Манаков Н.Л., Некипелов А.А., Овсянников В.Д. Задачи по теоретической механике
Подождите немного. Документ загружается.
(v∇)v = v
x
∂v
∂x
+ v
y
∂v
∂y
+ v
z
∂v
∂z
= v
z
∂v
∂z
= 0,
x, y, z
∂p
∂x
= 0 ,
∂p
∂y
= 0 ,
∂p
∂z
= η∆v
z
.
p = p(z)
z r
z r
(
dp
dz
= C
η∆v
z
= C
C = const.
p = Cz + C
1
.
p
0
z = 0 p
1
z = L p
0
−p
1
= ∆p C
C
1
p = p
0
−
∆p
L
z,
η∆v
z
= C,
v
z
r
∆
r
=
1
r
∂
∂r
µ
r
∂
∂r
¶
.
1
r
d
dr
µ
r
d
dr
v
z
¶
=
C
η
.
v
z
=
C
4η
r
2
+ C
2
ln(r) + C
3
.
C
2
r = 0
C C = −
∆p
L
C
3
v
z
(R) = 0,
C
3
=
R
2
4η
∆p
L
,
v
z
=
∆p
4ηL
(R
2
− r
2
).
dS =
2πr dr dV = v
z
dS v
z
r 0 R V
V =
Z
R
0
v
z
(r) dS =
Z
R
0
∆p
4ηL
¡
R
2
− r
2
¢
2πr dr =
πR
4
8η
∆p
L
R a
V =
π
8η
∆p
L
·
R
4
− a
4
−
(R
2
− a
2
)
2
ln(R/a)
¸
R
1
v
0
R
2
η
z
v = (0, 0, v
z
)
(
dp
dz
= C
η∆v
z
= C
C = const.
z
C = 0 p = const
∆v
z
= 0.
1
r
d
dr
µ
r
d
dr
v
z
¶
= 0.
v
z
= C
1
ln(r) + C
2
.
C
1
C
2
v
z
(R
1
) = v
0
, v
z
(R
2
) = 0,
v
z
v
z
= v
0
ln(r/R
2
)
ln(R
1
/R
2
)
.
Ω
1
Ω
2
R
1
R
2
R
1
< R
2
r, ϕ, z z
v
z
= v
r
= 0, v
ϕ
= v(r); p = p(r).
dp
dr
= ρ
v
2
r
,
d
2
v
dr
2
+
1
r
dv
dr
−
v
r
2
= 0.
r
n
n = ±1
v = ar +
b
r
.
a b
v(R
1
) =
R
1
Ω
1
, v(R
2
) = R
2
Ω
2
v =
Ω
2
R
2
2
− Ω
1
R
2
1
R
2
2
− R
2
1
r +
(Ω
1
− Ω
2
)R
2
1
R
2
2
R
2
2
− R
2
1
1
r
.
Ω
1
= Ω
2
= Ω v = Ωr
T
γ µ
c
2
=
µ
∂p
∂ρ
¶
s
,
p/ρ
γ
= C
c
2
= Cγρ
γ−1
= γ
p
ρ
.
p/ρ
c
2
= γ
RT
µ
.
c
2
= γ
kT
m
k m
