0125616 70095 charnyy i a podzemnaya gidrogazodinamika
DESCRIPTION
Charnyy i a Podzemnaya GidrogazodinamikaTRANSCRIPT
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1963
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1352 532. 529. 5 + 532. 546 (022)
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w wae
%CTB- .
dx . , , - ,
dt. dV, .
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/ dx.
dV = Qdt = mf dx,
Q dx= 171f - " ' dt
Ho dxldt , a Qlf .
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Re: Q ; JJ, -; ; - , -
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= (s, t), s (. I. 6); t .
(I. 1. 4),
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1 (1.1.5)
w gradfT. (I.1.6) -
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w = !
ds
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(j- ; Y ; , -! = \ . , z = 0.
[] = 2. , [] = /2, [] = 0,01 / . = 1 , [s] == , . ,
1 , , 1 , 1 /2 1 - , 1 2, 1 3/.
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(1.1.4) (1.1.7) =^-. (1 .1 .8)
== 100 - 1000 . , .
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ds - U '
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IT 1
(1.2.1)
. = const,
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, Q = const, (1.1.5)
Q = ~c4rf&> ( J - 2 - 3 ) s:
(1.2.4) s s = sx s, a
,
rw- (L2
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5)
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18 . I.
s.
2 su s2, (1.2.5)
H l~
f f i i
1~
(I 2 6)ds
cf(s)
i ? i v s2) = J4
dscf(s) (1.2.7)
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(I. 2. 6) . -
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1. / (s) = const, H = (s) -
. (1.2.2) :
" = 0 = 1$ + 2, (1.2.8). .
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2. - - ( ). , - . 1.7, h ;
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(I. 2. 2) s :
- f - - = 0 . (I.2.9). I. 7. -- -
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= In +
, (1.2.10) = 0 , -
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2. 19
= =
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=
= RK,
1 = 1 ! - (1.2.11)
( 1 . 1 . 5 ) ( I . 2 . 1 0 )wc = c ( I 2 1 2 )
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< 0 . , -
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20 . /.
(1.2.16) :
= ^ 1 + , (1.2.18) > 0 ;q < .
3. - .
/ = 4 2 (1.2.2) :
dr, (1.2.17),
^ ( ^ ) 0. (1 .2 .19)
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= ^ . + 2. (1.2.20)
w
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> 0 ,
< 0 .
Q = 4 21 w | = 4 |
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= ^ - + . (1.2.21)
Q ]> 0 Q
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Q .
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Q= 2 * ^ - > =
2
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__ 2/(
) _ 2/(0) = 2() ,j 9 2
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. . [25], i?0^ 1,5. , , r
c
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$ 3. 25
, G:
= - l W
, l ' ' --f(s)y(p)- (1.3.2) . . ,
Pi (p) =
(1.3.2) :
(1.3.3)
(1.3.4)
(I. 3. 4) (I. 2. 3) -
: / (s) - Q G , Pi, - ki (s), w yw . ,
.
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w =
dpdsdpds
apds
\ dp \\\\ ds \)
m l\ Vl
,1 /M
v \ ) -
dpds
ww\
(\\\II
dpdsdpds
(I.\
/
3.
3
5)
6)
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dp
W = dsdpds
r dpds
(1.3.7)
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[3, 19]. -
< (1.3.8)ds ' \w\ * \w\ g
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b [12, 20, 21].
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w (I. 3. 8) - , .
s
(1.3. 5)
(I. 3. 10) - , -
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- , / (s) - / = const, / = a s , / = C2S2, [8, 19].
(I. 3. 10) w (I. 3. 9):
dp _ ( ) G G* , , , 4ds )**() y(P)f(s) "*" g fHs)y(p) * V--i4
= (), = ki (s) kz (p) (I. 3. 11) , -
(1.3. ) - , -
.
| () = fi = const, = const (I. 3. 11) - , -
:
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4. 27
, (1.3.11) {),
dP
dpds
U.
GfU
G
0k G*g /2 (S)
i b G2
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1. . . . -, 1945.
2. . ., . . , . . . . , . 28, 5, 1940.
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3. . . . . . , 1960.
4. . ., . . ., 1955.
5. . ., . ., . ., . . - -
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. , 1947.8. . . . , 1961.9. ., ., .
. . . , 1962.10. . ., . . -
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. . 4, 1961.11. .
. . . , 1949.12. . . . -
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, 1949.20. . .
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, 1951.21. E n g e l u n d F. On the Laminar and Turbulent Flows of Ground
Water through Homogeneous Sand. Transactions of the Danish. Academy of Tech-nical Sciences, No. 3, 1953.
22. M . . - . . , . 118, 2, 1958.
23. . . - . . , , , 4, 1959.
24. . , . ., . . , - . ., . ., . . -
. , 1962.25. . .
.
, 1956.
-
II
1.
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[1]. [. I. 6, 7, 2].
,
(1.1.6)w= gradp,
= ; ; - . -
-
.
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, :
-
,
mdV, dV ; , , :
d^ ) = 0, (. 1.1)
w .
-
32 . II.
dpU =
dp
v '
7 = ~ .[. OZ
, v, w, p,Q, m , -
.
, ,
, ,
[3, 4, 5, 6]. -
= ,
= = const, . . .
, , v, w, p, Q,m (II. 1. 1) (II. 1. 2). - ,
Q = Q(P,T) (. 1.3)
m = m(p). (II. 1.4)
, .
-
, . . Q = const, m = const. :
&ivw=0. (. 1.5) .
( I I . 1.5) (II. 1.2) const, -
^ V2 ~ grad/cgrad p = 0. (II. 1.6) k = const,
V 2 P = 0, ( I I . 1 . 7 ). . = const . (1.2.17)
V 2 O = 0, (II . 1.8). . = const . (II . 1.8) -
-
2. 33
,
- -
= (), (1.2.9) (1.2.19) .
const (II. 1. 6) . ; = {, , ,, z, t) . ,
= (, , z), [II. 1. 6] .
= (, , z) . . [7 ] - . [8].
, , , z (II. 1. 2)
(, , , z, t), kv (, , , z, t), k2 (, , , z, t) .
, -
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. [.1.3]. -- ,
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( 3, . V). , , -
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2. . .
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:
, , , v
1 dpdt dz Q
dv , dv . dv . dv -,, i dp . . . . ..-dr+u4r + v-W + w^r = Y-TW^ ( - 2 )dw . dw , dw , dw 1 dp
-
34 . II.
, v, w ; X, Y, Z - , ; Q ; .
, -
, , z - (. II. 1, ),
V 7 / 9 9\ U, I = V, li g, (H. . )
g = 9,81 /2 .
Z'-c
. II . 1. ( -
).
( )(. II. 1, ),
X = 2 cos = 2 ,Y = (2 = (2/, (II. 2.3)
( z; r ; .
, -
,
. , ,
.
-
2. 35
,
.
, (II. 1. 1) , , -
:
X = X, + 2; Y = + 2; Z .-= Zx + Z2,
Yi, Zx ; 2, Y2, Z2
, .
, v, w -, , ,
1 1 / , . \ 1 . v
. v
-W+^{u + v + w j = + X + X1 dv 1 / dv , dv dv\ 1 dp ,
7. . T7- / T T ,.
Ur +i(u + v + wdi) = ~ T"^7 + ^ 1 + ' ( I L 2 " 4 )+ w j = ~ ~$~ ~dT~ + Zl + z*-
-
(. 2. 4) . , ,
, . (II. 2. 4) :
ur
ny,
n _ n / K\
oy x '
, Q X2, Q 2, Q ^2 -
:
QX2 = - ^ u ,
QY2 = --^V, (II. 2. 6)
e Z/
-
36 . II.
, -
:
I (
, -
(. II. 1, ). (II. 2. 2) (. 2.7) :
' ,
(
(-2-8)
, ,
(II. 2. 4), [9, 10].
, ,
, Yz, Z2 - .
. . [. I. 6, 7].
, 23 I, - . -
, ,
. . [11], . . [12] . - [. I. 21].
, -
.
, U (, /,z, t), . . \ Xi, Yi, Zt
F = gradZJ (II. 2.9)
, \i Q , =(), = (),Q = e(p). (II. 2. 7), Q (p),
-
2. 37
(. 2.10) Q , QV,w
* ( ) ( ) i *()'()
_ k(p)Q{p) dp A(p)Q(P)
(. 2.11) :
^ = () = - ^ . (II. 2. 12), = () -
.
. . -
. .() - .
(II. 2.11), , :/ , dU \ ( . dU \
v ' J' \ * ]
,^-(^- + . (.2.13,
, (II. 2.12)
.() " " ^ dp
dz dt_ () Q ()
dx, dy, dz, dt , dx, dy, dz, dt . (II. 2.10) (II. 2.13).
(.2.13) (II. 1.1), -
* . ^ , I~ dU ,dt * ' * ~ dz2 ~ I
-
38 . II.
(II. 1. 3) (II. 2. ), (. 2. 14), , , ,
, , Q. , - (II. 2. 12) = (). , = () = () =
()() () () ( ) ( ) ( [ ( ) ] ( )
() 1 ( ) ] ( ) () [ ( ) ]= @(). , (II. 2. 14)
Q. , , , , -
.
, ,
.
