Post on 22-Feb-2018
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, 2005.
7/24/2019 Mehanika Fluida KV
2/199
. ,
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:
, .
, .
.
:
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ISBN
: 250
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1. 1-
1
1.1
1
1.1.1 1
1.1.2 3
1.1.3 3
1.2
4
1.2.1 4
1.2.2 . 4
1.3. 6
1.3.1 6
1.3.2 . 9
1.3.3 . -
9
1.3.4 12
1.3.5
15
1.3.5.1
18
1.3.5.2
20
1.3.5.3 .
21
1.3.6 23
1.3.6.1 24
1.3.6.2 25
2. 2 28
2.1 . 28
2.2. - 33
2.3 42
2.4 . 68
2.5 92
2.5.1 92
2.5.2 104
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3. 3
114
3.1 114
3.2 1203.3 - 144
3.4 158
3.4.1 158
3.4.2 166
3.5 182
4. 191
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1
.
.
, . .
-
, .
1.1
1.1.1
, (, )
,
. , ,
,
.
.
:
1)
,
,
.
EvdivSpEP )3
1(2
(1.1)
: P , - ,
S - , . vdiv
- .
7/24/2019 Mehanika Fluida KV
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1 2
ijpP ijsS
.
(1.1)
,
3
)(
3
kk
k
k
zzyyxxii
sx
vvdiv
ppppp
(1.2)
ijs
1,2,3.ji,,)(2
1
i
j
j
ijiij
x
v
x
vss
(1.1)
),3
1(2 vdivspp ijijijij
(1.3)
ij -
ji
jiij
0
1 .
(1.3)
. ,
,
).(
i3
22
x
v
y
vpp
vdivx
vpp
yxyxxy
xxx
(1.4)
2) ,
,
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3
A
AdqQ ),(
q
dAnAd
- .
1.1.2
. :
,
.
.
,
,Re
vDvD (1.5)
.
.
1.1.3
,
.
:
, ,
, .
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1 4
, ,
.
,1
d
dpdp
dS
(1.6)
1.2
.
1.2.1
:
1)
2)
.
,
.
.
.
.
,ppppzyx
: .
1.2.2 .
V
.
: .
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5
,
V A ApddVF .0
, .
A V
gradpdVApd .
V
dVgradpF .0)(
V ,
,
,
,01
gradpF
.1
gradpF
V
dVF
A
Apd
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1 6
.
1.3
, , .
:
,
.
, .
1.3.1
m m mV V A
kjkijiji dAdVdVfDtD ,......... (1.7)
: hij hkij, h=f, ,, , .
, .
(1.7). Vm,
, Dt
D .
(1.7)
,
(1.7). f, . (1.7)
ij , , Vm, kij ,
, Am,.
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7
(1.7)
.
k
jik
jiji
xfv
tf
DtDf
......... (1.8)
fij, vk(xi,t)
, (1.9)
m m mV V A
kkji
ji
ji dAvfdVt
fdVf
Dt
D,...
...
... (1.9)
,
(1.7). -
(1.9)
m mV V k
kjiji
ji dVx
vf
t
fdVf
Dt
D)(
......
... (1.10)
(1.8)
m mV V k
kji
ji
ji dVx
vf
Dt
DfdVf
Dt
D.)( ...
...
... (1.11)
V , , t
Vm . ,
A=Am.
,
V V
ji
ji dVt
fdVf
dt
d.
...
... (1.12)
(1.9) (1.12)
V AA
kkjiji
V
ji
mm
dAvfdVf
dt
ddVf
Dt
D
,
......... , (1.13)
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1 8
:
V,
. .
,
m s ssmV V tAA
kkskjiji
s
ji dAvvfdVfdt
ddVf
Dt
D
)( )( )(
)(......
)(
... .)( (1.14)
(1.7)
(1.10)
mV k
jki
ji
k
kjijidV
xx
vf
t
f.0)(
...
...
...... (1.15)
,
. (1.16)
....
...
......
k
jki
ji
k
kjiji
xx
vf
t
f
(1.16)
(1.7) ,
, , . (1.16)
(1.12)
.)(...
...
...
... dVxx
vfdVf
dt
d
V V k
jki
j
k
kji
ji
(1.17)
,
, . (1.17)
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9
V A V A
kjkijikkjiji dAdVdAvfdVfdt
d............. (1.18)
(1.18) .
,
(1.17).
1.3.2 .
,
, .
mV
dVDt
D.0 (1.19)
(1.19) (1.7) :
,0
,
......
...
jkiji
jif
(1.16)
,0Dt
D0
i
i
k
k
x
v
x
v
t
(1.20)
(1.8).
(1.19),
(1.20) (1.18)
V
V A
iidAvdVdt
d, (1.21)
.
1.3.3 . -
, , :
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1 10
. ,
(1.7),
m m mV V A
jjiii dApdVFdVvDtD , (1.22)
Fi F
pji
,
(1.22) .
(1.22) (1.7)
,...
...
...
,
,
kijki
iji
iji
p
F
vf
(1.22), (1.16)
.j
ji
i
j
jii
x
pF
x
vv
t
v
(1.20) (1.8)
:
.)(j
ij
i
j
ij
i
x
pF
x
vv
t
v
(1.23)
.
