- Email: [email protected]
On configuration-dependent generalized oldroyd derivatives
47
Journal of Non -Newtonian Fluid Mechanics, 14 (1984) 47-65 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
ON CONFIGURATION-DEPENDENT DERIVATIVES *
GENERALIZED
OLDROYD
H. GIESEKUS Abteilung (F. R. G.)
Chemietechnik,
(Received
May 31, 1983)
Universitiit
Dortmund,
Postfach 50 05 00, D-4600
Dortmund
50
Summary After a short derivation of the upper and lower Oldroyd derivatives, linear combinations of them are examined with regard to their suitability for describing the non-affine motion of polymer structures occurring because of hampered flow. This leads to the introduction of a new type of convected derivative called a “configuration-dependent” derivative. While the combined Oldroyd derivatives can be considered to be upper convected derivatives supplemented by the addition of terms containing the rate of deformation tensor, in the new derivatives the irreversible rate of deformation tensor is substituted instead. This tensor, first introduced by Leonov, is strongly related to the configuration tensor, which is a measure of recoverable strain, at least in so-called one-mode models. The force-free motion of particles in steady simple shear flow as predicted by the respective generalized derivatives is analysed and it is found that the configuration-dependent derivative supplies a more realistic result for hampered flow than the combined Oldroyd derivative, except in the case of permanently isotropic structures for which both predictions are identical. For the special case of application of the configuration-dependent derivative to the configuration tensor, the expression is equivalent to that obtained by the application of the simple upper Oldroyd derivative itself. Thus it is concluded that the effects predicted by simple constitutive equations, when combined Oldroyd derivatives are used but not when only the upper Oldroyd derivative is used, may be more appropriately described by taking into account the non-isotropic character of the relative motion of structural elements occurring because of mutual interactions.
* Dedicated
to the memory
0377-0257/84/$03.00
of Professor
J.G. Oldroyd.
0 1984 Elsevier Science Publishers
B.V.
48 1. Introduction In Oldroyd’s pioneering paper on the formulation of constitutive equations [l] he introduced, amongst others, the extraordinarily fruitful concept of convected time derivatives. As shown in that paper these operators enable models of viscoelastic fluids originally restricted to infinitesimal strains to be generalized so that they are appropriate for finite deformations. In other words, they make these models “material-frame indifferent” or “objective”. At the same time Oldroyd demonstrated, by using the example of a Jeffreys model (i.e. a superposition of a Newtonian and a Maxwell model in parallel), that for each such infinitesimal model there exists not only one frame-indifferent model but an infinite manifold. These models can be classified into two main classes with respective members differing only for non-isochoric motions. However, the simplest representatives of each class, today commonly called Oldroyd fluid A and Oldroyd fluid B, show a distinctly different behaviour for isochoric motions as well. For example, their predictions for second-order behaviour in viscometric flows differ markedly although for first-order behaviour they are indistinguishable. This difference arises because in model A the “covariant convected derivative” (or “lower Oldroyd derivative”) is used whereas in model B use is made of the “contravariant convected derivative” (or “ upper Oldroyd derivative”). Comparison with experimental observations (such as the Weissenberg effect) showed that the predictions of model B agreed at least qualitatively with experiment whereas those of model A did not. The reason for this was not understood at once and some reflection on the structure of materials was required before any indication of the underlying physical difference between using one or the other of these derivatives was obtained [2]. Preference for model B was strengthened by the evidence that a dilute suspension of Hookean dumbbells in a Newtonian fluid under the combined action of rotational diffusion and frictional forces but neglecting hydrodynamic interactions behaves quantitatively like an Oldroyd fluid B, whereas it was not possible to find any analoguous structural model of an Oldroyd fluid A [3]. In addition, the simple network model of a concentrated polymer fluid proposed by Lodge [4] also leads to the model of an Oldroyd fluid B or, more precisely, to a linear superposition of these, called the Walters fluid B’ [5]. In spite of this, discrepancies between the predictions of this model and the observed behaviour of viscoelastic fluids (for instance the unrealistic prediction of a disappearing second normal-stress difference and constant shear viscosity in viscometric flows) motivated several authors and indeed first of all Oldroyd himself [6] to propose more general constitutive models to overcome those inadequacies. These equations can be considered to
49 include linear combinations of the two types of Oldroyd derivatives and not just one or the other. Later, Gordon and Schowalter [7] were motivated to introduce derivatives of this type by arguments from the theory of anisotropic fluids, and recently they have also been introduced in some more advanced network theories to allow for “non-affine motions” or “slippage” of the elements of the respective network structures, as for example by Johnson and Segalman [8], Phan-Thien and Tanner [9], Phan-Thien [lo], and Hsii and Schtimmer [ 111. Indeed, these generalized models, the so-called Oldroyd eight-constant model as well as the respective polymer solution and network models, provide a considerably improved description of real fluid behaviour and hence they are very often used for analysing more complicated flow situations by analytical and numerical methods. In spite of these improvements some important shortcomings remain, in particular in certain transient flow situations and as regards stability criteria. Whereas the Oldroyd eight-constant model is a purely empirical model, which can therefore only be evaluated by comparison with experimental results, the more recent polymer solution and network theories allow the hypotheses upon which they are based to be examined. This is attempted here in a continuation of a recent paper [ 121 in which the Oldroyd model B was generalized in a different way by introducing a configuration-dependent molecular mobility. The concept of a configuration tensor described therein (it can be understood as a tightening-up of the old idea of “recoverable strain”) together with the idea of Leonov [ 131 to decompose the total rate of strain into a reversible and an irreversible part are used here to introduce a new class of generalized Oldroyd derivatives. These take into account flow hampering by particle interactions within the deformed polymer coil and network structures and reduce to a quantity proportional to the simple upper Oldroyd derivative when applied to the “configuration tensor”. They are characterized in particular by force-free motion of particles in steady simple shear flow. In order to show more clearly the way in which one arrives at these derivatives a pertinent derivation is first given of the upper and lower Oldroyd derivatives. The relation with the original derivation starting. with the concept of convected coordinates is outlined in the Appendix. 2. Motion of a continuum and the Oldroyd derivatives Let the motion of an arbitrary particle imbedded in a material continuum be described by the one-parametric manifold of one-to-one mappings x = x(x,
t),
where X designates
(1) the position
of this particle
at a certain reference
time t,,
50
i.e. x(X, to) = X. The distance then given by
vector between
two neighbouring
particles
dx(t)=dX.vxx(t)=(vxx(t))+.dX=L’(t).dX.
is
(2)
Here vx is the gradient operator referring to the coordinates of the reference state and the dagger (t) indicates the transpose of the respective tensor. The one-to-one mapping (1) implies that eqn. (2) can be inverted to give dX=dx(r).V,X=(V,X)+.dx(t)=P-‘(t).dx(t),
(3)
where V, is the gradient operator referring to the coordinates of the actual state. In these mappings F(t) and F- '(t) are called the deformation gradients. From eqns. (2) and (3) it may be assumed that F(t) is a function of the independent variable X (“material deformation gradient”) whereas F-‘(t) has x as the natural variable (“spatial deformation gradient”). However, because of the existence and invertibility of the mapping (1) both may be interpreted as functions of either kind of variable. Using the velocity u(t) = &x(X, t)/& = Dx( t)/Dt and the velocity gradient L = ( v,u)+, which are usually considered to be functions of the spatial coordinates x, the following material time derivatives are obtained:
EF=L.F,
J&-l=
-F-I.L,
;dx=dx.L+=L.dx.
(5)
Let d A (t) be an oriented infinitesimal area embedded in the continuum and p(t) the local mass density. Then the associated volume element is given by dV( t) = dA( t) . dx( t) and the mass element by dm = p(t) dA( t) . dx( t). Because of the conservation of mass dm does not depend on time and therefore the vector dh(t)=p(t) depends ;
dh=
dA(t)
(6)
on time inversely -dh.L=
to the way in which dx( t) does:
-L+.dh.
