400
9 Fluid Dynamics
9.9 (a) Q = v throat a = v pipe A
∴ v throat = v pipe
A
a
= v pipe
D 2
d 2 = 10 ×
4 2
2 2 = 40 m/s
(b) p =
1
2
ρv
2
A 2
a 2 − 1
=
1
2
× 1000 × 40
2
4 4
2 4 − 1
= 12 × 10
6 Pa
9.10 Reynold’s number R =
ρ Dv
η
, where ρ is density, D diameter, v velocity and
η coefficient of viscosity.
(a) R =
1 × 0.1 × 300
0.018
= 1667
Flow is steady because R < 2200
(b) R =
1 × 0.1 × 300
0.008
= 3750
Flow is turbulent because R > 2200
(c) R =
1 × 0.1 × 300
0.004
= 7500
Flow is turbulent because R > 2200
9.11 Consider a mass element dm of the fluid at distance x from the vertical axis.
The centrifugal force on dm is
dF = dm ω
2 x = dm
dv
dt
= dm
dv
dx
v
vdv = ω
2 x dx
v dv = ω
2
x dx
v 2
2
=
ω 2
2
x
2
L
L – l
∴ v = ωl
2L
l
− 1
9.12 Applying Bernoulli’s equation to points A and B,
p A +
1
2
ρ g v
2
= P B
(1)
p A + ρ L gh = P B
(2)