TWO D I M E N S I O N L E S S G R O U P S RELEVANT IN T H E ,ANALY§IS OF S T E A D Y FLOWS OF VISCOELASTIC M A T E R I A L S GlANNl A S T A R I T A ' Chemical Engineering Department, University of Delaware, Newark, Del.
Five flow patterns (viscometric flow, pure shear, elongational flow, fourth-order flow, and a superposition of the first two) are analyzed in detail. The behavior of fluids characterized b y an essentially exponential rate of stress relaxation in such flows i s discussed. The discussion leads to the introduction of two dimensionless groups relevant for steady flows.
ISCOELASTIC flow problems have been analyzed in the Vliterature following two rather separate lines of thought. O n the one hand, theoretical analyses of completely rigorous character have been carried out for a few selected flow patterns, of such simplicity as to allow their analysis with a minimum of, if any, assumptions on the rheological nature of the fluid (8, 9, 7 7 , 74-77, 27). Matrix representation theorems (9, 76, 77), slow-flow asymptotic solutions (8, 27), and use of quantities expressed in a convected frame of reference (74, 75) have been the tools needed for carrying out such analyses. Asymptotic forms of the constitutive equation have been derived, and shown to be asyrnptotically equivalent to more general forms for the specific flow patterns considered. O n the other hand, such cogplicated flow patterns as boundary layer flows h;ve Eleen analyzed because of their engineering interest (3, 24). Often, asymptotic forms of the constitutive equation have been used without previous understanding of whether they might have a t least asymptotic validity for such flow patterns. A dimensionless group-Le., the Weissenberg number-has been used (3, 24), although its definition is not completely unequivocal outside of a restricted class of flow patterns ( 7 ) . Internal inconsistencies of such approaches have recently been realized, and a second dimensionless group, the Deborah number, has been defined and used in the literature
(7-5, 70, 73).
The first type of analysis suffers from limited scope; the second type, lack of rigorous foundation. I t is the main purpose of this paper t o contribute in filling the gap between the two approaches.
A convected frame, ti, is also considered, which coincides with the laboratory frame, x i , a t the instant of observation, as well as a convected frame, l', which coincides with Oi a t the instant of observation. The convention is taken that Greek letters identify quantities in the convected frame which are identified by the corresponding Latin letter in the laboratory frame. Thus, the rate-ofstrain tensor in the Et frame is q t j , and so on. The time lapse, r , is time measured backwards from the instant of observation; thus, by definition: at
7
= 0,
7 t j = eii
etj,
+
= l/Z ( W l
- U,,t)
1
(4)
The D operator does not commute with the raising and lowering of indices; thus, if Xiis a generic contravariant vector:
(5)
=
=
gi7g5se7s
-I/~
D,U
(6)
(1)
Two kinds of higher rate-of-strain tensors are considered : a covariant Rivlin-Ericksen tensor, (2)
All flow patterns are discussed with reference to a locally Cartesian laboratory frame, x i . A second Cartesian frame, 2 < ,is obtained by a rigid rotation of the former, such that, in this new frame, tensor i t , is diagonal. Thus, the 2i axes are the principal axes of the rate-of-strain ellipsoid. Present address, Chemical Engineering Department, Univer
sity of Naples, Naples, Italy.
= gij
I n particular, the contravariant rate-of-strain tensor is :
and the vorticity tensor, w i l , as: Wij
?.if
eij
= '/z Dgij
ecj
,il
q,t)
=
DXi # gijDXj
is defined as:
e t j = '/z
(3)
A
9ij
where yij is the metric of the ti frame, while g i j is the metric of x i , which numerically coincides with the Kronecker delta because x i is Cartesian. tijis, of course, diagnonal a t r = 0, when it coincides with ai,, but not necessarily a t any other value of 7 . T h e partial time derivative in the convected frame, transformed to the laboratory frame, is indicated by D. This is the convected derivative discussed by Oldroyd (74); the rate-ofstrain tensor may be shown to be:
Definition of Kinematic Tensors and Variables
The rate of strain tensor,
= xi
[I
(N )
At* =
(7)
D"-leij
and a contravariant White tensor, (.Y p
j
= DN-leil
(8)
The latter is not obtained by raising indices on the former. Definition 8 coincides with White's definition (23) in the case of incompressible liquids. Modified higher rate-of-strain tensors are defined as: VOL.