(7)
By means of the deformation gradients 1E and F- ’ two spatial measures deformation are derived, namely the Finger deformation tensor B=F.F+ and the Green deformation B-i
=F-‘t.F-1,
of
(8) tensor (9)
51 which satisfy the relations (dX)*
= dx .B-’
edx,
(dH)*=dh.B.dh
where dH = dh( to). It is easily deduced from eqns. (4) that the material these deformation measures are
(10) time derivatives
of
DB -=L.B+B.L+, Dt
(11)
and DB- ’ -= Dt
-(L+.B-l+B-‘.L).
(14
Since these formulae are strongly related to eqns. (5) and (7) respectively, is sensible to rewrite them by introducing two new operator symbols: &
dx=g dh=;
$
it
dx-L.dx=O, dh+L+.dh=O,
g=!$-(L.B+B.L+)=~, DB- ’ Dt+(L+.B-‘+B-‘.L)=O.
bB-‘bt
024
These are just the operators of convected differentiation, introduced by Oldroyd 111, applied to either a vectorial or a tensorial quantity. In this way it is at once obvious that the upper Oldroyd derivative of a vector a %a _=_Bt
Da Dt
(13)
L*a
describes the time rate of change of this vector relative to the deformation of a distance vector dx embedded in the continuum whereas the lower Oldroyd derivative ba -=bt
Da
Dt
+L+‘a
describes the time rate of change of the respective vector relative to the deformation of the product of an embedded area vector with the density. Correspondingly the associated Oldroyd derivatives operating on a (secondorder) tensor T
g=g-(L4-+
T.L+),
(15)
52 _=bT bt
(16)
E+(L~.T+T~)
describe the time rate of change of this tensor relative to the spatial measures of deformation, i.e. the Finger and the Green tensor, respectively. In this way the concept of convected differentiation has been introduced without explicit use of convected coordinates. The equivalence with Oldroyd’s original derivation, given, however, in a much more general way for tensors of arbitrary order, is sketched in the Appendix. As briefly mentioned in the introduction, a multitude of related derivatives can be associated with both kinds of Oldroyd derivatives which are deduced from the so-called relative tensors. But these differ from the derivatives presented above only by additional terms n(Tr L)a and n(Tr L) T respectively, n being a positive or negative integer associated with the “weight” of the relative tensors. Correspondingly these terms disappear in isochoric flows (Tr L = 0). This distinction can also be introduced by means of the above elementary formalism starting from the relations g
g=
dV=(TrL)dV,
but this shall not be considered
- (Tr L)p, further
(17)
here.
3. Combined Oldroyd derivatives It has been demonstrated in the preceding section that there is a strong relation between the two types of Oldroyd derivatives and the rate of change of characteristic vectors embedded in the deforming continuum, and that there are no other types of frame-indifferent derivatives for which analoguous relations are available. In spite of this a multitude of such derivatives as weighted averages of upper and lower may, of course, be constructed Oldroyd derivatives: ED E=
1
(18)
Let L be decomposed into its symmetric and skew-symmetric of deformation and the rate of rotation tensors w=:(L-L+),
D=;(L+L+),
(19)
respectively, and the corotational tensor T be introduced by 6ila -=_--
Da
3lt
Dt
6iJT -=_-
DT
‘9t
Dt
w.
a=e+a w ’ Dt
W- T+ T. W,
parts, the rate
or Jaumann
’
derivative
of a vector u and a
53 which represents the equally weighted average of the class of combined derivatives (18) and as such the sole member in which only the skew-symmetric part of L appears. Then the full class of derivatives (18) can be decomposed for a vector u and a tensor T as follows:
This is how the combined derivative was introduced [6] in his eight-constant model, in fact as +(T4l+Lb
T),
implicitely
by Oldroyd
(24)
which is equal to hKDT/iIDt with 5 = 1 - p/h, where T stands for the stress deviator as well as for the rate of deformation tensor. The special case of the Jaumann derivative was introduced much earlier by Zaremba [ 141 in connection with a fluid model of the Maxwell type and rediscovered independently some decades later by Fromm [ 151. 4. Configuration-dependent
convected derivatives
In the aforementioned recently published network theories [8- 1 l] combined Oldroyd derivatives (with 0 < 5 < 1) are used to allow for “non-affine network motion”, also called “motion with slippage”. The underlying idea is that the structural elements of the polymer network, usually called “beads”, are not stressed by the average motion of the continuum (interpreted in a statistical sense) but by a velocity field modified owing to shielding of the neighbouring beads. It is supposed that this effect reduces only the deformational component of the flow field, the reduction being signified by the parameter 5. This idea is well-known from the theory for dilute solutions of linear flexible polymers which are considered to exist as coils with hampered flow, the limiting cases being the “free draining” and the “non-draining” coil. The degree of hampering, generated in particular by hydrodynamic interaction, depends here on the statistical diameter and the density of the coils, i.e. on the molecular weight and the solvent quality. However, molecular structures such as these coils deform under flow conditions so that the question arises as to whether it is appropriate to signify the hampering of the draining motion in the vicinity of a bead by a constant shielding coefficient 6. Indeed, Hsti and Schtimmer [ 1 l] have proposed that this coefficient be replaced by a function of the invariants of the
54 rate of deformation tensor, which they term the “slippage function”. In contrast, Giesekus [ 121 recently intimated a less formal way of introducing structural points of view into the modelling of constitutive equations of polymer fluids, which is strongly related to his concept of configuration-dependent mobility [ 16,171. The combination of the structural deformation by a configuration tensor given here with Leonov’s notion of the irreversible rate of deformation tensor [ 131 leads straightforwardly to a configuration-dependent generalized Oldroyd derivative as will be demonstrated in the following. 4. I. Force-free motion under hampered flow conditions Let the distance of a representative bead from the centre of resistance of the respective polymer molecule be designated by the position vector r and let this quantity be considered as an infinitesimal vector like dx from the point of view of a macroscopic length scale. Then in an unhampered flow field Br/Q t is in general not zero but rather is related with the friction forces contributing to the excess stresses. If, however, in an ideal experiment all force-transmitting bonds were cut then the particle would move according to the condition sDr/at
= 0,
i.e. Dr/Dt=L.r=
On the other hand, differential equation Dr/Dt
(25)
(W+D).r.
if the flow is hampered by the surrounding beads the of force-free motion has to beesubstituted by
(26)
= i . r,
where i = (~6)~ corresponds to a modified velocity field 6 around the centre of resistance of the respective molecule. The principle of material-frame indifference allows only for the modification of the deformational part d, whereas I? = W must hold. The special choice leading to the aforementioned combined Oldroyd derivative is
ri= (1 -
[)O,
but a more general case. 4.2. The configuration
(27) choice is possible
which contains
this only as a special
tensor and the irreversible rate of deformation
tensor
It has been shown, for example by Giesekus [ 16,121, that the elastic excess stresses resulting from the connections between the beads as well as from entropic effects depend on the tensor ( w), at least in the case of a “one-mode model”. (The arrow-shaped brackets designate an ensemble
55 average.) If the state of rest is signified by the isotropic tensor (~#) = ( ( i . ~ ) / 3 ) 1 , a configuration tensor b * can be introduced using the definition (rr) = b.
=
3
b.
(28)
It represents a type of Finger tensor characterizing the deformation of the network structures corresponding to the notion of "recoverable strain". Leonov [13] introduced a decomposition of the velocity gradient into a reversible ("elastic") and an irreversible (" plastic") part L = Lie ) + Lip )
(29)
claiming that Lie ) is subject to a condition analoguous to that in eqn. (1 la) for the Finger tensor:
,e, = Dt
(L'e)'b+b'Ltte))=O"
(30)
As a result the complete upper Oldroyd derivative can be expressed as
Here Lip ) is understood to be an inner variable of the configurational state and therefore an isotropic function of b only. As a consequence Lip ) can be identified with its symmetric part D/p), i.e. W~p)= 0, and Dip ) must be commutable with b. Because of this eqn. (31) simplifies to ~b ~t - -2D/p)'b'
(32)
and the irreversible rate of deformation tensor Dip , is therefore expressible by the configuration tensor and its upper Oldroyd derivative in the simple form: Dip)= - ½ b - I "
~b ~t"
(33)
If eqn. (31) is transformed in the following way
D----~-(W.b-b. W) = -6~b ( O - D/p,)-b + b. ( O - Dip))= Db ~t'
(34)
it can be seen at once that under fully general flow conditions D and Dip ) are identical only when the configuration remains isotropic (b - 1). However, in motions with constant stretch history where Db/Dt can always be made zero by a suitable choice of the reference system, @b/6~t can in contrast only be made to disappear if the motion is irrotational ( W = 0). * In the papers by Leonov [13] and Giesekus [16] this quantity is denoted by the letter C.
