy 3= , ,
: )23)(3)(3(),( yaxyxyyx += (2.24.2)
, ),(),( yxByxuz = . (2.24.5)
(2.24.5) (2.24.1), , (2.24.1)
321
alpB
= . (2.24.6)
( 3,0 ayx == )
lpau
=2
max 361 . (2.24.7)
dyyxudxa
ua
xz
a
),(3
8 23
3
2
020 =
92
-
lpau
=2
0 801 . (2.24.8)
== 9200max uu = 2,22.
(2.24.8) - . (2.32) .
p 0u
Re= A , 3,53
3160 ==A . (2.24.9)
(2.24.9) ,
3
ad = , -
=d0Re u .
2.25. , - 0== yx uu , ,
(
)(yuz
0=
yp ), -
gdzdp
= . (2.10)
2
2
)(0 dyudg z+= . (2.25.1)
0)0( ==yuz . (2.25.2)
, -
93
-
0==y
z
dydu . (2.25.3)
(2.25.1) (2.25.2) (2.25.3)
=
2)()( yygyuz . (2.25.4)
3
0 3)()(
== gdyyuV zl ,
3
)(3=
gVl (2.25.5)
32
3)(
==
llz
VgVu . (2.25.6)
2.26. , , -, . -, 2.25.
4
33
31000,1
1020106,3106,22
=== d
VVl 2/.
3364
3
1031,0)2,998205,11(8,9
10003,1103)1(
3 ==
=g
Vl
( 610003,1 = 2/, 2,998= /3, /205,1 = 3).
32,01031,0
103
4===
lVu /.
94
-
( 015,0= d ), . :
39610003,1
1031,0432,04Re 63=
==
u .
. 2.27. F = 18149 .
2.28. 624
1074,42ln32
==
lDgd 2/.
2.29. -
0)(1 =+
z
ururr
zr , (2.29.1)
22
2110ru
zu
rur
rrrp rrr
+
+
= , (2.29.2)
2
2110zu
rur
rrzp zz
+
+
= . (2.29.3)
. , - ( rh ). (2.29.2) (2.29.3)
zp
-
(2.29.1) ,
0)()(1 ==+ hzuhurrr z
hrv , (2.29.6)
dzrzuh
uh
rhr
0
),,(1 .
-
== ddhhzuz )( , . -
(2.29.6) ,
21 r
ddh
hu hr = . (2.29.7)
(2.29.4). -
)(21 zhz
rpur
= (2.29.8) , (2.29.8),
2
121 h
rpu hr
= . (2.29.9) (2.29.7) (2.29.9) hru ,
rddh
hrp
=
3
6 , (2.29.10)
, - ,
0p
)(3 22030 rrddh
hpp
= . (2.29.11)
96
-
, , - - ( )
F
403
00 2
32)(0
rddh
hrdrppF
r
== . (2.29.12)
(2.29.12) - ) . (h
5,04
0
20
0)
341()( +=
r
Fhh
h . (2.29.13)
2.30. , , , , , ,
() gxp
= . -
, , -
0=+
y
ux
u yx , (2.30.1)
0)( 22
=+
yug x . (2.30.2)
(2.30.2) 0)0( ==yux
0=
=yx
yu
=
2)()( yygyux . (2.30.3)
97
-
3
0 3)()(
== gdyyuV x . (2.30.4)
y , ,
0=+
xV . (2.30.5)
(2.30.5) , = )(yuy .
(2.30.5) (2.30.4), , :
0)( 2 =+
xg . (2.30.6)
(2.30.6) : 0= . 0=x
0),0( == x . (2.30.7) -
(2.30.6), . -
)()(),( = Txfx . (2.30.8) (2.30.8) (2.30.6),
mddT
Tdxdffg ==
3
1)( . (2.30.9)
(2.30.9) , , , , , .
m
(2.30.9) - (2.30.7)
)(xf
98
-
gxm
xf)(
2)(
= . (2.30.10)
)(T (2.30.9)
cmT += 2
1)( . (2.30.11)
)()(),(
1 cgx
x += , (2.30.12)
mcc 21 = . (2.30.12)
x 1 .
1 ,
c
0 0 .
0x
020
01 )(
=
gx
. (2.30.13)
0 M l ( M (
0 l
x0
),() ), =
0
2
01 )(3
2
=
gl
Mlc . (2.30.14)
2.31. - (. (2.13)) ),( ru
22
2 11ru
ru
rruu
+=
, (2.31.1)
99
-
0)0,( == ru , (2.31.2) 000 ),( rurru == . (2.31.3)
2.18
00)( r
ruru = . (2.31.4)
),(),( 10
0 = rurruru . (2.31.5)
),(1 ru -
211
21
21 11
ru
ru
rruu
+=
, (2.31.6)
001 )0,( r
ruru == , (2.31.7) 0),( 0 == rru , (2.31.8)
.