. -
, , (II. 2. 13)
2 = 0, (II. 2.15)
-
. , ,
, ,
. . [. I. 8], , , ,
,
w -
Q W. Q W
w = g Q W. -
() = = const, k(p) = k= const{
)1 = ^-, (.2.17)
P = fy(p)dp, (. 2.18) -
.
-
2. 39
, , ,
(. 2.14) ^- = -~^
- (. 2.19), , -
(. . ) . (II. 2. 12) (II. 2. 18) - J, , -
.
.
, , =
= Y (, )
, (. 2.20)
R ; ; z
-
40 . II. ,
3.
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4. 41
[13, 14, 15], -
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/ (s). , , ,
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, , -
f2 (. II. 2).
. ,
>/, w - J1 ; .
. II . 2.- ,
.
t + t (. . 2)., :
( 1 ) , . (II. 3.1)
,
, -
/
At, ( 1 ) 0 d t
, -
, ,
.
-
.
4. :
C i - G 2 = ^ - JmydV, (II.4.1)dt
( y ) Gi G2 /2; V /2.
, dV = f(s)ds (II. 4.1) - :
G(s, t)-G(s + ds, t)= 2) fWd*
_ [_[.. . .
-
42 . II.
,
==
~1~ f(s)- (. 4. 3)as at
: (yw
x) ( w
v) ( 2) "1 ()
| dz I dt ' w
x, Wy, wz w , , z.
5.
,
/2. -
-
, , . , -
.
:
I v 1 / 1 17 I I D I 7> I ] rp | rp i (- \
(V) _^ _^
/ 2 = I Qdq; / = I Q dq (II. 5.2)(/2) (/1)
-
/2 / ; dq = wndf df, w
n df -
;
2
/ l f /2; Rn - ; R
n
, ; W , , , -
. . 7\ ; 2 .
, -
dV , - .
.
(II. 5. 1) , . , , , -
,
. w/m 4 I ,
.
-
5. 43
, , (II. 5. 1) s , - /i /
ds:
-dTls-dT2s, (. 5.3)
| - ,
[16, 17]
jwndf
(-5. 5)
(.5.6)
w . , [16, 17], % = 1 / 3 ,
=: 0,03. I 0,2. - ,
.
% (II. 5. 5) . I. 4.
i. (. 5. 7)
0 , gj, .
(1.4.3) (1.4.4)
(II . 5.8)
h = H1H2 . (. 5.9) -
. . .
1 S = QQ* 1 gh
\A v I 4
-
44 . II.
,
J r*v(r)dr f( I I . 5 . 1 1 )
[J0
,
, , -
.
, z (. II. 1, ). , (. 5. 3), , , :
dP = d[pmf(s)], ( I I . 5. 12)(dR
n)s = Pd{mf(s)], (. 5. 13)
dzdWs=-ymf(s)ds.~ , ( I I . 5 . 14)
(dR'n)
a = O. (II. 5. 15)
-
/ (s) ds. .
(II. 5. 15) . . (dR
n)
s. (d/?
n)
s
-
s - .
1. , s.
(i-m)f(,s)p. (a)2. , s -f- ds.
()
3. , .
Pd[(l-m)f(s)]. (),4. ,
. , ds , - , -^ (lm)f (s)ds, (1 ) / (s) ds . , - ,
(d)
-
$ 5. 45
,
'
).=. () dTis dT2s : dTla -
/ (s)ds - .
, (II. 5. 3)
= dp-mf (s) pd [mf (s)]+pd [mf (s)] Y mf (s) ds dT2s =
= ~^-dsmf(s)-ymf(s)ds p dTZs. ( I I . 5. 16)os ds
(II . 5.3) ds :
dT2s -, . . ,
flS *.
(II. 2. 6) -, , ,
u w,
kQdT2s=
r v mf(s)ds = -Zjmf(s)ds. ( I I . 5. 18)
( 3, . I ) , - ( I . 3-8), -
^^ __ w ( I I . 5 . 19)ds * | | '
Z, , . ,
, -
, , , -
, . . . ( II. 5. 19) 1 , [24] . . , -
,
1 1 U W
-. ,
( I I . 2. 7). 124] , , , [2] , -
-
46 . II.
, -
-. , , [. I. 11] ==
-
$ 5. 47
(II. 4. 3) ,, ,
[ 1 + | 1 1 1 + | w dG i (y dw i wi ( ig m ds+\g dt + g dt
:
1 -j- w d dw wg m dt g dt g
(II . 5.22) :
, mf (s),. . \
m ds \ g m j g m dt
j- , - .
5 = 0 \ - (. 5.23) . | = 0 < 1 (II . 5.23) .
,
:
gm dt n| w
x \ 9y _ 1 dp 1 l\i\w\
g m dt dx i T B / I " 7
gm dt +-m-\w*-di[g-
m-) + wy-dj[g-
m-)+W*dz\ g m
I wy I dy = 1 dp 1 / p, | w | e - Wg m dt dy \ ^
-
4 . II.
6. , ,
, -
Y = Y () (II. 4. 3) (II. 5. 22) (II. 5. 23) Y, , w.
-~ (, ) - , -
, .
-
, , -
.
-
, .
/2,
(/)
(V) (/l)
jLE- + i\ydq_. fyZmdv+LBH + -^f- . (.
(/2) (V)
(II. 6. 1) , ~ (, ) , ;
, /2 ; ,
.
-
, , -
.
QBH
1 : = j~= /- .
,
.
LBH = 0, ,
, ,
( ) -
-
.
-
6. 49
, LBB = 0. ,
~\, (II. 5. 1) (II. 5. 3). , - 2,
. ,
.
, -
, ,
, LBH = 0 - ,
. 1.
.
-
50 . II.
, , -
(II. 5. 10) (II. 5. ) , :
jyif ' (IL6'4) v () (. 1.4)
J rv(r)dr[ f)
[ f r* v (r) dr]30
, (II . 5.11) (II. 5. 12), 0 .
\2 w mdf, 771 / WI
(/) .
(II . 6.6) w /(). w
n, | ' = , | -
(II. 5.5). , , g ' . / I I . 6.2) ds (II . 6. 3) (II . 6.6) :
(II. . 7) 7"i -
. -
,
.
,
. ,
, -
. ,
, -
, .
.
1 = 1, (. 6. 8) Cj / .
-
6. 51
:
( l - m ) (), (.6 .9)
Xi . (II. 6. 9)
. , -
(II. . 7). - s, - .
(II. 4. 3), (. 5. 22) (II. 5. 23), (II. 6. 7) (II. 6. 9) = Y (> ) - w, , , , 7\.
(II. 6. 7) . -
i = i(p,T)=u + A ^ . ( II . 6. 10)
-
[%)]p, ( . 6 . 1 1 )
; v = v(p, 7") =1 , ( dv\
= -.=- ; \~^f) v , , .
(II. 6. 10) (II. 6. ) (II . 6. 7) . (II . 6. 11) ,
, ds dt,
di . . / \ ]
(II . 6. 12)
(. 6. 7) :
-
52 . II.
{ +
. (.6). . [. I. 19]
, i == i (, ) = const, . ., -, , -
. -
[. I. 19]. (II. 6. 13) , i (, ) const .
,
.
i (, ) = const .
(II. 6. 13) - (II. 4.3) (II. 6.12),
( ) 4 ^ +
-
S 6. 53
, -
(II. 5. 23). (II. 6. 14) mf(s), -
[ 1 J dv\
">)*
2g \ m ) dt ^ dt [ ^ 2g \ m ) J [ )p dt
'
1 1 f 9T\ \mf (s) ds I ds w J \ dn Jcp f (s) J '
mj - , (II. 6.15), (II. 5. 23), -
w > 0:
_
F-Hf i^- < " " 1 6 ) (. 6.15) :
ffs\ g m
g dt g m dt J ~ ds L^ \ 2 i ay , 1 + 6 ' ( ' | [\ dp__m) dt "^ a t L 2g \ m j J \dT jp dtA dt ' 2g
f cr, a r I i / ay
Jw\ (w)y w I 1 + g u > \ ] 1 \ , m j 1
/ ^ \ Y \ m ) ,m ds \ g m ) \ g dt ^ \dT jp g dt """
-
$4 . II.
HLT() LJL?y + JL\S+Aygmdt \ jp g m dt + ds[Z+ 2g
CP ^ | E ' - i ()' i ay , 1 + 6 ' / f f t A _4 + 2g \ m / Y di "*" dt [ "* 2g [m I \ \ dT )p dt ~~
w dT , 9 T
~m~ ds + d t J [y~1 [ d T ) p \ m \ k^{__1+61 N Y 4 j _ A ( 1 + 1 \ ,
,( 9v\ \wy_dz- jvy_J_ ' * N a |\dT jp{m ds ' m ds \ g m j ' mg 9
1 1 v"/W
,
1 / \ 1 (\ 1\,,(\]
/ ^ \ 2 i _ay , \%> tw\*] ( d y \ [_^dz_g \m) dt ^ dt [2g \m) J y\dT)p\mds^
w_ 9_ I J _ + | _ w\ _w_ d_ lw\ _1_(2__1__)==
m ds \ g m j mg dt \m j g\mj dt dt j
m Y Ym/ (s) 5 s L 3 s J V d n /cp Y / (s) J ,
-
6. 55
-
.