56 4.3. Definition
of configuration-dependent
generalized
Oldroyd derivative
It is implicit in the concept of the irreversible rate of strain tensor that the beads of a polymer structure follow the elastic part of the deformation gradient L.(,) 2 L - DC,, without any frictional loss. This suggests the hypothesis that flow hampering effects arising from hydrodynamic or molecular interactions should be dependent on the irreversible rate of deformation tensor DC,,, rather than on the complete tensor D = Dee, + Dcp,. According to this hypothesis a new type of convected derivative is proposed in which instead of eqn. (27) ”
D = D - (*DC,,)
5* =D++-‘.g
(35)
is introduced. This configuration-dependent derivative is, correspondingly, defined for a vector a and a tensor T, respectively, as follows: D*a
Da
-=6r+[*Dc,,.a,
(36)
Dt
z=g+<*(Dc,,.T+ The subsequent lrD*z -= Dt
T.D&
(37)
special cases are easily verified:
-2(D-s$*D&
The latter, which follows at once from eqn. (31), is of basic significance for establishing constitutive models, as will be discussed briefly in Section 6. However, our primary interest here is to compare relevant properties of this configuration-dependent convected derivative with the combined Oldroyd derivative and to argue that this derivative is appropriate to model structural changes generated by deformational motions of polymer fluids. 5. Force-free motion of a particle in steady shear flow with configuration-dependent hampering It has been demonstrated by eqn. (34) that the various types of convected derivatives lead to different predictions even in flows with constant stretch history if these are not irrotational. This suggests that just such a flow should be analysed, the most appropriate choice being simple shear flow. To elucidate the basic properties it is helpful to study the force-free motion of a bead in the respective flow field in which the mode of
57 hampering is given by the applied derivative. The differential equation this motion is obtained by introducing eqn. (35) into eqn. (26) to give: Dr/Dt
= (L - <*DCr,)) .
r.
of
(40)
5. I. Kinematics of hampered flow in simple shear As the neutral direction plays no part in simple shear flow the problem is formulated in two dimensions only with x, designating the direction of flow and x2 that of the shear gradient, y being the rate of shear. As is well known, the following matrix representations then hold:
(41) With
it follows that (43) Introducing
these expressions -i//2
6,
into eqn. (33) results in
b,,b*, (44)
+ h,b22
-
b:2
IIb,,b,,
-
2b:2
However, the additional condition that Zb/%t has to be an isotropic tensorial function of b and therefore commutable with this tensor means that D cpj must be a symmetric tensor, i.e. bz2 = b,,b,, - 2b:,. The resulting simplifications lead to the following expressions for b and DC,,:
b1
II
1+2z2
1 +z2+zjl+
*wA
:u-
Y/2 1 +
MS-
l/S)]’
/I
l/S)
1
*z
1 -i(C-
=l/S)1
ii
j//2 1+z2
z 1 IIII ’
II
(45)
z
1
1)
1
--z/i (46)
with s=
Jb,/b,,=cot+,
z = +(I - l/S)
= cot 2+,
(47)
58
where b, and b,, are the principal values of b, associated with the major and minor axes, and 9 is the angle between the first principal direction and the flow direction. It is seen from the above expressions that the configuration is fully determined by either the ratio of the principal values or the orientation angle I$, the dependence of these two quantities on one another resulting from the symmetry condition utilized above. Substitution of eqn. (46) into eqn. (35) results in 6
II
f/2
A
1 +z*
-5*z
1 +z*-[5*
1+z*-_l*
#$*z
with principal
L31.11
=
values + z* - #$*)‘+ #$**z*
++$J(l -
and the angle \k between given by cot 2 \k = -
(48)
Ii ’
the first principal
direction
5*z 1+z*-<*
and the flow direction
(50)
*
As is well known (see, for example, Giesekus [ 181) the streamline pattern of homogeneous plane flow fields is ei_ther hyperbolic or elliptic d:pending on whether the deformational 1 part D or the rotational part W of the associated velocity gradient L predominates; the intermediate case is given by the straight line pattern of simple shear flow. Because @= W the rate of rotation is determined by j1/2, whereas it can be seen from eqn. (49) that the rate of deformation b, is less than y/2 for 0 < [* -C 2 and equal to y/2 for c* = 0 and 2. Therefore the path lines of beads moving without the action of external forces in hampered simple shear flow are ellipses oriented so that the common angle between the direction of the major axis and the flow direction is Cp= \k - s/4 with cot 2 cp = 1+ z* - t* 5*z and the common axes a,, 1s
I
ratio of the magnitude
(1 +z*)+/(l
z=?I= =
(51)
+z*-<*)*+<**z*
(1 + z”) - J(1 + z* - <*)‘+
cot(2+-
a) cot
of the major axis a, to the minor
I’* t**z*
I (52)
59
Fig. 1. Comparison of a path line of the force-free motion of a bead in hampered flow (full line) with the shape and orientation of the average configuration (dashed line). The chosen values were t* = l/3 and + = 30” (i.e. { = 1.732), resulting in @ = 5.45” and Z = 2.732 by using eqns. (51) and (52).
An example of such an elliptical path line is given in Fig. 1 together with the shape of the tensor ellipse of bIi2 which according to eqn. (47) is specified by the orientation angle C#Iand the ratio of the major to the minor axis 5. Before analysing eqns. (51) and (52) in the general case two special cases will be discussed. 5.2. Path lines with isotropic structure This corresponds to z = 0 (i.e. [ = 1 and + = 0) and is just that case in which the flow hampering is described by the combined Oldroyd derivative defined in Section 3, i.e. DCP,= D, 5* = 5. Under these circumstances eqns. (5 1) and (52) simplify to give @=O,
Z=@?PC
(53)
This means that the principal axes coincide with the axes of the coordinate system, more precisely that the major axis is aligned with the flow direction, and that the ratio of the axes, 2, decreases monotonically from infinity (i.e. straight line pattern of simple shear flow) to one (i.e. circular pattern of pure rotational flow) as the shielding coefficient .$ increases from zero to one, the latter limiting case corresponding to completely hampered flow. Figure 2 shows some examples of such flow patterns associated with various values of the parameter 5.
60
Fig. 2. Path lines of the force-free motion of a bead in hampered flow in the limiting case of isotropic structure (+ = 0, J = 1) for various values of the shielding coefficient t* = 5.
5.3. Path lines with complete shielding The special case t* = 1 corresponds to complete shielding in non-isotropic structures. Now eqns. (51) and (52) simplify to give
a=+,
Z=[
i.e.
a/a,,
= :b,/b,,.
