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On the converging flow of viscoelastic fluids through cones and wedges
Journal of Non-Newtonian Fluid Mechanics, 14 (1984) 219-247 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
ON THE CONVERGING FLOW OF VISCOELASTIC THROUGH CONES AND WEDGES +
219
FLUIDS
A.M. HULL * and J.R.A. PEARSON ** Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, Prince Consort Road, London (Gt. Britain) (Received March 15, 1983)
Summary A novel approximation method for obtaining solutions to the slow flow of viscoelastic fluids through converging cones and wedges is presented. The method involves considering two distinct regions in the flow field divided by a streamline: an inner irrotational sink flow occupying most of the flow field and an outer highly sheared self-lubricating layer adjacent to the walls. This procedure greatly simplifies the governing equations and allows analytic expressions to be obtained. Since the two flow regions are separated by a streamline, different constitutive equations can be used for the inner and outer flow fields. This enables constitutive models to be used which model polymer melt rheology reasonably accurately and still retain analytic solutions. The method is applied to both wedges and cones; extensions to converging annuli are straightforward. Finally, an example flow field is considered in relation to a typical die-entry flow in polymer processing.
1. Introduction The situation to be considered is that of the slow flow of viscoelastic fluids through converging wedges and cones (see Fig. 1). Part of the justification for this is to arrive at a better understanding of die-entry flows + Dedicated to the memory of Professor J.G. Oldroyd * Present address: Du Pont de Nemours International S.A., Polymer Products Department, 28 Rue Alexandre Gavard, 1227 Carouge, Geneva (Switzerland). ** Present address: Schlumberger Cambridge Research, PO Box 153, Cambridge CB2 3BE (Gt. Britain) 0377-0257/84/$03.00
0 1984 Elsevier Science Publishers B.V.
220 in polymer processing. By assuming that the die-entry has a large contraction ratio occuring over a short distance or that the flow rates are high, the entrance geometry is idealised to one of infinite extent upstream with a point or line efflux downstream.
Fig. 1. Geometry
and coordinate
system of the flow problem.
It has proved to be rather difficult to obtain solutions for flows in converging sections. In fact, for a viscoelastic fluid involving at least one objective time derivative of the stress tensor (in the differential formulation of the constitutive equation) only one exact solution has been discovered [ 11. This solution is for a 180” wedge (half-angle 90’) and is for the General Linear Upper convected Viscoelastic Fluid defined in [2] as
(14 in which 7 is the stress tensor, yIol the strain tensor, both tensors in the upper convected formalism, and M is the memory function. expansion solutions Otherwise, inverse coordinate (l/r) or “indirect” have been obtained using various constitutive equations. These are constructed either as perturbation or series expansions [3-61. They are only valid in regions far from the apex of the cone or wedge. This is as expected; however it should be mentioned that the expansions result in asymptotic diverging series [7,8]. It would seem to be obvious that for solutions valid in the neighbourhood of the apex, direct expansions in r are required. However, as shown in [8] there is a significant difficulty in doing this due to the effect of the upstream history of deformation. That is, the direct expansion or downstream flow involves the history. of the flow in a non-trivial way and it does not seem possible to resolve this with a matched solution. For these reasons a different approach is explained in this paper. The Extensional Primary Field approximation (EPF), which was originally pro-
221 posed independently by Metzner [9] and Pickup and Pearson [ 10,111, is used as a starting point. The EPF results from diagonalising the stress tensor, 7
7 =
i
711
[ol
M
[ol
722
PI
[ol LOI 733
I
(l-2)
by neglecting the shear stress components T,~, T,~, 723. Of course, 7 must be defined relative to some non-rotating coordinate system. The EPF cannot satisfy the no-slip boundary condition so it can only apply to some core region not adjacent to the walls (see Fig. 2). The object here is to develop and extend the EPF showing how to treat the remainder of the flow field and, in particular, satisfying the no-slip boundary condition. The kinematic justification for the EPF is from experimental observation of die-entry flows, surveys of which are given in [8,12]. Provided that the flow remains slow, a condition satisfied except for high fluxes of polymer solutions, most of the flow field in the converging region has almost radial streamlines, i.e. v0 -=xv, almost everywhere. More important, most of the outflow comes from the central portion of the flow whether recirculating vortices are formed or not. This means that the velocity field is proportional to l/r or l/r2 for a wedge or cone, in which the coordinate systems are cylindrical or spherical polar, respectively. Furthermore, there is considerable evidence that over most of a central core region there is little variation in the magnitude of the velocity across the core region. Consider now the whole flow field which is illustrated schematically in Fig. 3. Part a shows flow towards the apex of a cone or wedge with half angle 0. If the radial velocity component is smooth as in part b (i), then
core
of extent of EPF
Fig. 2. Die-entry
showing
streamlines
for core region.