=
=
012
0
2
100 exp),( r
rJr
Crruru kk
kk , (2.31.9)
= 1
0
21
1
0
21
0
)(
)(
dJ
dJuC
k
k
k , k 0)(1 =J .
3.1. 200 )( udldufp
= . - (2.32).
100
-
3.2. 2200 )( dudufF
= . (2.28). 3.3. . -
[ ] = /, [ ] = , [ ] = /(), [),,( 0 = dufF
0u d F ] = /2. - 4=n , , - .
3=k
= duF 0
2
= , , , , - ,, :
+=1 , =1 , = 2 , , 1=== . , . (. 2.17)
= duF 0= 3 .
3.4.
=
2025
0r
frM .
3.5.
=
yFu .
3.6. . , x , ),( yux :
2
2
yuu xx
=
, (3.6.1) 0)0,( ==yux , (3.6.2)
0)0,0( uyux == > . (3.6.3)
101
-
(3.6.1) (3.6.3) , -
),,,( 0uyfux = . (3.6.4) (3.6.4)
=2
0
yFuux . (3.6.5)
, - . , - (3.6.5), (3.6.1) (3.6.3) .
3.7. . , x , (, ) :
2
2
yuu xx
=
, (3.7.1)
0)0,( ==yux , (3.7.2) 0)0,0( uyux == > , (3.7.3) 0)0,( == >hyux . (3.7.4.)
(3.7.1) (3.7.4) , -
),,,,( 0uhyfux = . (3.7.5) (3.6.4)
= hyyF
uux ,
2
0. (3.7.6)
. (3.7.1) (3.7.4) .
3.8. )(2
3
2
ms
fgsT
= . 3.9. )(3 = AfgE .
102
-
3.10. 512 )( = ECr .
3.11. 3ghCM = . 3.12.
)()(
=g
fD .
3.13. g
V
= .
3.14. sg
mCT = .
3.15. . 2AgCE =3.16. , 0=y
. -
z
0=+
z
uy
u zy , (3.16.1)
xp=0 , (3.16.2)
)( 22
2
2
zu
yu
yp
zu
uy
uu yyyz
yy
++
=+
, (3.16.3)
)( 22
2
2
zu
yu
zp
zuu
yuu zzzzzy
++
=+
. (3.16.4)
0)0,0( == zyuz , (3.16.5) = uzyuz )0,( .
(3.16.1) (3.16.5) , (, ) -
yu zu
),,,,(1 = uzyfuz , 103
-
),,,,(2 = uzyfuy . (3.16.6) (3.16.6) -
, . [ ]yu = /, [ ]u = /, [ ]y = ,
[ ]z = , [ ] = /3, = /(). (3.16.7) [ ] (3.16.7) :
6=n , - 4=l ,
. 3=k - ( 3= kn ) - , ( 2= ln ) - ( ). k , .
1= kl
l , , m (3.16.8)
(3.16.6)
),,,,( 31
=
AAu
AAA
AAzAyAf
AAu l
l
m
l
mll
lz . (3.16.9)
1,1,1 3 === AAu
AAzA l
l
ml . (3.16.10)
(3.16.10) 31,,1z
Az
uAz
A ml === .
(3.16.9), - : zu
=
zz
zyfzu
zyf
uu Re,, 33 . (3.16.11a)
: yu
104
-
=
zy
zyfzu
zyf
uu
Re,, 44 . (3.16.11)
0
==
y
z
yu . (3.16.12)
(3.16.12)
(3.16.11)
zu
)(Re5 zfuz =
,
: zRe
)(Re2
zFu=
. (3.16.13)
, -
: fc
zRe
2)(Re
2
= uc zf . (3.16.14)
3.17. , - ,
u
2
2
yu
zuu
yuu zzzzy
=+
, (3.17.1)
0=+
z
uy
u zy (3.17.2)
0)0,0( == zyuz , (3.17.3)
= uzyuz )0,( . (3.17.1) (3.17.3) ,
yu zu
),,,,(1 = uzyfuz , (3.17.4) 105
-
) . (3.17.5)
106
(3.17.4) (3.17.5)
,,,,(2 = uzyfuy ,
, , , z , y, z. zu .
