fluid_mechanics

p
A fluid contained in
a tank that is rotating
with a constant
angular velocity
about an axis will
rotate as a rigid
body.
z = constant
dp
dr
dp ––– = rw 2 r
dr
r
2.12.2 Rigid-Body Rotation
2.12 Pressure Variation in a Fluid with Rigid-Body Motion 75
After an initial “start-up” transient, a fluid contained in a tank that rotates with a constant angular
velocity v about an axis as is shown in Fig. 2.30 will rotate with the tank as a rigid body. It is
known from elementary particle dynamics that the acceleration of a fluid particle located at a distance
r from the axis of rotation is equal in magnitude to rv 2 , and the direction of the acceleration
is toward the axis of rotation, as is illustrated in the figure. Since the paths of the fluid particles
are circular, it is convenient to use cylindrical polar coordinates r, u, and z, defined in the insert in
Fig. 2.30. It will be shown in Chapter 6 that in terms of cylindrical coordinates the pressure gradient
§p can be expressed as
Thus, in terms of this coordinate system
and from Eq. 2.2
(2.29)
(2.30)
These results show that for this type of rigid-body rotation, the pressure is a function of two variables
r and z, and therefore the differential pressure is
or
§p 0p
0r ê r 1 r
a r rv 2 ê r a u 0 a z 0
0p
0r rrv2
0p
0u 0
0p
0u ê u 0p
0z ê z
dp 0p 0p
dr
0r 0z dz
dp rrv 2 dr g dz
(2.31)
On a horizontal plane (dz 0), it follows from Eq. 2.31 that dp dr 2 r, which is greater than
zero. Hence, as illustrated in the figure in the margin, because of centrifugal acceleration, the pressure
increases in the radial direction.
Along a surface of constant pressure, such as the free surface, dp 0, so that from Eq. 2.31
1using g rg2
dz
dr rv2
g
Integration of this result gives the equation for surfaces of constant pressure as
z v2 r 2
2g constant
0p
0z g
(2.32)
Axis of
rotation
z
ω
r
ω
2
a r = r ω
x
θ
r
e z
e
y
θ
e r
(a)
F I G U R E 2.30
(b)
Rigid-body rotation of a liquid in a tank. (Photograph courtesy of Geno Pawlak.)
(c)