Q . \ - .
1
[6, 18] *. , \ = . - (II. 6. 9) . mf(s):
1- CiYi 9TX _aQ (T-TJ 1 ,~~^ ~ Tj~H + Aymf(s) W [1~
A {i7m)f^ (II. 6- 10> / m/ (s)
(. 6. 19) (. 6. 15), (II. 6. 19) (II. 6. 18). , -, . \ = , ^
(w \ 1- ClYi w (dv \ 1_1 _
A \m ds "*" dt / "*" Am dt m [ \ ) J ds^
dv\
, _ , 1 1 ( \ n w( vM-^ ^L ^
, Am ,
(.6.20,
[mycp-\-(l m)c1y1] m%-\-(i m) %
.
4-(1 )11 = /-^, ml-\-(l m)%1 = A, (. 6. 21)
* , , -
, , , -
,
1 -
-
56 . II.
, (II. 6.17),\ I \
w-AmT[-r)-T
) Tff (" 22)JCp f (s) v '
Bj :_ l + J ! / u > V ] S ' - E x / \ 2 1 a Y
-
, (II . 6.19) (. 6.18)
T (\ I dp 1 __\ _ 1 / \ (s)
1 a L 9 1 / \ (1-)(1)+ ^Ym/W "
( )/()J~l^'*r/' ^v/W , Amy (II. 6. 21),
_ ( A * L ) " M - . (. 6.24)V an/cp /(s) '
e t
w { \i+^ ()] w
2 m I a* L 2g \ m) \ m ds g m
( aY a [ l ' . _ N ^ 1
-
7. 57
-
. , ,
:
&% w -- , , :
,
* + 9 + (II. 6. 27)3s dz
, ,
, I
\ , I. \ , (. \ -- - + -%- -% + -~ ~
\ I \ J dz \ dz = const ^2 - .
-
, , -
. : .
, -
. ,
Y = Y ( ^) -, .
-
.
-
. , -
, , -
, ,
-
.
7. , -
, , -
ei ^ , ,
. (II. 6. 22) (II. 6. 24) ,:
.
-
-5S . II.
(II. 7. 3) , :
,
f = const, 6 = ,
iI_ = il/'ji^-(-i.Ou)2) = *L (II 7. 5)
(II. 7. 5) - , -
,
. ,
(II. 7.6) , -
.
pv RT, R - ,
( * ) , 5 = - ^ _ = - ^ - = 0. (II. 7. 7)v
, ,
J- 0, . , ot
.
,
, ( \ I -^=-1 ~~
Y \ / ot
Y RT ' \ }
. _ ( \ dp Amp dp dp+ C ^ j = = _ = y l m . (II. 7.8)
.
-
7. 59
[19, 20]. (II . 6. 11) di , ,
- * * . . (.7.9)
-
[20, 21]. -
, ~ 0. , , 0, > 0 < 0 [19, 20, 21,22]. (II. 7. 1) (II. 7. 2), -
, \ , \af) " (II. 7. 9)
(\_ (
\Y \dTJ- V )
(II. 7. 1) (II. 7. 2) : . . dp AmT
=
6
>-- d s
( \ = _ e Y C p U , [-JL^-L., (. 7. 10)
^ -~ I - ( )
(, ) (5
, (II. 7.1) (II. 7. 2)
:+ = - ev CPW i + A m T P w =^ . . ( II . 7. 12)
(II. 7. 9) (II . 7. 11)
-
60 . II.
( I I . 7.12)
Y^."L+ cwL=__i_ [ 1_p ( P ir ) n Y e p I P-OS Ot Cp
^ T^(P,T)^- . (. 7. 14)
(II . 7.13) ,
^ (. 7. 15)
, , ,
-
8. 61
mf(s)ds,
-
62 . II.
ds _ dTv cpw ~~ dp
- * = ^ . (II. 8.4, Q -
, . .
p = p(s). (II. 8.4)
d
(. 8. 5)
[(const)j (const)2] = 0 . , ,
(consl)3 = i
i); V . s V (),
(II . 8.2):^^. (.8.7,
Y .
- ( = 0 Q - (s) , , (V).
(II. 8.7) () + ^-=^()- (.8. 8)
|5 - p(V). (.8.7) ( II . 8. 8)
" ~ ' P [ V ^J?I ()- ( I I . 8 . 9) 1 6 /
, (V) - V, (V) -
= v1 v =
-
S 8. 63
, s = s b V=V1 7 (V
x, t). (. 8.9)
, Fj ,
Vx, , ' ?, .
(II. 8. 10), ,
, ,
. ,
-
64 . II. ,
-
-
, . (
) - , -
.
. (II. 7. 17) - . ,
, :
W , a V = V(s) ( I I . 8. 11). s = 0 F = 0
=
(') ( I I . 8.13)
T(O,t) = Tc(t) = W(t), ( I I . 8. 14)
( I I . 8. 15)
(h) tj At V,
CV
= ^1 (. 8. 16)
, -
,
.
(II. 6. 21)
ycvQ _ ycpQ Q 1 ( 8 1 7 )
_
)eiYi m 1- Ci Yi"" m cp Y
. (II. 8. 17) , -
. ,
.
. . [5, 6],. . [23], . . [3, 4] .
: -.
-
8. 65
1. H u b b e r t M. . Darcy's Law and the Field Equations of the Flowof underground Fluids. Petroleum Transactions AIME, 1956.
2. - . . - . , 1952.
3. . .
. . , 1959.4. . ., . .
. , 4. 1958.5. . . . .
- . . , , 1961.6. . .
. .
, 1962.7. . .
. . , . 105, 6, 1955.8. O r o v e a n u . Asupra miscarii unui fluid incompressibil printr-un
mediu poros neomogen, comunicarile Academiei Republicii populare R mine,I. VIII, No. 1, 1958.
9. . . . , 1948.10. . .
. , . III ,1958.
11. . . , . . . ., . IV, . 1, 1940.
12. . . . . . ., . XIII, . 5, 1949.
13. . . . -, 1957.
14. . . . - - , 1958.
15. . ., . . , . . , . II. , , 1948.
16. . . . -, 1937.
17. . 3. . , 1956.18. . .
. . . 2, 3, 1953.19. . . . , 1956.20. . X. . -
, 1953.21. a n d b k of Chemistry and Physics, 37 ed. Chemical Rubber
Publication, vol. I, II, 19551956.22. . . . , 1961.23. . . -
. -, . II, 1958.24. . .
. . . . , 2, 1961.25. . .
. , . I I I . -, 1963.
-
III
1. . .
,
, -
, -
-
.
. :
z w = 0, = = .
-
:
, v (III. 1.1) (III. 1.2), -
V-g + -0. ( I I I . 1 . 8 ) .
-
1. 67
. ,
, .
, -
. , -
, . -
,
. -
, -
, .
. -
, . .
, (12, , , V.
dx cos = = -^~
its \V\
cos a, = =
dy vCOS do = = -=r-
ds \V\dz_ds
(III. 1.4)
ds dx, dy dz; \V\ . :
dx dy=
^ ds | f | ' ds
dx
vdx dy = 0.
(III. 1.5)
(, ) = . (III. 1.6) , -
. -
-
(. III. 1). Y = W (, ) ,
,
.
. W (, ) .
. III. 1. .
-
68 . III.
= (, ). ? (, ) = const. , -
dXF = - S - d x + ^ T ^ = - (. 1.7)
(III. 1.5) (III. 1.7), , . ,
dW = ~dx + ^-dy = vdx udy = O. (III. 1. 8)
(III. 1.8) dx dy,
v (III. 1. 9) - (III. 1.1).
' '
-J = -J , -3 = -S . ( I I I . 1.10)dz ' '
(III. 1.10) .
, (, ) .
(III. 1.10), , - ,
*
' * '
,
I 1
(III. 1.10) - .
. -
z = -\- iy, i2 = 1 .
1
z + iy z.
-
1. 69
,
z = + iy, (, ) -j-+ i V (, ).
:
F (z) = F ( + iy)? (, ) + iN (, ),
(, ), N (, ) , z = x -\-iy. , , .
F(z) = z2 = (x + iyl)* -,
F (z) = z2 = 2 2 + i2xy. ,
(, ) = >- \ N (, ) = 2, (, ) + iN (x, ) =
2
2 + , -
, 2
2 +
z = x + iy. - ,
,
(, ) = 2 -f- 2. Mi (, ) + iN (x, ) = 2 -\-
+ 2 + 2zr/, , - z x-\-iy.
, (III. 1.10) ,
(, ) -f- i Y (, ) ( ; W ) - z = -f- iy.
, -
, , ,
z = x -f- iy. , -\- i W ,
,
(III. 1. 10). : + i W z = x -f- iy,
0. , (III. 1. 12) .
( ,
-
70 . III.
z = + iy), :dz .
-
1. 71
,
.
F (z) = (, ) +-\- i W (, ), z = + iy, ,
: , .
, ,
, , -
. .
,
, . .
(. III . 1). :
{, ) = const, W (, ) = const. |(. 1.16)
* + * * _ . * + *-. (.1.17) \--\ = kt -
\" :/= const ( I I I . 1.17):
(
-
72 . III.
2. . . ,
, -
, ,
. ,
(I. 2. 18):
-. (III. 2.1)
= const, = const == const.
