Mechanics of Fluids

108 The principles governing fluids in motion
Fig. 3.8
is therefore
(p + δp)δA − pδA + W cos θ = δpδA + ϱgδAδr cos θ
where θ is the angle between the radius and the vertical.
By Newton’s Second Law this force equals the product of the mass of
the element and its centripetal acceleration, an. Noting that δr cos θ = δz,
z being measured vertically upwards from a suitable datum level, and using
eqn 3.2, we then have
δpδA + ϱgδAδz = ϱδAδran = ϱδAδr
�
u 2
r
�
∂un
+
∂t
Dividing by δAδr and taking the limit as δr → 0 now gives
�
∂p ∂z u
+ ϱg = ϱ
∂r ∂r 2
�
∂un
+
r ∂t
(3.17)
If the streamlines are straight and not changing direction with time, the
right-hand side of eqn 3.17 is zero since r =∞while un = 0 and is not
changing. For a constant-density fluid, integration in the direction r then
gives p+ϱgz = constant, that is, the piezometric pressure is constant normal
to the streamlines. Where r is not infinite the exact manner in which p varies
across the streamlines, even for steady flow, depends on the way in which u
varies with r. Two special cases, the free and forced vortex, are discussed in
Section 9.6.4 and 9.6.5.
An important consequence of the pressure variation perpendicular to
curved streamlines is the tendency of a jet of fluid to attach itself to a convex
solid body. This is known as the Coanda effect, after the Romanian engineer
Henri Coanda (1885–1972) who made use of it in various aeronautical
applications. It may be simply demonstrated by using a solid cylinder (e.g. a
finger) to deflect the flow from a water tap (Fig. 3.9). The curvature of
the streamlines between sections AA ′ and BB ′ requires a net force towards
the centre of curvature, and, as the outer edge of the stream is at atmospheric
pressure, the pressure at the surface of the cylinder must be below
atmospheric. Consequently the flow does not continue vertically downwards
from BB ′ , but bends towards the cylinder. The sub-atmospheric pressure