222 (ii)
e=B / *
ll (iI
‘“r
0
e-,
0
Fig. 3. (a) Geometry of converging wedge or cone of half angle 8 with polar coordinates (r, 0). A -core region; B-self-lubricating region. 8, (r) refers to position of interface between lubricating and core regions. (b) Radial velocity component profiles as functions of B (at some given r): (i) smooth profile with no lubrication; (ii) kinked profile with sharp change in shear rate (l/r)au,/M at t9 = t?,(r), indicating lubrication effect.
r- ‘Clv,/SJ is of the same order of magnitude as av,/& over most of the flow region, and so neither simple shear flow nor irrotational flow are suitable approximations. If, however, a cone region where there is little radial variation in v, is assumed to arise so that av,./ib B r-‘av,/M, then there must also be a strongly sheared self-lubricating layer close to the wall with r-‘av,/ahJ z=- av,./ib, as in part b (ii). Thus the flow may be approximated by a simple shear flow within 8, < ]0] < 0, and an irrotational, sink flow within 0 G 181~ 8,. (The velocity profile in part b (ii) should not be taken as invariable, as it is possible that recirculation occurs in the wall layer implying a positive radial velocity in the neighbourhood of the wall.) In this manner, the no-slip boundary condition at the wall can be satisfied. Metzner [9] has proposed, as a criterion for the existence of an EPF,
N,,=J$L sh
where NTr is the Trouton ratio and the simple shear viscosity The assumed kinematics for tion for dynamic consistency, particular strain rate be much of magnitude greater than the
(1.3)
between the uniaxial extensional viscosity Q qsh. the extended model require a stronger condinamely that the extensional stresses at a greater than the shear rate, possibly an order strain rate. In crude terms this is
223 N,
>> l/(0
- 6,) z+ 2/a.
(1.4)
Recent experimental data on the stress in uniaxial extension have been reported for various polymer systems including low-density polyethylene (LDPE) [ 131, polystyrene (PS) [ 141 and two fully compounded rubbers, poly(isobutene-co-isoprene) (Butyl rubber) and poly(chloroprene) (Neoprene) [15]. Although extremely high strains were recorded, no steady state was reached, which means that a true viscosity in uniaxial extension could not be measured by filament extension. However, a measure of the stress likely in extensional flow is given by the maximum stress achieved in the experiment. In turn, this will be taken as a measure of the stresses achieved in the sink flows occurring in the core region. For the shear layer, the stress involved either in the start-up of simple shear or at steady values, can be taken as representative. From the experimental data shown in Hull [ 151, it can be seen that the extensional stresses are one or two orders of magnitude greater than the shear stresses at comparable deformation rates. This represents sufficient dynamical justification. To a first approximation the interface 8, separating the flow field will be treated as a streamline, i.e. there is no interchange between fluid undergoing simple shear and fluid undergoing sink flow. In sections 2 and 3 explicit calculations are presented in which the fluid in the core is described by a UCMF (Upper Convected Maxwell Fluid) and the fluid in the shear layer by the Ellis Viscosity Equation. Such a model where the two regions are not linked by the same constitutive equation will be termed a Two-Fluid Model (2FM). In section 4, the same calculations are done for a certain “naive” rheological model. Here the two regions are linked by the same constitutive equation. Such a model will be termed a Two-Zone Model (2ZM). The 2ZM does not rely upon 8, being a streamline and so should open up possibilities of calculating higher-order terms in the approximation method. The term 2FM is only meant to imply that different rheological aspects of what may well be one single fluid are being considered. A higher-order approximation scheme would have to be based on a matching of an almost viscometric flow field to an almost extensional flow field. Pipkin and Owen [ 161 have considered the form taken by a general constitutive equation in the first case and Pountney and Walters [ 17,181 in the second. 2. Wedge flow The equations of continuity and stress equilibrium with respect to a cylindrical polar coordinate system for steady, planar flow in a wedge ( r,B),
224 with corresponding
velocity
components
(u, U) are
(2-l) (2.2) (2.3) The boundary
conditions
for no slip at the wail, i.e. at 0 = + 0, require
u(r,f
0) = u(r,+ 0) = 0. (24 In the steady state, the flux per unit width 2@of fluid crossing every section of the wedge must be the same, namely
Q=
(2.5)
fLde
The convention chosen is that the flux in the converging direction is negative. The flow is now separated by an interface defined at 8 = e,(y). At this stage of the approximation 0, is considered to be constant locally and at most slowly varying with radial distance: thus it is required that r
ae,/ib e
1.