6 NO. 2
MAY 1967
257
Y/2 Tlij
where 0 is a constant scalar having the units of time. Physical interpretations of 0 have been discussed (7, 2, 5, 6, 73, 79) (see also the discussion following Equation 68). The physical components of the modified rate-of-strain tensors all have the same units, so that they can be directly compared, in order of magnitude analyses, with each other and with the physical components of the rate-of-strain. The symbol Zz is used to identify the invariant of a tensor X i j defined by:
The analysis in this paper is limited to incompressible liquids, so that the trace of e i j is zero by definition.
=
(i/2
;Tar
;) 0
(23)
Couette flow, helical flow, flow down a circular pipe, and flow caused by rotation of solids of revolution (75) are examples of viscometric flow. Fourth-order flow is described, in the x i frame, by: VI
= pyx2x3
v2
= 7x3
v3 =
0
or, equivalently,
t
The relevant kinematic variables are:
Flow Patterns
Four basic flow patterns are discussed here: viscometric flow (VF), . . . fourth-order flow (FOF), pure shear (PS), and elongational flow (EF). Viscometric flow is described, in the xi frame, by: VI
=
7J2
= 03 i
7x2
0
or, equivalently, by (75): x1 =
51
x2
=
52
x3
= E3
- ypr
3
The relevant kinematic variables are:
0
0
r/2
ze = y2 z,
=
(34) For the sake, of brevity, the expressions for y i l and t f t j are omitted; these are easily obtained from Equation 25. Oldroyd (75) concisely refers to helical flow with an axial gradient of angular velocity as a FOF. Truesdell and No11 (20) show that the vanishing of some higher rate of strain tensors implies that tensor v i , j is nilpotent-Le., ( ~ * , ~ ) = 3 0. This in turn implies that either Equation 21 or Equation 34 holds true; hence, VF and FOF are basically the only two flow patterns for which some higher rate-of-strain tensors vanish. Pure shear is described, in the x i frame, by:
y2
(N) ( N ) Aij
=
Bij = 0, N
1 Yij
258
=
(-iT
>2
-77
1
lLEC F U N D A M E N T A L S
Y2T2
8)
or, equivalently (75):
VI
= yx2/2
v2
=
2.3
= 0
yx'/2
1
(35)
x1 = xa = x3 =
- p sinh ( y r / 2 )
cosh (‘y7/2) -E1 sinh (yr/2)
€1
+ p cosh (yr/2)
$3
T h e relevant kinematic. variables are
gi’ !)
= ei5 = (:/2
Wf5
t
so that only three “viscometric functions” are to be determined (8, 76, 77, 27):
Tzz
= 0
z* = y2 zw = 0
pi-
(53)
= $2(r2)
The viscometric viscosity, q,,, and the viscometric normal stress coefficients, $1 and $z, are even functions of y on the basis of symmetry considerations. I n FOF, there are five independent components of stress, all of which are functions of the only two kinematic parameters -i.e., y and @. In PS, symmetry indicates that there are only two independent components of stress, Tlz and T11 T33 = Tzz T33. Thus two pure shear functions can be defined as:
-
-
F 2 )
&, =
- T33 Y2
At,
cosh ( y r ) 0 ( y r ) -sinh (77)
-sinh ( y r )
(54)
0
(55)
Elongational flow is described, in the xt frame, by: v‘ = yx’ v!l = -+/2 UB = -7%3/2
I n EF, symmetry indicates that there is only one independent component of stress-namely, T i l - Tzz = T l l - Ta3. Thus, only one elongational function is defined :
or, equivalently: ==
$1
x2 ==
p
%3 ==
$3
x1
exp (-77) exp ( y r / 2 ) exp (y7/2)
Slow Flow Approximations The simple fluid constitutive equation can be written as:
The relevant kinematic: variables are:
0 ut,! = ef5 =
e{j
(57)
=
wt:, = 0,
-Y/2 zw =
where F i j ( ) is a tensor-valued functional of the tensor function vmn, and s is a dimensionless time lapse:
0 s =
ze = 3 Y2
f/e
(58)
The functional Ft, can be expanded in a series of integrals as follows :
e2
ImIm
+ .}
~ ’ ( s ~sz)~tn(sl)tl~(sz)dsldsz , ..