:
,)(t
v
k
k
x
pFvvDivPF
Dt
vD
(1.24)
p
,
),,(zyx
nnnn
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11
.p
,
ji jjij
zzyyxxjjiijjn
dApdAn
npnpnpnpepnnPp
(1.25)
(1.18) (1.22)
V A V A
jijijjii dAnpdVFdAnvvdVvdt
d. (1.26)
,
.3
1)(
2
12 ijij
k
kij
j
i
i
j
ijij px
v
x
v
x
vpp
(1.27)
(1.27) (1.23),
, ,
,
),(3
1)(
2
2
k
k
ik
i
ii
k
i
k
i
x
v
xx
v
x
pF
x
vv
t
v
(1.28)
.vdiv3
1
gradvpFDt
vD (1.29)
- ,
(1.26),
A A
k
i
k
k
ii
V A V
ikkii dAx
v
x
vpdAdVFdAvvdVv
dt
d.)(
1
(1.30)
(1.30), ,
,
, . .
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1 12
1.3.4
,
, . , (
2
2v)
,
m m m m mV V A V Aiijijiii dAqQdVdAvpdVvFdV
v
eDt
D
,)2(
2
(1.31)
Q , qi
i-.
, (1.31)
(, )
( ) .
(1.31)
f, (1.7).
,
,)(
),2
(
ki...j
i...j
2
...
kiki
ii
ji
qvp
QvF
vef
(1.16) (1.20)
.)()2
(2
i
i
j
iji
iix
q
x
vpQvF
ve
Dt
D
(1.32)
(2.23) vi
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13
.)2(Dt
D
)
2(
2
2
DivPvvFv
x
pvvF
Dt
vD
j
ji
iii
(1.33)
(1.33)
(1.32).
j
iji
x
vpvP
,
)(z
p
y
p
x
pv
x
pvDivPv z
yx
j
ji
i
.
, . , (1.33)
.
,)(2
1
,
jiji
j
i
i
j
ji
i
j
ji
j
iji
j
iji
j
jii
j
ji
i
spx
v
x
vp
x
vp
x
vp
x
vp
x
pv
x
pv
(1.34)
. (1.34)
m m mm V A V
jijijijiii
V
dVspdAvpdVvFdVv
Dt
D.
2
2
(1.35)
(1.35)
. , Fi
U(x,t),
,F,i t
U
Dt
DU
x
Uvv
x
UF
iiiii
(1.36)
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1 14
(1.33)
j
ji
ix
pv
t
UU
v
Dt
D
)2
(2
(1.37)
(1.34) (1.35)
, .
V (1.13), (1.35) (1.37),
A V
jijijiji
V A V
ii
dVspdAvp
dVt
UdAvU
vdVU
v
dt
d
.
)2()2(
22
(1.38)
(1.27)
.
n ,
, (1.27) :
.,
,pT-pEP n
iizzyyxxnn
nijijij
nnnnnT
nppp
(1.39)
,
- (1.30),
(1.35).
(1.27)
)0x
v(
k
k
(1.35)
(1.13)
V A V A A V
j
i
j
j
iiiiiiii
22
,dVdA)x
v
x
v(vdAv
pdVvFdAv
2
vdV
2
v
dt
d
(1.40)
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15
ijijs
(1.27)
.)(2
1
22
1 222
i
j
j
i
ijijjiji x
v
x
v
ss
(1.41)
Fi(1.36) (1.38) (1.40)
.
1.3.5
,,
.
, ,
,
, ,
.
,
,
.
,
.
,
v(x,t), p(x,t) (x,t).
. ,
,
R
s rdrtrxvRA
Vtxv0
2,),,(21),(
(1.42)
: R , r . A
V
v(x,t)
vs(x,t).
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1 16
R
s rdrtrxpR
txp0
2,),,(2
1),(
(1.43)
R
s rdrtrxR
tx0
2.),,(2
1),(
(1.44)
,
.
, .
.
,'fff (1.45)
: f -
f - f.
, (1.45),
).,('),(),( txvtxvtxv (1.46)
,
T
dttxvT
txv
0
.),(1
),( (1.47)
T
dttxpT
txp0
,),(1
),(
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17
T
dttxT
tx0
.),(1
),(
,
(1.44) (1.47)
, , ,
.
A
2 .dAv
.
, , :
As
dAvAv
.1 22
(1.48)
A A
T dAvdAvAv
).(1 2'22
(1.49)
A
3dAv
.
,
vs3A,
, , :
As
dAvAv
.1 33
(1.50)
+2. ,
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1 18
,
.
,
,)32)(3(4
)12()1(4
33
ttttt
Tnnn
nn (1.51)
nt .
, ,
,
A
sAppdAP . (1.52)
(1.52)
.
,
,
,
, .
.
1.3.5.1.
,
.
,
.
x (1).
1.
1
x1=x
2
x
V
v(x,t)
ydx
x1=x+dx
n
1n
2n
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19
dx.
(1.21)
2
1
.)( 222111
x
x
AvAvdxAt
(1.53)
(1.53) dx.
dxxx 12 (1.54)
,)(,,
dxvAx
vAvAtxtdxx
(1.55)
, (1.54) (1.55),
:
.0)()(
x
vA
t
A (1.56)
(1.8), (1.56)
:
,011
Dt
DA
Ax
v
Dt
D
(1.57)
, (1.8)
Df/Dt= ,x/fvt/f f,A (1.58)
, .
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1 20
1.3.5.2
(1.26) (1.27)
.