[ ]
zyx z
zu = , [ ] yy = , [ ] zz = , [ ] = , [ ]
zx
y
= , [ ]
u z= . (3.17.6)
- (3.17.6),
. (3.17.1) , - .
(3.17.6) 6=n , 5=l , 4=k - ( 2 = kn ),
( 1= ln ) ( 1= kl ). -
(3.17.6), , iA :
y
yy
,
z
zz A
, A
zyxzyx
,
A
uzz A
. (3.17.7)
(3.17.
4)
),,,,( 21uz AuAAAAAzAyAfAu = . (3.17.8) uzyuzy (3.17.8)
-
107
= uAu . (3.17.9) (3.17.9) ,
(3.
,1 ,1 ,1 === zy AzAyA 1 iA
17.8), zu
uzu
=
= z
yfuy
fu
z321 1,,1,1,1 . (3.17.10)
(3. ,
17.5). yu , zu , (3.17.10)
=
zuyf
yuzuy
4 . (3.17.11)
z
uy ,
yu :
,
=
z
uyfzuuuy
4 . (3.17.12)
0
== zu .
yy (3.17.13)
(3.17.13) (3.
zu 17.10),
)0(4 fzuu
= . (3.17.14) (3.17.14)
-
2)(Re
2
= uc zf , (3.17.15)
zzzf
CfcReRe
)0(2)(Re 4 == , =zu
zRe . (3.17.16)
4.1. ) 16,5 , ) -
5 . 4.2. 1,49 . 4.3. -
1,54 . 4.4. 10,8 . 4.5. -
5,5ln5,2 0 += v
uruum , (4.5.1)
)5,52
Relg3,25,2(000+=
uu
uu
uum
, 80 = uu ,
)82lg75,55,5Relg75,5(8
0+=
uum . (4.5.2)
+= 8,015,0Relg
. (4.5.3)
(4.5.3) (4.5.2),
0
25,101,1 +=uum . (4.5.4)
108
-
: 410Re = 22,10 =uum , 510Re =165,10 =uum .
4.6. 917,0 =VV . 4.7.
-
==
N
i
ii
N
iN
i
N
ii puF
Fuu
15,0
5,0
1
1
1 2 ,
FFii = - . -
i
22
222
15,0
uupN
i
i
=
, (4.9.1)
2
15,0
1
=
N
i
i .
4.10. - -
i
2)(316,0 2
25,0
i
iii
udl
dup = 25,1
75,1
i
i
ducp = ,
i -
75
74
ii dpu
= , (4.10.1)
2
316,0 25,0= lc . -
==N
iii
N
iN
i
N
ii
dpu
F
Fuu
1
75
74
1
1
1 , (4.10.2)
i = Fi /F - . -
i
110
-
47
1
75
47
=N iid
ucp . (4.10.3)
(4.10.3) 25,1d , (4.10.3)
2)(316,0 2
25,0
udl
duAp = , (4.10.4)
4
7
1
75
25/1
= N iiddA .
=== N
i
iN
i
i
N
i
N
i
N
i
ddF
FFd
1 1
1
1
1
14
44. (4.10.5)
(4.10.4) (4.10.5)
0 = , (4.10.6) 25,0
0 )(
316,0 = du -
, -
4
7
1
75
25.1
1
1
= N iiN
i
i dd
.
4.11. .2
(Re)01,120
up =
111
-
4.12. ) 1,01 , ) 3,90 , ) - 1,71 , - 0,12 10-6 2/.
4.13. ) 1,01 , ) 1,06 . -
4.14. 588=dldp /. 4.15. 1413 707 . =F4.16. 0,151. =4.17. 0,064 . =h4.18. h =0,62 , =h 0,51 . 4.19. 4,210=Re 6, , = 0,29. 4.20. = 2, 4, 6, 8, 10 =0max uu 2,0, 1,50, 1,33, 1,25, 1,20 -
. 4.21. 0,165 , =l =xRe 6,3103, . 4.22. 0,86 , =l = 6,9 . 4.23. x = 50 = 0,71 , (z)y=0 = 3,17104 /,
(z)y==0. x = 200 = 1,42 . 4.24. 4,94.105 .
5.1. , -
+
++= ijij uPT )divcossin(32 22 r
+ ( )
+
+ i
j
j
i
xu
xu22 cossin ,
+
+
+=i
j
j
iijij x
uxuuPT
div32 r .