(59)
Consider the dimensionless group, N1, defined by: Components of Stress
N~ = f i e
I n any incompressib1.efluid, there are at most five independent observable components of stress-namely, three tangential stresses and two normal stress differences. T h e symmetry of the stress tensor yields the other three tangential stresses, and the trace is undetermined in incompressible fluids. Symmetry considerations show that, in effect, the number of independent observable components may be less than five, as discussed below. I n VF, the direction of the x3 axis is arbitrary, and thus T13 and Tz3 are zero. There are only three independent comT I Z ,Til T Z Zand , T Z Z- Tar. ponents of stress-namely, Only one kinematic parameter, y, completely defines the flow,
-
(60)
A flow pattern will be defined as “slow” if N1 < 1. When a slow flow is analyzed, Equation 59 can be approximated within an “order” m by considering only terms of order Nlmor larger. For m = 3, the following equation is obtained: - = (2
et,)
f [4 aoOeikejn -
clo
(3) 2
BAr
(2)
PoAtjl
f { 8 alPeine,nc{ -I+
- (2)
(2)
+ 2 82e(e1d5’ f &e?))
(61)
where the terms in round brackets are of order 1, the terms in square brackets of order 2. and the terms in curled brackets VOL. 6
NO. 2
MAY 1967
259
of order 3. Coefficients a t and pi are related to the functions G, G’ . . . . . . in a known way. The first-order approximation is the Newtonian constitutive equation ; the second-order approximation gives the so-called second-order fluid. An equivalent equation can be written in contravariant form, by substituting modified White tensors for the modified RivlinEricksen tensors. The two forms are equivalent if the approximation is carried out to the same order. If, on the other hand, coefficients C Y ( are taken arbitrarily as zero, as is usual in “linear” theories, the two forms are not equivalent. This is in some sense a trivial conclusion, in view of the fact that White tensors are not obtained by raising indices on Rivlin-Ericksen tensors. Yet, the nonequivalency of covariant and contravariant forms of “linear” theories has sometimes been regarded as surprising. I t should not be surprising that the arbitrary assumption a i = 0 has different physical implications in the covariant and contravariant forms. Functions q v , $1, $2, qs, G8, and 7 8 obtained from Equation 61 are: Tu =
Go)
+ [OI +
((CY1
+
8*)PoY2Pf
(62)
Not all functionals F i j { ] in Equation 57 define a fluid, or, in other words, not every set of functions G, G’, . . . . in Equation 59 allows steady flow with finite stresses. I n the case of VF and FOF, the q ( j tensors in Equation 59 are polynomials of finite order in T (2 for VF, 4 for FOF). Truesdell and Noll’s theorem proves that there are no basically different flow patterns sharing this property. If the functions G, G’, , , . , are assumed to be essentially exponential, lim G(s)
0:
exp(-s)
I
S~,SZ+rn
all integrals in Equation 59 exist for VF and FOF. The form 68 is, of course, not implicitly required, but much experimental evidence supports it. When Equation 68 holds, V F and F O F may occur at any value of N1, resulting always in finite stresses. Equation 68 provides a physical interpretation of l / O as a pseudofirst-order kinetic constant for the stress relaxation phenomenon. Equation 68 assumes that stress relaxation is asymptotically of first-order kinetics. If, on the other hand, flow patterns such as PS and E F are considered, for which q i j is an exponential function of T , Equation 59 will predict a finite value of the stresses (when Equation 68 holds) only u p to a limiting value of y . Even if only the leading integral in Equation 59 is considered, and Equation 68 is assumed to hold, the integral is seen to approach infinity at some limiting value of y. For example, in the case of PS, from Equation 43: lim q i j = 0 yr+m
1;
- exp (77)
1
(6%
and thus, considering Equation 68, the leading integral in Equation 59 is seen to diverge when Within the first-order approximation (Newtonian), qv, qs, and are constant; $1, $2, and $a are zero; and qs = 3 q v (78). Within the second-order approximation, q v , qa, 11.1, 11.2, and 1. are constant; q., $1, and $2 are mutually independent, as are q , and &; is shear-dependent, and increasing with y if $2 is positive, as in most real fluids. Within the third-order approximation, $1, $2, and $d are constant; q v ! qa, and q E are shear-dependent.