, q -
- (-) - (-).
,
. . -
,
.
.
.
0 (. III . 2). ; , , 0 :
W = AQ + B, (III. 2. 2) .
:
-f- i =-^\nr + iAQ + iB + C. ,
--:
. I I I . 2. .
+ i = J-(1
4- i6) + const,J- (III. 2.3)
-
2. 73
const = iB -\- .
lnr + 9 = ln(re1 9) . z
z = -f- iy, , = cos 9, = sin 9,
z = r (cos 9 + isinS). -
cos 9 + i sin 9 = e i . ,
In ( 4 ' ) = In [r (cos 9 + i sin 9)] = Inz. (III. 2.4) , , -
,
:
F(z) = ^-\nz + C. (III. 2.5)
,
. -
, : , .
, , ,
, -
- , zo = ++ iyo.
,
, (III. 2.5) z - (z z0).
F (z) = .in (z-zo) + . (III. 2.6)
.
(. III. 3). = , = 0 --, = , = 0 -. , -
. ( )
^ ^ = -5 (. 2. 7)
-
74 . III. -
\ , , ;
, - \ .
( )
2 = - ^ - 1 2 , 2 = ~ ^ - 8 2 , (III. 2. 8) 2 ; 2 , 2 .
. III. 3. .
,
F(z)=^-ln^^ + C. (III. 2. 9), ,
-
. (III. 2. 9) - (III. 2.8), - :
=
+ 2 = - in - + const, (III. 2.10)W = 4'1 +
1 F 2 = ^ - ( 6 1 - 0 2 ) . (III. 2. )
-
2. 75
*1 = const ,
1 2 = const. (III. 2. 12), , 6i 62 , [, ], . ,
, , ,
,
92 = const. ,
. ,
6
02 - .
, ,
.
, ,
, -
, (. III . 3). -
.
, (III. 2.10) = const -, = const.
= const , -
:
.
:
{ - a f + 2 = ( + af + 2,(1 ) 2 + (1 ) 2 (1 + ) + (1 ) 2 = 0,
2 + 2 2 | ^ + 2 = 0. (III. 2.14)
(III. 2.14)
( { ^ ) ' [ ( e ) " ] ^ - (-25) -
2aj/c/(l ) (1 + )/(1 ), / = 0. , , -
- .
-
76 . III.
= RK ,
.
(III. 2. 9) RK. q ,
. RK -
.
RK , - '
(III. 2. ) . III. 3 :r1 = Rli-6; r2 = 2a-(RK )
.
,
.
\ =
. 2 -
-,
. 2
2
< 2 < 2 + .
2, 2.
:
= - ^ - 1 + . (III. 2.17) (III. 2.17) (III. 2.16),
^ ^ ^ _ . (.2.18) , ,
, 6. ,
' " , ,
:
2-(-6)
-
2. 77
2.
/ / 7? | \ I Si . / / 7? ^ |
2 RK + = 2 + ,2 2
2 = - ^ . (III. 2.19) 2
(III. 2.18). , (III. 2.18):
2 2
(III. 2.18) :
_ -. / \ 1
(III. 2.20)
|
-
78 . III.
F Fi . , F, Fx ().
,
z = z (), F (z) z - . , z == z (Z) z - .
-
,
,
, ,
.
-
-
, , -
.
.
. . . .
[1]. - -
. . . [2]. ,
.
[. . 2, 3, 4, 5], 3, 4. 3.
, z == -f- iy F (z). = | + i , - z z = z () = (z).
z = z (,) ,
*() = *(5 + )=*( , | ) + (.),
x = x(l,i\), y = y(l,r\),1 = 1(,), = (*.*/)
(III. 3. 1) z. , z = z (), Z, - z.
-
S 3. 79
. ,
, . . ,
-
. , ,
\ .
rfz,'Hz,
. I I I . 4 .
dzldt, ,
^ .
,
lim r- = lim Ax + i
|. , () z dz dt, . di,dt,2, ... z dzi, dz2, ..., z, - (. III. 4).
^- -
,
dz1 (III. 3. 2)
-
80 . III. ,
(III. 3. 2)
-
. arg dzx
dzx .
,
arg dzx arg dz2 = arg dx arg dt,2,
. . dzi, dzz di, dt,2 . z() (z) ,
-
.
z
.
QC,
,
(III. 3. 2)dz (III. 3.3)
, -
.
z - I, . dn dl - I z dv dX , .
| Q \ z
dn\dl\, (HI. 3.4)
wn= -j -
.
, -
, (III. 3. 2)
\dn\ =
I d / I =(.3.5)
-
3. 81
(III. 3.4), dz
=- ,
( -
3-
6
(III. 3. 6) (III. 3. 4) -
, - -
z. -
, -
-
F(z)=*Az. (III. 3.7) -
. -
,
F (z) = + i = { + iy),
const (L)
= , (III. 3. 8)
. I I I . 5. - -
--
-
.
,
= = const , (. III. 5), W = == const , .
, v
(III. 3. 9)
, F (z) = Az -
= .
z = In t, (III. 3.10)
-
82 . III.
= | +-ir\ = Q e . Q, 0 .
z = + iy = In (Q e i e) = In Q + i 0, (III.3.11), ,
3
= 1 , = 0. (III. 3.12)
= const - z In Q = const, Q = const,. .
, = const - 0 = const (. III. 5).
,
= = const, ~ = Ay = const z Q = const 0 = const, . . > 0 -
q = 2 .
-
-
.
-
-
: 1,- I /
. III . 6. 2(III. 3.13)
.
z = 0, =
,
=
,
-
=
(. I II . 6). z
.
= (z) z == z (), - ZB Q = QK , zc = ia z,
,
= 0 , .
-
3. 83
, ,
-
-
.
:
f l - ( in. .14), z = ia, (III. 3. 14) -
t, = 0, . . z - = 0 .
z - Q = QK %. , (III. 3. 14) z = ,
arctg _
xia l/za + a ' e x - i 2arctg
1 * - = g l ' e ' < - 3 - 1 5 > \ = .
, z = x QK , z = ia = 0. , (III. 3. 14) - .
(III. 3. 3) Qc =
( I I I . 3 . 1 4 ) (z-\-ia)(zia) I 2ia
z
= 4-^ '.' +, (III.3.17) ,
'= +-^j-InQK. (III.3.18)
2(-
)4 1 8
-
84 . III.
Qc (III.3.16),
2(
-)in^L In
=2(
)2 (III. 3.19)
, (III. 3.19) , .
(III. 3.14). z la* "
z-\-ia '
, z -
= 0, = , , = 0, , b =f= (. III. 6).
, z = iy, > z , - QK, Q K < i < QK, = 0. -, (III. 3. 14) z = iy,
iy~iaiy + ia
(III. 3.20)
\ - , 0 S QK ^ 5 ^ QK, = 0.
z ( = 0, = ) :
^ = 5
= Q K - ^ f ' (. 3.21). . Z, .
(III. 3. 3) QC:
Qc = dz * = 0
= QK
2ia
* = 0
(ib + ia2 Q K (III. 3.22)
t, Qc 6 (III. 3.21) b ,
QC, -
- = + ),
(III. 3.23)
-
3. 8S
2 Q K _
1 -QK
2 QK 1 \2
QK
. III . 7.
z , , , , , -
(III. 3. 19), - :
_ 2(
-
)In 2
(III. 3.25)
(III . 3.23) ( III . 3.24), - (III. 3.25) - Qc, - Q1{ 6 (. I I I . 6):
_ 2(
)In \
L 2 QK
- ,2:
( -
^-(1..2) 6 / Q K 2a QC
-1 inlJMJ L ec
(III. 3. 26) (III. 2. 21), .
, (III. 3. 26),
(. III . 7). z RK
Ri
-
86 . III.
/re-;
, -
.
-
= 2 /.
= zm. (III. 3.27)
z =
= Q ,
(III. 3.28)
, (III. 3.27) = z = 2 , . . -
z .
(III. 3.28) = Rx -
Q =i?. QC (III. 3.3)
dz (III. 3.29)
QK (III. 3. 27) = . (III. 3.30)
, Z, , QK R - 6 = R .
(III. 3.26) QC, QK,
2(
-), | (
1 \i J ^ i I ~ R2m
> 5 ~\ -" /
(III . 3.31) :
\ 2
2(-) ^ (. 3.32)
-
3. 87
(. III. 8) 12
= ' (III. 3.33) z = 4- iy.
( + )
.
=-QKe
8 = (III. 3.34).
. III . 8.
(III. 3. 34) , = z Q = QK .
= 2 , = L, = 0, 1, 2, ... (III. 3. 34) Z,
L
Q = QKe~ , 9 = 2 , == 0, 1 , 2 , . . . (III . 3. 35) (III . 3.3)
Qc =J t L
rc.
( I I I . 3.36) (III. 3.26), -
_
:LL. (III. 3.35) 6" = QKe a :
2(-)2 L
L
sh
-
US . III.
, ,
-
5. 89
.
-
,
,
,
, , ,
= 2
* < ^ > = , ^ t K w * ! L , (. 4.4>In
] n / aK~bK (-)
V
2 2 ,2 2 .2 =
=
.