112
-
5.2. 0== zy uu , )( yuu xx = . (5.13) (5.22) (5.23)
yu
yxp
du x
x
+
= 2
20 . (5.2.1)
(5.2.1)
00
=
=ycxx
lu
yu . (5.2.2)
u= - : 2xu
2
22
2
220
yu
xp
du xx
+
= . (5.2.3)
, 0=
yux ,
=
2dxpu .
= uuU xx -
2
2222 10
yUbU xx
++= (5.2.4)
=
dCb .
(5.2.4), (5.2.2),
+= by
bluyu
cx exp21
11)( . (5.2.5)
0c =l
= byuyux exp1)( ,
cl
= uyux )( .
113
-
5.3. 0== zy uu , )( yuu xx = - (. (5.13) (5.21)) -
uC =2xu
2
22
2
220
yu
xp
du xx
r +
= . (5.3.1)
2
22
22 110
YU
kU xx
+= , (5.3.2)
: hyY = ,
=
uuU xx ,
=r
dxpu
2 ,
dChk r
= . (5.3.2)
kYCkYCYU x chsh1)( 212 ++= . (5.3.3)
.
1C 2C
kkYuYux ch
ch1)( = , (5.3.4)
= uYux )( . (5.3.5) 5.4. 5.3 ,
=1
02
1 )(ch
ch1 kdYk
kYuu . (5.4.1)
-
dChk r
= . 10.
114
k C, - , .. . , -
-
- .
1 10 100 1 .1030.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
. 10
0,95 0,9 0,85 0,8 0,75 0,7 0,65 0,6 1 10 100 1103
5.5. [5]
2
2
= , , .
mu
1 : x1 = 1,1 7; x1 < x2; 0= 0,06 1
( ) 2,05,01 Re1 1 2= mx ; (5.5.1) x1 = 1,1 7; x1 > x2; 0= 1 8
( ) ( ) 202,059,005,01 /Re94,01380 ,= 1 mx , (5.5.2)
115
-
0 = (x11)/( x21); x1 = s1/d; x2 = s2/d - ; d .
Re - .
- r
2p
2
udl
r= (5.5.3)
.
(5.15). k
r
( ) .Re)(
14
1
Re 0,23,22
2,30,2-
xx
xxxr
1=)(
3
(x) -, - . 5.5.1.
5.5.1
(x)
1,02 1,12 nx
bxax)1(
)( += a = 2,168 b = 1,495
n = 1,5 1,13 1,30 cxbxax ++= 2 )( a = 348,2
b = 904,5 c = 602,1
1,30 1,80 bxax += )( a = 2,323 b = 17,14
1,80 10,0 bxax += )( a = 1,762 b = 8,111
116
-
5.6. [5] :
2
2mu = , (5.6.1)
, .
mu
1 :
.ReC 27,01 = m (5.6.2) . 5.6.1.
. - .
mRe
r
2p
2
udl
r= (5.5.3)
.
5.6.1
* x1 C 0,1 1,7 1,44 C = 3,2 + 0,66 (1,7 - )1,50,1 1,7 < 1,44 C =3,2+0,66(1,7 )1,5 +
11,044,1 x [0,8+0,20(1,7-)1,5 ]
1,7 6,5 1,44 3 C = 0,44 ( + 1)21,7 6,5 < 1,44 C = [0,44 + (1,44 x
1)]( + 1)2
1,7 3 10 C = 0,062 + 0,21 (10 x1)
-0,24
* = (x1 1)/( 1),
/x222
212 4 x/xx
/ += - - .
117
-
k (5.15). r
( ) 0,2773,12
27,327,0
0,27- Re
)(1
132324ReC x
xxxr
1+1
=
3
, (5.6.4)
x 1,44 , C = 3,587 ; x < 1,44 ,
C = 3,587+ 8,338 (1,44 x). (x) -
, - . 5.6.2.
5.6.2
(x)
1,02 1,06 nx
bxax)1(
)( +=
a = 2,50, n = 0,5
1,06 1,40 bxax += )( a = 7,91, b = 1,01 1,40 6,0 bxax += )( a = 17,48, b = 13,79 5.7. -
, ..
0=
xui , . (5.7.1) zyxi ,,=
, , -
z
0=
zui , , (5.7.2) zyxi ,,=
z 0=zu .