(1.26)
V A V
ijijijjii dVFdAnpvvdVvdt
d )( (1.59)
V A V
dVFdAnTnpnvvdVvt
)( (1.60)
(1.60) (1.39).
, 1,
(1.59), (1.60) . .
dx x(i=1, v1=vx=v) (1.59) (1.60):
A
wnxx
V
x
Atdxxtx
x
V
x
dxdxx
ApdxpA
xdApn
Adxdx
dy
gdVF
dxAvx
AvAvAvAvdAnvv
vAdxdVv
,)()(
;
;)(
;
2
,
2
,
2
2
2
221
2
11
(1.61)
: y ,
O=D - , v = v(x,t) w .
(1.61) (1.60), (1.59),
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21
.)()( 2 Ddx
dygA
dx
dApAvp
xvA
tw
(1.62)
(1.56)
vvv
fw
82
2
(1.63)
(1.62)
02
1
vvDdx
dy
gx
p
x
v
vt
v
(1.64)
= (Re, /D) : Re=vD/- - . (1.64)
( 0t/v ) (=const.) (dy/dx=0) .
1.3.5.3 .
(1.38) (1.40) (1.36) (1.39)
:
V A A A V V j
iij dV
x
vdVFdAvnTdAnvpdAnv
vdV
v
t.)(
22
22
(1.65)
, 1,
.
1,(1.65) :
.2
()2
(22
dx
dygvA
vpvA
x
vA
t
(1.66)
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1 22
(1.56) gy
(1.66)
, ,
.0)2
()2
(22
gypv
vAx
gyv
At
(1.67)
(1.67) ,
(1.56)
.2
2
xCgypv
, , (1.65).
(1.64). , (1.64) dx 1-1 2-2
:
2
1
2
1
.222
22
2
221
1
2
11
x
x
x
x
dxt
vdxvv
Dgy
pvgy
pv
(1.68)
(1.53).
2
12
x
xg dxvvDY
(1.69)
, . .
(1.68)
2
1
f(x). vza)(Y12in
x
xin dt
dvL
dt
dvxxdx
t
vY (1.70)
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23
1.3.6
(1.68) (1.69).
,
.
. ,
, , , , .
.
, ,
., , :
Y Y.
Yg= Y+ Y. (1.71)
.
2
2v
Y (1.72)
- v .
,24
2v
R
lY
h
(1.73)
: - , l , Rh
Rh= A/O.
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1 24
1.3.6.1
.
.
,
.
.
. , ,
,
, ,
, .
. :
1.
2.
.
. ,
.
, .
.
. .
. .
.
-
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25
,)1( 2
1
2 A
A (1.74)
: 1 2
. ,
.
, , ,
. ,
, , ,
.
1.3.6.2
(1.73)
.
(, , ),
.
),(Re,D
(1.75)
: - D .
.
.
,
(1.73)
.
Re
64 (1.76)
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1 26
,
. .,
,
.
.
. 4000
.
.8,0)log(Re21
(1.77)
4000 < Re < 40 000
.Re
316,04
(1.78)
40000
).7,3log(21
D (1.79)
, (1.79) ,
.
,
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27
.)Re
68(11,0 4
1
D
(1.80)
(1.80)
.
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2
, :
,
.
, ,
.
2.1 .
,
:
1.
2. .
.
:
pm ()
pv ()
P.
, ,
. ,
.
,
(),
:
p = pa+ pm
:
p = pa pv
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29
pA,
Psti{ljivog fluida= pA.
(
),
,
.
Pnesti{ljivog fluida=pcA,
pc
.
.
,
pa
. 1, . 2. :
,
. pm,
.
pa+ pm.
pv,
.
pa pv.
2.
pm
Pv
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2 30
,
,
.
().
1. h=?
p1p2. : hA, hB, , m. (3)
3.
,
. m .
m .
(I-I). , ,
. :
pA+ ghA+ mgh = pB+ ghB. (1.1)
.)(
g
hhgpph
m
ABAB
(1.2)
pA
pB
hAhB
I I
m
h=?
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31
.
.
4.
4. II II.
II-II :
pA+ ghA= pB+ ghB - mgh. (1.3)
(1.1) (1.3) mgh (1.3). '''' '' '' .
(1.3)
,)(
g
hhgpph
m
ABAB
(1.2) .
.
pA
pB
hAhB
II II
m
h=?
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2 32
2. h2=?
12. : 1, 2, , H1, H2, H3, h, h1.(5.)
5.
, .
I-I 5.
:
pa+pm+1gH1+2gH2+mgh2= pa+ pm+1g(H1+H3)-mgh1+1gh.(2.1)
(2.1)
'''' (1.3).
(2.1)
.)( 122
31
2 hHhHhmm
I I
pa+ pm
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33
3.
1 2
h1h2. : pv, h1, h2, H1, H2, m. (
6)
6.I-I :
pa-pv+g(H1+h1) = pa+gH2+mgh2-g(H2+h2-H1)+ mgh1.
g(H1+h1-h2+H1) =pv+mg(h1+h2),
.)(2
121
211
hh
g
p
hhHm
v
,
, .
2.2
().
Pv
H1
H2
h1h2
I I
m m
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2 34
( ) .
,
. , ,
: p, .
F(p,,T) = 0
.
pv = RT,
: v R-
R= 287 J/kgK.
p1V1= p2V2,
: p1 p2
V1V2 .
:
1.
( )
2.
.
.