(2.6)
The flow within the region 181G 8, is assumed to be a sink flow, locally, with a velocity field satisfying the continuity equation (2.1) given by ti=
-K/r,ti=O
K a constant.
(2.7)
Quantities designated by “ A ” refer to the core but will only be used where possible confusion with the shear layer may arise. The formal boundary conditions at the interface are the continuity of velocity and of the normal and tangential tractions. Neglecting the possible slight curvature of 8,, these are Li = u, (6 = U), (2.8) B + +e*=p + ree,
(2.9) (2.10)
&I = rre 7
all evaluated at 0 = 8,. Because the core is a sink flow, +,.* will vanish identically. However, it is to be expected that rre f 0, so to the first approximation (2.10) will not be satisfied. However, a measure of the accuracy of the method is provided by the ratio of the tractions in (2.9) and (2.10). Thus it will be desirable for
brel c=
IP+%el +B,=
I%4 IP+ reel e=8, K
I.
(2.11)
225 Since 8, is a streamline there is an extra kinematic condition on the flux which is sufficient to determine a unique solution. If the flux in the core is 20 and the flux in the shear layers is 2Q, then
e=i,+Q,
(2.12)
0 = - K8,, constant,
(2.13)
Q = 4”“~ dr, constant.
(2.14)
I
The effect of the core flow on the shear layer will be shown to be through a( p + fee)/& determined at the interface from (2.9), and also through the radial velocity of the interface, -K/r. The expression a($ + feo)/i3r for the UCMF given by (2.15) can be calculated, by first evaluating kinematics of (2.7): (2K.s/r2
P[o]=
0
,O
0
0'
(1 +ZKs/r”)-’
0
0
O/
and then integrating
the strain
tensor
for the line sink
ys=t-t’
(2.16)
(2.15) to give
7TT= -P./h5,
+ee = (P/M( 1 - &%W),
(2.17)
in which all other stress tensor components vanish, E, is the exponential integral [25,p. 2281 and 5 = r*/(2AK) is a dimensionless local Deborah Number. The required expression is: (2.18) where (2.19)
Ap* = 1 - 5 - 25* + (3 + 2<)c2eEE,(<).
A graph of Ap* as a function oft is shown in Fig. 4 having asymptotes AP* - 2.0, 1.0 for large, small 5. Since Ap* does not vary substantially with 5, a suitable order of magnitude for Ap defined in (2.18) is (2.20) The shear layer equations
may be found
from the continuity
and stress
226 2 -
AP*
I--/
0
0
2
4
6
8
4
10
Fig. 4. Graph of Ap* defined in (2.19) as a function of 5.
equilibrium equations (2.1)-(2.3) once a new variable p = 0 - 8 (which measures the angle from the wall to any given point in the shear layer) is introduced. The transformed equations are (2.21)
(2.22) (2.23) Let x = 0 - 13,, so that TX is the thickness of the shear layer and assume the usual order of magnitude requirements for lubrication theory, i.e. r-u/K= O(l),
Y-’ a/@? = 0( l/rx)
and a/h
= 0( l/r).
According to the usual theory it is expected that 2) = 0( Kx/r). If v is larger than this, then retaining the dominant term in the continuity equation (2.21) gives au/@3 = 0 and the boundary conditions at the wall cannot be satisfied except trivially for 0 = 0 (in which case au/i!@ is not the dominant term). The largest term in the strain rate tensor 9 is (2.24) It is now appropriate to specify a constitutive equation for the stress r. Assume a power-law type dependence (which is usual for polymers at high shear rates) with 7rl3= -M&In,
(2.25)
r,, - ree = -Pl?J”
p, q, m, n being constants.
Using (2.24) to give an order
of magnitude
for (2.25), the order
of the
227 terms in eqns. (2.22)-(2.23)
in order of appearance are {eqn. (2.22))
and
4 -K
i xr2
r
n -j.x
1,
{eqn. (2.23))
xr3
The shear-stress term dominates the normal-stress term in (2.22) provided
-.&
K
4
1.
