Page 275 - Bird R.B. Transport phenomena

Problems 259

(b) Verify that the result in (a) simplifies to the Newtonian result when n goes to unity.
(c) Show that the mass flow rate in the annular region is given by

(d) What is the mass flow rate for fluids with n = I?
(e) Simplify Eq. 8B.3-2 for the Newtonian fluid.

8B.4 Flow of a polymeric liquid in a tapered tube. Work Problem 2B.10 for a power law fluid,
using the lubrication approximation.
8B.5 Slit flow of a Bingham fluid. 1 For thick suspensions and pastes it is found that no flow oc-
curs until a certain critical stress, the yield stress, is reached, and then the fluid flows in such a
way that part of the stream is in "plug flow." The simplest model of a fluid with a yield value
is the Bingham model:

J ^ T o (8B.5-1)
when t > r
0
in which т 0 is the yield stress, the stress below which no flow occurs, and д 0 is a parameter
V
with units of viscosity. The quantity т = |(T:T) is the magnitude of the stress tensor.
Find the mass flow rate in a slit for the Bingham fluid (see Problem 2B.3 and Example
8.3-2). The expression for the shear stress r xz as a function of position x in Eq. 2B.3-1 can be
taken over here, since it does not depend on the type of fluid. We see that \r \ is just equal to
xz
the yield stress r at x = ±x , where x is defined by
0 0 0
r = ° L x (8B.5-2)
o L 0
(a) Show that the upper equation of Eq. 8B.5-1 requires that dvjdx = 0 for \x\ ^ x , since
Q
T XZ = -r\dv /dx and r xz is finite; this is then the "plug-flow" region. Then show that, since for
z
x positive, у = -dv /dx, and for x negative, у = +dv /dx, the lower equation of Eq. 8.3-5
z z
requires that
{-fio(dv /dx) + T for +x < x < +B
= z 0 0
TY7 ~^ / J / J A - T 0 f o r - B < x < - ;
(b) To get the velocity distribution for + j < x < +B, substitute the upper relation from Eq.
0
8B.5-3 into Eq. 2B.3-1 and get the differential equation for v . Show that this may be integrated
z
with the boundary condition that the velocity is zero at x = В to give

+ s
What is the velocity in the range \x\ ^ x ? Draw a sketch of v (x).
z
0
(c) The mass flow rate can then be obtained from
f +B f B f B ( dv\
w = Wp v dx = 2Wp v dx = 2Wp x[ --± )dx (8B.5-5)
J z J z
-в o J Xo \ ax/

1
E. C. Bingham, Fluidity and Plasticity, McGraw-Hill, New York (1922), pp. 215-218. See R. B. Bird,
G. C. Dai, and B. J. Yarusso, Reviews in Chemical Engineering, 1,1-70 (1982) for a review of models with a
yield stress.