Mechanics of Fluids

380 The flow of an inviscid fluid
✷
Example 9.2 A ventilating duct, of square section with 200 mm sides,
includes a 90 ◦ bend in which the centre-line of the duct follows a
circular arc of radius 300 mm. Assuming that frictional effects are
negligible and that upstream of the bend the air flow is completely uniform,
determine the way in which the velocity varies with radius in the
bend. Taking the air density as constant at 1.22 kg · m −3 , determine
the mass flow rate when a water U-tube manometer connected between
the mid-points of the outer and inner walls of the bend reads 11.5 mm.
Solution
If the upstream flow is uniform, all streamlines have the same Bernoulli
constant and if friction is negligible all streamlines retain the same
Bernoulli constant.
∴ ∂
∂R
�
p ∗ + 1
2 ϱq2
�
= 0
∴ ϱq ∂q
∂R =−∂p∗
∂R =−ϱq2
R
∴ dq
q =−dR
R
(by eqn 9.16)
which on integration gives qR = constant = C, the equation for a free
vortex.
∂p ∗
∂R
and integration gives
p ∗ A − p∗B =−1
2 ϱC2
�
= ϱq2
R
1
R 2 A
= ϱC2
R 3
− 1
R2 �
B
1000kg · m −3 × 9.81N · kg −1 × 0.0115 m
=− 1
2 × 1.22kg · m−3 C 2
whence C = 3.141 m2 · s−1 .
∴ Mass flow rate = ϱQ = ϱ(0.2 m)
�
1 1
−
0.42 0.22 � 0.4m
0.2m
qdR
�
m −2
� 0.4m dR
= ϱC(0.2 m) = ϱC(0.2 m) ln 2
0.2m R
= 1.22 kg · m −3 3.141 m 2 · s −1 0.2 m × ln 2
= 0.531 kg · s −1