XP i xr2 i
(2.26)
For the Butyl rubber examined n = m and q/p = 0( 1). This is also the case for a number of melts including HDPE, PP and PS [ 191. In addition n = m at high shear rates is a feature of the White-Metzner Model which was developed specifically to model the shear stress and the normal stress difference in simple shear flow [20]. Their model is confirmed by data for LDPE and HDPE. In any case, provided m - n < 1 it will always be possible to choose x small enough so that (2.26) is satisfied in some region not including the origin. As the origin is approached, the normal stress difference may have to be taken into consideration. However, in what follows it will be assumed that the shear stress terms dominate. Written in terms of (r,6), (2.22) becomes upon retaining the largest terms: (2.27) Of the terms in (2.23) only the normal stress term is significant when n -C 1. Therefore in the variables ( r,O), (2.28) and p + roe does not vary across the shear layer but is determined at the interface. For consistency the terms in (2.27) must have the same order of magnitude. Thus x
=
o((;)( ;)‘-n)“~‘+n~.
(2.29)
If n = 1 and writting q = p, eqn. (2.28) is obtained only if p/P=
1
(2.30)
and the consistency relation (2.29) is x = 0( ~/fi)‘/2 -=x 1, a stronger require-
228 ment than (2.30). The 2FM with a Newtonian shear layer is a possible model for wedge flow of a melt in at least two particular cases. (i) The melt contains a low molecular weight phase which is sufficiently mobile to migrate to the surface at the wall to form a lubricating layer. Rubber compounds exhibiting wall slip, particularly EPDM and Nitrile compounds [21] and polymers such as HDPE [22] and PVC are examples of such materials. (ii) The die-entry is lubricated by an external agent of low viscosity which is pumped into the region adjacent to the wall. Winter et al. [23] have used this technique in an experiment intended to achieve pure extensional motions in the entry region, although Caswell [unpublished, reported at 1981 workshop on extensional flow, Ross Priory] disputes the possibility of achieving such motions for 0 /K 1. Otherwise if the melt is the same material in both regions then the zero-shear-rate viscosities will be identical, p = fi. However as discussed in [24] temperature rises may occur at sufficiently high flux in the die-entry. Most of the temperature variation occurs in a thin layer either adjacent or very close to the wall. In such cases, it should be possible to contain the temperature layer within the shear layer, which would lower the effective viscosity p. For Butyl rubber a temperature rise of about 15°C causes the viscosity p to be halved with a corresponding decrease in Q [8]. So to this level of approximation temperature rises increase the suitability of the 2FM. This conclusion is valid for all values of n. Rather than using a Power-Law model, a solution of (2.27) is obtained for the more realistic Ellis model given by [2] for the viscosity n (2.31)
/J/11 = 1 + IQ/.$-‘,
where s and (Yare constants; (Yis equivalent to l/n in eqn. (2.25). This model has both a Newtonian constant-viscosity region at low shear rates and a power-law region at high shear rates. An expression for the shear rate y,./,Bis obtained from (2.24) and (2.31) (2.32) The shear stress results from an integration 7re = -rBAp
- K,(r),
where K,(r) Let
is a function
71
=
-t-Ape,
-
K,
of (2.27) using (2.18) (2.33)
satisfying
the boundary
conditions.
(2.34)
229 be the shear stress at the interface r, = -rApO
and (2.35)
- K,
be the shear stress at the wall, Then since Ap < 0, (2.36)
7i<7W,7i
the velocity
U, it is necessary
to distinguish
two possibili-
Case I Backflow does not occur in the shear layer, in which case 7W< 0 and rre < 0 everywhere within the shear layer. Case II Backflow does occur, in which case rW> 0 and Q = 0 for some 8 E (8,,0]. For Case I, substitution $=i(rApe+K,)(l Integrating UC
of rre in (2.33) into (2.32) gives
+ [rApB+Kl/s]LI-‘).
to obtain
u satisfying
(2.37)
the boundary
condition
(2.4) gives
&(t[(rAp@+K,)2-(rApB+K,)2]
-
1 +( (Y+ l)s”-1
EcrAp0
+ K,)a+’
- (rAp8 + IL,)“‘]).
