Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
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∂ρ
∂t
+ div j = 0, j = ρv
∂(ρs)
∂t
+ div j
s
= 0, j
s
= ρsv
∂(ρv
i
)
∂t
+
3
X
k=1
∂Π
ik
∂x
k
= 0,
Π
ik
= pδ
ik
+ ρv
i
v
k
i
x
k
t = t
0
s(p, ρ) = s
0
= const,
∂v
∂t
+ (v∇)v = −∇w
∂v
∂t
− [v × rot v] = −∇
µ
w +
v
2
2
¶
.
w
∇p = ρg.
µ
∂v
∂t
= 0
¶
v
2
2
+ w + gz = const ≡ B
B
ρ = const
v
2
2
+
p
ρ
+ gz = const.
rot v = 0
v = ∇φ φ
∂φ
∂t
+
v
2
2
+ w + gz = f(t)
f(t)
µ
∂φ
∂t
= 0
¶
v
2
2
+ w + gz = const ≡ F,
F
∆φ = 0.
∂φ
∂n
¯
¯
¯
¯
S
= v
n
¯
¯
¯
¯
S
= 0
S
Π
ik
σ
ik
Π
ik
= Π
ik
− σ
ik
,
σ
ik
= η
Ã
∂v
i
∂x
k
+
∂v
k
∂x
i
−
2
3
δ
ik
X
l
∂v
l
∂x
l
!
+ ξδ
ik
X
l
∂v
l
∂x
l
,
η ξ η ξ
ρ
µ
∂v
∂t
+ (v∇)v
¶
= −∇p + ρg + η∆v +
µ
ξ +
1
3
η
¶
grad div v.
v|
S
= 0
S
v
0
v|
S
= v
0
F
n
F
i
= pn
i
−
X
k
σ
ik
n
k
.
Ω
ρ
p
0
z
r v = [Ω × r]
v
x
= −Ωy v
y
= Ωx v
z
= 0
1
ρ
∂p
∂x
= Ω
2
x,
1
ρ
∂p
∂y
= Ω
2
y,
1
ρ
∂p
∂z
= −g.
p =
1
2
ρΩ
2
(x
2
+ y
2
) − ρgz + p
0
.
z = 0
p =
1
2
ρΩ
2
(x
2
+ y
2
) + p
0
.
p = 0
z =
Ω
2
2g
(x
2
+ y
2
) +
p
0
ρg
,
r ϕ z
v = Ωre
ϕ
, ∇ = e
r
∂
∂r
+ e
ϕ
1
r
∂
∂ϕ
+ e
z
∂
∂z
.
∂
∂ϕ
e
ϕ
= −e
r
(v∇)v = −Ω
2
re
r
1
ρ
∂p
∂r
= Ω
2
r,
1
ρ
∂p
∂z
= −g.
a
U
ρ φ
p
∞
S
0
r ϑ ϕ
U φ
0
S
0
∂φ
∂r
¯
¯
¯
¯
r=a
= 0, φ
0
¯
¯
¯
¯
r=∞
= −Ur cos ϑ.
φ
0
φ
0
= R(r)P (ϑ).
1
r
2
∂
∂r
µ
r
2
∂φ
0
∂r
¶
+
1
r
2
sin ϑ
∂
∂ϑ
µ
sin ϑ
∂φ
0
∂ϑ
¶
+
1
r
2
sin
2
ϑ
∂
2
φ
0
∂ϕ
2
= 0
d
dr
µ
r
2
dR
dr
¶
− λR = 0,
1
sin ϑ
d
dϑ
µ
sin ϑ
dP
dϑ
¶
+ λP = 0,
λ
λ = l(l+1) l = 0, 1, 2, . . .
P
l
(cos ϑ) P
0
= 1, P
1
=
cos ϑ λ = l(l + 1)
R
l
= A
l
r
l
+ B
l
r
−l−1
,
A
l
B
l
φ
0
=
∞
X
l=0
(A
l
r
l
+ B
l
r
−l−1
)P
l
(cos ϑ).
A
l
= −Uδ
1l
;
A
l
la
l−1
− B
l
(l + 1)a
−l−2
= 0,
B
l
= −
U
2
a
3
δ
1l
.
φ
0
= −U(r +
a
3
2r
2
) cos ϑ.
v
0
|
r=a
= v
0
ϑ
|
r=a
=
3
2
U sin ϑ.
ϑ = 0, π, v
0
= 0
p
p
∞
+
ρU
2
2
= p + ρ
9
8
U
2
sin
2
ϑ, p = p
∞
+
ρU
2
2
(1 −
9
4
sin
2
ϑ).
S
U S v = U v
0
φ v φ
0
φ = Ur + φ
0
φ = −
a
3
U
2
cos ϑ
r
2
.
a
v = ve
r
, v < 0.
∂v
∂t
+ v
∂v
∂r
= −
1
ρ
∂p
∂r
1
r
2
d
dr
(r
2
v) = 0.
r
2
v = F (t),
F (t)
v
F
0
(t)
r
2
+ v
∂v
∂r
= −
1
ρ
∂p
∂r
.
r R =
R(t) 6 a
−
F
0
(t)
R(t)
+
V
2
2
=
p
0
ρ
,
V = dR(t)/dt p
0
F (t) = R
2
(t)V (t),
F (t)
−
3V
2
2
−
1
2
R
dV
2
dR
=
p
0
ρ
.
V = 0 R = a
V =
dR
dt
= −
s
2p
0
3ρ
µ
a
3
R
3
− 1
¶
.
τ =
r
3ρ
2p
0
Z
a
0
dR
p
(a/R)
3
− 1
.
B
τ =
s
3a
2
ρπ
2p
0
Γ(5/6)
Γ(1/3)
= 0.915a
r
ρ
p
0
.
h
v =
p
2gh
S
σ ¿ S τ
τ =
S
σ
s
2H
g
.
R
σ
v
h
p
0
ρ
+ 2gR =
p
0
ρ
+ gh +
v
2
2
v =
p
2g(2R − h)
dt
dh =
vσdt
S
,
S = π
£
R
2
− (R − h)
2
¤
= π(2Rh − h
2
)
h
t =
π
σ
√
2g
Z
2R
0
h
√
2r − h dh =
16πR
2
15σ
s
R
g
.
p p
0
h ρ
v =
s
2(p − p
0
)
ρ
+ 2gh
p
p
0
T µ
γ
w
1
= w
2
+
v
2
2
v
1
' 0
v =
p
2(w
1
− w
2
).
w
w = u +
p
ρ
.
u = c
v
T
c
v
p = ρ
RT
µ
R µ
c
p
− c
v
=
R
µ
,
w = c
p
T
c
p
c
p
/c
v
= γ
v =
q
2c
p
(T
1
− T
2
).
T
2
p
γ−1
/T
γ
= const.
v =
r
2
µ
γ
γ − 1
RT
h
1 − (p
0
/p)
γ−1
γ
i
.
v
T
∞
M = v/c c
γ
T
max
= T
∞
µ
1 +
M
2
(γ − 1)
2
¶
.
u
l
p = const v
x
=
u
l
y xz
x
rot v = −
u
l
k 6= 0.
F
x
= η
u
l
, F
y
= −p
l
∆p/L L
∆p
p = p
0
−
∆p
L
x, p
0
x = 0 v
x
=
1
2η
∆p
L
y(l − y)
R
∆p/L L
∆p
z v = (0, 0, v
z
)
div v = 0
∂v
z
∂z
= 0,
v
z
= v
z
(x, y) v
z
= v
z
(r) r
ρ(v∇)v = −∇p + η∆v.