_ 4 (
) _ 2 (
) / J J J ^
+
^ ' ' 'In v " K y K ' , : , .. In
q= 2 ( - )
t ; = + ^ ; =
+ . ( . 4.6)
, >
RK Rc, (III. 4. 6) -.
, -
,
, . . [6].
5. -
. . -
[. II. 2]. ,
- , , -
.
, -
, , -
. , , -
. .
z ABCDA, - (. III . 10, ), (z) Q = 1 . Q
-
90. III.
:, ABCDA ,
1
1 z1 + z
( I I I . 5.1)
L | L >
. III. 10.
= 1, = \.
= - (III. 5. 2)
Y -.
z (. III. 1, ) - .
Q < 1 - . -
, -
,
.
= t h . (III. 5.3)
,
(III. 5. 1) (III. 5. 3) . -
dz
dz r
, (III. 5. 1), dz 1 1 z2
(III. 5. 4)
(III. 5. 5)
dz
(III. 5. 1) ,
(III. 5. 6)
(III. 5. 7)
-
8 5. 91
Vli-(\+yl)f+4yl
[1 (*2 + 2)]2 + 4!/2- . t fl.
2
-
92 . III.
(III. 5. 3) ,
sh x sin l =
c h z + c o s j , ' T 1 = c h z + cosy (III. 5. 16) z
. (III. 5. 16) , (III. 5.3)
r __e
z - 1
~
2 + 1 '
2 = - ^ | - . (III. 5.17)
(III. 5.17)
2= *
, (III. 5.18)
/ ( 1 Q2)8 + 4 Q 2 sin2 61 2QC
^ ? (.5.19)
f (III.5.20) .
6. .
.
- .
QK == ), TV . .
(, -,
= const), (. . III . 6). - , , .
zc = ib -
( )
'w=itln-ii-- (-6)
-
6. 93
, (III. 3. 14), - z , z = ia t, 0, z = ib = 6, 6 (III. 3. 23). - .
(III. 3. 14) z ,'
.
( I I I . 6. 2)
, z ( I I I . 6. 2) 6 ( I I I . 3. 23) ( I I I . 6. 1):
& _
ie-
-
9 in
2 l
~ 6) (III . 6.3)
(III. 6.3) : / >'
(III. 6. 4)
const = -~ In
In ( S) - -, -
= 6 .
2
-
-,
t=P- ( , . I I I . 11). - : -, -
= 6 , -,
Z, = -- ( - ), [ | = QK -.
-
94 . III.
( I I I . 6. 4) , = Qe* = Q (COS -)-i sin 8), -:
- ^ - R e J L l ng zn Q
- Q (cos 8 + i sin6)--2.g (QCOSB ) + 2 6
| _
"to [ ?IQCOSB j^-l + Q 2 s i n a 6
6
=
Q = QK:
e
a
K + 62- 2 Q l l 6 c o s 6
(III . 6. 5), * * , 1 (Q2 + 6 2 -2Q6cose)o 2 . +
^1
eK 2Q6 COS 8
(III. 6. 6) , .
6 . ' (. III . 12)., 9 ' 0 ' = 6 . (III. 6. 6)
4 62 .
-
.? . 95
(III. 6. 7) , . QK . ' q\, g2,...
-
(III. 6. 7):N
^ p 2 + 6 j 2 6 ; cos ( 0 . )
' - X gj In 2 3 * . ( I I I . 6. 8)
, 2 . 2 Q 6 J c o s ( 6 { )4JT
. III . 12. . I I I . 13.
,
9. (III. 6. 8) ,
(III. 6. 8) Q = QK
(
= const) , 6.
- ~ 2 Q 6 { C O S ( 0 at
Q = Qu ( ):2
. .
_ _ 1 2 J
(,6) , , , .
(Q, 0) -
(6).
-
96 . III.
[ 7 ]
( 6 ) = + 2 ( cos 6 +^ sin 8), ( I I I . 6 . 11) = 1
1 2 1 ?" = _ I (8) cos 8(Q,8) Q = Q K ( I I I . 6.11)
4>(QK'9) = 2 (
-
6. 97
n = l
N '
2
n = l
iV (') 2 ' , i = l i=JV
i = l = /.
,
Qi, ,
( I I I . 6 . 1 5 )
(.) =
+ ^ , (Pncosra6 + P^sinn8)l-^-
n = i
V2 ' '* 206,003(8 ,)
=. (III. 6.18)
-
V8 . III.
( I I I . 6.16) ( I I I . 6.17)
n = l
~
6
2 kh Qc j ( I I I . 6. 19)
, , 2 4 V, Qei , -.
.
- ,
. kh/yi -
(III. 6. 18) (III. 6.19) .
-,
Q K(. III. 14). ,
.
-
, ,
QK, , , -
(9) G. (III. 6. 12) -
[7] -
'
. ,
(Q, 6) Qi, (III. 6. 18)
.
,
III . 14. .
khl\i. - Q, 6
, Q,(III . 6. 18) khl\i.
, , , -
Q, 6 khl\i. , khl\i Q = 0, .
-
6. 99
( I I I . 6.18)
_
2 (
- )
( i n . 6 . 2 0 )
(III. 6. 20) khl\x,
> (3. 6. 19) Q
cj.
, , -
, -
.
1. . . , . . . , 1958.
2. . . - . .- ,
. II, 1956.3. . ., . .
. , 1953.4. - . .
. - , ., 1947.5. . .
. , 1939.6. . .
, . . . . -,
1955.7. . . , . 1, 2, 3. ,
1948.
-
IV
1. .
III - -
.
--
. -
, , .
-
, ,
.
.
.
1. , ,
.
,
.
, ,
, -
, .
2. : , .
, .
, -
, . . ,
.
-
1. 101
- ,
.
.
, . . , -
: , ; , -
.
-
.
, -
,
.
,
.
, -
, . .
,
, . ,
.
-
,
.
, , ,
.
-
.
-
.
, -
, -
.
, -
. h - . -
.
,
. IV. 1, . , -
, , -
, , -
= .
-
102. IV.
, -
(. IV. 1, ).
.
.
, .
/. -,
-
,
-
-
.
(I. 2.18)
= - ^ 1 + const (IV. 1.1)
, -
, q = Qlh , -
.
-
.
, -
-
, -
.
-
. IV. 1. . -
. -
,
.
-
. ,
.
(. IV. 2), qi, qz, ...
, , -
-
2. 103
, , -
(IV. 1.1):
(IV. 1.2)
, "2, ", ... , . . ; &, , , ... (const), (IV. 1.1), - . f
M
-
. , -
.
,
1 = 1(IV. 1.3)
; * t-ro , -.
. IV. 2.
. 4) Qi t- h.
. -
.
(\ = 0) -.
2. .
(IV. 1. 3) - .
, .
, ,
.
-
104 . IV.
III -
.
.
,
. .
= 0
(. IV. 3).
.
. IV. 3. . IV. 4. -
.
,
(. IV. 4, ). (. IV. 4, ) ,
,
.
.
,
,
! / > 0 = 0 .
-
2. 105
, -
(IV. 1. 1), (IV. 1. 1) -
.
=
(IV. 1. 1) =
-
.
= 0 (IV. 1. 1) -,
(. IV. 3). , (IV. 1. 1) - ,
, -
. ,
.
, ,
: - -, . .
.
.
, -
, (IV. 1. 3), = 2, qi = q, q% = q,
n - -; - -
-.
,
(IV. 2. 1) , = .
, - -
, - .
(IV. 2. 1) = 0,
= 21 = = . (IV. 2. 2) ,
q , . .
.
= + - ^
1 , (IV. 2.4)
-
106 . IV.
,
, ^ 2. , 2 - -
- =
. -
=
(2
) < < (2 -)--f- r
c).
2 ^ 2 , (IV. 2. 4).
< 7 = 2 ( - ) . (IV. 2. 5)In
-
,
.
-
, -
, 2.
RK , RK - . , -
,
, , .
3. .
. -
,
.
,
. IV. 5, , , .
. - ,
, RK .
, -
.
. ,
, ,
.
.
,
.
-
3. 107
, -
.
, .
.
.
8 , , , . -
.
, , -
, [1 ]. (IV. 1. 3)
= - 5 2 + . (IV. 3 . 1 )
, i - 9i-
.
. (IV. 3.1)
i = " ^ ( ? 1 1 1 +? 1 2, 1 +9 I n r 3 , 1 + - + * 1 , i) + c . (IV. 3. 2)
4 ; rc i ; 2 1, 4 , . . ., rn i - * , , . . ., - .
.
.
, r 2 4, r 3 v...,rn i , , .
,
300500 , 0,1 , . :
-
108 . IV.
, -
.
,
, , -
.
(IV. 3. 5) ( + 1) . ,
(IV. 3. 2) (IV. 3. 5), .
, .
(IV. 3. 2) (IV. 3. 4) - (IV. 3. 5). .
(IV. 3. 2) (IV. 3. 5): R
,
^ + ft^ + ? 3 + + ffn1,2 2 , 2 , 2
_
= . l n ^ J L + ftin_?2L + ftln-^ + . . . + ? n l n - ^ - ) . ( I V . 3 . 8 )
\ ri,n 2, 3, )
: , , ,, , .
- ,
- . , -
(IV. 3. 1), - , .
. -
, -
(IV. 3. 5) . ,
, .
, -
,
-
, .
.
.
