Лекции по гидрогазодинамике
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x
y
¯v
x
= f(y)
y
y + l
¯v
x
(y)
¯v
x
(y +l)−¯v
x
(y)
¯v
x
(y + l) − ¯v
x
(y) =
∂¯v
x
∂y
l.
v
0
x
v
0
x
v
0
y
v
0
x
v
0
y
v
0
y
v
0
x
v
0
y
v
0
x
v
0
x
v
0
y
σ
yx
∼ ρ
l
∂¯v
x
∂y
!
2
.
l
“
l
l = κ y, κ = 0.4.
l = κ y
s
1 −
y
R
, R − .
σ
yx
= ρ l
2
∂¯v
x
∂y
!
2
.
∂¯v
x
/∂y
∂¯v
x
/∂y
σ
yx
= ρ l
2
∂¯v
x
∂y
∂¯v
x
∂y
.
σ
yx
= µ
∂¯v
x
∂y
, µ ≡ ρ l
2
∂¯v
x
∂y
.
µ
R
“
x
y
0 6 y 6 R
r L
p S (p+∆ p) S
S = π r
2
σ
yx
F = σ
yx
S S = 2 π r L
∆ p π r
2
= σ
yx
2 π r L, σ
yx
=
∆p r
2 L
.
r = R σ
0
=
∆p R
2 L
σ
yx
= σ
0
r
R
.
σ
yx
= σ
yx
+ σ
yx
= µ
∂¯v
x
∂y
+ ρ l
2
∂¯v
x
∂y
!
2
.
σ
0
r
R
= σ
0
R − y
R
= µ
∂¯v
x
∂y
+ ρ κ
2
y
2
1 −
y
R
!
∂¯v
x
∂y
!
2
.
y ∼ 0
σ
yx
∼ σ
0
σ
0
= µ
∂¯v
x
∂y
,
¯v
x
|
y=0
= 0
¯v
x
v
∗
=
v
∗
y
ν
, v
∗
≡
v
u
u
t
σ
0
ρ
.
σ
yx
= σ
0
R − y
R
≈ σ
yx
= ρ κ
2
y
2
1 −
y
R
!
∂¯v
x
∂y
!
2
,
∂¯v
x
∂y
=
v
u
u
t
σ
0
ρ
1
κ y
=
v
∗
κ y
.
¯v
x
v
∗
=
1
κ
ln
v
∗
y
ν
+ const = 5.75 lg
v
∗
y
ν
+ const.
const = 5.5
δ
y = δ
v
∗
δ
ν
= 5.75 lg
v
∗
δ
ν
+ 5.5.
v
∗
δ
ν
= N ≈ 11.5.
δ
d
=
11.5
v
∗
d/ν
.
v
∗
=
q
σ
0
/ρ
σ
0
= ∆ p R/2 L
∆ p = λ L/d ρ v
2
/2
δ
d
=
11.5
q
λ /8 Re
,
Re = v d/ν v
Re
Re
∆ < δ
λ = f(Re)
4000 < Re 6 20 d/∆
λ = f(Re, ∆ /d) 20 d/∆ < Re 6
500 d/∆
λ = f(∆ /d), Re > 500 d/∆ .
Re
µ = 0 Re → ∞
v
x
v
max
=
y
R
!
n
=
1 −
r
R
!
n
,
v
max
n
Re Re = 4 · 10
3
1/6 Re = 32 · 10
5
n
“
v
x
v
max
=
y
R
!
1/7
=
1 −
r
R
!
1/7
.
¯v
x
= 0.816 v
max
Re
Re = v
∞
·R/ν R v
∞
Re