Page 69 - Bird R.B. Transport phenomena

54 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow

Fig. 2.4-1 The momentum-flux distribution
and velocity distribution for the upward
flow in a cylindrical annulus. Note that the
momentum flux changes sign at the same
value of r for which the velocity has a
Velocity maximum.
distribution

Shear stress
or momentum-
flux distribution

the tube—that is, in the direction opposed to gravity. We make the same postulates as in
§2.3: v z = v (r), v e = 0, v r = 0, and p = p(z). Then when we make a momentum balance
z
over a thin cylindrical shell of liquid, we arrive at the following differential equation:

f(rr ) = (2.4-1)
rz
r
This differs from Eq. 2.3-10 only in that & = p + pgz here, since the coordinate z is in the
direction opposed to gravity (i.e., z is the same as the h of footnote 1 in §2.3). Integration
of Eq. 2.4-1 gives
, Q
г (2.4-2)
2L г
just as in Eq. 2.3-11.
The constant Q cannot be determined immediately, since we have no information
about the momentum flux at the fixed surfaces r = KR and r = R. All we know is that
there will be a maximum in the velocity curve at some (as yet unknown) plane r = \R at
which the momentum flux will be zero. That is,

(2.4-3)

When we solve this equation for C } and substitute it into Eq. 2.4-2, we get

(2.4-4)

The only difference between this equation and Eq. 2.4-2 is that the constant of integration
Q has been eliminated in favor of a different constant A. The advantage of this is that we
know the geometrical significance of A.
We now substitute Newton's law of viscosity, r = -yXdvJdr), into Eq. 2.4-4 to ob-
rz
tain a differential equation for v z
dv, - g > ) . (2.4-5)
L
dr