(54)
This means that the orientation of the path line ellipses coincides with that of the configuration tensor and the ratio of its axes is equal to the square root of the ratio of the respective principal values. Indeed, b has the properties of a Finger tensor and so it is b 1’2 that characterizes the shape of the average configuration of the deformed network structures. Thus, in the limiting case of completely hampered flow the force-free motion of a bead follows exactly the motion of the respective structure itself. 5.4. Path lines in the general case In Fig. 3 the dependence of @ and Z on the orientation angle C#B is shown. For increasing values of C#Ithe angle Cp first increases to a maximum, with
61
0' 0’
0"
10"
2o"
I
I
10’
20'
3o" Ql 40' 45O I
II
30' (0 4o" 45'
Fig. 3. Orientation angle Cp and ratio of the axes Z of path line ellipses as a function of the orientation angle + of the configuration tensor. The dashed line designates the locus of maximum values of Q given by eqn. (55).
magnitude 2J1-5* Q,max = f arccot
5*
= 2( 9 - 48)
(55)
at (56)
62 45. 10
30'
20@
10' 10
15'
6"
Z 5
2
1 1
2
5
5
1
Fig. 4. Ratio of the axes Z of path line ellipses as a function { of the average molecular structure.
of the ratio of the principal
axes
and then decreases to zero at + = 9r/4 (i.e. [ = 1) if <* < 1. In contrast Z decreases monotonically as the orientation angle + increases and approaches its minimum value, given by eqn. (53), with horizontal tangent if <* < 1. In Fig. 4 this latter dependence is shown once again in a double-logarithmic plot of Z versus { illustrating the gradual approach to a straight line with increasing values of { as well as of the shielding coefficient t*. 6. Discussion The configuration-dependent convected derivatives introduced here have been shown to represent a generalization of combined Oldroyd derivatives with these contained as a special case for structures with permanently isotropic configurations. However, under normal conditions, in which fluid motion results in non-isotropic configurations, it has been demonstrated that these generalized derivatives are based on more appropriate suppositions about the reference velocity, that is the velocity of a particle not subjected to any forces. This was seen most clearly when the limiting case of complete shielding was analysed, for then the force-free motion of a structural element was found to be identical with the motion of the respective structure itself, whereas using combined Oldroyd derivatives it would be purely rotational motion like that of a rigid sphere. If it is accepted that the generalized configuration-dependent derivatives are more appropriate for including shielding effects than the usual combined
63 derivatives, and they are applied to the configuration tensor (or an equivalent quantity), it follows that the respective constitutive equations essentially reduce to those of an Oldroyd fluid B. This is evidenced at once by eqn. (39). Hence, effects such as those mentioned in the introduction, which were included when the upper Oldroyd derivative was substituted by a combined one, can no longer be attributed to simple flow hampering or shielding at all. At first this seems to be a disappointing result, but actually it emphasizes once again the fact, already well-known in the theory of dilute polymer solutions, that interaction of polymer beads under deformational motion always results in non-isotropic properties. These affect the mechanism of relative motion of structural elements so that they cannot be taken into account using a pre-averaging procedure without loss of essential features. However, if this anisotropy is properly taken into consideration it will lead to a more satisfactory prediction of the respective flow effects than by using combined Oldroyd derivatives. In conclusion, although the above investigation of the relative motion of structural elements under conditions of hampered flow may have questioned the usefulness of combined Oldroyd derivatives for describing the respective effects, it has also emphasized once again the special significance of Oldroyd’s original concept of convected time derivatives. In particular, the equivalence relation (39) proves that the upper Oldroyd derivative maintains its central role in constitutive equations even when shielding effects are included, and this in spite of earlier conjectures that it would have to be replaced by a more complex combined derivative.
Appendix Let a convected coordinate dent coordinates 5’ so that dx(t)
system be defined
by a set of time-indepen-
= d&,(t),
(Al)
where dx( t) is given by eqn. (2) and the gi( t) designate a time-dependent basis by means of which the metric tensor is determined as gikCt)
=giCt)
The reciprocal
‘gk(f).
basis gk( t) is then given by
g;(t)*g”(t)=&Y This basis can be used to connect the vector d/z, introduced time-independent components dqk: dh(t)
642)
= dqkgk(t).
(A3) by eqn. (6) with
(Ad)
64 It is seen from eqns. (Al) and (A4) that the co- and contravariant bases gi( t) and gk( t) move in the same way as dx( t) and dh( t), respectively; and according to eqns. (5) and (7) (A5) Material-frame indifferent time derivatives are understood to be composed of material time derivatives of the respective co- and contravariant components in a convected coordinate system but without inclusion of the time derivatives of the base vectors. This means that for a vector a(t> = ai(t)g,(t)
=
646)
ak(f)gk(r)
and a tensor T(t)
= +;‘(t)g;(t)g;(t)
= CTkk’(t)gk(t)gk’(t)
the upper and lower Oldroyd G(t)
Da+)
-=-
Dt
at
Dt
Dt
Making
b&) gi(t)y
are defined Dt
bT(t) bt=
gi(t)gi’(t)?
as follows:
Dak(t)
bt=
D?‘(t)
W(t)
-=
derivatives
(A?
*,
648)
g”b),
h/&) Dt
(A91
gk(t)gk’(t)’
use of the expressions
Da(t) Dt
Dar’ Dt gi(t)
DT( t) Dt
D?’
-=_
.Dgi(t) +&’
Dak
Dt
+gl(‘)-D~
m”(t) +7kk’
v
Dgi’(t)
+gi’(l)
Drkk’ D,s”@h”+)
Dgk(t) Dt
(t)+ak
Dg-0)
--==g,(t)g,Jt)+Ti”
=
k
=Dtg
Dt
&?“‘(d
+gktf)
Dg;;”
)
(Al 1)
as well as eqns. (A5) enables eqns. (A8) and (A9) to be rearranged to give eqns. (13- 16) of Section 2. It is, however, felt that the derivation given in Section 2 facilitates a real understanding of the basic difference between the upper and lower Oldroyd derivatives, whereas the more formal derivation using convected coordinates, despite its elegance, makes the reason for this difference obvious only for those who are thoroughly versed in this formalism.
* The representation interest.
of a tensor by its mixed components
is not included
here as it is not of
65 Acknowledgment The author English.
is indebted
to Dr. M. Hibberd
for his help in improving
the
References 1 J.G. Oldroyd, On the formulation of rheological equations of state. Proc. Roy. SOC., A200 (1950) 523-541. 2 H. Giesekus, Der Spannungstensor des visko-elastischen Korpers. Rheol. Acta, 1 (19581961) 395-404. 3 H. Giesekus, Die rheologische Zustandsgleichung. Rheol. Acta, 1 (1958-1961) 2-20. 4 A.S. Lodge, A network theory of flow birefringence and stress in concentrated polymer solutions. Trans. Faraday Sot., 52 (1956) 120-130; Elastic liquids, Academic Press, New York, 1964. 5 K. Walters, Non-Newtonian effects in some general elastico-viscous liquids. In: M. Reiner and D. Abir (Eds.), Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Macmillan, New York, 1964, pp. 507-519. 6 J.G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Sot., A245 (1958) 278-297. 7 R.J. Gordon and W.R. Schowalter, Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions. Trans. Sot. Rheol., 16 (1972) 79-97. 8 M.W. Johnson, Jr. and D. Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation. J. Non-Newtonian Fluid Mech., 2 (1977) 255-270. 9 N. Phan-Thien and RI. Tanner, A new constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech., 2 (1977) 353-365. 10 N. Phan-Thien, A nonlinear network viscoelastic model. J. Rheol., 22 (1978) 259-283. 11 Y.T. Hsii and P. Schiimmer, Some general rheological properties of dilute polymer solutions predicted from intrinsic viscosity measurements. Rheol. Acta, 2 1 (1982) 5 1 l-5 13; Quantitative description of rheological properties of dilute polymer solutions. II. Estimation of rheological material functions at moderate shear rates using the Rouse-Zimm model with slippage. Rheol. Acta, 22 (1983) 348-353. 12 H. Giesekus, A unified approach to a variety of constitutive models for polymer fluids based on the concept of configuration-dependent molecular mobility. Rheol. Acta, 21 (1982) 366-375. 13 AI. Leonov, Nonequilibrium thermodynamics and rheology of viscoelastic polymer media. Rheol. Acta, 15 (1976) 85-98. 14 S. Zaremba, Sur une forme perfection&e de la thttorie de la relaxation. Bull. Int. Acad. Sci. Cracovie (1903) 594-614; cf. also ibid., pp. 614-621. 15 H. Fromm, Stoffgesetze des isotropen Kontinuums, insbesondere bei zahplastischem Verhalten. Ing.-Arch., 4 (1933) 432-466. 16 H. Giesekus, Die Elastizitat von Fltissigkeiten. Rheol. Acta, 5 (1966) 29-35. 17 H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newtonian Fluid Mech., 11 (1982) 69- 109. 18 H. Giesekus, Stromungen mit konstantem Geschwindigkeitsgradienten und die Bewegung von darin suspendierten Teilchen. Teil II: Ebene Stromungen und eine experimentelle Anordnung zu ihrer Realisierung. Rheol. Acta, 2 (1962) 112-122.
