Page 167 - Bird R.B. Transport phenomena

Problems 151

Fig. 4D.3. Rotating disk in a circular tube.

4D.3 Flows in the disk-and-tube system (Fig. 4D.3). 9
(a) A fluid in a circular tube is caused to move tangentially by a tightly fitting rotating disk at
the liquid surface at z = 0; the bottom of the tube is located at z = L. Find the steady-state veloc-
ity distribution v (r, z), when the angular velocity of the disk is П. Assume that creeping flow
e
prevails throughout, so that there is no secondary flow. Find the limit of the solution as L —> «>.
(b) Repeat the problem for the unsteady flow. The fluid is at rest before t = 0, and the disk
suddenly begins to rotate with an angular velocity ft at t = 0. Find the velocity distribution
v (r, z, t) for a column of fluid of height L. Then find the solution for the limit as L —> oo.
Q
(c) If the disk is oscillating sinusoidally in the tangential direction with amplitude П , obtain
о
the velocity distribution in the tube when the "oscillatory steady state" has been attained. Re-
peat the problem for a tube of infinite length.
4D.4 Unsteady annular flows. 10
(a) Obtain a solution to the Navier-Stokes equation for the start-up of axial annular flow by a
sudden impressed pressure gradient. Check your result against the published solution.
(b) Solve the Navier-Stokes equation for the unsteady tangential flow in an annulus. The fluid
is at rest for t < 0. Starting at t = 0 the outer cylinder begins rotating with a constant angular
velocity to cause laminar flow for t > 0. Compare your result with the published solution. 11
4D.5 Stream functions for steady three-dimensional flow.
(a) Show that the velocity functions pv = [V X A] and pv = [(Vi/^) X (V^ )] both satisfy the
2
equation of continuity identically for steady compressible flow. The functions ф ф and A
и ъ
are arbitrary, except that their derivatives appearing in (V • pv) must exist.
(b) Show that, for the conditions of Table 4.2-1, the vector A has the magnitude -рфк 3 and
the direction of the coordinate normal to v. Here h is the scale factor for the third coordinate
3
(see §A.7).
(c) Show that the streamlines corresponding to Eq. 4.3-2 are given by the intersections of the
surfaces ф = constant and ф = constant. Sketch such a pair of surfaces for the flow in Fig. 4.3-1.
х
2
(d) Use Stokes' theorem (Eq. A.5-4) to obtain an expression in terms of A for the mass flow
rate through a surface S bounded by a closed curve C. Show that the vanishing of v on С does
not imply the vanishing of A on С

9
W. Hort, Z. tech. Phys., 10, 213 (1920); С. Т. Hill, J. D. Huppler, and R. B. Bird, Chem. Engr. ScL, 21,
815-817(1966).
10
W. Miiller, Zeits. fur angew. Math. u. Mech., 16, 227-228 (1936).
11
R. B. Bird and C. F. Curtiss, Chem. Engr. Sri, 11,108-113 (1959).