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35
4. F=?
h,
. : pa, a, b, h, d, D. (7)
7.
F, h.
. , p1
p2(7).
p0V0= p1V1:
.4
))()((4
)(22
1
22 dDhappdDap aa (4.1)
.
(4.1)
,1ha
appp aa
).1(1
ha
app a (4.2)
p1< 0,
.
, , ,
p2>0.
F
h
d D
a b
F
h
d D
a b
) )
pa
pa
p1
p2
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2 36
p0V0= p2V2:
.
4
))((
4
2
2
2 DhbppbDp aa (4.3)
(4.3)
,2hb
bppp aa
).1(2
hbbpp a (4.4)
. 8. .
8.
, , 8. P1
P2.
, :
,4
))(1(4
)(2222
11
dD
ha
apdDpP a
(4.5)
.4
)1(4
22
22
D
hb
bpDpP a
(4.6)
F + P1= P2. (4.7)
F
P1P2
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37
(4.5) (4.6) (4.7)
h:
4))(1(
4)1( 222
dD
ha
apD
hb
bpF aa
.
5. -
. F, .
,
. : F,G, D, pa, H. (9.)
9.
9, F
,
x.
, y. x y= ?
, 9) 9)
.
4
,4
2
1
2
0
DpFG
DpG
m
m
F
x
G
H
y
D
pm0
F
GGPm0 Pm1
)
) )
7/24/2019 Mehanika Fluida KV
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2 38
:
.)(4
,4
21
20
D
FGp
D
Gp
m
m
, ,
,1100 VpVp
).(4
)(4
)(2
1
2
0 yxHD
ppHD
pp mama
.
,)())(( 011 Hppppyx mmma
,)(
1
01 Hpp
ppyx
ma
mm
,
:
.)(4
42 FGDp
Fyx
a
6. h=?
, D, H
m. , pa
7/24/2019 Mehanika Fluida KV
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39
. : pa,D,H, m,.
(10)
10.
9).
.
( ) p.
h. ,
10)
,21 hhh
.
z
,
,pzHpa (6.1)
p . (6.1)
.Hp
pz a (6.2)
, 10)
:
,mgPP a
H
D
pa
z
h2
h1h
P
Pa
mg
II
) ))
p
7/24/2019 Mehanika Fluida KV
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2 40
,44
22
mgD
pD
p a
.4
2D
mgpp a (6.3)
(6.3) (6.2),
.42
H
D
mgp
p
za
a
(6.4)
z,
,1 zHh
, (6.4)
).4
1(
2
1
D
mgp
pHh
a
a
(6.5)
h2
I-I.
:
,2 pghpa
.2g
pph a
(6.3)
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41
,
42
2g
pD
mgp
haa
.4
22D
mh (6.6)
(6.5) (6.6)
).4
1(4
2
2
D
mgp
pH
D
mh
a
a
.
7. D
H d pv
. H=?
. : D,
d, , pv.(11)
11.
11.
Pv G.
G
Pv
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2 42
.GPv
,
4)( 22
dDpP vv ,
.4
2
HD
ggVG
d
.
.44
)(2
22
DgHdDpv
.)(
2
22
gd
dDpH v
,
2.3
,
. 2.1
.
, .
. 12.
7/24/2019 Mehanika Fluida KV
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43
12.
12.
,
.
.
P
.
A
Pm
pm c
APv
pv c
Pm= pmA
Pv= pvA
PC
D
pm
zc
zc
AgzP c
Px CD
pm
zcx
zcx
xxcxAgzP ,
.22 zx
zz
PPP
gVP
Vz
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2 44
,
.
.
,
.
.
8. FC
, H.
a b; . (
13)
13.
;gH0yp .aHgaypB
A=ab:
.222
Hagab
abpp
ApP BAC
C() :
;0xC
.H2a3
H3a2ay
3
a
pp
p2py C
BA
BAC
yC.
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45
9. AB
O, .
A. A O b.
. H (
O) .
. (14)
14.
ghpgHpA ; .
:
;ba
a
p
p
A
0
;p
ba
ap A0
;ap2
1P 01 .bpp
2
1P A02
7/24/2019 Mehanika Fluida KV
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2 46
:
.3
b
pp
p2pa
A0
A0C
M0 0 ; F b Pa
P aA C 1 230; F b P
aP aA C 1 23
.
( A) ,
FA=0, :
;aP3
aP C21 ;
3
b
pp
p2pbpp
2
1p
6
a
A0
AOA00
2
AA2
A
2
p2pba
a
6
bp
ba
a
6
a 323A b2ab3ap: . (9.1)
:
sin
sinbHasinbaH
. (9.2)
(9.1) (9.2) :
.b2sin
sinbHb3
sin
sinbsinHb3sinbH3H 323
332223
:
.sinb3H0sinbH3H 23
:
H3bsin.
,
.
7/24/2019 Mehanika Fluida KV
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47
.
10. e
- . L. : pv, H, L, , 1, 2.
(15)
15.
- : - pv,
1 2. Pv, P1, P2, .
Cv, .
, ,
pv.
,vvv ApP
Av .
,
sin
H, ,
L
,
sin
LH
Av
H
H
2H
pv
1
2
A
A
H
H
2H
pv
1
2
A
A
Pv
P1
P2
zc1
zc2
C1D1
C2D2
Cv
1
2
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2 48
.sin
HLpP vv (10.1)
1 P1
,1111 AgzP c (10.2)
: zc1 C1
1 ( 1), 1 1. zc
C1, 16.