Journal of Non -Newtonian Fluid Mechanics, 14 (1984) 47-65 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
ON CONFIGURATION-DEPENDENT DERIVATIVES *
GENERALIZED
OLDROYD
H. GIESEKUS Abteilung (F. R. G.)
Chemietechnik,
(Received
May 31, 1983)
Universitiit
Dortmund,
Postfach 50 05 00, D-4600
Dortmund
50
Summary After a short derivation of the upper and lower Oldroyd derivatives, linear combinations of them are examined with regard to their suitability for describing the non-affine motion of polymer structures occurring because of hampered flow. This leads to the introduction of a new type of convected derivative called a “configuration-dependent” derivative. While the combined Oldroyd derivatives can be considered to be upper convected derivatives supplemented by the addition of terms containing the rate of deformation tensor, in the new derivatives the irreversible rate of deformation tensor is substituted instead. This tensor, first introduced by Leonov, is strongly related to the configuration tensor, which is a measure of recoverable strain, at least in so-called one-mode models. The force-free motion of particles in steady simple shear flow as predicted by the respective generalized derivatives is analysed and it is found that the configuration-dependent derivative supplies a more realistic result for hampered flow than the combined Oldroyd derivative, except in the case of permanently isotropic structures for which both predictions are identical. For the special case of application of the configuration-dependent derivative to the configuration tensor, the expression is equivalent to that obtained by the application of the simple upper Oldroyd derivative itself. Thus it is concluded that the effects predicted by simple constitutive equations, when combined Oldroyd derivatives are used but not when only the upper Oldroyd derivative is used, may be more appropriately described by taking into account the non-isotropic character of the relative motion of structural elements occurring because of mutual interactions.
* Dedicated
to the memory
0377-0257/84/$03.00
of Professor
J.G. Oldroyd.
0 1984 Elsevier Science Publishers
B.V.
48 1. Introduction In Oldroyd’s pioneering paper on the formulation of constitutive equations [l] he introduced, amongst others, the extraordinarily fruitful concept of convected time derivatives. As shown in that paper these operators enable models of viscoelastic fluids originally restricted to infinitesimal strains to be generalized so that they are appropriate for finite deformations. In other words, they make these models “material-frame indifferent” or “objective”. At the same time Oldroyd demonstrated, by using the example of a Jeffreys model (i.e. a superposition of a Newtonian and a Maxwell model in parallel), that for each such infinitesimal model there exists not only one frame-indifferent model but an infinite manifold. These models can be classified into two main classes with respective members differing only for non-isochoric motions. However, the simplest representatives of each class, today commonly called Oldroyd fluid A and Oldroyd fluid B, show a distinctly different behaviour for isochoric motions as well. For example, their predictions for second-order behaviour in viscometric flows differ markedly although for first-order behaviour they are indistinguishable. This difference arises because in model A the “covariant convected derivative” (or “lower Oldroyd derivative”) is used whereas in model B use is made of the “contravariant convected derivative” (or “ upper Oldroyd derivative”). Comparison with experimental observations (such as the Weissenberg effect) showed that the predictions of model B agreed at least qualitatively with experiment whereas those of model A did not. The reason for this was not understood at once and some reflection on the structure of materials was required before any indication of the underlying physical difference between using one or the other of these derivatives was obtained [2]. Preference for model B was strengthened by the evidence that a dilute suspension of Hookean dumbbells in a Newtonian fluid under the combined action of rotational diffusion and frictional forces but neglecting hydrodynamic interactions behaves quantitatively like an Oldroyd fluid B, whereas it was not possible to find any analoguous structural model of an Oldroyd fluid A [3]. In addition, the simple network model of a concentrated polymer fluid proposed by Lodge [4] also leads to the model of an Oldroyd fluid B or, more precisely, to a linear superposition of these, called the Walters fluid B’ [5]. In spite of this, discrepancies between the predictions of this model and the observed behaviour of viscoelastic fluids (for instance the unrealistic prediction of a disappearing second normal-stress difference and constant shear viscosity in viscometric flows) motivated several authors and indeed first of all Oldroyd himself [6] to propose more general constitutive models to overcome those inadequacies. These equations can be considered to
49 include linear combinations of the two types of Oldroyd derivatives and not just one or the other. Later, Gordon and Schowalter [7] were motivated to introduce derivatives of this type by arguments from the theory of anisotropic fluids, and recently they have also been introduced in some more advanced network theories to allow for “non-affine motions” or “slippage” of the elements of the respective network structures, as for example by Johnson and Segalman [8], Phan-Thien and Tanner [9], Phan-Thien [lo], and Hsii and Schtimmer [ 111. Indeed, these generalized models, the so-called Oldroyd eight-constant model as well as the respective polymer solution and network models, provide a considerably improved description of real fluid behaviour and hence they are very often used for analysing more complicated flow situations by analytical and numerical methods. In spite of these improvements some important shortcomings remain, in particular in certain transient flow situations and as regards stability criteria. Whereas the Oldroyd eight-constant model is a purely empirical model, which can therefore only be evaluated by comparison with experimental results, the more recent polymer solution and network theories allow the hypotheses upon which they are based to be examined. This is attempted here in a continuation of a recent paper [ 121 in which the Oldroyd model B was generalized in a different way by introducing a configuration-dependent molecular mobility. The concept of a configuration tensor described therein (it can be understood as a tightening-up of the old idea of “recoverable strain”) together with the idea of Leonov [ 131 to decompose the total rate of strain into a reversible and an irreversible part are used here to introduce a new class of generalized Oldroyd derivatives. These take into account flow hampering by particle interactions within the deformed polymer coil and network structures and reduce to a quantity proportional to the simple upper Oldroyd derivative when applied to the “configuration tensor”. They are characterized in particular by force-free motion of particles in steady simple shear flow. In order to show more clearly the way in which one arrives at these derivatives a pertinent derivation is first given of the upper and lower Oldroyd derivatives. The relation with the original derivation starting. with the concept of convected coordinates is outlined in the Appendix. 2. Motion of a continuum and the Oldroyd derivatives Let the motion of an arbitrary particle imbedded in a material continuum be described by the one-parametric manifold of one-to-one mappings x = x(x,
t),
where X designates
(1) the position
of this particle
at a certain reference
time t,,
50
i.e. x(X, to) = X. The distance then given by
vector between
two neighbouring
particles
dx(t)=dX.vxx(t)=(vxx(t))+.dX=L’(t).dX.
is
(2)
Here vx is the gradient operator referring to the coordinates of the reference state and the dagger (t) indicates the transpose of the respective tensor. The one-to-one mapping (1) implies that eqn. (2) can be inverted to give dX=dx(r).V,X=(V,X)+.dx(t)=P-‘(t).dx(t),
(3)
where V, is the gradient operator referring to the coordinates of the actual state. In these mappings F(t) and F- '(t) are called the deformation gradients. From eqns. (2) and (3) it may be assumed that F(t) is a function of the independent variable X (“material deformation gradient”) whereas F-‘(t) has x as the natural variable (“spatial deformation gradient”). However, because of the existence and invertibility of the mapping (1) both may be interpreted as functions of either kind of variable. Using the velocity u(t) = &x(X, t)/& = Dx( t)/Dt and the velocity gradient L = ( v,u)+, which are usually considered to be functions of the spatial coordinates x, the following material time derivatives are obtained:
EF=L.F,
J&-l=
-F-I.L,
;dx=dx.L+=L.dx.
(5)
Let d A (t) be an oriented infinitesimal area embedded in the continuum and p(t) the local mass density. Then the associated volume element is given by dV( t) = dA( t) . dx( t) and the mass element by dm = p(t) dA( t) . dx( t). Because of the conservation of mass dm does not depend on time and therefore the vector dh(t)=p(t) depends ;
dh=
dA(t)
(6)
on time inversely -dh.L=
to the way in which dx( t) does:
-L+.dh.
