312 Chapter 6—Solution of Viscous-Flow Problems
16. Parallel-disk rheometer—M. Fig. P6.16 shows the diametral cross section
of a viscometer, which consists of two opposed circular horizontal disks, each of
radius R, spaced by a vertical distance H; the intervening gap is filled by a liquid
of constant viscosity µ and constant density. The upper disk is stationary, and the
lower disk is rotated at a steady angular velocity ω in the θ direction.
There is only one nonzero velocity component, v
θ
, so the liquid everywhere
moves in circles. Simplify the general continuity equation in cylindrical coordi-
nates, and hence deduce those coordinates (r?, θ?, z?) on which v
θ
may depend.
Now consider the θ-momentum equation, and simplify it by eliminating all
zero terms. Explain briefly: (a) why you would expect ∂p/∂θ to be zero, and (b)
why you cannot neglect the term ∂
2
v
θ
/∂z
2
.
Also explain briefly the logic of supposing that the velocity in the θ direction
is of the form v
θ
= rωf(z), where the function f(z) is yet to be determined. Now
substitute this into the simplified θ-momentum balance and determine f(z), using
the boundary conditions that v
θ
is zero on the upper disk and rω on the lower
disk.
Why would you designate the shear stress exerted by the liquid on the lower
disk as τ
zθ
? Evaluate this stress as a function of radius.
17. Screw extruder optimum angle—M. Note that the flow rate through the
die of Example 6.5, given in Eqn. (E6.5.10), can be expressed as:
Q =
c(p
2
− p
3
)
µD
,
in which c is a factor that accounts for the geometry.
Suppose that this die is now connected to the exit of the extruder studied in
Example 6.4, and that p
1
= p
3
= 0, both pressures being atmospheric. Derive
an expression for the optimum flight angle θ
opt
that will maximize the flow rate
Q
y
through the extruder and die. Give your answer in terms of any or all of the
constants c, D, h, L
0
, r, W , µ, and ω.
Under what conditions would the pressure at the exit of the extruder have its
largest possible value p
2max
? Derive an expression for p
2max
.
18. Annular flow in a die—E. Referring to Example 6.5, concerning annular
flow in a die, answer the following questions, giving your explanation in both cases:
(a) What form does the velocity profile, v
z
= v
z
(r), assume as the radius r
1
of the
inner cylinder becomes vanishingly small?
(b) Does the maximum velocity occur halfway between the inner and outer cylin-
ders, or at some other location?
19. Rotating rod in a fluid—M. Fig. P6.19(a) shows a horizontal cross section
of a long vertical cylinder of radius a that is rotated steadily counterclockwise with
an angular velocity ω in a very large volume of liquid of viscosity µ. The liquid
extends effectively to infinity, where it may be considered at rest. The axis of the
cylinder coincides with the z axis.