,
,
, . . . ,
(. IV. 5, ). ,
1 27-! '
2, 3, . . ., u>n , . . -,
-
:i. 109
, -
, ,
(. IV. 5, ), - :\ + ~w* -\- 3-*-...+
-
110 . IV. ,
4.
.
-
, -
, .
. -
, -
.
/ -.
-
-
,
-
.
-
. -
(. IV. 7).. IV. 7. , -
,
, , , -
.
.
, -
.
-
.
,
. .
N ( N 2, 3, 4), .
(IV. 3. 1) -
= 4 2
-
5. IIP
qu i = 1, 2, . . .,
2JJ (
. q -. .
-
112. IV.
(III. 3. 38) :
9 = L . \ . |
2 2
*ft_l. I
2
2
(IV. 5.3)
= ' . (IV. 5. 4)
(IV. 5. 1) (IV. 5. 3) :
-
5. 113
, , R -
, , ,
.
,
q , .
h - .
(IV. 5. 3)
1 -In
2 mh 2 mh rc
, 1 . "r In
1 m h
(IV. 5.8)2a mh 2 mh r
c
?
|(
- _
" " - - " -
. /
/ = 2 mh . (IV. 5. 8)
Q , , ;
Q' = -- ^ - 1 (IV. 5. )4 2 mh r
c
v '
,
. ,
(III. 3. 31) (III. 3. 37) (III. 3. 32) (III. 3. 38), , L .
-
- (. IV. 11, ) -- (. IV. 11, ).
txZ jib(iv. a. w)Q~ kf ~ * * '
/ ; L -; ; hCp L.
= - 4 In^r-. (IV. 5. 13)
/ (IV. 5. 11), h . , , -, .
-
114. IV.
v
2 , . . . 4 2
(. IV. 11).
1 >
2 - , - ,
. , ,
,
/ 2 3
, 1 I1 1
1
1
{t
1
t|
\
1
;
. IV. 11.
. . , -
. IV. 12 . . IV. 12, a Qi, Q2, Q3> 64 , : - (. IV. 11, )
3 h3 4 " 4 (IV. 5. 14)
fe1Cp. 2, /i3Cp. 4. J
^1' ^
2 l -^ ' -^4
, . IV. 11, .
, : *
10=0
. IV. 12. -.
- -
(. IV, )
Qi = ah1
- ^ . = - 4->.
= L_1 kahiCV
:/3
(IV. 5. 15)
-
5. 115
-
(IV. 5. 11):
2itmj/1
' 2 I
2 '
In- 0" (IV. 5. 16)
mi, m2, 3, 1, r c 2 , r c 3 , 2ai, 2"2, 2 3 , , , -
; 1, 2, h3 .
- -
. IV. 13. .
, ,
.
, .
. IV. 14. .
, . . IV. 12, - , .
pK 2i . IV. 12, , - S Q 0.
,
.
,
-
116 . IV.
[. II. 9, 2] , , -
-
(. IV. 13, ) -, (.. IV. 13, ).
-
,
,
I (. IV. 14). (IV. 5. 10) ( I I I . 4. 5) (III . 4. 6)
= 1
-
- ^ '
(lv-
5-
17)
4a Q' (IV. 5. 11).
. . ,
[3], , - 23%.
1. . ., . . . , , 1939.
2. . ., . ., . ., . ., . .
. , 1948.3. . .
. , 2, 1961.
-
V
1. .
,
.
, .
-
.
, , .
-
-
,
.
-
.
, .
,
, ,
.
,
, -
.
, . . , ,
.
, (.V. 1).
-
118. V.
RK - , -
.
.
. .
-
.
:
-
1. 119
, , -
.
, q (,) dt,, :
(V. 1.2)4
-
(V. 1. 2).
. V. 2.
, -
: , =
-
=
,
-
=
.
.
qdl, (. V. 2). -
, q dt, , . .
, .
, -
, .
-
120 . V.
.
qdt, - . ,
qdt, . ,
-
.
=
=
=
.
-
. -
-
-
q ().
-
q (Z),
V
Q =0.2 0,6
. V. 3.0,8
q () - ,
. V- 2, .
[. 1.11]:
(V.1.3)
g
- ln - - (V.1.4)(V.1.5)
.
q>(h) . V. 3.
-
1. 121
.,
-:
() = J" 4-1 e~xdx.
( = 1, 2, . . .), :
1
-
( -\- 1) = (), . - -
[1]. ,
. -
, . V. 3. q () = const
-
(. V. 2): < 2h
Q [ ]
\ \ (1 )(-
W
(2,1 - w + x) + I (2,1 +w-x)- (2,1 + w + x)] ++ (Q4) + const, (V. 1.7>
(V.I. 8>
t, (s, ) - , [2];
.> 2h
= ._ __^ _ T_L V -
(2 Q) COS (2 nw) sin (2 ;) +*- 1
+ const; (V. 1.9) In 1
-
.
-
122 . V.
. . [3, 4] [5, 6], -
, , -
.
(. V. 2) q () = q == const . . :
z < ,
i = l
z > b,
z h h ~ b T (r \
|? sh (ijX p / (fi.j)
ft z b . r
R RK Vi?K ^ ( V . I . 11)2 , ftOr2/4
x4 sh -- J (Hj)
y? = k/kz, kz. , JQ (-^- \. , Jx (\Xi) -
; (., /() = 0- ,
, -
. . [7; . I. 16]. ,
, [8, 9]. .
. -
, . -
, -
.
; - .
, . . , -
,
-
1. 123
, -
.
. . [10], - -
. . . ,
, -
.
-
. .
.
. -
.
, , . . / 0., :
,
.
.
. . , -
.
, ,
.
, -
,
3. . . , -
.
, -
, -
, ,
, .
. (V. 1. 3) ,
h1. , , RK >
xUh, - .
. . [11] - (. V. 4):
, 1,6 bIn
1 .
-
124 . V.
. V. 4. - -
.
-
, -
(V. 1.2),
.
(V. 1. 12) -
,
q () = const, .
(V. 1. 12) >
.
,
( , ) -
-
,
( ) [12, 13, 14].
2. .
-
-
.
. .
115, 16]. . .
,
[. II. 9], .
-
. , .
V. 5.
.
-
Ro, : > h. ; . R . . -
-
Ro, .
-
2. 125
, -
(. V. 6).
. V. 5. - .
vrv
. ,
.
.
-
-
-
Ro, -
-
-
. -
-
-
-
. -
-
- . V. 6. .
(IV. 3. 6) (IV. 3. 8),
2(-0)< ? = (V.2.1)
-
126 . V.
, , -
.
,
:
0 (V. 1.4) , RK = R0^h. (V. 2.1) (V. 2. 2.),
_
2(
) ...
.
, -
Ro -,
, | 0 . (V. 2. 3) - 0 :
lo = \n^- + l
o-ln^- = ln^ + C, (V.2.4)
' ' '
C = go- l n - ^ - (V.2.5>
, -
.
(V. 1.4),
2 ( ) i n .
, , In Ro
:
-&'
, . .
, [. IV. 2; . I. 8. I. 16).
, . . -
, ROr
-
2. 127
, , Ro > h
Ro>i-h. (V. 2. 4), -
(V. 2. 3)
) 2 ft (
) (V. 2. 7)
(V. 2. 3) (V. 2. 7)
.
-
,
, -
(V. 2. 6) - .
(V. 2. 6) . V. 7 - = (h).
"(h)
16
12
4
\\
1
\>
\
0.2
\
\
0,6,
.
0.8-
h
. V. 7.
Q . . '^ - -
= 100. .
, ,
(-L l) In '01 h .\ ft j rc
. .
[7, . I. 16; . I. 8] , , , , -
. 4, 5. (V. 2 . 7 ) . :
~
= . (V.2.8)
~ =
. ,
-
128 . V.
.
In \- = In .
, 6
, -
'
.
-
, , .
. . -
, .
,
.
i?o . , ,
Q= 2",-) . ( V . 2 < 9 )
ln^JL +e
Ro , -
, , Ro ^> ir-h, .
QCOB= 2 "*( - > . (V.2.10)
-
= - # - - , (V.2. )VCOB
. V. 8.
Q (? .
,
Ro/rc
,
Ro/rc .
-
S. 129
, ), .
(V. 2. 8), (V. 2. 9) (V. 2. 10)
) =
,
(V.2. 12)
(V.2.13) -
) , - R
o
, -
.
-
,
-
, , -
'
,
,
,
.
( ,
[17]), .
. . [18] . . [19].
3. ,
-
,
-
.
- , ; - w :
0,2
. V. 8. -.
=
-
13Q . V.
kr ; kz -
z . ,
d u . d v . d w
-bj + -dj + ~ = 0
+ 0-)+^S = O. (V.3.2) , -
. (V. 3. 2)
xz = z' (V. 3.3)
' ^ dz't ~ U - (V. 6. )* '
.
-
h, , .
= 0, z = h, ( rc< r < i ? 0 ) , - - ^ = 0; (V.3.5)
=
; (V.3.6) =
-:
(V.3.7) g (z)
, q (z) [.. 7].
(V. 3. 4) - . -
, = 1, z = z'. -
:
0.
-
3. 131
(V.3.8) O = i?(r)Z(z), (V.3.9)
R (), Z (z) , - .