16.
,
. I-I.
C1
D1.
zc1, vc1. zc1.
, pa pv,
pa.
I-I :
pv
zc1H/2
1
I IC1D1
pa-pvpa1
P1
zc1vc1
vc1
7/24/2019 Mehanika Fluida KV
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49
1112
cava gzpH
gpp (10.3)
(10.3)
.211
H
g
p
zv
c (10.4)
1
.sin
1 LH
A
(10.5)
(10.4) (10.5) (10.2)
.sin
)2
(1
11 LHH
g
pgP v
(10.6)
P1 zc1 1. ,
11
11
c
cc
zA
Iz ,
Ic1 1.
,12
)sin
(3
1
LH
Ic
:
pa-pv
1H/2!
:
pa
1zc1!
!
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2 50
.sin12
sin12
sin
1
2
2
1
3
3
1
cc
cz
LH
zH
LH
z
P2. , P2
, D2.
,2222 AgzP c (10.7)
: zc2 C2
2 ( 2), 2 2. zc
C2, 17.
17.
zc2. ,
II-II
C2. ,
,
1.II-II
pv
H
1
II II
C1
D1
pa-pvpa1
P1
C2
D22
H
P2
2
zc2
2
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51
22212
cava gzpH
ggHpp (10.8)
(10.7)
.2
2
2
1
2
2
HH
g
pz vc
(10.9)
1zc22
1
.
, 1
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2 52
2(17 2/).
(10.9) (10.7)
,2)2
( 22
1
2
22 HLH
Hg
pgP v
(10.10)
.22 HLA
-.
x z ,
15.
x-
,coscos 21 xv PPPP
z
.sinsin 1 zv PPP
.)sinsin()coscos( 212
21
22 PPPPPPPP vvzxu
. ,
.
.
11. L
. 1 2.
. . : h1, h2, 1, 2, , L.(18)
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53
18.
, :
F, 1 P1, 2 P2.
1
,1111 AgzP c (11.1)
.sin
11 L
hA
(11.2)
10, zc1
1.
,2
111
1 caa gzph
gp
.21
1
hzc (11.3)
(11.2) (11.3) (11.1)
O
K
1
2
h1
h2
A
O
K
1
2
h1
h2
A
2
1
F
zc1
zc2
P1
P2
vc1
vc1
vc2
vc1
C1
C2
D2
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2 54
.sin2
1 2111 L
hgP
(11.4)
2
,2222 AgzP c (11.5)
.sin
22 L
hA
(11.6)
zc2
2.
,2
222
211 caa gzph
gghp
.22
1
2
12 hhzc
(11.7)
(11.6) (11.7) (11.5)
.sin
)2
( 222112
Lhh
gghP (11.8)
. ( ),
,
vc1, vc2, vc1, vc2.
,11
11
c
cc
vA
Iv
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55
.sin1212
)sin
(
3
3
1
31
1
LhL
h
Ic
vc1,
.sin2sin
111
hzv cc
.sin6
11
hvc
,22
22
c
cc
vA
Iv
.sin1212
)sin
(
3
3
2
32
2
LhLh
Ic
vc1,
.sin2sin
222
hzv cc
.sin6
22
hvc
,
.0O
M
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2 56
18
,0)sin2
()sin2sin
(sin
)(2
221
121
21
cc vh
Pvhh
Phh
F
.)sin2
()sin2sin
(sin
22
2112
1
21
cc vh
Pvhh
Phh
F
. , .
12. -
Pu.
: R, 1, 2, Pu, pm. (19)
19.
, ,
x
z . - x-
: 1-2
, 2 3
1 3 4 2. ,
R
2R
R
pm
1
2
A
A
R
2R
R
pm
1
2
A
A
Pmx
P1xzc1x
z
x
A1
A2
A3
A4
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57
,
.
pm. , , x ,
.mxm PP
,mmm ApP
RLAm ,
.RLpP mm (12.1)
, 1
.1111 AgzP c (12.2)
RLA 21 .zc1
1
,111 caam gzpgRpp
.1
1 Rg
pz mc
(12.3)
(12.2) (12.3) (12.1)
.2)(1
111 RLRg
pgPP mx
(12.4)
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2 58
dPx dPx
dPz dPz
dP dP
R
dPxdPx
dPz dP dPR
, x
, R.
2. 20.
20.
dA, dP.
. , ,
,
dPx,
. 2 z , ,
20 .
21.
. ,
.
:
1. z ,
x .2.
,
()
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59
,
().
, x
.x
,1PPP mx
, (12.1) (12.4) :
.232
1 LgRRLpP mx (12.5)
z . 1-22-3,
z- .
.
,2 zz gVP (12.6)
Vz
.
2 .
22.
22.
H
RL
2
I I
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2 60
Vz,
,kvadrarapolucilindz VVV
.2
1 2 LRHRLVz
H.
I-I,
.2 21 gHpRgpp ama
.22
1
2
Rg
pH m
H Vz
).
2
12(
2
1
2
RR
g
pRLV mz (12.7)
(12.7) (12.6)
.2
12
2
2
2
1 LgRLgRRLpP mz (12.8)
,
1.
2.
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61
,22 zx PPP
, (12.5) (12.8)
,)2
12()23(
22
2
2
1
22
1 gRgRRpgRRpLP mmu
.
