Page 122 - Bird R.B. Transport phenomena

Problems 107

Disk at 2 = В rotates (a) First consider the problem where the annular region is
Fluid with viscosity with angular quite narrow—that is, where к is just slightly less than
/л and density p is velocity П unity. In that case the annulus may be approximated by a
held in place by thin plane slit and the curvature can be neglected. Show
surface tension Disk at z = 0 is fixed that in this limit, the velocity distribution is given by
Both disks have
radius R
and R » В 1 - к +1 (3B.6-1)

Fig. 3B.5. Parallel-disk viscometer. where f = r/R.
(b) Next work the problem without the thin-slit assump-
tion. Show that the velocity distribution is given by
(a) Postulate that for small values of П the velocity pro-
2
files have the form v r = 0, v z = 0, and v e = rf(z); why does (1 1 - к 2 I) - (1 - к ) In \
this form for the tangential velocity seem reasonable? Pos- (3B.6-2)
tulate further that 2P = ^(r, z). Write down the resulting
simplified equations of continuity and motion.
(b) From the ^-component of the equation of motion, ob- 3B.7 Momentum fluxes for creeping flow into a slot
tain a differential equation for /(z). Solve the equation for (Fig. 3.B-7). An incompressible Newtonian liquid is flow-
f(z) and evaluate the constants of integration. This leads ing very slowly into a thin slot of thickness IB (in the у di-
ultimately to the result v 0 = Clr(z/B). Could you have rection) and width W (in the z direction). The mass rate of
guessed this result? flow in the slot is w. From the results of Problem 2B.3 it can
(c) Show that the desired working equation for deducing be shown that the velocity distribution within the slot is
the viscosity is /л = 2BTJTT£IR\
(d) Discuss the advantages and disadvantages of this in-
strument.
at locations not too near the inlet. In the region outside the
3B.6 Circulating axial flow in an annulus (Fig. 3B.6). A slot the components of the velocity for creeping flow are
rod of radius KR moves upward with a constant velocity v 0
through a cylindrical container of inner radius R contain- 2w (3B.7-2)
2 2
ing a Newtonian liquid. The liquid circulates in the cylin- (X 2 + y )
der, moving upward along the moving central rod and 2w
moving downward along the fixed container wall. Find x 2 + y ) (3B.7-3)
2 2
the velocity distribution in the annular region, far from the
end disturbances. Flows similar to this occur in the seals of v=0 (3B.7-4)
some reciprocating machinery—for example, in the annu- Equations 3B.7-1 to 4 are only approximate in the region
lar space between piston rings.
near the slot entry for both x > 0 and x < 0.

Rod of radius KR
moves upward with
velocity v 0 \

4 V

It 11
Cylinder of length L
" and inner radius R
(with L » R)

Fig. 3B.6. Circulating
flow produced by an
axially moving rod in a Fig. 3B.7. Flow of a liquid into a slot from a semi-infinite
closed annular region. region x < 0.