Лекции по гидрогазодинамике
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•
•
a = a(M) = lim
V →M
A
V
m
V
.
A a
M
A
V
A V m
V
V V M
V
0
S
0
V S
A V
0
A
V
0
A S
0
Q
A
A Ω
A
dA
dt
= Q
A
+ Ω
A
.
A V
0
dV
0
dA = ρ · a · dV
0
ρ
[ /
3
]
Z
V
0
dA =
Z
V
0
ρ · a · dV
0
.
A
dA
dt
=
d
dt
Z
V
0
ρ · a · dV
0
=
Z
V
0
∂
∂t
(ρ · a · dV
0
) =
Z
V
0
∂
∂t
(ρ · a) dV
0
,
d/dt
∂/∂t V
0
dV
0
dQ
A
dS
0
Q
A
=
Z
S
0
dQ
A
=
I
S
0
−→
I
0
A
·~n · dS
0
,
−→
I
0
A
A ~n
A
A
Q
A
−→
I
0
m
= ρ·~v ~v
−→
I
0
A
= ρ ·~v
A
· a,
~v
A
A S
0
ω
0
A
A V
0
dV
0
dΩ
A
= ω
0
A
· dV
0
A A V
0
Ω
A
=
Z
V
0
ω
0
A
dV
0
.
A
Z
V
0
∂
∂t
(ρ · a) dV
0
= −
I
S
0
−→
I
0
A
·~n · dS
0
+
Z
V
0
ω
0
A
· dV
0
.
−
A V
0
A
−
S
0
V
0
I
S
0
−→
I
0
A
·~n · dS
0
=
Z
V
0
div
−→
I
0
A
· dV
0
.
Z
V
0
"
∂
∂t
(ρ · a) + div
−→
I
0
A
− ω
0
A
#
· dV
0
= 0.
V
0
∂
∂t
(ρ · a) + div
−→
I
0
A
= ω
0
A
.
A
V
dA
dt
= Q
A
+ Ω
A
A V
A =
Z
V
dA =
Z
V
ρ a dV.
V
x
i
= x
i
(ξ
k
, t) t
V
ξ
d/d t V
ξ
J =
∂ (x, y, z)
∂ (ξ
1
, ξ
2
, ξ
3
)
d
d t
Z
V
ρ a dx dy dz =
d
d t
Z
V
ξ
ρ a J d ξ
1
d ξ
2
d ξ
3
=
Z
V
ξ
d
d t
(ρ a J) d V
ξ
.
Z
V
ξ
d
d t
(ρ a J) d V
ξ
=
Z
V
1
J
d
d t
(ρ a J) d V =
=
Z
V
1
J
d J
d t
ρ a +
d
d t
ρ a
!
d V.
1
J
d J
d t
= div ~v,
dA
dt
=
Z
V
ρ a div ~v +
d
d t
ρ a
!
d V =
Z
V
ρ
d a
d t
d V.
A dS
−→
I
A
−→
I
A
−→
I
0
A
−→
I
A
=
−→
I
0
A
− ρ ~v a = ρ a ( ~v
A
−~v)
A
a = 1
−→
I
A
~v
A
= ~v
ρ
da
dt
+ div
−→
I
A
= ω
A
.
ω
0
A
, ω
A
A
a
ω
A
= 0, ρ
A = m, a = 1, ~v
A
= ~v
m
= ~v
∂ρ
∂t
+ div(ρ ~v) = 0.
dρ
dt
=
∂ρ
∂t
+ (~v ∇) ρ.
div ρ ~v = ∇ρ ~v = ρ(∇~v) + (~v ∇ )ρ
dρ
dt
+ ρ div ~v = 0.
div ~v
ρ =
div ~v = 0.
∂ρ/∂t = 0
div ρ ~v = 0.
k n
k
k
= ρ
k
/ρ ρ
k
k ρ
ρ =
n
X
k=1
ρ
k
.
k
k
a = c
k
k
ρ
dc
k
dt
+ div
−→
I
k
= ω
k
.
−→
I
k
= ρ
k
c
k
(~v
k
−~v) k
−→
I
k
−→
I
k
= −D · grad c
k
;
D
ρ
dc
k
dt
= ρ
"
∂c
k
∂t
+ (~v ∇) c
k
#
= div(D grad c
k
) + ω
k
.
D =
dc
k
dt
= D
0
∇
2
c
k
+
ω
k
ρ
,
D
0
≡ D/ρ, ω
k
k
∇
2
= ∂
2
/∂x
2
+ ∂
2
/∂y
2
+ ∂
2
/∂z
2
~v = 0
∂c
k
∂t
= D
0
∇
2
c
k
+
ω
k
ρ
.
~v a = ~v
ρ
da
dt
≡ ρ
d~v
dt
,
•
•
−→
F
m
−→
F
s
ρ
d~v
dτ
=
−→
F
m
+
−→
F
s
.
−→
F
m
−→
F
s
~
f
m
~
f
m
=
~
f
m
(~r, τ).
dV
d
−→
φ
m
= ρ
−→
f
m
dV
V
−→
Φ
m
=
Z
V
ρ
−→
f
m
dV.
−→
F
m
= ρ
−→
f
m
−→
F
m
= ρ ~g g =
9.81 /
2
-
6
x
3
x
2
x
1
A!
!
!
!
!
!
!
!
!
!
!
!
B
@
@
@
@
@
@
@
C
~n
dV
dx
1
, dx
2
, dx
3
~
σ
−1
,
~
σ
−2
,
~
σ
−3
,
~
σ
n
dS
1
, dS
2
, dS
3
−
ρ dV
d~v
dτ
= ρ dV
−−→
f +
3
X
k=1
−−→
σ
−k
dS
k
.
dS
n
~n dS
n
dS
k