0=+
+
zu
yu
xu zyx , (5.7.3)
118
-
(5.7.1) (5.7.2) 0=
yuy 0const ==yu ,
. , ,
)( yuu xx = . (5.7.4)
(5.7.4) (5.13) :
yu
yxPuk xxxx
+
= 10 , (5.7.5)
yPuk xyx =0 . (5.7.6)
(5.7.5), (5.7.6) (5.19):
2222 )()cossin( xx ucpuccpP +++= , (5.7.7) (5.20):
+= 221 cossin , (5.7.8) (5.16), (5.17):
xx
xx udu
dk )(
2)cossin(
2 22
+= , (5.7.9)
xxyx udus
dk )(
2cossin)(
2
= . (5.7.10) (5.7.5), (5.7.6)
yu
uyxpu
dx
x
+
=2
12
2)(
20 , (5.7.11)
yuc
ypu
dx
x
=2
2
)()(2
0 . (5.7.12)
(5.7.11) - . (5.7.12) - .
)(yux
.
119
-
) 0,0
=
= hyx
yu . -
(5.7.11)
)(2)( 0
= dxpuyux . (5.7.13)
)( .
(5.7.12) .
)()(
=
xp
yp . (5.7.14)
) 0)()0( ==== hyuyu xx . (5.7.11)
bYb
bYbuYux ch1shsh)1ch()( 0 += , (5.7.15)
hyY = ,
1
2
)()(duhb
= , )(
2 0
= dxpu .
(5.7.12) +=+ )0()()()( 2 pYucYp x
+
+
bYb
YbYbb
bd
hu sh1)1ch(sh
1ch2
20 . (5.7 16)
= )0()1( ppp
+
bbb
bbdhu sh1)1ch(
sh1
22
20 . (5.7.17)
5.8. : 5.7. 5.9. :
- , .
)(yux
120
-
5.10. ( ) - :
),( tzrt =
rrrr
rzrr
r mkm
rrPmm
zm
+
=+
2, (5.10.1)
zzzz
rzrz
r mkm
rg
zPm
zmm
+
=+
2, (5.10.2)
0=+ zrr mzm , (5.10.3)
vrzrr qrcm
zcm +
=+ , (5.10.4)
2 d
mk zzz = , 2 dmk rrr = . (5.10.5)
rr rr
= 1 . -
, (5.10.1) (5.10.5) -
zz
dglPPm
0
2
= ,
lzrft
lz
rrkkttzrt rr )()1(2),( 020
2
00 =
= . (5.10.6)
-, .. - . .
rrRr
Z MKMRA
RZMA
+=
, (5.10.7)
121
-
)(2 RZfMR
MAZZ
MA ZZRZZ +
+
=
, (5.10.8)
0=+
ZMM ZrR , (5.10.9)
, iM; 01 zi mmM = ,
= ltgp 0 ; , , . lrR = lzZ = zrK =
0 1 q
rr
rrzcmz +
=
, (5.10.10) , 0 -
rcm
zcm
rr
rrq rz
= 000 1 , (5.10.12)
(5.10.7)(5.10.9) -
, zm prm , . , ,
ZA A
RKM r
= 1 , (5.10.13)
))((21
ZRfZM Z
= , (5.10.14)
)(12 22
RfZR
RRRK
=+
(5.10.15)
(Z = 0) = 0, (Z = 1) = 0, 00
=
=RRR. (5.10.16)
122
-
(5.10.13) (5.10.16) .
= rk 1 + 2 ( - 1) Lrk 2 2 , (5.10.17)
1 = )(21 2 ZZ , 2 (R, Z) = ,
=
..5,3,1)sin()(2
kk kZRP
)()(
2)(
4)(
8)(
2(R) 01
04
0223 RI
RI
kk
RRkkk
Pk
kk
+= ,
2Kkk = ,
3..5,3,1 001
12
)()sin(
)()(16)1(
kkZ
RR
RIRIL
kkM
k k
krr
= =
, (5.10.18)
=
= ..5,3,122 )cos()(2)1(
4 kkr
rZ kZRkPZRLk
kM , (5.10.19)
(5.10.17). )( RPk k . 11 . 12
(. 11) (. 12) -1000 (1 , 2 ).
0.8
0.9
1
1.1
-1 -0.5 0 0.5 1R
U/U0
2
1
. 11
123
-
0.8
0.9
1
1.1
-1 -0.5 0 0.5 1R
U/U0
2 1
. 12
1. .. , -
. .: , 1970. 2. : -
/ .. , .. , .. . .: , 1993. 8081.
3. .. . .: , 1968, .42-43.
4. .., .., .., .. . .: -, 1988.
5. .., .., .. ( , -, ). .: , 1984.
6. .., .., .., .., .. - // - - . 29 1 2007 ., , . . . 124
. . , , () , . (2.6) , , 4.
((( ((( . , (2.31) : . (2.22.1)