(V. 3.9) (V. 3.8),
Z (R' + - ') + RZ" =
^ ^ ( ) 2 . (V.3.10)
(V. 3. 10) z, r, (V. 3.10) , -
X2, .
(V. 3. 10) :Z"X*Z = 0, (V-3.ll)
R"+1R' + X2R =0. (V.3.12)
(V. 3.10) X2 , - (V. 3. ) (V. 3.12) :
Z = e~Xz, e*z; R = J0(Kr), Y0(Xr), (V.3.13) J
o, Y
o
.
(V. 3.10) X2 , - (V. 3.11) (V.3.12)
Z = sinXz,coslz; R = I0(Xr), K0{Xr). (V.3.14) /0, Ko [20].
X = 0Z = C!Z + c2, R = 3 In + 4. (V. 3 5)
z = 0 -^- = 0, (V. 3.13), , -. (V. 3.14) - cos X z.
-
132 . V.
(-^- = 0 z = h) , ,sin h= 0,
= 0 + In +
+ 2j [ \~~h~=1
^ 1
C 0 S ~
= 0 + Cj In +
)]
h' = nh; (V. 3.18)
,
,
, -
=
, Ro.
q(z) 0 < z < f t :
q(z) = qo+ V 9 n C o s - ^ - , (V. 3. 19)
() ^ j ^ - d z . (V.3.20)
,
, :
Rn , Xh \ . \ / _ ( 3 211
n = l
- / . ( ^ (^1), (V.3.22)
(V.3.23)
-
3. 133
, . .
,
. ,
.
(V. 3. 16) .
. . [10], 10 -, .
-
.
,
, -
,
.
q(z) :q(z) = -|- = const, 0 < z < ; q (z) = 0, < z < h. (V. 3. 24)
Q , . . (V. 3-20) -
(V. 3, 21) :
(V. 3. 26)
n = i
0 < z < 6 1 , , = 6, =:
in ILJlh.. (V. 3. 27)
n = l
-
134 . V.
:
n = l
,X Sin Sin -^-
(V.3.28)
(V. 3.29)
U
n = l
\ Y.h
sin re sin
n = l
(V.3. 28)
,
, ,
.
= 1, =
(V. 3.30)
\1 ( V ; (V.3.32)
.
1.
, -*0. (V. 3. 30)
"
(.3.33)
-
$ 3. 135
2. = bv
)~) sin2 n
451 ""' J?- (V.3.34)
q>x = -j-^ - .
,
( = 0) (b = bi) - ,
bi . ,
bi Q, -
bi
-
. -
. .
[21], -
q (z), . . - . V. 9. [45]. - ,
, [22, 23], . . -
(. V. 9). -
, -
, ,
5 VI:Q = 5 - c s 1 - . (V. . 35)
; s - (. V. 9); | (V. 1. 4).
. .
. . . -
. 1. . . h = 12 , Ro = 30 .
. 1 -
.
-
136 . V.
1
,
0,764
0,56
0,33
,
9,67,24,89,67,29,69,67,24,89,67,29,69,67,24,89,67,29,6
h
0,80,60,40,80,60,80,80,60.40,80,60,80.80,60,40,80,60,8
S,
2,42.42,44,84,87,22,42,42,44,84,87,22,42,42,44,84,87,2
Qlc
-
39,335,631,072,658,792.438,432,724,565,452,885,033,229,121,558,747,573,0
2
-
(V. 3. 35)
44,137,727,275,860,194,540,133,623,768,753,686,034,828,419,659,645,474,5
% --
12,25,9
12,254,32,382,34,32,3
-3,35,01,51,187,84
2,36,53
1,534.322,061,82%
, kz = 0, = \ (V. 3. 31), , -
.
' nnRK\ . *
\ 1
. ) ~* 2 %h
. \ / , ,
\
~ith~
= 0,577. . . .
-
S 3. 137
(V. 3.22) (V. 3. 23) ->
\ , I , ,
\
.
2 xfc
(V. 3. 34) 00
_ 2x/i
,
V l n s i n ! ! "
= 1
(V. 3. 37) [24]
V s i n 2
2
sin2 12 2
I cos2rc(p 122
1"
2"
cos 2
, (V. 3. 37) (V. 3. 31), (V. 3. 29), = 2In . | ( - i) = / _
^L , 2 \ j \
'
=
+ f i2(
In ^ 2 -
(V. 3. 39)
.
-
- -
, h h' = % h. (V. 3. 34), -
(V. 3. 22) (V. 3. 23) Ro > 0,5 h, - R
o .
(V. 3. 34) . . [25], , - . 2 h = 10 ,
= 0,1 , xh/rP == 100.
-> (V. 3. 39) = ( 4 l) In -^ 2- . (V. 3. 40)
, . 2, - , -
-
138. V. ,
2
0 ,
10510,50,2
100501052
0,125
19,3619,1114,0710,4034,500
h =
0,25
9,3619,1656,4824,6032,642
b/h
0,5
3,3183,2282,1701,5500.6682
0,75
1,0401,0190,72030,51160,2936
. = 588,11 ~ 590 > 10 ,h = 0,0032.
, . /0 100, (V. 3. 34), . 2, - .
-
h 0, = 100.
, ,
li, - 0 -,
(V. 3. 39), 1
. -
,
. 2, /, | (% rjj) |, (V. 3. 40)
fi-lW . (V.3. 41)h j r
c
:
r0 x
= i - 1 Inu ) 0,01 (V.3.42) , =f= 100
'
. 2 (V.3.42).
. 2 ,nb -
= j h .
-
4. 139
(V. 3. 34) :
2r (h) = ''"" ^ ' v 5 * . . _ 1 _ s i n 2 [ ( ) ] ( (^ J I xAn=i \
(V.3.43)
(h), (I h) b b' = h b. (1 h) h -
. 2, (1) = 0. -
. -
, , -
, , -
.
[26, 27, 28, 29 .]. , - ,
[8, 9]. , - -
, . , -
. , -
, , -
. -
,
.
-
[30, 31], -, [32].
, , ,
-
10%.
4. -
.
-
140 . V.
,
.
-
, . .
(. V. 10).
(V. 1.4) , - .
.
,
. V. 11. , - q
-
. V. 10. - . V. 11. . .
,
; , -
.
. -
.
, -
: /0 = 2 - 3. -
,
(. V. 12). '
.
.
. -
(V. 1. 4).
-
4. 141
,
.
,
-
-, . .
.
, 5, 6, , -
.
-
\ \2rc
: // // / /Jj s/s////// ///////. /////// //////////////J J
UL :_-^' !
//////////////////////////////////////////''//7
, -
.
-
,
-
,
,
.
-
[. I. 8, 16; . IV. 2, 33]. ,
b h. b^=h (V. 4. 1)
,
, hlb. , , h,
. V. 12.
/it =
~~'
(V. 4. 2)
-\- . . - -
.
(V. 1. 4) , " ,
. . [34]:C'~=f, (V.4.3,
D ; 1 ; h = blh; -.
-
142 . V.
. . =s 0,4 .
-
, 23 - ^ 0,15. -
.
[34, 35, 36, 37].
5. ,
-
.
- z' =
= z = -1- z, . . kz , ,
. , .
'
Q = , Q -- ; Q' . -,
Q=,2n }*=**. Cr^dz> = ^ . (V.5.1)
J J v 'J J0
(V. 3. 3).
In il-(V.5.2)
, -
;
,
-
= Ro . (V. 5. 2)
, .
. z' == xz . - -
, -
, , -
.
-
5. 143
-
. -
.
,
[12, 38]. , ai, ,
as,
(V.5.4)
(V. 5. 4) - . -
,
,
, = 0. ,
,
Q :
Q
4 J
--
(+!)(V. 5. 5)
. V. 13. -. -
a=Yaia2 - - ,
= 14 = _L, (v.5.6)t ia (V.5.5) (V. 5.6) -
:
f- =
(
) , = ^ ,Qo 2
(V.5.7)
(V. 5. 7) . () (V. 5. 7) . V. 13,
1
-
144 . V.
, (V. 5. 4)
0 = 4 ( 0 - ) ] / ^ (V. 5. 9)
-
h, , d, - .
RR > h . -, RK ^> ^>Ro -, R *= h.
Q' - ' = h. r
c 1.
(V. 5. 10) =
^ ^
N 2dYx 2 h' NdVv. (V. 5. 10) (V. 5. 11)
=R
o
= -%-f In ^ + % 1
(-1) - In^s- l . (V. 5. 12)2 h I r
c Nd r
c \
lni?0/rc
(V. 5.12)
{) (V.5.13), ,
. . -
, ( 2,53 ) . -
-
6. 14.4
, , , -
(V. 5. 13) 3 h ( 1 (V.5.14)
6.
.
, -
, , -
.
. V. 14.
.
. V. 15. -
,
.
,
, , . .
Q , (. V. 14):= C0Ap, (V. 6.1)
.
-
2nkh
'"
(V.6.2)
,
(. V. 15).
-
146 . V. ,
-
2 = % \. ,
~-
2 G.
2
(?, - ,
.
, (?
= (&2), ,
:
-
6. 147
2 =
-
148 . V.
.
.
(. . V. 6). -
300500 .
. V. 10 , - ,
.
, , -
. .
( 2, . V), > /, , , -
(. . V. 12). , :
R ( - ) , ,
-
.
