Therefore,
T’ = 328.0’ K.
(3) .iris. R.. Rudd. D. F.. Amundson, N. R., Chem. En?. Sci. 12, 88 (1960). (4) Bellman! R., “Dynamic Programming:’‘ Princeton University Press, Princeton. N. J.. 1957. (5) Chang, S. S. L.; “Dynamic Programming and Pontryagin’s Maximum Principle.” Proceedings of Dynamic Programming It-orkshop (Second Annual Pre-JXCC it’orkshop); pp. 109-83, Boulder, Colo., June 1961. (6) Converse. A. O., “Computer Optimization of a Multi-stage .Allocation Problem by Means of a Non-imbedding Technique.” 55th annual meeting A.1.Ch.E. (Preprint 116). (7) Fan, L. T., iVang, C. S.. ”The Discrete Maximum Principle,” Chap. 8: iViley, New York, 1964. (8) Katz, s.,ISD. E N G . CHEM.F U N D A M E N T A L S 1 , 226 (1962). (9) Pontryagin, L. S., et al.. “The Mathematical Theory of Optimal Processes,’’ tr. by K. N. Trirogoff, Interscience, New York, 1962. ,
T 2 = 291.5’ K . T 3 = 274.1’ K . Acknowledgment
T h e authors are indebted to Stanley Katz and R. .4ris for their suggestions which clarified many major points of this paper. literature Cited (1) Aris, R., “Discrete Dynamic Programming,” Chap. 11, Blaisdell Publishing Co., New York, 1963. (2) Aris, R., “Optimal Design of Chemical Reactors,” Academic Press, New York. 1961.
RECEIVED for review January 8, 1963 ACCEPTED October 28, 1963
OSCILLATORY BEHAVIOR OF NORMAL STRESSES IN VISCOELASTIC FLUIDS M I C H A E L C. W I L L I A M S A N D
R. B Y R O N B I R D
Chemical Engineering Department, L‘niuersity of Wisconsin, Madison, Wis. This paper deals with the time-dependent behavior of normal stresses exhibited b y fluids in small-cytplitude oscillation. The equations of motion for a cone-and-plate system are solved to relate the amplitude and phase relationships of the oscillating stresses to experimental measurements. The results are expressed in terms of a “complex normal stress coefficient” and a “normal stress displacement function.” Finally, the predictions of several phenomenological models (Oldroyd, and Coleman and NOH)are compared to suggest ways in which data may ultimately be analyzed. A three-constant Oldroyd model i s particularly helpful in estimating orders of magnitude of various effects. A new relation has been found between the normal stress displacement function and the imaginary part of the complex viscosity.
may be classified as to the way in which they restrict the material movement while measuring stress : A N Y RHEOLOGICAL EXPERIMENTS
SHEARSTRESS Steady motion with large deformation Unsteady motion with small deformation NORMAL STRESS
Steady motion \vith large deformation Unsteady motion \vith small deformation Phenomena under the first category of shear stress have been widely studied under the name ”non-Nebvtonian” flow, and those under the second category are usually described by “linear viscoelasticity.” Phenomena of the first type under normal stress received little attention before about 1947. The second type under normal stress does not yet seem to have been exploited, either as an analytical tool for categorizing high polymeric substances or for further testing of molecular or continuum rheological models. Only three experiments of this type have come to our attention. \Yeissenberg ( 2 3 ) . using 6 cone-and-plate geometry, first demonstrated the character of the phenomenon. Later, Marvin ( 7 7) conducted an exploratory investigation measuring stress a t right angles to the direction of shear for samples strained sinusoidally in simple shear. I n addition, Tt’ard and Jenkins (22) studied oscillatory motion of rubbery solids benveen rotating circular disks and related their results to Rivlin‘s equations for the elasticity of solids. The only theoretical consideration of fluid phenomena of this type seems to be that of Coleman and Sol1 ( 3 ) ,\vho have explored the behavior of their “simple fluid” model. 42
I&EC
FUNDAMENTALS
I n the folloiving development we propose a method for measuring and analyzing the time-dependent behavior of normal stresses in liquids. Complex Normal Stress Coefficient and Displacement Function
Consider a fluid placed betxveen two large flat plates; the lower one is fixed and the upper one is made to execute a sinusoidal motion in the x-direction with frequency w (in radians per second), If the coordinate direction perpendicular to the plates is the z-direction, then in the region between the plates. L’, = v , ( z . t ) . ~l~ = 0:and u , = 0. T h e only nonvanishing elements of the rate of deformation tensor eik = (l,’Z) ax&) (bc,,’d.~fi)]will be e,,(z,t) = e , , ( z > t ) = ( l 1 bo, dz). \\.’hen the amplitudes of vibration are small, after the initial transients have damped out, all quantities will be oscillating sinusoidally:
[(at.,,
+
p
=
Re
{G
f p0e2”‘}
(4)
Here, p is the isotropic pressure and T ~ ,= x , ) - PS,,. where ri,is the pressure tensor and a,, is the Kronecker delta. T h e quantities u ith the superscript zero are complex amplitudes.
