Page 278 - Bird R.B. Transport phenomena

262 Chapter 8 Polymeric Liquids

8C.3 Verification of Giesekus viscosity function. 5
(a) To check the shear-flow entries in Table 8.5-1, introduce dimensionless stress tensor com-
ponents Tj = (\/T) )TJJ and a dimensionless shear rate Г = Ay, and then show that for steady-
t 0
state shear flow Eq. 8.5-4 becomes
T - 2TT - a(T 2 + 7 p = О (8С.З-1)
xx yx xx
T - a(1% + T] ) = О (8С.З-2)
yy y
T ~ TT - aT (T 4- T ) = - f (8C.3-3)
yx yy yx xx yy
There is also a fourth equation, which leads to T zz = 0.
(b) Rewrite these equations in terms of the dimensionless normal-stress differences N^ = T xx
- T and N 2 = T yy - T , and T .
yy
yx
zz
(c) It is difficult to solve the equations in (b) to get the dimensionless shear stress and normal-
stress differences in terms of the dimensionless shear rate. Instead, solve for N v T , and Г as
yx
functions of N :
2
Г„,Ш^» (SC.3-4,
a{\ - Щ
(1-2.)N P
+ 2
a(l - N ) 4
2
(d) Solve the last equation for N as a function of Г to get
2
N 2 = fix) = (1 - *)/[! + (1 " ^OL) \ (8C.3-7)
X
where
л + 1 ^ Г 1 ^ 1 _ _ . . . .
= = x 4 a ( 1 a ) r 2 + ( 8 C 3 8 )
8 ( l - a)T 2
Then get the expression for the non-Newtonian viscosity and plot the curve of 17(7).
8C.4 Tube Flow for the Oldroyd 6-Constant Model. Find the mass flow rate for the steady flow
in a long circular tube using Eq. 8.5-3.
6
8C.5 Chain Models with Rigid-Rod Connectors. Read and discuss the following publications:
M. Gottlieb, Computers in Chemistry, 1, 155-160 (1977); O. Hassager, /. Chem. Phys., 60,
2111-2124 (1974); X. J. Fan and T. W. Liu, /. Non-Newtonian Fluid Mech., 19, 303-321 (1986);
T. W. Liu, J. Chem. Phys., 90, 5826-5842 (1989); H. H. Saab, R. B. Bird, and С F. Curtiss,
/. Chem. Phys., 77, 4758-4766 (1982); J. D. Schieber, /. Chem. Phys., 87, 4917-4927, 4928-4936
(1987). Why are rodlike connectors more difficult to handle than springs? What kinds of prob-
lems can be solved by computer simulations?

5
H. Giesekus, /. Non-Newtonian Fluid Mech., 11, 69-109 (1982).
6 M. C. Williams and R. B. Bird, AIChE Journal, 8, 378-382 (1962).