(7)
By means of the deformation gradients 1E and F- ’ two spatial measures deformation are derived, namely the Finger deformation tensor B=F.F+ and the Green deformation B-i
=F-‘t.F-1,
of
(8) tensor (9)
51 which satisfy the relations (dX)*
= dx .B-’
edx,
(dH)*=dh.B.dh
where dH = dh( to). It is easily deduced from eqns. (4) that the material these deformation measures are
(10) time derivatives
of
DB -=L.B+B.L+, Dt
(11)
and DB- ’ -= Dt
-(L+.B-l+B-‘.L).
(14
Since these formulae are strongly related to eqns. (5) and (7) respectively, is sensible to rewrite them by introducing two new operator symbols: &
dx=g dh=;
$
it
dx-L.dx=O, dh+L+.dh=O,
g=!$-(L.B+B.L+)=~, DB- ’ Dt+(L+.B-‘+B-‘.L)=O.
bB-‘bt
024
These are just the operators of convected differentiation, introduced by Oldroyd 111, applied to either a vectorial or a tensorial quantity. In this way it is at once obvious that the upper Oldroyd derivative of a vector a %a _=_Bt
Da Dt
(13)
L*a
describes the time rate of change of this vector relative to the deformation of a distance vector dx embedded in the continuum whereas the lower Oldroyd derivative ba -=bt
Da
Dt
+L+‘a
describes the time rate of change of the respective vector relative to the deformation of the product of an embedded area vector with the density. Correspondingly the associated Oldroyd derivatives operating on a (secondorder) tensor T
g=g-(L4-+
T.L+),
(15)
52 _=bT bt
(16)
E+(L~.T+T~)
describe the time rate of change of this tensor relative to the spatial measures of deformation, i.e. the Finger and the Green tensor, respectively. In this way the concept of convected differentiation has been introduced without explicit use of convected coordinates. The equivalence with Oldroyd’s original derivation, given, however, in a much more general way for tensors of arbitrary order, is sketched in the Appendix. As briefly mentioned in the introduction, a multitude of related derivatives can be associated with both kinds of Oldroyd derivatives which are deduced from the so-called relative tensors. But these differ from the derivatives presented above only by additional terms n(Tr L)a and n(Tr L) T respectively, n being a positive or negative integer associated with the “weight” of the relative tensors. Correspondingly these terms disappear in isochoric flows (Tr L = 0). This distinction can also be introduced by means of the above elementary formalism starting from the relations g
g=
dV=(TrL)dV,
but this shall not be considered
- (Tr L)p, further
(17)
here.
3. Combined Oldroyd derivatives It has been demonstrated in the preceding section that there is a strong relation between the two types of Oldroyd derivatives and the rate of change of characteristic vectors embedded in the deforming continuum, and that there are no other types of frame-indifferent derivatives for which analoguous relations are available. In spite of this a multitude of such derivatives as weighted averages of upper and lower may, of course, be constructed Oldroyd derivatives: ED E=
1
(18)
Let L be decomposed into its symmetric and skew-symmetric of deformation and the rate of rotation tensors w=:(L-L+),
D=;(L+L+),
(19)
respectively, and the corotational tensor T be introduced by 6ila -=_--
Da
3lt
Dt
6iJT -=_-
DT
‘9t
Dt
w.
a=e+a w ’ Dt
W- T+ T. W,
parts, the rate
or Jaumann
’
derivative
of a vector u and a
53 which represents the equally weighted average of the class of combined derivatives (18) and as such the sole member in which only the skew-symmetric part of L appears. Then the full class of derivatives (18) can be decomposed for a vector u and a tensor T as follows:
This is how the combined derivative was introduced [6] in his eight-constant model, in fact as +(T4l+Lb
T),
implicitely
by Oldroyd
(24)
which is equal to hKDT/iIDt with 5 = 1 - p/h, where T stands for the stress deviator as well as for the rate of deformation tensor. The special case of the Jaumann derivative was introduced much earlier by Zaremba [ 141 in connection with a fluid model of the Maxwell type and rediscovered independently some decades later by Fromm [ 151. 4. Configuration-dependent
convected derivatives
In the aforementioned recently published network theories [8- 1 l] combined Oldroyd derivatives (with 0 < 5 < 1) are used to allow for “non-affine network motion”, also called “motion with slippage”. The underlying idea is that the structural elements of the polymer network, usually called “beads”, are not stressed by the average motion of the continuum (interpreted in a statistical sense) but by a velocity field modified owing to shielding of the neighbouring beads. It is supposed that this effect reduces only the deformational component of the flow field, the reduction being signified by the parameter 5. This idea is well-known from the theory for dilute solutions of linear flexible polymers which are considered to exist as coils with hampered flow, the limiting cases being the “free draining” and the “non-draining” coil. The degree of hampering, generated in particular by hydrodynamic interaction, depends here on the statistical diameter and the density of the coils, i.e. on the molecular weight and the solvent quality. However, molecular structures such as these coils deform under flow conditions so that the question arises as to whether it is appropriate to signify the hampering of the draining motion in the vicinity of a bead by a constant shielding coefficient 6. Indeed, Hsti and Schtimmer [ 1 l] have proposed that this coefficient be replaced by a function of the invariants of the
54 rate of deformation tensor, which they term the “slippage function”. In contrast, Giesekus [ 121 recently intimated a less formal way of introducing structural points of view into the modelling of constitutive equations of polymer fluids, which is strongly related to his concept of configuration-dependent mobility [ 16,171. The combination of the structural deformation by a configuration tensor given here with Leonov’s notion of the irreversible rate of deformation tensor [ 131 leads straightforwardly to a configuration-dependent generalized Oldroyd derivative as will be demonstrated in the following. 4. I. Force-free motion under hampered flow conditions Let the distance of a representative bead from the centre of resistance of the respective polymer molecule be designated by the position vector r and let this quantity be considered as an infinitesimal vector like dx from the point of view of a macroscopic length scale. Then in an unhampered flow field Br/Q t is in general not zero but rather is related with the friction forces contributing to the excess stresses. If, however, in an ideal experiment all force-transmitting bonds were cut then the particle would move according to the condition sDr/at
= 0,
i.e. Dr/Dt=L.r=
On the other hand, differential equation Dr/Dt
(25)
(W+D).r.
if the flow is hampered by the surrounding beads the of force-free motion has to beesubstituted by
(26)
= i . r,
where i = (~6)~ corresponds to a modified velocity field 6 around the centre of resistance of the respective molecule. The principle of material-frame indifference allows only for the modification of the deformational part d, whereas I? = W must hold. The special choice leading to the aforementioned combined Oldroyd derivative is
ri= (1 -
[)O,
but a more general case. 4.2. The configuration
(27) choice is possible
which contains
this only as a special
tensor and the irreversible rate of deformation
tensor
It has been shown, for example by Giesekus [ 16,121, that the elastic excess stresses resulting from the connections between the beads as well as from entropic effects depend on the tensor ( w), at least in the case of a “one-mode model”. (The arrow-shaped brackets designate an ensemble
55 average.) If the state of rest is signified by the isotropic tensor (~#) = ( ( i . ~ ) / 3 ) 1 , a configuration tensor b * can be introduced using the definition (rr) = b.
=
3
b.
(28)
It represents a type of Finger tensor characterizing the deformation of the network structures corresponding to the notion of "recoverable strain". Leonov [13] introduced a decomposition of the velocity gradient into a reversible ("elastic") and an irreversible (" plastic") part L = Lie ) + Lip )
(29)
claiming that Lie ) is subject to a condition analoguous to that in eqn. (1 la) for the Finger tensor:
,e, = Dt
(L'e)'b+b'Ltte))=O"
(30)
As a result the complete upper Oldroyd derivative can be expressed as
Here Lip ) is understood to be an inner variable of the configurational state and therefore an isotropic function of b only. As a consequence Lip ) can be identified with its symmetric part D/p), i.e. W~p)= 0, and Dip ) must be commutable with b. Because of this eqn. (31) simplifies to ~b ~t - -2D/p)'b'
(32)
and the irreversible rate of deformation tensor Dip , is therefore expressible by the configuration tensor and its upper Oldroyd derivative in the simple form: Dip)= - ½ b - I "
~b ~t"
(33)
If eqn. (31) is transformed in the following way
D----~-(W.b-b. W) = -6~b ( O - D/p,)-b + b. ( O - Dip))= Db ~t'
(34)
it can be seen at once that under fully general flow conditions D and Dip ) are identical only when the configuration remains isotropic (b - 1). However, in motions with constant stretch history where Db/Dt can always be made zero by a suitable choice of the reference system, @b/6~t can in contrast only be made to disappear if the motion is irrotational ( W = 0). * In the papers by Leonov [13] and Giesekus [16] this quantity is denoted by the letter C.