and c and d, are complex time-independen t functions \vhich In a n eadirr depend upon the frequency of oscillation LO publication ( 3 6 ) . the displacement dJ of the normal stress T , was not included, and consequently some of the results given t1iei.e Lvere slightly in error. Note that the isotropic pressure b, and the normal stresses T , vary ~ with a frequency twice that of the o h e r quantities. 'l'lnis comes about because the stresses normal to a plane should not depend on whether the motion of that pldne is in the positive or negative x-direction. Hence, the normal stress must execute a full cycle of oscillation during the same time that the shear stress performs a half-cycle of its oscillation, 'l'hese frequencv relationships were observed in 12:eisserihrrg's original experiment (23) and \vere also deduced by Coleman and Noll ( 2 ) in their continuum theory of simple fluids. /Ye \vi11 follo\v the lead of the linear viscoelasticians l e g . , Ferry (.,I)], \vho for rnany years have characterized their Huids in terms of a complex viscosity q* defined for small-amplitude oscillatory motion by : 7r2(I
=
- 2 O*Pr20
~
(0)
T h e complex viscosity is ( h e n conventionally broken down into real and imaginary parts:
- ill"
q* =
(7)
T h e study of experimental curves of ?'(a) and ~ " ( w has ) been helpful in elucidating and classifying polymeric structures. Similarly, it seems use~'u1to define several quantities associated with the normal stress behavior. \'$'e propose to define a normal stress displacement function, Cd$ as follows :
Re i d , - d,{
=
-4('ie,,0;2
(8)
We further propose the follo\ving definition of a coniplex normal stress coefficient, ::*! thus: T,,'
-
T,,'
=
-4(* ( e Z r @ Y
19)
Next: it is assumed that the arb terms i n Equation I 5 are small compared \vii.h the other terms and that the ncwmal componcnts of x perpendicular to the flow direction a r c eclu:il 'L'he first assumption might be Innde intuitivvly. sincr the trimsport of 6-niomentuni in the r-dircctjon is ~w1~tv-trrlto hr. irnimportmt. but it call be further justified b y 'in order-ofiriagriitude analvais, as sho\Yn i n Appendix 4. 'I'he uwonr! assumption is still controversinl k o r iteody flow \ < r have thr following infimnatinn: Several independent e x p r r i m e n t r r . ~ ~ Kotaka, Rurata. and 1 dmiura (5); Pilpel ( , I T ) , Phili1,poR [ / J ) ; R o b e m i 77)- - h d v e presented evidericc. that the second assumption is either valid 01 clt Irnst verv good for Inan\' fiuids O n the other hand; Lodge (SI,Markovitz J n d R r w v n ( I O ) . and Sdkiadis i 7,Y) have given evidencr t o the contrarv. althoiigh the Inst-Inenticinrti work hns hren sharplv criticized b\- il'hilr (2.1), 'I he genrrnl continuum theories, su[:h as that 0 1 C ' ( . i l ~ m a n and Sol1 ( 2 , J'), permit inequalitv of the normal strrys components in question but d o not specifically excludr the possibility of equdht) , -1l'he rnolecdar theory develolird hv Yamnmoto ( ~ 1 7lrdds ) to the conc!urion that "although normal comporirnts of the Stress are in genrrdl different from rack other, those along the directions orthogonal to thta flow ciiiwtion are almost isotropic." If the second aswmption \cere not made, it would be necessary to modif! the entire annlvsis ti!. introducing additiorial functions to describe the totnl strc-ss distribution-e.g.. Re { I f , , -- d,] = --.lbd ' p I 2 @ I 2 and T~~~ T,," = - 4 3 * (e>..O,i*. / V e do not explore that possihjlitv further here. M'ben the t\\Io assumpticins arc. m d r , the equations or motion become :
T h e quantity { * is then broken down into real and imaginary components : (* = ( ' - j i " (10) This kind of definition of a complex normal strrss coefficient was suggested earlier by {.hi=authors (26). l h e problem is to determine equations for evaluating I": and i" from experimentally measurable quantities dnd then to studv the frequency dependence of these three material functions as predicted by various theories.
k?=
r',
Oscillatory Cone-and-Plate System
-
at
! aR!? 7
2 [VV
+
re
h$
OSULIATING INPUT VELOCITY V+ ATe=$
I
f
= em, =
I I
A
2
118)
( V v ) + ] are
(see Table I ) :
e,*
Red
T h e cot 9 term in Equation 18 cannot he neglected in thi!: unsteady-state problem as is customary i n the strady-state analvsis of the cone-and-plate system This is not inconsistrnt
Consider a viscoelastic material in the cone-and-plate system depicted in Figure I . It is postulated that t$ = u 6 ( r , 0, t ) and that u , = 0 and i ~ g= 0. For these postulates, the only nonvanishing components of e
2 cot .P.