)2
12()23( 222
2
1
22
1 gRgRRpgRRp
PL
mm
u
13. -
.
-. L.
: 1, 2, R, L. (23).
23.
. x-
z- .
,
z ,
x .
R L,
.RLAx
R
RA
A1
2
R
RA
A1
2
2
1
z
x
zc
PxPz
R
7/24/2019 Mehanika Fluida KV
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2 62
,2 xcx AgzP
zc.
,
. :
.2
221 caa gzpR
ggRp
.
,22
1 RRzc
.)
2
(2
12 RL
RRgPx
z
,2 zz gVP
, , Vz
,
2., :
24.
y
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63
, ,
.
24,
. ,
. 25.
25.
kvadracilindraz VVV 4/3 ,
,4
3 2 RLyLRVz
y
, .
,221 gypgRgRp aa
7/24/2019 Mehanika Fluida KV
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2 64
.2
1 RRy
),(4
3
2
12 RRRLLRVz
).14
3(
2
12
2
LgRPz
.22 zx PPP
14.
, L.
h. : h, R, L, , . (26)
26.
.
,
.
,xcx AgzP
x
Px
C
D
zc
Vz
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65
.hLAx zc
,
,2
caa gzph
gp
.2
hzc
,
.2
1 2LghPx
.
.zz gVP
, Vz.
:
.2
1kvadracilindrasec manjegkaikvadraz VVVV
(26)
,Rcos xcos R
x
.cos
2
1
360
2
hLRLR
RhLVz
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2 66
).cos
2
1
360
(
hR
hgRLPz
,22 zx PPP
.)cos21
360(
41 24 hRhhgLP
15.
, L, R, h.
: R, h, , L.(27)
27.
,
.22 zx PPP
,xcx AgzP
.hLAx
Px
zc
R-h
x
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2 68
.2)( 222 hRhhRRx
:
),arccos(360360
22
cilindrasecR
hRL
RL
RV kai
.2)(2
1)(
2
1 22/1 hRhhRLhRxLV kvadra
,2)(21)arccos(
360
22
hRhhRLR
hRLRVz
.2)(2
1)arccos(
360
22
hRhhRL
R
hRLRgPz
2.4 .
'' ''. , ,
, , .
.
, 28?
28.
pm
R
Vz
Pn
G
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2 70
30.
.
30.
, .
G (
G ,
0G ) 0P
:
00 PG ,
00 PG
00 PG .
,00 gVP
V0 .
,00 gVG
V0 .
.
G
P0
V0
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2 72
G01
P02.
, ,
.
().
Pn1, ,
,1111 AgzP cn
1()
.21 RA
zc1
,
,111 caam
gzpghpp
.21
1
h
g
pz mc
.)2
(2
1
11
Rh
g
pgP mn
Pn2, ,
,2222 AgzP cn
2
.22 RA
7/24/2019 Mehanika Fluida KV
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73
zc2
,
,222 caam gzpgHpp
.2
2
Hzc
.2
2
22 R
H
gPn
G0je
0110 gVG .
1
,21 301 RV
.2
1 310 gRG
0je
0220 gVP .
2
,2
1 302 RV
7/24/2019 Mehanika Fluida KV
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2 74
.2
1 320 gRP
(
.
32.
l?
,
. ,
,
.2
1 30 RV
, ,
,
,31 2|
0 lRV
. .
, ).
-
. -
, .
P0
G0
R
l
7/24/2019 Mehanika Fluida KV
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75
,
2.
33.
33. x .
:
.0coscos 0201 PPGP nn
.0cos2
1cos
2
1
2
1
2
1 22
2
1
2
2
2
1
2 gRRgRRgHRghRRpm
2R2
,0)(cos2 2121 gRgHghpm
.cos)(2
2
211
g
gRghpH m
17. d 2H, m
.
D D0.
,
b.
.
P0
G0
RPn1
Pn2
x
y
7/24/2019 Mehanika Fluida KV
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77
,2
43
1 2
0 Hd
V
.6
1 20 HgdP (17.1)
,mmm gVG
,6
1 20 HdVVm
.
.6
1 2HgdG mm (17.2)
,111 AgzP cn
,164
)2(2
2
1
d
d
A
.2
d x
2
:2:
H
dHx
HxHd
7/24/2019 Mehanika Fluida KV
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2 78
11 caam gzpghpp
.11 hg
pz mc
.16
)(2
11
dh
g
pgP mn (17.3)
Pn2.
,222 AgzP cn
.164
)2
( 22
2
dd
A
22 caa gzpghp
.22 hzc
.16
2
22
dghPn (17.4)
,
pm,
7/24/2019 Mehanika Fluida KV
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79
.4
)( 202
DDpP mm (17.5)
.
.cbFc (17.6)
.
,
.
zc1 Pn1.
,11
11
c
cc
zA
Iz
.102464
)2
( 44
1
dd
Ic
.
)(64)(16
1024
1
2
1
2
4
1
hg
p
d
hg
pd
d
zmm
c
,
.64 2
2
2h
dzc
,0 OM
7/24/2019 Mehanika Fluida KV
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2 80
.0)()(33
22110 lFlPFlzlPzlPH
GH
P cmcncnm
Fc
.)()(3
)( 22110 lPFlzlPzlPH
GPlF mcncnmc
(17.1), (17.2), (17.3), (17.4), (17.5) (17.6)
,
,4
)()64
(16
)
)(64
(16
)()(18
1
2
0
2
2
22
2
1
22
1
22
lDDpFlh
dl
dgh
hg
p
dl
dghpHgdcb
m
mmm
.4
)()64
(16
)
)(64
(16
)()(18
11
2
0
2
2
22
2
1
22
1
22
lDDpFlh
dl
dgh
hg
p
dl
dghpHgd
bc
m
mmm
18. : A-A,
B-B .
a.