+
The group N1 is, within the second-order approximation, proportional to y$~/q,,--i.e., to the ratio of the first normal stress difference to the tangential stress in VF. This is by no means true in other flow patterns; thus, the N 1 group, although in some sense similar to the Weissenberg number (3, 23), should not in general be regarded as expressing a ratio of normal to tangential stresses. Rapid Flows
If flows are considered for which N1 > 1, high order contributions cannot be neglected, and expansions such as Equation 59 become futile. Yet, for steady flows such as discussed above, matrix representation theorems can be used to infer the form of the constitutive equation which is apt for the specific flow pattern: V F and EF have been discussed (7, 9, 72, 76, 77,27). The problem may also be approached from Oldroyd’s viewpoint (75), by writing out explicitly the history of deformation, y f j ( T ) , and the history of stress, T t j ( 7 ) . Whichever approach is chosen, substantially no new information is obtained beyond what is contained in Equations 50 to 56, which have been obtained by simple geometric considerations. 260
l&EC FUNDAMENTALS
ye 4
(70)
1
The conclusion reached above by considering the expressions for q i j given by Equations 23, 43, and 50 can of -course also (N)
be reached by considering the expressions for A , , given by Equations 21, 42, and 49; higher rate-of-strain tensors are of order (ye)”, and thus they become exceedingly large as N increases if y0 exceeds some upper bound. This is true unless the higher rate-of-strain tensors vanish, a possibility exhausted by considering V F and FOF. Indeed, consideration of either Equations 23, 43, 50, or 21, 42, 49, is equivalent: One is in the one case considering the function q t l ( r ) itself, in the other case the terms of its series expansion. The matter may be considered from a different viewpoint. If the 2i frame is chosen for observation, i i j is, of course, diagonal. ? j b j is, for VF and FOF, diagonal only at T = 0Le., the line of material points along which the principal stretching takes place at the instant of observation is not the same as that along which it takes place a t any other time (the same fact may be phrased by saying that the transformation matrix needed to make q f j diagonal is time-dependent, and thus the rate of strain cannot be made diagonal a t all T in a convected frame). As a consequence, an exponential rate of stress relaxation is sufficient to overcome the stress buildup due to steady flow at any value of the shear rate. I n contrast with this, in the case of PS and EF, +j;, is diagonal at all r values; thus, the principal stretching has taken place at all previous times along the same line of material points. An exponential rate of stress relaxation is able to keep the stress finite only if the rate of stretching does not exceed some critical
value. Figure 1 gives a geometrical representation for PS and VF. Rheological models which imply Equation 68 predict vE, qs, and to become infinity a t some value of y (7, 72). I n particular, the “Maxwell” constitutive equation :
Til
+ t’DTi5 = 2 p ~ ‘ j
(74)
(71)
predicts that T~ and $s become infinity when y8 + 1, and qE becomes infinity when y0 + l / 2 . O n the other hand, the “fluid of convected elasticity” (22), which has G( ) zero a t s > 1-i. e., has an infinite rate of stress relaxation a t s = 1always predicts a finite value for qs, $s, and q E .
