Mechanics of Fluids

9.6.5 Forced (rotational) vortex
This type of motion is obtained when all particles of the fluid have the same
angular velocity about some fixed axis. That is, the fluid rotates about that
axis like a solid body. Because an external torque is required to start the
motion, the term ‘forced vortex’ has been used, although ‘rigid-body rotation’
might be preferable. Such motion may be produced by rotating about
its axis a cylinder containing the fluid. Alternatively, rotation of a paddle in
the fluid will produce forced vortex motion within its periphery (although
beyond the periphery conditions are more nearly those of an irrotational
vortex). Once steady conditions are established, there is no relative motion
between the fluid particles and thus no shear forces exist, even in a real fluid.
The velocity at radius R from the centre is given by ωR where ω represents
the (uniform) angular velocity. Substituting q = ωR in the centripetal force
equation (9.16) gives
from which integration yields
δp
ϱg + δz = ω2R δR
g
p
ϱg + z = ω2R2 + constant
2g
that is, p ∗ = ϱω2R2 + constant (9.19)
2
where p ∗ = p + ϱgz.
Thus the piezometric pressure p ∗ increases with radius. Fluid may be supplied
to the centre of a forced vortex and then ejected at the periphery at a
much higher pressure. This principle is the basis of operation of the centrifugal
pump, which will be considered in Chapter 13. When the discharge valve
of the pump is closed, the blades of the impeller cause the fluid to rotate with
substantially the same angular velocity ω as the pump shaft, and the increase
of piezometric pressure from the inlet to outlet radius is given by
p ∗ 2 − p∗ 1
= ϱω2
2
�
R 2 2 − R2 �
1
If a forced vortex is produced in a liquid in an open container, the pressure
at the free surface of the liquid is atmospheric and therefore constant. Thus,
for the free surface, z = ω 2 R 2 /2g + constant. If z = z0 when R = 0 then
z − z0 = ω 2 R 2 /2g
and if R is perpendicular to z (i.e. if the axis of rotation is vertical) the surface
is a paraboloid of revolution (Fig. 9.19).
That the forced vortex is rotational is readily seen from the general expression
(9.12) for vorticity where q/R + (∂q/∂R) = ω + ω = 2ω which is not
zero. The fluid particles rotate about their own axes. Consequently, although
Basic patterns of flow 381