= 13,6, 8,6
ba .
A-A B-B, H=2b h=b/3. : , b, d.
(35)
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2 82
PNA, .
,
,AcANA AgzP
.4
2dAA
zcA ,
-(), ,
p0,
.
,0 cAaa gzpgHpp
.HbzcA
H=2b,
.3bzcA
.4
3 2gbdPNA
,
,00 AA
gVP
7/24/2019 Mehanika Fluida KV
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83
.16
1)
2(
2
1 330 d
dVA
.16
1 30 gdPA
- ,
.43 2gbdPP NAAistezanja
,
, , , .
.16
1 30 gdPP AAsmicanja
-:
Gz, PNB,
.
,BcBNB AgzP
.4
2dAB
zc,
-,
7/24/2019 Mehanika Fluida KV
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2 84
,0 cBaa gzpghpp
.hbzcB
h=b/3,
.3
4bzcB
.3
1 2gbdPNB
,
,zz gVG
.2
7
4
7
)4)4((12
1
22
22
bdHdV
ddddHV
z
z
.2
7 2gbdGz
-
,
.6
19
3
1
2
7
2
22
gbdP
gbdgbdPGP
Bistezanja
NBzBistezanja
7/24/2019 Mehanika Fluida KV
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85
,
, ,
.0BsmicanjaP
19. , ,
L.
=0,5D.
. : L, D, , . (36)
36.
36.
,
P0,
.00
gVP
.
,4
)2(
2
0
2
0
LDV
LDV
D
F F
P0
7/24/2019 Mehanika Fluida KV
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2 88
20. G
2 d1.
1 1 2
. : G, d, D, d, d1, l.(38)
38.
. ,
p2 D p3
d
7/24/2019 Mehanika Fluida KV
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95
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. b.
, h.
.
, b=h .3
41.
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,
gF
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x
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97
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2
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,
,
.
,2
hhHz
bx
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),2
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g
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b
gha
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R , h.
a.
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. :D
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101
. :
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Rx
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0
2
zhHD
Vcilindra
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4
32
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1(
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.
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7/24/2019 Mehanika Fluida KV
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103
,
,1
ma ppp
hz
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,)( 111 chgapp ma
.)( 111 hgappc ma
1
).)(( 111 zhgappp ma (24.4)
, 2,
, 2. ,
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,
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11 hgappp
z
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2
2
7/24/2019 Mehanika Fluida KV
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2 104
).)(( 22112 hhgappp ma (24.5)
- 2. ,,
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a (24.6)
: zc, xc
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,
,
42
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c
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x
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7/24/2019 Mehanika Fluida KV
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105
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:
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y
x
),( dzFdyFdxFdp zyx
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7/24/2019 Mehanika Fluida KV
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2 106
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7/24/2019 Mehanika Fluida KV
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2 108
),( dzFdyFdxFdp zyx
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22
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.
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113
, r
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7/24/2019 Mehanika Fluida KV
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3
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.
1.
,
.
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27. ,
h, .
. : pm, pv, ,
h,L, D, u, v, k, f, . (48)
48.
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0 0
pv
B
1 1
v
F
k
k k
k
L,D
u hv
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7/24/2019 Mehanika Fluida KV
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115
(1.68),
0-0 1-1.
0-0 :
,
mp
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, .02
2
1 v
1-1 :
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2
2
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2
vDLpghp fkvuvm
:
u ,v- ,- ,f- ,1 - ,
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7/24/2019 Mehanika Fluida KV
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3 116
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(=1) !
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4
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3 118
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121
. : L, l1, l2, D, d1,
d2, (), h, u, v, v1, v2, u, R1, R2, k. (50)
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50 ,
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0-0 1-1 :
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A
B C
h
v1 v2
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k
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3 122
()
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)(2
1
1
111
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d
lv
D
Lpgh
pvkRvu
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.
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)(2
2
2
222
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d
lv
D
Lpgh
pvkRvu
aa
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.
:
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51.
51.
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444
2
22
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11
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vd
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1V 2V
7/24/2019 Mehanika Fluida KV
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123
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: v, v1v2.
,
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,
2
2222
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1111
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lC
d
lC
D
LC
vkR
vkR
vu
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2
11
2 vC
vCgh (24.4.)
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2
22
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vCgh (24.5)
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(29.4) (29.6)
.
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2
2
1
1
v
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v
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,
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112
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3 126
52.
52
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.
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, .
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2
2
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d
lLv
d
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2
2
2
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d
lLv
d
lLgH
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(30.1) (30.2)
H
0 0
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l,d, /2
L,D,
v1v01
v2
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129
31.
, h .
.
. :K,h, D, D1, D2, l, l1, l2, l3, u, , , v, v1, , p.
(53)
53.
,
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v2,
,4
2
22
DvhK
.4
22
2 D
hKv (31.1)
B
0 0
1 1
v1
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Yp
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131
.2
33
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l
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vu (31.5)
gH, (31.4),
,
I II.
