Лекции по гидрогазодинамике
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I
0
~v
= I
~v
+ ρ ~v ~v = −σ + ρ ~v ~v.
~
I
0
~v
= p ~n + ρ ~v (~v ~n).
∂ ρ ·~v
∂ t
+ Div I
0
~v
= ρ ·
~
f
m
,
a = e e [ / ]
e
.
= u + v
2
/2 + ψ
u, v
2
/2, ψ
∂
∂t
ρ ·
u +
v
2
2
+ ψ
+ div
−→
I
0
e
= 0,
ρ ·
d
dt
u +
v
2
2
+ ψ
+ div
−→
I
e
= 0
−→
I
0
e
,
−→
I
e
−→
I
0
e
,
−→
I
e
~v
ρ ·~v ·
d~v
dt
= ρ ·
d
dt
v
2
2
= ρ ·~v ·
−→
f
m
+ ~v · Div σ.
−→
f
m
= −gradψ;
∂ψ
∂t
= 0.
ρ ·~v ·
−→
f
m
= −ρ ·~v · gradψ = −ρ ·
"
∂ψ
∂t
+ (~v · ∇)ψ
#
= −ρ ·
dψ
dt
.
ρ ·
d
dt
v
2
2
+ ψ
= ρ ·
de
.
dt
= ~v · Div σ.
e
.
≡
v
2
/2 + ψ
~v · Div σ = Div (~v · σ) − σ · (∇~v)
ρ ·
de
.
dt
− Div (~v · σ) = −σ · (∇~v).
−−→
I
e
.
= −~v · σ;
ω
e
.
= −σ · (∇~v).
ω
e
.
= p · div ~v − µ D.
µ D = σ
0
·(∇~v)
µ D
ρ = Const
µ D = µ
∂v
i
∂x
k
+
∂v
k
∂x
i
!
·
∂v
i
∂x
k
=
µ
2
∂v
i
∂x
k
+
∂v
k
∂x
i
!
2
ρ ·
du
dt
+ div
−→
I
u
= ω
u
−→
I
u
=
−→
I
q
−→
I
q
= −λ · grad T λ
[ /( ·K)]
λ
λ
ρ ·
du
dt
= div (λ · grad T ) + ω
u
ω
u
ω
u
= σ · (∇~v),
ω
u
[ /
3
]
−→
I
q
= 0 σ
0
= 0
div ~v = 0
σ · (∇~v) = −p · div ~v = 0,
du
dt
= 0, u = Const.
du = C
V
· dT, λ = λ(T )
ρ ·
V
·
dT
dt
= ρ · C
V
·
"
∂T
∂t
+ (~v · ∇)T
#
= div (λ · grad T ) + ω
u
.
C
V
ω
u
≡ σ ·(∇~v)
C
V
= C
p
, ~v = 0
ρ · C
p
·
∂T
∂t
= div (λ · grad T ) + ω
u
.
λ =
∂T
∂t
= a · ∇
2
T +
ω
u
ρ · C
p
.
−→
I
e
.
−→
I
e
=
−→
I
q
+
−−→
I
e
.
= −λ · grad T −~v · σ.
−→
I
0
e
−→
I
0
e
= −λ·grad T −~v ·σ + ρ~v ·e = −λ ·grad T +ρ~v ·
u +
v
2
2
+ ψ +
p
ρ
−~v ·σ
0
.
u + p/ρ = h
−→
I
q
= 0
σ
0
= 0
−→
I
0
e
= ρ~v ·
h +
v
2
2
+ ψ
[∂(ρe)/∂t = 0, div(ρ~v) = 0]
div(
−→
I
0
e
) = div
ρ~v ·
h +
v
2
2
+ ψ
= ρ~v · grad
h +
v
2
2
+ ψ
= 0
h +
v
2
2
+ ψ = .
[d~r ×~v] = 0
d x
v
x
=
d y
v
y
=
d z
v
z
.
p
ρ
+
v
2
2
+ ψ = .
ψ = g·z z
p
ρ
+
v
2
2
+ g · z = .
ψ ≈
h +
v
2
2
= .
ρ ·
d
dt
u +
v
2
2
+ ψ
= −div
−→
I
e
= div (λ · grad T + ~v · σ)
−→
I
q
= 0, σ
0
= 0
−
3
X
k=1
3
X
i=1
∂
∂x
k
[v
i
(p δ
ik
)] = −div(p ~v) = −div
ρ
p
ρ
~v
!
,
∂p/∂t = ∂(ρ p/ρ)/∂t
∂(ρ p/ρ)
∂t
+ div
ρ
p
ρ
~v
!
= ρ ·
d (p/ρ)
dt
ρ ·
d
dt
u +
p
ρ
+
v
2
2
+ ψ
=
∂p
∂t
V
Z
V
ρ ·
d
dt
u +
p
ρ
+
v
2
2
+ ψ
dV =
Z
V
∂p
∂t
dV
t+T
Z
t
Z
V
ρ
d
dt
u + p/ρ + v
2
/2 + ψ
dV dt =
t+T
Z
t
Z
V
∂p
∂t
dV dt,
t+T
Z
t
∂p
∂t
dt = 0
d
dt
u + p/ρ + v
2
/2 + ψ
= 0.
u + p/ρ + v
2
/2 + ψ
= ,
h
c
∼ 10
−2
v
0
h
x
y
x l y h
h l
x v
x
∼ v
0
ρ (~v ∇) ~v = −grad p + µ ∇
2
~v.
x, y
ρ
v
x
∂v
x
∂x
+ v
y
∂v
x
∂y
!
= −
∂p
∂x
+ µ
∂
2
v
x
∂x
2
+
∂
2
v
x
∂y
2
,
ρ
v
x
∂v
y
∂x
+ v
y
∂v
y
∂y
!
= −
∂p
∂y
+ µ
∂
2
v
y
∂x
2
+
∂
2
v
y
∂y
2
.
v
y
∼ v
0
· h/l v
x
v
y
∼ 0 x
h l
µ
∂
2
v
x
∂y
2
=
∂p
∂x
,
∂p
∂y
= 0,
∂v
x
∂x
= 0.
v
x
(y) =
1
2 · µ
∂p
∂x
!
y
2
+ C
1
y + C
2
.
C
1
, C
2
y = 0 v
x
= v
0
,
y = h v
x
= 0.
C
2
= v
0
; C
1
= −
v
0
h
−
1
2 µ
∂p
∂x
!
h.
v
x
(y) =
1
2 µ
∂p
∂x
!
(y
2
− y h) + v
0
1 −
y
h
!
.
Q =
h
Z
0
v
x
dy =
v
0
h
2
−
1
12 µ
∂p
∂x
!
h
3
= const.
∂p
∂x
=
6 µ v
0
h
2
−
12 µ Q
h
3
.
h =
∂p/∂x =
p |
(
x = 0
x = l
)
= p
0
,
h = h(x)
x h = h(x)
p = 12 µ
v
0
2
Z
d x
h
2
− Q
Z
d x
h
3
!
+ C.