(2.38)
Let {=r,+,-ri=
-rApx
(2.39)
and rAp0
k=
+ K,
- rA px
z---z TV T1- 7w
It can easily be shown satisfies KAPP
r
=
xl’
r
(2.40)
that the boundary
-k-i+&
k
condition
ka+l
The flux Q is calculated
i&Y=
-‘w>o. s (2.8) requires
- (k + l)‘+‘]j.
that k
(2.41)
from (2.14) as
1
1
5 a-’
-xc2-z-6+-s a+1 0
[
ka+l
+
1
-k lx+2
i
u+2 - (k + 1)-+2)],
(2.42)
230 and the interface
shear stress pi is
7i = (1 + k)rApx.
(2.43)
In principle, it is possible to obtain x and k from (2.41), (2.42) once Q and Q (i.e. K) are given. The corresponding results for Case II can be shown to be
KAW-
Xl2
i
r
QAPP -=
__-#
r
2a+1s i
-s a-’
1
j-L+-
0
.ol+1
(J
- (1 -i,“‘)),
(2.44
;-~+_L(~)“-’ 1 .a+2 + (1 _jy+2 (Y+2 ( J
j( ja+Li
)
(2.45)
N
7I = (1 -j)rApx,
(2.46)
where j is defined
as (2.47)
The asymptotic behaviour of x and k and therefore 7, for large and small values of l/s is readily obtained. For large Z/S, corresponding either to small 5 or a Power-Law fluid, eliminating x and l/s from (2.41) (2.42) and (2&l), (2.45) kU+‘+(ka+2-(k+
Q
--_ K
1)a+2)/(a+2)
CaseI 3
k a+’ _ (k + l)“+’ --
J
.a+1
_
( J ‘u+2 + (1 +“+‘)/(a .a+1 _ (1 _j)a+l J
+ 2) Case II,
(2.48)
(2.49)
The critical value of Q/K signifying the transition between Case I and II is given when k = -j = 0; thus (Q/K),,i, = - l/(a + 2). A graph of k as a function of Q/K, for various values of (Yis shown in Fig. 5. The asymptote as Q/K + m is clearly k = -j - - l/2, and by using the binomial theorem it can be shown that the asymptote as k + 00 is Q/K - - l/2. (When Q/K < - l/2 no meaningful solution is obtained since it requires x < 0). The important conclusion to be drawn as far as the question of uniqueness is concerned is that k is a single-valued function of
Q/K-
231
k
no reverse
_
Fig. 5. Graph of k as a function of Q/K as given in (2.48) and (2.49).
In terms of k, x and 71 are given by
(-ku+’ + (1 + k)a+l)-’ ( -’ .a+1 + (1 -j,,C’)-’
Case I,
Case II,
(2.50)
(2.51)
1
(2.52)
I__
PK r2
4(a + 1)Ap”;
X (((1 -j)“+l
-ja+1)-1)“0+2(1
-j)
Case II,
(2.53)
where the asymptote of Ap as 5 + 0, which corresponds to l/s + 00, has been used in place of Ap itself.
232 For small l/s, corresponding either to 5 + cc or to a Newtonian fluid, in the shear layer the equations for Case I and II are identical. So from (2.41) (2.47)
Q k’2 + ‘I6
--
K
k+
as
l/s
+ 03
l/2
with the explicit solution k=
-(Q/K+
This expression X3
r* -
-j-&Q/K+ -
(2.54)
for k
1/3)/(2Q/K+
1).
(2.55)
for k is used to obtain x and 7r:
I>,
E(48Ap**/~/fi)“~(Q/K+ r*
(2.56) 2/3)(1
+ 2Q/K)-2’3
as[/s-,O.
(2.57)
Because it is required that rr be finite, thus eliminating the possibility of 1 + 2Q/K + 0, the thickness of the shear layer being small depends essentially on p/fi being small as suggested previously using order of magnitude arguments. For reasonable internal consistency of the approximation method the expression in (2.11) is evaluated. The normal traction p + roe at the interface is calculated from (2.18) by a straightforward integration with Ap* replaced by its asymptotic value as S/s -+ 0. Clearly p + roe - fiK/r* in this limit, thus the internal consistency parameter C-
(/&‘3(48)“3(Ap*)2’3(Q/K+
as Z/s -+ 0.