.
MN, - - .
.
,
.
s . (V. 6. 5)
dp = -^ - w ds + Q w2 ds.
.
,
i i
\
(V. 6. 9)
I . (V. 6. 9),
.
' -
.
.
-
6. 149"
, . .
i
(V.6.10)
,
,
,
.
,
,
.
^ ^ 8 =
1 7 ' (V-6-12)
-
(V. 6.5), =
-
:i
- = - ^ In - ^ + Q fw* ds. (V. 6. 13)2 J
, Q = Q (} (V. 6. 13).
, ,
, .
(V. 6. 9), .
, -
,
, -
(V. 6. 9) ., , ,
.
: 1) ; 2) ; 3) .
-
150 . V.
1.
-
-.
-
/
= 2 - i- , - -
, . . -
:
< < 0 , (V.6.14)
wc .
(V. 6. 13)
0,
0 .
,
1 1
,- 1 1 -
-
6. 151
2, , /2, /3,
(V. 6. 16) :
.
2.
.
D. -
- .
,
.
-
:
.
< <
+ nD, = 2-=-4.
, . . -
, 6 ^ 0 , 4 / ) [43]. (V. 6. 13) :
i rn + nD 2
1 ) I J -
.
j ^ - (V'6-20> / .
(V. 6.13), = AQ + BQ2.
-
152 . V.
(V. 6. 8)
= 1
-^' (V-6.21)
=
4 0 / ^ \2 D
\ V ,
, 0,15 < < 0,40. ess0,4 .
3.
-
w = -^ , (V.6.22)
, t dz^et, 0,25 < < 0,50.
, ,
1 1- 2 ;
120 , 1
nkh
1 rfa
-
6. 153
.
. . .
[. I. 20; 44]; (V. 6. 6) (V.6. 25)
-?, Q, (V. 6. 25) (. V. 16).
, - /Q ^~,
Q . .
.
-
-
(V. 6. 26)
. V. 16.
"
-
154 . V.
, ,
. IV. 12 , - / / + Q" ( 0 , Q" ((?) = BQ . , , , -
, -
, -
. -
.
1. . ., . . - . . , 1948.
2. w 11 . . A table of the generalized Riemann zeta function in a par-ticular case. Quarterly Journal of Mechanics and Applied Mathematics (Oxford),vol. 5, No. 1, 1952, pp. 116123.
3. . ., . . - . . . ,
. . . ., 5, 1961.4. . . ,- - -
. . . 1962 1963.
5. . . . , 1956.
6. . . , 1955.7. -
. , 1954.8. i n e R. L. Well Productivity increase from Drain Holes as measu-
red by Model Studies. Petroleum Transactions AIME, vol. 204, 1955.9. L a n d r u m R . L., C r a w f o r d P. B. Effect of Drain Holle Dril-
ling on Production Capasity. J. Petroleum Technology, Febr., 1955.10. . . ,
. . , . 11, 1951.11. . . .
, 1950.12. . . . . ,
1954.13. . . . , 1943.14. . .
. . , 1948.15. . . -
. . , . ., 10, 1946.16. . . . .
, . ., 16, 1952.17. . .
. , 1961.18. . . -
,
-
. . , . 62, 1959.19. . . .
. , 1957.20. . ., . . -
. , 1953.
-
6. 155
21. . . - . . .-. -
. , 1955.22. . . -
. . . , . XVII, 1953.23. . . -
. . , , 2,1953.
24. . ., . . ,, . , 1951.
25. . . . . -
, 1955.26. . .
. . , , 3, 1957.27. . . -
. . , , 7, 1958.28. . , . ., . ., -
. . -
. . , . .-. .
, 5, 1954.29. . ., . . -
. ,
, 1960.30. . .
. . , , 17, 1950.31. . .
. . . . , . 12,1953.
32. . ., . . . . ., 9, 1955.
33. . . . -, , 1956.
34. . . - . .
, , 6, 1950.35. . . .
. .
. , 1953.36. . .
. .
. . , 1954.37. . . -
. -
- .
. . , , 1953.38. . . , . 1. , 1951.39. . ., . . -
. . ., 23, 1945.40. . -
. . , 1961.41. . . , . .
. . ,
, , 1, 1962.42. . , . , . .
. . .
, . . . ., 3, 1961.
-
156 . V.
43. . . . . , , 2, 1955.
44. . . , . ., . ., - . ., . ., . ., . .,
. X . . ,1955.
45. . . - , . -
, , 8, 1957.
-
VI
1. .
, -
-
, .
, . . -
.
. -
, ,
.
,
, -
, .
, ,
.
, , , -
. -
-
-
. , ,
,
.
.
-
,
-
158 . VI.
, -
.
-
. ,
.
-
(-), . VI.1.
-
,
-
.
-
.
Hi ,
. VI. 1. -
.
.
,
, ( ABC). -
.
-
-
. -
-
-
.
-
.
. VI. 2.
-
.
,
.
''', RK , .
. VI. 2.
- .
, -
-
1. 159
,
,
.
-
-
, .
, -
-
,
-
.
-
,
-
, -
.
-
-
.
. VI. 3. ,
, . -
,
.
. . [. . 2] ,
.
.
-
.
.
h, - i, i = sin (. VI. 3). - :
1) -;
2) , . . = z -{- = (, ). , ,
, , .
[1], , 2
-
160 . VI.
-
, -
.
: -
,
. . . -
[2].,
-
,
.
, -
, -
. ,
, -
.
( ,) - , - , , ,
.
, ,
-
.
.
dh / \>= , ( V I . 1.1)
; ds - .
# = z + ^ - . (VI. 1.2)
, , , . . -
. ,
, h:
H = h, ( V I . 1 .3) ( V I . 1 . 1 ) .
-
2. 161
(VI. 1. 2) , , = ,
,
z. z .
=- ( V I . 1 . 4 ) , -
. -
, . . q - h
q = uh-l = ch-^. (VI. 1.5)
2.
,
t. (VI. 1. 5) .
,
qdx = chdh,
qx = ~ + const. (VT. 2.1) (const) (VI. 2.1)
.
-
(. . VI. 1):
i = 0 (VI. 2.1), ~
const = ,
qx = (
' ~
. (VI. 2.2)
(VI. 2.2) . .
-
162 . VI.
- , -
/, 2 . - (VI. 2. 2)
ql = - |- {\ \) (VI. 2.3)
9="^7"(.-"i ")- (VI.^.4)
(VI. 2. 2)
(\ ^ -. (VI. 2. 5) q
(VI. 2. 4), -
' ~
* . (VI. 2. 6) ,
,
. VI. 1 ( ). (VI. 2. 5) (VI. 2. 6), . .
", , ,
.
. (VI. 2. 6) % = ( = /) h = 0 , , q/h, . , &*=;> Hz,. . .
(VI. 2. 4), , ,
.
3.
-
(. . VI. 2). RK . -
.
, h.
, (VI. 1. 4) - :
dhw c
-
S 4. 1 163
.
,
-:
Q = \wr\2arh = c~2nrh. (VI. 3.1) ,
h h () ,, , .
Q, h(r) . (VI. 3. 1) :
Qy- = 2nchdh.,
Q In = ch2 + const. (VI. 3. 2) (const)
:
Q In i?K = -\- const,
Q\n^ = nc(Hl h\ (VI. 3.3) ,
( '').
(1 Hi)Q > . (VI. 3.4)In -
(VI. 2. 4) (VI. 3. 4) . ( 7, . VI), -
.
. . .
4.
, (VI. 2. 4) (VI. 3. 4) - [. I. 111.
(VI. 3. 3) (VI. 2. 5)
. -
.
-
1U . VI.
(VI. 2. 4) (VI. 3. 4) .
1951 . [3]. .
-
h, . z -
dz (. VI. 4). - -
dz
. VI. 4. - () -
().
(VI. 4.1)
-
-
. -
h:h
g=fudz. (VI. 4.2)
-
= z + ~ .
,
=
(VI. 4.3)
(VI.4.4,
.
; .
[. I I I . 7] :() (
J f{z, a)dz= J1 () 1 ()
-
# 4. 16
() 2 ()
^ 1 / (J(J /1 ( ) >1 ( )+ /( ) - ^ - . (VI. 4. 5)
(VI. 4.4) (VI. 4.5). , h = h(x) (VI. 4.4) ; -, (VI. 4. 5) = ,
-
166 . VI.
,
(VI. 4. 8) . , -
.
, q . (VI. 4. 9),
g dx = dP,
qx = + const. (VI. 4. 10)
const =
= P-L = 0.
Y
q = C ~ ) . (VI. 4. 11) 2 = I,
q = ( 7 ) . (VI. 4.12) ,
:
(VI. 4.13)# 2 - 1 =
- z 2 - - 2
1 2 (VI. 4. 12)= _ {\-\)
= ^_
( 2 _ 2 )
(VI. 2. 4), , , .
(VI. 3. 4), - ,
.
, - , -
:h
+ z]dz, (VI. 4.14)
-
4. 167
,
: rc
-
168 . VI.
hB , . . [. II. 2]:
df- = 1 > 3 5 i ( V I - 4> 1 7 > , = h (. VI. 4),
, -
(VI. 2. 2) (VI. 3. 3).
[. I. ; . .