+
Let y indicate the sum, y1 yz. The flow pattern described above includes V F and PS as limiting cases: When y1 = yz, PS is obtained; when y2 = 0, VF is obtained. For such a flow, the invariants Z, and Zw and the groups N1 and N z are given by:
More General Two-Dimensional Flow Pattern
Focusing the attention on V F and PS, the following considerations may be made. Both flows are completely identified by only one kinematic parameter-Le., the shear rate, y. Yet, as discussed above, these two flows are in some sense essentially different. The dimensionless group N1 defined in Equation 60, which is undoubtedly relevant in the analysis of both types of flows, does not, however, suffice for a complete characterization, being unable to discriminate VF from PS. The polynomial nature of v i j ( 7 ) , or the nondiagonality of r j I j ( 7 ) ,which are characteristic of VF, are related to the fact that is nilpotent. The nilpotency of v i , , implies that Ze and Z, are equal (see Equations 17, 18, and 30). I n contrast with this, the exponential nature of v , j ( ~ ) , and the diagonality of ~ i j ( ~which ) , are characteristic of both PS and EF, are due to the fact that w i j = 0 (see Equations 38 and 47). iij(7)is nondiagonal a t 7 # 0 whenever I, # I,. These considerations suggest the formulation of a second dimensionless group, N,, as :
Nz = d Z e - I , 8
(73)
>
\
PURE A
A
FLOW A
X2= CONST
= CONST
E*. x z =
=
2dy,Yze
The rate-of-strain tensor is the same as in VF and PS:
(79) but all other kinematic tensors are not:
(72)
which is identically zero when v i , , is nilpotent. The significance of the N P group is best understood by considering a fifth flow pattern, which can be described, in the x i frame, by:
VISCOMETRIC
N~
V
\
0
0 (82)
Equations 81 and 82 clearly show that, if a n exponential rate of stress relaxations is assumed, stresses will tend to infinity when N z --*. 1 ; the value of Arl may be as large as one wishes without resulting in infinite stresses. I n particular, the Maxwell equation predicts, as obvious, infinite stresses when N z --*. 1. The group N z is defined in a properly invariant fashion. I n V F and FOF, NZ = 0 identically, and infinite stresses never develop if Equation 68 is assumed. I n PS and EF, Nz = N l , and infinite stresses develop when N1 = N z + 1. I n the flow pattern described by 73, N 2 5 N1, and infinite stresses develop when N z + 1, immaterial of the magnitude of N1. These considerations suggest the following generalization: Fluids for which the rate of stress relaxation is essentially exponential may flow at any value of N1 without developing infinite stresses, but cannot flow at values of N P exeeeding some critical upper limit of the order of unity.
SHEAR
CONS7
N1 and Nz Groups
d€ T =0
d€
r f o
Figure 1. Geometric representation of pure shear and viscometric flow
The N 1 group is related to the “Weissenberg number.” T h e definitions of the latter in the literature do not clearly indicate whether it should be regarded as the product of shear rate and natural time, or as the ratio of normal to tangential stresses: T h e two things are equivalent only in VF, a t low shear rates, but not under other flow conditions. The definition given in Equation 60 for group N1 is general in character, VOL. 6
NO. 2
MAY 1967
261
properly invariant, and unequ’vocal. By definition of “slow flow,” N 1 is seen to be a dimensionless measure of the fastness of flow. When N1 < 1, expansions of the form of Equation 61 may be used; when N1 > 1, only matrix representation theorems may be of some help in deciding the form of constitutive equation to be used. The N Z group may perhaps in some instances express the same ratio of physical quantities as the Deborah number (3-5, 73). Yet, the latter has been suggested essentially by considering unsteady flows (73), while the definition of NZ given here is based on the analysis of steady flows. Should the Deborah number be defined in the present context, attention should be focused on unsteady flows for which 7
> r*,
‘lij
= 0
(83)
Le., suddenly accelerated flows. For such flows, the integrals in Equation 59 need to be evaluated only over 0 6 s 6 r*/O. Suppose now that O/r* 1 ; the G functions could be approximated with constants, because s >
- I
.-
