Page 277 - Bird R.B. Transport phenomena

Problems 261

8C.1 The cone-and-plate viscometer (Fig. 2B.11). 3 Review the Newtonian analysis of the cone-
and-plate instrument in Problem 2B.11 and then do the following:
(a) Show that the shear rate у is uniform throughout the gap and equal to у = -у вф = С1/ф .
0
Because of the uniformity of y, the components of the stress tensor are also constant through-
out the gap.
(b) Show that the non-Newtonian viscosity is then obtained from measurements of the
torque T and rotation speed £l by using
z
£&

(c) Show that for the cone-and-plate system the radial component of the equation of mo-
tion is

if the centrifugal force term -pv\/r can be neglected. Rearrange this to get

0 = -дтг /д In r + (Тфф - т ) + 2(тде - т, ) (8С.1-3)
00
гг
т
Then introduce the normal stress coefficients, and use the result of (a) to replace dir /d In r by
n
дттоо/д In r, to get
дщ /д lnr=-(%+ 2^ )y 2 (8C.1-4)
2
о
Integrate this from r to R and use the boundary condition ir (R) = p to get
a
n
Щв(г) = irooiR) ~ (% + 2^ )y 2 In (r/R)
2
= Pa~ % 7 2 - (% + 2^ )y 2 In (r/R) (8C.1-5)
2
in which p a is the atmospheric pressure acting on the fluid at the rim of the cone-and-plate
instrument.
(d) Show that the total thrust in the z direction exerted by the fluid on the cone is
^=1*1 [Що(г) ~ Va\ rdrdO = lirR %y 2 (8C.1-6)
2
Jo Jo
From this one can obtain the first normal-stress coefficient by measuring the force that the
fluid exerts.
(e) Suggest a method for measuring the second normal-stress coefficient using results in part
(c) if small pressure transducers are flush-mounted in the plate at several different radial loca-
tions.
8C.2 Squeezing flow between parallel disks (Fig. 3C.1). 4 Rework Problem 3C.l(g) for the power
law fluid. This device can be useful for determining the power law parameters for materials
that are highly viscous. Show that the power law analog of Eq. 3C.1-16 is

/ и
Г 1/„, ,o C 2 л )

3
R. B. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics,
Wiley-Interscience, New York, 2nd Edition (1987), pp. 521-524.
4
P. J. Leider, Ind. Eng. Chem. Fundam., 13, 342-346 (1974); R. J. Grimm, AIChE Journal, 24, 427-439
(1978).