56 4.3. Definition
of configuration-dependent
generalized
Oldroyd derivative
It is implicit in the concept of the irreversible rate of strain tensor that the beads of a polymer structure follow the elastic part of the deformation gradient L.(,) 2 L - DC,, without any frictional loss. This suggests the hypothesis that flow hampering effects arising from hydrodynamic or molecular interactions should be dependent on the irreversible rate of deformation tensor DC,,, rather than on the complete tensor D = Dee, + Dcp,. According to this hypothesis a new type of convected derivative is proposed in which instead of eqn. (27) ”
D = D - (*DC,,)
5* =D++-‘.g
(35)
is introduced. This configuration-dependent derivative is, correspondingly, defined for a vector a and a tensor T, respectively, as follows: D*a
Da
-=6r+[*Dc,,.a,
(36)
Dt
z=g+<*(Dc,,.T+ The subsequent lrD*z -= Dt
T.D&
(37)
special cases are easily verified:
-2(D-s$*D&
The latter, which follows at once from eqn. (31), is of basic significance for establishing constitutive models, as will be discussed briefly in Section 6. However, our primary interest here is to compare relevant properties of this configuration-dependent convected derivative with the combined Oldroyd derivative and to argue that this derivative is appropriate to model structural changes generated by deformational motions of polymer fluids. 5. Force-free motion of a particle in steady shear flow with configuration-dependent hampering It has been demonstrated by eqn. (34) that the various types of convected derivatives lead to different predictions even in flows with constant stretch history if these are not irrotational. This suggests that just such a flow should be analysed, the most appropriate choice being simple shear flow. To elucidate the basic properties it is helpful to study the force-free motion of a bead in the respective flow field in which the mode of
57 hampering is given by the applied derivative. The differential equation this motion is obtained by introducing eqn. (35) into eqn. (26) to give: Dr/Dt
= (L - <*DCr,)) .
r.
of
(40)
5. I. Kinematics of hampered flow in simple shear As the neutral direction plays no part in simple shear flow the problem is formulated in two dimensions only with x, designating the direction of flow and x2 that of the shear gradient, y being the rate of shear. As is well known, the following matrix representations then hold:
(41) With
it follows that (43) Introducing
these expressions -i//2
6,
into eqn. (33) results in
b,,b*, (44)
+ h,b22
-
b:2
IIb,,b,,
-
2b:2
However, the additional condition that Zb/%t has to be an isotropic tensorial function of b and therefore commutable with this tensor means that D cpj must be a symmetric tensor, i.e. bz2 = b,,b,, - 2b:,. The resulting simplifications lead to the following expressions for b and DC,,:
b1
II
1+2z2
1 +z2+zjl+
*wA
:u-
Y/2 1 +
MS-
l/S)]’
/I
l/S)
1
*z
1 -i(C-
=l/S)1
ii
j//2 1+z2
z 1 IIII ’
II
(45)
z
1
1)
1
--z/i (46)
with s=
Jb,/b,,=cot+,
z = +(I - l/S)
= cot 2+,
(47)
58
where b, and b,, are the principal values of b, associated with the major and minor axes, and 9 is the angle between the first principal direction and the flow direction. It is seen from the above expressions that the configuration is fully determined by either the ratio of the principal values or the orientation angle I$, the dependence of these two quantities on one another resulting from the symmetry condition utilized above. Substitution of eqn. (46) into eqn. (35) results in 6
II
f/2
A
1 +z*
-5*z
1 +z*-[5*
1+z*-_l*
#$*z
with principal
L31.11
=
values + z* - #$*)‘+ #$**z*
++$J(l -
and the angle \k between given by cot 2 \k = -
(48)
Ii ’
the first principal
direction
5*z 1+z*-<*
and the flow direction
(50)
*
As is well known (see, for example, Giesekus [ 181) the streamline pattern of homogeneous plane flow fields is ei_ther hyperbolic or elliptic d:pending on whether the deformational 1 part D or the rotational part W of the associated velocity gradient L predominates; the intermediate case is given by the straight line pattern of simple shear flow. Because @= W the rate of rotation is determined by j1/2, whereas it can be seen from eqn. (49) that the rate of deformation b, is less than y/2 for 0 < [* -C 2 and equal to y/2 for c* = 0 and 2. Therefore the path lines of beads moving without the action of external forces in hampered simple shear flow are ellipses oriented so that the common angle between the direction of the major axis and the flow direction is Cp= \k - s/4 with cot 2 cp = 1+ z* - t* 5*z and the common axes a,, 1s
I
ratio of the magnitude
(1 +z*)+/(l
z=?I= =
(51)
+z*-<*)*+<**z*
(1 + z”) - J(1 + z* - <*)‘+
cot(2+-
a) cot
of the major axis a, to the minor
I’* t**z*
I (52)
59
Fig. 1. Comparison of a path line of the force-free motion of a bead in hampered flow (full line) with the shape and orientation of the average configuration (dashed line). The chosen values were t* = l/3 and + = 30” (i.e. { = 1.732), resulting in @ = 5.45” and Z = 2.732 by using eqns. (51) and (52).
An example of such an elliptical path line is given in Fig. 1 together with the shape of the tensor ellipse of bIi2 which according to eqn. (47) is specified by the orientation angle C#Iand the ratio of the major to the minor axis 5. Before analysing eqns. (51) and (52) in the general case two special cases will be discussed. 5.2. Path lines with isotropic structure This corresponds to z = 0 (i.e. [ = 1 and + = 0) and is just that case in which the flow hampering is described by the combined Oldroyd derivative defined in Section 3, i.e. DCP,= D, 5* = 5. Under these circumstances eqns. (5 1) and (52) simplify to give @=O,
Z=@?PC
(53)
This means that the principal axes coincide with the axes of the coordinate system, more precisely that the major axis is aligned with the flow direction, and that the ratio of the axes, 2, decreases monotonically from infinity (i.e. straight line pattern of simple shear flow) to one (i.e. circular pattern of pure rotational flow) as the shielding coefficient .$ increases from zero to one, the latter limiting case corresponding to completely hampered flow. Figure 2 shows some examples of such flow patterns associated with various values of the parameter 5.
60
Fig. 2. Path lines of the force-free motion of a bead in hampered flow in the limiting case of isotropic structure (+ = 0, J = 1) for various values of the shielding coefficient t* = 5.
5.3. Path lines with complete shielding The special case t* = 1 corresponds to complete shielding in non-isotropic structures. Now eqns. (51) and (52) simplify to give
a=+,
Z=[
i.e.
a/a,,
= :b,/b,,.