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I
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44
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3
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3
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l
gHv
vu (31.6)
,
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,
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133
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2
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l
gHDV
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V (m3/s) .
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P2 . 5
5V .
a) P1, C4 C5
4V 5V
.
b)5 ,
2 4V (Ci
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3 134
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7/24/2019 Mehanika Fluida KV
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135
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7/24/2019 Mehanika Fluida KV
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3 136
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7/24/2019 Mehanika Fluida KV
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143
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145
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34.
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l, (ppa=pbp)
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7/24/2019 Mehanika Fluida KV
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3 150
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153
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1
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2
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2
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7/24/2019 Mehanika Fluida KV
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3 154
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7/24/2019 Mehanika Fluida KV
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3 156
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7/24/2019 Mehanika Fluida KV
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163
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7/24/2019 Mehanika Fluida KV
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3 164
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7/24/2019 Mehanika Fluida KV
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165
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7/24/2019 Mehanika Fluida KV
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3 166
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7/24/2019 Mehanika Fluida KV
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3 168
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7/24/2019 Mehanika Fluida KV
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3 170
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3 172
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3 174
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7/24/2019 Mehanika Fluida KV
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175
R
A(z)
z
dz
l,d,
v1(z)
,
l,d, v2 v1k k
u
v(z)
v2(z)
L,D,
Rr(z)
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4
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. : R, D, d, L, l, , u, v1, v2, k, .
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77.
77. D
.
,)(
)(2
R
RzV
dzzAT
,
)(zV ,
D. ,
v(z) A(z).
.
:
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181
(40.1)
,
2
)(
2
2
24
1
2
2
2 v
d
dvgz su
.
)(
2
4
1
22
d
d
gzv
su
,
)(
1
4
1
2
2
d
dsu
.222 gzv (40.3)
(40.3)
.24
)(2
222 gzd
zV
v1(z) (40.3)
,22
1
2
221 gzd
dv
,)( 2
1
221d
d
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183
.
79.
79.
, , , s x-.
,
p
v
c
vM (3.5.1)
: .v
p
c
c
-
,3
1vgraddivvpF
Dt
vD
, ,
dx
1
1
2
2
p2
v2
2T2
p1
v1
1T1
p
v
T
p+dp
v+dv
+dT+dT
Dx
dx
l
+dA=const.
M(x)
Re(x)
=(x)
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3 184
,)(3
1x
ivdivvdx
dp
Dt
Dv
, vdtdx
.0(3
dxivdivv
dpvdv x
(3.5.2)
, ()
.0)2
(2
rdYdpv
d
,2
2v
D
dxdYr
, ,
.2
22
D
dxM
v
dvM
p
dp (3.5.3)
,0v
dvd
(3.5.4)
.0
T
dTd
p
dp
(3.5.5)
dx. dYr.
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185
(3.5.2) (3.5.5)
.
, .
41.
(=const.). ,
1M .
: l,D,R,T, , . (80)
80.
, . ,
,
,
.
, ,
, .
, ,
:
,0v
dvd
(41.1)
,2
22
D
dxM
v
dvM
p
dp (41.2)
D
v
lm1
M1 M=1/
T=const =s
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3 186
.0T
dTd
p
dp
(41.3)
(41.3)
(=const.),
.
d
p
dp
(41.1)
,
v
dvd
.v
dvd
p
dp
(41.4)
(41.4) (41.2),
,2
22
D
dxM
v
dvM
v
dv
, ,
.21 2
2
D
dx
M
M
v
dv
(41.5)
(41.4)
.21 2
2
D
dx
M
M
p
dp
(41.6)
,
, ,
, :
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187
.21 2
2
D
dx
M
M
v
dv
p
dpd
(41.7)
,.MconstcMv
,M
dM
v
dv
.21 2
2
D
dx
M
M
M
dM
(41.8)
(41.8) ,1
2
2
M
M
.2
12
2
D
dx
M
dM
M
M s
,
,
.2
112
1 02
2
D
dx
M
dM
M
M slM
M
m
(41.9)
,
,lnln2
1
2
1
2122
1
2
2
1 MMMMD
lms
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3 188
.ln2
1
2
1
2 2
1
2
1
2
2
1
M
M
MMD
lms
(41.10)
).ln2
1
2
1(
2
2
1
2
1
2
2
1M
M
MM
Dl
s
m
(41.11)
,1
2
M
).ln11(2 2121
1 MM
Dls
m
42.
(const.). ,
=1.
: l,D,R,T, , . (81)
81.
, .:
,0v
dvd
(42.1)
Mk=1Dv
l
M1
Tconst, dq=0 =s
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3 190
, ,
.
2
1
T
dT
M
dM
v
dv (42.8)
(42.5) (42.7) (42.8)
,
)2
11(
)1(
2 23
2
MM
dMMdx
D
(42.9)
, .
=1,
.
(42.9) x
.2)1(
2)1()(ln
2
1)
11(
12
22
22
M
M
M
M
MMD
l l
ll
s
( Ml)
.
,)1(2
)1(ln
2
1)1
1(
12
2
2M
M
MD
lks
.)1(2
)1(ln
2
1)1
1(
12
2
2 M
M
M
Dl
s
k
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4.
1 ., ., , , ,
, , 1990.
2 ., .,
,, 1997.
3 Roberson I., Crowe R., Engineering Fluid Mechanics, Boston, 1993.
4 ., ,
, 1978.
5 ., - ,, , 2000.