2/3)(1
- 2Q/K)-2’3 (2.58)
The approximation is therefore reasonably accurate or internally consistent either for a Newtonian fluid in the shear layer or for an Ellis fluid far from the origin when p/ii -=x 1. This is precisely the condition for x -GZ1. (That Q/K should approach - 2/3 is inadmissable since x would then be negative). From the expression (2.58) for x and the graph of Ap* in Fig. 4, it can be seen that if shear thinning does not occur, the relative thickness of the shear layer, x/O, increases as Ap* decreases and that the maximum variation x max/X,in = (0.5) - “3 - 1.3. It has been shown that the 2FM is a reasonable method when a true lubricating agent exists between the main body of the melt and the wall, but when this is not the case i.e. p = ,ii the 2FM breaks down upstream (r + 00). This means that it is not possible to determine Q and consequently x and k
233 from the upstream conditions. So far in the analysis Q, the flux in the shear layer, has been treated as an arbitrary constant, but in view of the above remarks it might be desirable to involve some principle, preferably physical, which enables Q to be determined from the downstream flow. Returning to the original argument for the 2FM, it was stated that the elongational stress should be much greater than the shear stress, so that the interface boundary condition on the shear stress (2.10) is insignificant compared to that for the normal traction (2.9). This can now be made stronger by requiring that the value of Q should be such that the interface shear stress is a minimum. In terms of the accuracy parameter C, C should be a minimum with respect to k in the limit as l/s -+ cc. From (2.52) and (2.53) and the asymptotic value of p + ree calculated from (2.18) (1 +k) [(1 + k)a+’ _ ,k,,+l]‘Aa+2)
Ci
(2.59)
as S/s + cc
for Case I and II with kc[ - l/2, cc). It can be shown numerically that the minimum in C lies on the envelope of the set of curves in Fig. 5 as indicated. For all values of (Yin the range of interest, Q/K is negative (a net inflow in the shear layer) and the wall shear stress rW is positive indicating that backflow occurs with the velocity being positive adjacent to the wall. The upstream flow field can be calculated using an indirect (l/r) expansion by perturbing about Newtonian flow, the first term of which determines the flux Q. This is matched to the downstream 2FM through Q. Successive terms in the indirect expansion can only be matched once higher-order terms are calculated downstream. However there is sufficient information to map out the essential features of the whole flow field. Before doing this, further examination of (2.54), apart from the value of k, shows that 2
cc0
ST
(a-
l)Aa+2)
as{/s+
cc.
(2.60)
i PK i
Since (Y> 1 for a shear thinning fluid, (2.60) confirms the order of magnitude arguments which suggested that the shear stresses become negligible in their influence on the core flow as the origin is approached. 3. Cone flow The analysis for two-fluid flow in a cone follows that for flow in a wedge. For this reason the method will not be described in such detail as in Section 2 except where variations occur.
234 The equations of continuity and stress equilibrium with respect to a spherical coordinate system for steady axisymmetric flow (r, t?, $) and velocity components (u, 0, w) are
(3.4) The boundary u(r,
conditions
+O) = v(r,f
are
0) = 0,
(3.5)
and the flux Q is given by e=
2n/@r2u
dr.
(3.6)
0
For an extensional d = -K/r2,
flow field in the cone, the core velocity
is of the form
6 = $ = 0.
(3.7)
Again a Maxwell Fluid is taken for the inner region with a constitutive equation in the integral upper convected formalism as defined in (2.15). The strain tensor ytoI is calculated from the velocity field (3.7) to be /
\
0 A
Y[O]
0
=
0
2/3
0
\
I (3-g)
Evaluation
of the stress tensor components
yields (3.9) (3.10)
235 (3.11) in which I?( a, 5) is the incomplete gamma function [25, p. 2601 and = r3/(3M) is a dimensionless local Deborah Number or cube of radial 5 distance. In order to carry out a complete order of magnitude evaluation of the shear layer equations, the normal traction gradient is calculated to be
-$;[r’(i,,-i,,)]=
Ap=$(P+fos)= -413
+
4t-
l/3
_
4t
-w
e[lY(1/3,()
+ 2 +3++. (3.12)
Asymptotic
expansions
for large and small 5 are obtained (3.13)
and
Ap-
-
(3.14)
Thus Ap shows a character in conical flow different from that in wedge flow. A graph of Ap* is shown in Fig. 6. Although far from the origin the behaviour is similar to the wedge, i.e. it conforms to the Newtonian asymptote, at the origin it has a singularity. It is now shown, using an order-of-magnitude analysis for a power-law fluid similar to that in Section 2, how to obtain the shear layer equations. Introduce the variable /? = 0 - 8 and transform eqns. (3.1), (3.3) to (3.15)
“0
1
2
Fig. 6. Graph of Ap*([)
3
4
defined in (3.12).