bQ
\
I-
br
T h e approximate expres:;ion for eRq is valid because 0 is very nearly r . ' 2 (see Appendix -4).For these same postulates the equation of continuity is identically satisfied and the equations of motion [e.g.. Bird. SteTvart. and Lightfoot ( 7 ) 1 become:
Figure 1. *(I
OSCILLATING OUTPUT NORMAL STRESS rr, AT e =
$
Oscillating cone and plate system with
= 1@ VOL. 3
NO. 1
FEBRUARY 1964
43
with the previous neglect of a cot 0 term in eo,, as is demonstrated in Appendix A. M'e are interested in a sinusoidal solution of these equations mhich is valid after the initial transient disturbance has abated. The boundary conditions to be satisfied are: At 0 = r / 2 , At 0 = 00,
0
L+ =
u, = rR
(1 9)
X Re { e z w f )
(20)
From Equation 17 it is found that is nearly independent of 0, since cot 0 G= 0. [Actually, combination of Equations
16 and 17 shows
a;;o/?2: ; -
-
~
-
- 8. which is at most 0.02
-
in this problem. \$''e shall see that (droo,'d In r ) is the quantity to be measured.] Next, Equation 18 may be \vritten for sinusoidal small amplitude motion, and? in so doing, the components of roemay be given in terms of eoo and the complex viscosity q*, \\'hen the approximate expression for eod in Equation 11 is used, one finally gets the follouing ordinary differential equation for the complex velocity amplitude u g o :
This equation may be rebvritten in the following form: a2 -O@.(
ap
cos $) -
CY2
(umO
cos $) = 0
where a 2 e ( i w pr Z / q * ) - 1 and 4 z ( ~ , ' 2 )- 0. solved with the boundary conditions that : At $ = 0: At $ =
$0,
o$O =
(22)
*
- (T~,O
- ~ o o ~ ) l + = o sinh a$o
(29) - R2cos2 $0 Actually, what one measures at I$ = 0 is the time-dependent behavior of roo as a function of radial distance, but r o ois simply related to (roe- roo) at i: = 0 according to Equation 16: {
. (-7)
-
_aroo _
~
b l n r +=o
=
( T ~ O- T o o )
= ( T O O - 708) *=0
, *=o
0
(23) (24)
= rCl
! OL pcos p $0 sinh a$ rR cos 4 sinh 040
(25)
I\ hich is the complex amplitude for the velocity oscillations; it should be kept in mind that this expression contains the complex viscosity q*. Next \ve apply the definitions of {' and j-* and use Equation 11 to get:
I t should be pointed out that nowhere in the above analysis is it necessary to know how the pressure tensor x,, is decomposed into pS,, and T?,. In this geometry: the experimental observation of ~ 8 as 0 a function of r leads directly to information about (s,$ - 7 8 0 ) in terms of which {* and cd were defined. In a simpler system, such as the parallel plates in the first section, measurement of the oscillatory behavior of s,, does not seem [For that to yield data from \vhich [* can be found. system, a n analysis similar to that given for the cone-andplate leads to the following analog to Equation 29:
where d 2 = i o p , q* and b is the distance between the parallel = ~ plates. If it \sere possible to measure both T , , ~ ~ ~and rzro1 r = ~ : then Equation 29a could be evaluated experimentally; it is not clear to us that such a measurement is feasible. If the second assumption is used in conjunction Ivith the condition T,, T~~ T~~ = 0 then (r,,O - a r r 0 )a = ~ can be replaced by - 3TZz01 r = ~ . Even so? { * cannot be obtained from the measurbecause the function p ( t . t ) is unknown.] The able rrz~r=O axially oscillating coaxial cylinder arrangement suffers from the same disadvantage. I t is postulated that:
+
+
where T , 2 < @ < 3r,'2. T h a t is, one measures experimentally a displacement D ? an amplitude A , and a phase shift @: all as a function of the frequency w. From this information it is easy to reconstruct :
b$ rR
Re { d , I N PUT OSCILLATION,
\
-
do)'+=^)
=
D
v+ = mRe(eilYt) A T w k
Figure 2. Phase relations among the measurable quantities
wt
OUTPUT OSCILLATION
44
I & E C FUNDAMENTALS
(30)
This is to be
T h e solution is:
""
Substitution of Equation 25 into Equations 26 and 27 then gives expressions for cd and { * ; when these expressions are evaluated at 4 = O? \se find:
T,,
ea]= e cos ( w t - X 1 ATI/I=O
(32)
{ (Tmq0
Re
Im
-
{ (TqO0 -
Teeo)
=o]
)!,
=
A
COS
2@
(33)
2~
(34)
= - A sin
TBQ0)i!,=0)
The phase relations among the various quantities are shown in Figure 2. Finalli- \ve express the material functions I d , {’, and { ” in terms of the measurable quantities D,A , @ %and $ 0 : {d
=
- [D:R2fi-K2
cos2
$0]
I’ =
+
[ and Meter (72). For this model, sample values of the model constants are available. \Ve (26) have found that a threeconstant modification of the Oldroyd model is convenient here, the three constants being a lower limiting viscosity 70. a relaxa9 A B , In tion time A i , and a retardation time As. with X I terms of these constants: the model is (in Cartesian tensor notation, with summation convention) :
