Page 191 - Bird R.B. Transport phenomena

Problems 175

Measured velocity profiles suggest that this assumption for b is reasonable, at least for high
3
Reynolds numbers. Assume further that к in Eq. 5.3-3 is the same for the inner and outer
walls.
(a) Show that direct application of Eq. 5.3-3 leads immediately to the following velocity pro-
files 4 in the region r < b (designated by <) and r > b (designated by >):

vt 1 - aR)vi
(5C.2-2)
Vz 1 ln - aR)vi (5C.2-3)
vi K v v
in which u« =
(b) Obtain a relation between the constants A*^ and A > by requiring that the velocity be con-
tinuous at r = bR.
(c) Use the results of (b) to show that the mass flow rate through the annulus is

2
w = - « )Vl - b 2 (5C.2-4)
in which В is
2
(b 2 - a ?' 2
B = + + (5C.2-5)
i l 2
5C.3 Instability in a simple mechanical system (Fig. 5C.3).
(a) A disk is rotating with a constant angular velocity П. Above the center of the disk a
sphere of mass m is suspended by a massless rod of length L. Because of the rotation of the
disk, the sphere experiences a centrifugal force and the rod makes an angle в with the verti-
cal. By making a force balance on the sphere, show that

cos в = (5C.3-1)
2
U L
What happens when П goes to zero?

Mass of
sphere = m

Fig. 5C.3. A simple mechanical system for illustrating concepts
in stability.

3
J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York (1958); R.
R. Rothfus (1948), J. E. Walker (1957), and G. A. Whan (1956), Doctoral theses, Carnegie Institute of
Technology (now Carnegie-Mellon University), Pittsburgh, Pa.
4 W. Tiedt, Berechnung des laminaren u. turbulenten Reibungswiderstandes konzentrischer u. exzentrischer
Ringspalten, Technischer Bericht Nr. 4, Inst. f. Hydraulik u. Hydraulogie, Technische Hochschule,
Darmstadt (1968); D. M. Meter and R. B. Bird, AlChE Journal, 7, 41-45 (1961) did the same analysis using
the Prandtl mixing length theory.