fluid_mechanics

314 Chapter 6 ■ Differential Analysis of Fluid Flow
y
g
v z
v z
(a)
R
r θ
z
dr
(b)
r
z
F I G U R E 6.34
The viscous flow in a horizontal,
circular tube: (a) coordinate
system and notation used
in analysis; (b) flow through
differential annular ring.
V6.13 Laminar
flow
The velocity distribution
is parabolic
for steady, laminar
flow in circular
tubes.
is only a function of the radial position within the tube—that is, v z v z 1r2. Under these conditions
the Navier–Stokes equations 1Eqs. 6.1282 reduce to
and integrated 1using the fact that 0p0z constant2 to give
r 0v z
0r 1
2m a 0p
0z b r2 c 1
Integrating again we obtain
(6.143)
(6.144)
(6.145)
where we have used the relationships g r g sin u and g u g cos u 1with u measured from the
horizontal plane2.
Equations 6.143 and 6.144 can be integrated to give
p rg1r sin u2 f 1 1z2
or
p rgy f 1 1z2
(6.146)
Equation 6.146 indicates that the pressure is hydrostatically distributed at any particular cross
section, and the z component of the pressure gradient, 0p0z, is not a function of r or u.
The equation of motion in the z direction 1Eq. 6.1452 can be written in the form
1 0
r 0r ar 0v z
0r b 1 0p
m 0z
v z 1
(6.147)
4m a 0p
0z b r2 c 1 ln r c 2
Since we wish v z to be finite at the center of the tube 1r 02, it follows that c 1 0 3since
ln 102 4. At the wall 1r R2 the velocity must be zero so that
c 2 1
4m a 0p
0z b R2
and the velocity distribution becomes
0 rg sin u 0p
0r
0 rg cos u 1 0p
r 0u
0 0p
0z m c 1 0
r 0r ar 0v z
0r bd
v z 1
(6.148)
4m a 0p
0z b 1r2 R 2 2
Thus, at any cross section the velocity distribution is parabolic.
To obtain a relationship between the volume rate of flow, Q, passing through the tube and the
pressure gradient, we consider the flow through the differential, washer-shaped ring of Fig. 6.34b.
Since v z is constant on this ring, the volume rate of flow through the differential area dA 12pr2 dr is
dQ v z 12pr2 dr