(54)
This means that the orientation of the path line ellipses coincides with that of the configuration tensor and the ratio of its axes is equal to the square root of the ratio of the respective principal values. Indeed, b has the properties of a Finger tensor and so it is b 1’2 that characterizes the shape of the average configuration of the deformed network structures. Thus, in the limiting case of completely hampered flow the force-free motion of a bead follows exactly the motion of the respective structure itself. 5.4. Path lines in the general case In Fig. 3 the dependence of @ and Z on the orientation angle C#B is shown. For increasing values of C#Ithe angle Cp first increases to a maximum, with
61
0' 0’
0"
10"
2o"
I
I
10’
20'
3o" Ql 40' 45O I
II
30' (0 4o" 45'
Fig. 3. Orientation angle Cp and ratio of the axes Z of path line ellipses as a function of the orientation angle + of the configuration tensor. The dashed line designates the locus of maximum values of Q given by eqn. (55).
magnitude 2J1-5* Q,max = f arccot
5*
= 2( 9 - 48)
(55)
at (56)
62 45. 10
30'
20@
10' 10
15'
6"
Z 5
2
1 1
2
5
5
1
Fig. 4. Ratio of the axes Z of path line ellipses as a function { of the average molecular structure.
of the ratio of the principal
axes
and then decreases to zero at + = 9r/4 (i.e. [ = 1) if <* < 1. In contrast Z decreases monotonically as the orientation angle + increases and approaches its minimum value, given by eqn. (53), with horizontal tangent if <* < 1. In Fig. 4 this latter dependence is shown once again in a double-logarithmic plot of Z versus { illustrating the gradual approach to a straight line with increasing values of { as well as of the shielding coefficient t*. 6. Discussion The configuration-dependent convected derivatives introduced here have been shown to represent a generalization of combined Oldroyd derivatives with these contained as a special case for structures with permanently isotropic configurations. However, under normal conditions, in which fluid motion results in non-isotropic configurations, it has been demonstrated that these generalized derivatives are based on more appropriate suppositions about the reference velocity, that is the velocity of a particle not subjected to any forces. This was seen most clearly when the limiting case of complete shielding was analysed, for then the force-free motion of a structural element was found to be identical with the motion of the respective structure itself, whereas using combined Oldroyd derivatives it would be purely rotational motion like that of a rigid sphere. If it is accepted that the generalized configuration-dependent derivatives are more appropriate for including shielding effects than the usual combined
63 derivatives, and they are applied to the configuration tensor (or an equivalent quantity), it follows that the respective constitutive equations essentially reduce to those of an Oldroyd fluid B. This is evidenced at once by eqn. (39). Hence, effects such as those mentioned in the introduction, which were included when the upper Oldroyd derivative was substituted by a combined one, can no longer be attributed to simple flow hampering or shielding at all. At first this seems to be a disappointing result, but actually it emphasizes once again the fact, already well-known in the theory of dilute polymer solutions, that interaction of polymer beads under deformational motion always results in non-isotropic properties. These affect the mechanism of relative motion of structural elements so that they cannot be taken into account using a pre-averaging procedure without loss of essential features. However, if this anisotropy is properly taken into consideration it will lead to a more satisfactory prediction of the respective flow effects than by using combined Oldroyd derivatives. In conclusion, although the above investigation of the relative motion of structural elements under conditions of hampered flow may have questioned the usefulness of combined Oldroyd derivatives for describing the respective effects, it has also emphasized once again the special significance of Oldroyd’s original concept of convected time derivatives. In particular, the equivalence relation (39) proves that the upper Oldroyd derivative maintains its central role in constitutive equations even when shielding effects are included, and this in spite of earlier conjectures that it would have to be replaced by a more complex combined derivative.
Appendix Let a convected coordinate dent coordinates 5’ so that dx(t)
system be defined
by a set of time-indepen-
= d&,(t),
(Al)
where dx( t) is given by eqn. (2) and the gi( t) designate a time-dependent basis by means of which the metric tensor is determined as gikCt)
=giCt)
The reciprocal
‘gk(f).
basis gk( t) is then given by
g;(t)*g”(t)=&Y This basis can be used to connect the vector d/z, introduced time-independent components dqk: dh(t)
642)
= dqkgk(t).
(A3) by eqn. (6) with
(Ad)
64 It is seen from eqns. (Al) and (A4) that the co- and contravariant bases gi( t) and gk( t) move in the same way as dx( t) and dh( t), respectively; and according to eqns. (5) and (7) (A5) Material-frame indifferent time derivatives are understood to be composed of material time derivatives of the respective co- and contravariant components in a convected coordinate system but without inclusion of the time derivatives of the base vectors. This means that for a vector a(t> = ai(t)g,(t)
=
646)
ak(f)gk(r)
and a tensor T(t)
= +;‘(t)g;(t)g;(t)
= CTkk’(t)gk(t)gk’(t)
the upper and lower Oldroyd G(t)
Da+)
-=-
Dt
at
Dt
Dt
Making
b&) gi(t)y
are defined Dt
bT(t) bt=
gi(t)gi’(t)?
as follows:
Dak(t)
bt=
D?‘(t)
W(t)
-=
derivatives
(A?
*,
648)
g”b),
h/&) Dt
(A91
gk(t)gk’(t)’
use of the expressions
Da(t) Dt
Dar’ Dt gi(t)
DT( t) Dt
D?’
-=_
.Dgi(t) +&’
Dak
Dt
+gl(‘)-D~
m”(t) +7kk’
v
Dgi’(t)
+gi’(l)
Drkk’ D,s”@h”+)
Dgk(t) Dt
(t)+ak
Dg-0)
--==g,(t)g,Jt)+Ti”
=
k
=Dtg
Dt
&?“‘(d
+gktf)
Dg;;”
)
(Al 1)
as well as eqns. (A5) enables eqns. (A8) and (A9) to be rearranged to give eqns. (13- 16) of Section 2. It is, however, felt that the derivation given in Section 2 facilitates a real understanding of the basic difference between the upper and lower Oldroyd derivatives, whereas the more formal derivation using convected coordinates, despite its elegance, makes the reason for this difference obvious only for those who are thoroughly versed in this formalism.
* The representation interest.
of a tensor by its mixed components
is not included
here as it is not of
65 Acknowledgment The author English.
is indebted
to Dr. M. Hibberd
for his help in improving
the
References 1 J.G. Oldroyd, On the formulation of rheological equations of state. Proc. Roy. SOC., A200 (1950) 523-541. 2 H. Giesekus, Der Spannungstensor des visko-elastischen Korpers. Rheol. Acta, 1 (19581961) 395-404. 3 H. Giesekus, Die rheologische Zustandsgleichung. Rheol. Acta, 1 (1958-1961) 2-20. 4 A.S. Lodge, A network theory of flow birefringence and stress in concentrated polymer solutions. Trans. Faraday Sot., 52 (1956) 120-130; Elastic liquids, Academic Press, New York, 1964. 5 K. Walters, Non-Newtonian effects in some general elastico-viscous liquids. In: M. Reiner and D. Abir (Eds.), Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Macmillan, New York, 1964, pp. 507-519. 6 J.G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Sot., A245 (1958) 278-297. 7 R.J. Gordon and W.R. Schowalter, Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions. Trans. Sot. Rheol., 16 (1972) 79-97. 8 M.W. Johnson, Jr. and D. Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation. J. Non-Newtonian Fluid Mech., 2 (1977) 255-270. 9 N. Phan-Thien and RI. Tanner, A new constitutive equation derived from network theory. J. Non-Newtonian Fluid Mech., 2 (1977) 353-365. 10 N. Phan-Thien, A nonlinear network viscoelastic model. J. Rheol., 22 (1978) 259-283. 11 Y.T. Hsii and P. Schiimmer, Some general rheological properties of dilute polymer solutions predicted from intrinsic viscosity measurements. Rheol. Acta, 2 1 (1982) 5 1 l-5 13; Quantitative description of rheological properties of dilute polymer solutions. II. Estimation of rheological material functions at moderate shear rates using the Rouse-Zimm model with slippage. Rheol. Acta, 22 (1983) 348-353. 12 H. Giesekus, A unified approach to a variety of constitutive models for polymer fluids based on the concept of configuration-dependent molecular mobility. Rheol. Acta, 21 (1982) 366-375. 13 AI. Leonov, Nonequilibrium thermodynamics and rheology of viscoelastic polymer media. Rheol. Acta, 15 (1976) 85-98. 14 S. Zaremba, Sur une forme perfection&e de la thttorie de la relaxation. Bull. Int. Acad. Sci. Cracovie (1903) 594-614; cf. also ibid., pp. 614-621. 15 H. Fromm, Stoffgesetze des isotropen Kontinuums, insbesondere bei zahplastischem Verhalten. Ing.-Arch., 4 (1933) 432-466. 16 H. Giesekus, Die Elastizitat von Fltissigkeiten. Rheol. Acta, 5 (1966) 29-35. 17 H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newtonian Fluid Mech., 11 (1982) 69- 109. 18 H. Giesekus, Stromungen mit konstantem Geschwindigkeitsgradienten und die Bewegung von darin suspendierten Teilchen. Teil II: Ebene Stromungen und eine experimentelle Anordnung zu ihrer Realisierung. Rheol. Acta, 2 (1962) 112-122.