k
5
236
%)I+& +%)+ la7
+7ee-%4
-J ap
re
cot(0 - p)7 rB r
o = 7
r
~~(r3r~~)-~~(p+r~e)+cot(q-~)r~S=0. Assume
the following
@-8,=x~
(3.16) (3.17)
orders of magnitude
1,
r-la/ap=o(i,,hx), a/ar=o(i/r), u=O(Kx/r2), Tre=O(q(K/Xr3)"),
r2u/K=
rrr
O(l),
?e and
%e - 7++ = o(
p( K/xr3)“).
The second normal stress difference, 7ee - Q+, is chosen to have the same order of magnitude as the first, 7rr - rse, on the basis of the limited experimental results available. Han [26] reports that for a wide range of polymer melts 7ee - T++,varies between -0.1 and - 0.7 of q, - res. The shear stress term r-l a( rrs)/M will dominate over the normal stress terms in (3.16) provided 4
K
xP i xr3 i
“-* >> 1 >
(3.18)
which is similar to the condition (2.26) and will be assumed to be valid, The normal stress term r- ’ a( p + ree)/M will dominate the other terms provided g
4
‘-?I
K
l-1 xr3
22~
1 asr+
cc
(3.19)
and g 47 i
K
4’3--nA,,3X-l+n
>>
asr+()
(3.20)
i
Thus, far from the origin where the flow field must be Newtonian, the 2FM is valid if p/p z+ 1, the same condition as for the wedge. At the other extreme, close to the origin the 2FM is valid at least for all values of n G 1, which is slightly different from the wedge result. So to first order the pressure does not vary across the shear layer. For consistency it is expected that x will have the order of magnitude (3.21)
231
(3.22) Retaining the largest terms the shear layer equations in terms of (r,B),
become,
when written
(3.23) The shear stress rre is to be determined shear rate yre given by
using the Ellis model (2.31) and a
1 au Y,e = ; ae . The boundary
(3.24) conditions
at the interface
8 = 8, are
ti= u,
(3.25)
B + +*~BB = P + 7003
(3.26)
and the fluxes &,Q in the cone and shear layer are Q = 2rr2/Oz4 Q=
(3.27)
de,
(3.28)
-2n:8,.
Noting the strong similarity between the equations for cone and wedge flow, the equations for the flux and the shear stress at the boundary of the shear layer are readily obtained for both Case I and II corresponding to negative and positive r,,,. They are given by QAPP ---=
2rr2
_xs2 i
ck”+2-(k+
where k is a solution KAPP r2
---=
l)a+2)])
CaseI,
(3.29)
CaseII,
(3.30)
of
XC2-k-1/2+--$
kU+‘-
(k + l)a+l])
Case I, (3.31)
a-1
j-1/2+&
0
S
[
J .a+’ - (1 -j)a+‘])
Case II,
(3.32)
238 and r, = (1 + k)rApx
Case I,
(3.33)
= (1 -j)rApx
Case II,
(3.34)
in which ri, l, k, j have the same meaning as in section 2. The asymptote for small c/s is identical to the wedge but with Q/K in (2.54)-(2.57) replaced by Q/(2rK) (once Ap* has been defined for the core). However the asymptote may correspond either to 5 --, cc, or to a Newtonian fluid throughout the shear layer (S + 00). As for the wedge, as 5 + cc, the 2FM is valid only when p/p z+ 1. But for s + cc and close to the origin Ap* has a singularity according to 5-i. Using the asymptote of Ap (3.14) and the definition of Ap* (3.12), the equations for x and ri corresponding to (2.56) and (2.57) are x3
71
_
-
4P(l/3>(3XK)“’ 3r -
(3.35)
~[~P(l,3)(3hK)“3]2’3(48;)“3(&+2/3)(l
+$)-2’3 (3.36)
as l/s + 0 though in such a manner that s + cc and l+ cc. The interface angle 6, approaches 0 as r + 0. The accuracy parameter C can be evaluated once p + ree is calculated from (3.14). Close to the origin P + ree -
2!J(l/3)(3XK)4’3 9hr4
as r ~ o
(3.37)
and C=0(r413)
asr+O.
