Mechanics of Fluids

378 The flow of an inviscid fluid
Fig. 9.18
two streamlines shown in Fig. 9.18, the flow being assumed steady. At radius
R from the centre of curvature the pressure is p, at radius R + δR it is p + δp;
consequently there is a net thrust on the element, towards the centre of
curvature, equal to
(p + δp)(R + δR)δθ − pRδθ − 2
�
p + δp
�
δR sin
2
δθ
2
(for unit thickness perpendicular to the diagram). As δθ → 0, sin(δθ/2) →
δθ/2 and, with higher orders of small magnitudes neglected, the thrust
reduces to Rδpδθ. (Shear forces, even if present, have no component
perpendicular to the streamlines.)
The component of the weight acting radially inwards
= RδθδRϱg (δz/δR) = Rϱg δθ δz
where δz is the vertical projection of δR so that arccos (δz/δR) is the angle
between the radius and the vertical. Thus the total inward force is
Rδpδθ + Rϱgδθδz = Mass × Centripetal acceleration
Division by Rϱgδθ gives
= ϱRδθδR(q 2 /R)
δp q2 δR
+ δz =
ϱg R g
Now, by Bernoulli’s theorem for steady flow of a frictionless fluid:
p q2
+ + z = H
ϱg 2g
(9.16)
where H is a constant along a streamline (although it may differ from one
streamline to another). Differentiation gives
δp 2qδq
+ + δz = δH (9.17)
ϱg 2g