Conversely, if V F is considered, but y is allowed to be a function of time (withy = 0 at r > r*),a finite value of the Deborah number would be calculated, while the value of N z would still be zero. Nonetheless, for flows which are steady in an Eulerian sense but not in a Lagrangian sense, the values of Nz as calculated from Equation 72 and the value of ND, as calculated from Equation 84 may coincide; an example of this case is the boundary layer flow which will be discussed in a later communica tion. Acknowledgment
Discussions with A. B. Metzner were extremely helpful in obtaining a physical understanding of the problems discussed. Careful analysis of the implications of Oldroyd (75) and of Truesdell and No11 (20) has been the basis for the present analysis. Nomenclature (N)
= Rivlin-Ericksen tensor, sec.-N
A ij W“
= modified Rivlin-Ericksen tensor, sec. -l
Aij
(N)
Bij -
=
(N)
Bii
D eij
Fi,{ git
1
G( ), G‘( 262
White tensor, sec.-N
= modified White tensor, set.-' = convected derivative operator, sec. -l = rate-of-strain tensor, sec.-l = tensor-valued functional, g cm.-l set.+ = metric of x i frame, dimensionless ) = dimensionless memory functions
I&EC FUNDAMENTALS
N Ni, Nz
= = = =
ND,
=
I, m
s =
Ti j Vi Ui.j Wij
Xi jji
Xi Xij
r/e
= = = = =
= = =
=
defined in Equation 11, units depend on Xij an integer an integer dimensionless numbers Deborah number dimensionless time lag stress tensor, g. crn.-l set." velocity vector, cm. set.-' covariant derivative of velocity, sec.-l vorticity tensor, set.-' a Cartesian frame a Cartesian frame where d i j is diagonal a generic vector a generic tensor
GREEKLETTERS = Reiner-Rivlin coefficients, dimensionless = dimensionless coefficients = dimensionless constant ;i = shear rates, set.-' 7, 71, YZ Yii = metric of {(-frame, dimensionless Itii = convected rate of strain, set.-' Itv = viscometric viscosity, g. cm.-1 sec.-’ Its = shear velocity, g. cm.+ sec.-l qE = elongational viscosity, g. cm.-1 sec.-1 e = natural time, sec. PO = zero-shear viscosity, g. cm.-1 sec.-1 = convected frame .? = convected frame T = time lag, sec. = time elapsed from startup of flow, sec. T* = normal stress coefficients, g. cm.-1 $1, $2, $. Lyi
ti
For tensorial quantities, dimensions of the physical components are given. For operators, the dimensions are given which formally satisfy the principle of dimensional homogeneity when operators are regarded as factors. The word “frame” is used in the sense of a triple family of surfaces mapping euclidean space. The summation convention and comma notation are used throughout. literature Cited
(1) Astarita, G., Can. J . Chem. Eng. 44, 59 (1966). 4, 354 (1965). (2) Astarita, G., IND.ENG.CHEM.FUNDAMENTALS (3) Ibid., 5 , 548 (1966). (4) Astarita, G., XXXVIth International Congress of Industrial Chemistry, Bruxelles, 1966. (5) Astarita, G., Metzner, A. B., Atti Accad. Lincei, Rome VIII-46, 74 (1966). (6) Bird, R. B., Can. J . Chem. Eng. 43, 161 (1965). Phys. Fluids 8, 52 (1965). (7) Bird, R. B., Spriggs, T. W., (8) Coleman, B. D., Noll, W., Arch. Rat. Mech. Anal. 3, 289 (1959); 6, 355 (1960). (9) Coleman, B. D., Noll, W., Phys. Fluids 5 , 840 (1962). (10) Denn, M. M., Division of Industrial and Engineering Chemistry, 151st Meeting, ACS, Pittsburgh, 1966. (11) Ericksen, J. L., in “Viscoelasticity. Phenomenological Aspects,” Academic Press, New York, 1960. (12) Lodge, A. S., “Elastic Liquids,” pp. 229-81, Academic Press, New York, 1964. (13) Metzner, A. B., White, J. L., Denn, M. M., Chem. Eng. Proor.. in mess. 0 ’ .> --- r -
(lij-oldroyd, J. G., Proc. Roy. SOC.A-200, 523 (1950). (15) Ibid., A-283, 115 (1965). (16) Rivlin, R. S., J . Rat. Mech. Anal. 5 , 179 (1956). (17) Rivlin, R. S., Ericksen, J. L., Ibid., 4, 323 (1955). (18) Trouton, F. T., Proc. Roy. SOC.A-77,426 (1906). (19) Truesdell, C., “Symposium 11, Order Effects,” Haifa, 1962, Pergamon Press, London, 1964. (20) Truesdell, C., Noll, W.,“Non-Linear Field Theories of Mechanics,” in “Handbuch der Physik,” 111/3, SpringerVerlag, Berlin, pp. 439-40, 1965. (21) Ibid., pp. 429-75. (22) Zbid., p. 480. (23) White, J. L., J . Appl. Polymer Sci. 8, 1129 (1964). (24) White, J. L., Metzner, A. B., A.1.Ch.E. J . 11, 324 (1965). (25) White, J. L., Tokita, N., J . Appl. Polymer Sci.,in press. RECEIVED for review July 12, 1966 ACCEPTEDOctober 31, 1966
