COMPARISON O F CONSTITUTIVE E Q U A T I O N S FOR VISCOELASTIC F L U I D S D . C . B O G U E A N D J. 0 . D O U G H T Y Department of Chemical and Metallurgical Engineering, University of Tennessee, Knoxuille, Tenn.
Constitutive equations for viscoelastic fluids are reviewed in the following framework: (1 ) theories involving a time derivative of stress (Oldroyd theories, especially those of Bird and coworkers and of White and Metzner), and (2) theories involving an integrated deformation history (especially those of Pao; Bernstein, Kearsley, and Zapas; and Bogue). Emphasis i s on theories explicit enough to yield predictions for steady-state viscosities and normal stress functions, for linear dynamic behavior, and for recoil phenomena. The formation of dimensionless groups for correlating data i s also discussed.
HE central problem in engineering rheology is that of findT i n g explicit constitutive equations, with a small number of constants, to correlate the variety of viscous and elastic phenomena observed experimentally. Once such a n equation (or equations) is found, one will be able to use it to solve simple problems directly and to suggest dimensionless groups for treating complicated problems indirectly. Ideally one would hope to construct a theory starting with basic microscopic variables: particle size, particle shape, etc., in the case of suspensions; chain geometry, molecular weight, etc.: in the case of polymeric materials. For dilute suspensions of well defined particles there has been good progress along these lines (23). For dilute polymer solutions (and in some theories for pure polymers) there has also been good progress, developing from the early theories of Rouse and of Bueche (8, 25). These theories apply to linear deformations or to steady shearing deformations a t low shear rates. As to nonlinear behavior, one has shear-dependent viscosities in the theories of Bueche ( 4 ) , Yamamoto ((34,Pao (76, 77), and Takemura (22). The theories of Yamamoto and Pao also treat normal stresses. With the possible exception of Pao’s continuum theory, however, the framework of these theories is not such that one can undertake the analysis of complicated flow geometries. Turning to theoretical continuum mechanics one finds the important result that there are no more than three independent material functions [the three “viscometric functions” of Coleman and No11 (5, 73)]for correlating simple shearing (viscometric) flows. But the theory does not preclude the possibility of interdependence of the functions. For polymer solutions one has a notable similarity between the frequency-dependent dynamic viscosity (in a linear deformation) and the sheardependent steady-state viscosity (in a large deformation) ; in the turbulent flow of concentrated polymer solutions Meter ( 7 7 ) has supplied the missing “elastic” parameter by considering only the viscosity function; and, as a final example, high normal stress effects are usually associated with strongly non-Newtonian viscosities. These diverse facts remind us that the phenomenological functions which give rise to viscous and elastic effects are not completely independent, as one knows they will not be, from molecular considerations. This is the motivation for considering the kinds of theories discussed here. The discussion is rnostly oriented toward polymer systems, with a few remarks on applications in suspension rheology.
Clarsiflcation of Theories
Any classification of theories is rather arbitrary, since many of them are, under the appropriate conditions, special cases of others. I t seems reasonable, however, to define two major groups: those involving (1) a n explicit time derivative of stress, and (2) a n integral of the deformation history. The former includes Oldroyd-type theories and the latter those of the Coleman-No11 and Bernstein-Kearsley-Zapas type. (For certain kinds of flows the latter type can be expanded into a series to give the familiar Rivlin-Ericksen tensors.) The arbitrariness of the classification is evident when one observes that the original Oldroyd formulation (74) and subsequent developments by Walters and coworkers (24, 26, 27) are in terms of a convected integral of strain rate. However, the working form of these equations involves a time derivative of stress and that is the criterion for the classification. Also, because of the variety of equations available, one has to establish some guidelines as to what constitutes a suitable theory. For the present purposes the desirable properties of a theory were set down as follows: (1) I t should be coordinate-invariant-Le., cast in tensor notation, (2) it should be capable of predicting shear-dependent (non-Newtonian) viscosities, (3) it should predict normal stresses in steady shear, (4) it should be able to reconcile the dynamic experiments of linear viscoelasticity, (5) it should explain stress relaxation phenomena, and (6) it should be explicit, with definite constants to be fitted to experimental data. Not all of these requirements were satisfied in all cases, however. Theories Involving a Time Derivative of Stress
One of the earliest tensor theories for viscoelastic fluids was that of Oldroyd (74, 75), who developed the concepts of convected coordinates and convected time derivatives and used them to generalize linear viscoelastic (spring and dashpot) models to nonlinear deformations. In subsequent work Walters showed in a compact formulation how the Oldroyd theories may be generated from integral-type expressions by a proper choice of the relaxation function (26). To introduce non-Newtonian viscosities, however, it is necessary to add cross terms (products of stress and strain rate with themselves or each other). I n the most general form of this sort Oldroyd introduced eight material constants (75),while Walters has investigated a form involving five material constants and a relaxaVOL. 5
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tion function (27). Two other forms which seem the most promising for practical applications are discussed in detail below. Throughout the discussion it is understood that while convected coordinates play a part in the motivation of the model, the equations actually presented are referred to a fixed coordinate system and may be written out in any working coordinate system of interest. Three-Constant Oldroyd Model. A three-constant form has been studied extensively by Bird and coworkers, particularly Williams and Bird (32, 33) :
1
ZGi A ,
o l d r o y d ~ ~ E t n - , i l 3- Constant \,--,E-- using --small - X vertical placement not si nificant
BKZ(specia1 case)
70
$
K , shear rate Figure 1 . Viscosity function
where 90, h l ,
A2
1
log
= material constants
5
p
I
Oldroyd-/ 3- Constant
\ \i
-BKZ(special case)
derivative 1
E -
2
I
( g k j a i , , - dit), the vorticity tensor
vertical placement not significant
L i f i e d Coleman-Noll
In steady shear, with K as the shear rate, this equation reduces to : a . Steady Shearing Flow
711 ~
- 722 KZ
=2to(y) 1 722
-
733
+ - X?K2
= 0
(3)
b. Recdl Following Steady Shearing Flow Stress Deformation Shear Rate c,,(t)
(4)
Thus the Oldroyd equation predicts two limiting viscosities: shear rates and ~ 0 ( h ~ / ha1t )high shear rates. Typical curves for the predictions of this model are shown in Figures 1 and 2 . The Oldroyd equation also has the capability, not shown here, of correlating linear dynamic experiments with the same three constants (32). As these authors emphasize, this important property allows one to take advantage of the theoretical and experimental work of the physical chemists and others concerned with a molecular description. I n recoil experiments (Figure 3) one suddenly removes the shearing stress (but not the normal stresses) and observes the deformation. For such flows one has 712 = 0, but 711, 7 2 2 , 7 3 3 = nonzero. Also dlz K ( t ) , d l l = d22 = d33 = 0. To avoid physical arguments about the initial conditions it is easier to use
v
I
9 0 a t low
I
F i g u r e 3. tions
S c h e m a t i c r e p r e s e n t a t i o n of d e f o r m a -
analytically. However, for low shear rates one can neglect the cross terms and write
(5) The boundary condition is K ( 0 ) = K OE tion (with b -+ 0) is
K(t) = -KO
(A1
-
X2)
rl20/so.
The solu-
e-t;,,‘
A2
(6)
Thus the recoverable strain is given by where the time constant, b, is vanishingly small ( b