(3.38)
This confirms the validity of the 2FM for a Newtonian shear layer sufficiently close to the origin. The asymptote for large S/S, corresponding to a Power-Law fluid, is similar to the wedge provided the correct asymptotic value of Ap* is used. Figure 5 with Q/K replaced by Q/~TK is the graph of the parameter k. Again the value of Q/(2rK) minimising the interface shear stressr, lies on the envelope in Fig. 5. The asymptotic results corresponding to (2.50)-(2.53) for x and r1 are ((1 + k)a+’
- Ikla+‘)-’
Case I & II,
(3.39)
239 l/la+Z) 71
--
-@Y
r3 (3.40)
Case I &II, with AP* - 4lY( 1/3)(3XK)“3/9r and the accuracy
or internal
as Z/S + cc, consistency
parameter
a-1
C = ( u3/$K)x(
r3/3hK)30.
(3.41)
By way of an example of the flow field expected for the 2FM consider a standard injection nozzle which is used for testing moulding machine performance. Referring to Fig. 7, the nozzle has the dimensions [24] D = 2.38 mm;
D, = 15.9 mm;
L=25.4mm;
20 = 90”.
The parameters for the Ellis fluid modelling the Butyl and Chloroprene rubber compounds are given in Table 1. Two choices of flow rate are made which are intended to correspond roughly to flows typical in injection moulding and extrusion. Defining a characteristic velocity U as in Table 2, choose (i) U = 10 ms- ’ for injection moulding and (ii) U = 0.1 ms-’ for extrusion. The parameter k is chosen such that C is a minimum with respect to k in the limit as c/s --) 00. The asymptotic forms of x, (S/S)“-’ and C are calculated for the mean velocity U at the exit of the converging region and are given in Table 2 for the Butyl rubber compound. The flow in a wedge and a cone are not completely similar; the large l/s asymptote depends on the fluid relaxation time X in the case of cone flow, whereas the wedge flow is independent of A. In the calculations for Table 2, A is taken as 12.5 s, a value derived from experimen-
Fig. 7. Typical cylindrical die geometry.
240 TABLE
1
Ellis fluid parameters
for rubber compounds
Ellis fluid parameters
p(Nsm-‘)
s (Nm-*)
(Y
Butyl Chloroprene
1.05 x 106 2 x106
4.4x 104 7 x104
7.7 5.0
TABLE 2 2FM parameters for Butyl rubber for flow in a nozzle at given mean velocities; in metres, r0 = 1.7 X 10e3 m.
(9 U(m s-‘) K=U(r,cos45°)2(m3s-‘) k X (s/s)“-’ C Q
(m3 s-‘)
8 (m3 s-l)
TABLE
c (UsY-
(ii) 0.1
10.0 1.4x 10-s -0.41 1.9X lo3 r2.9 1.2X 10K3 r-l4 4.4X lo3 r2.9 -3.5x1o-6
1.4x -0.41 1.5 X 1.0x 3.6 X - 3.5 x
lo5 r2.9 102’ r-14 IO5 r2.9 10-s
-6.9x
-6.9x
lo-’
1O-5
10-7
3
Values of the 2FM parameters
X
r is measured
at core extrance,
r =
r,
(9
(ii)
0.23” 9.2 x 1O-3 3.2x 1O24
18” 0.75 2.6 x lOI
tal data [8]. The values of r, and r, as defined
in Fig. 3 are
r0 = 1.7 X 10m3 m, r, = 1 .l X 10m2 m Using the expressions in Table 2, the angle subtended by the shear layer and the values of C and ( {/s)~- ’ at r = r, are as in Table 3. From the results in Tables 2 and 3, it is apparent that the 2FM becomes increasingly consistent with increasing flow. rates. Also the large l/s asymptote is valid at these rates throughout the converging region and the shear layer extends back along the whole core. However, as the flow rate decreases, the shear stress at the interface becomes increasingly important at the beginning of the conic section. It may prove necessary to modify the model to accommodate this feature. The streamlines of the flow field are sketched in Fig. 8. The “known”
241
B
-
known
-----
hype
streamlines _
-__-__--_---_-
Fi’g. 8. Sketch of streamlines
in converging
region.
streamlines refer to the sections of the flow field which can be calculated using the 2FM in the conic section and the well-known solution in the upstream tube flow. The “hypothetical” streamlines refer to the region where some matching remains to be done. Since the flux through the shear layer is small compared to that through the core, the interface streamline, when traced back upstream, is very close to the wall. The core streamlines show a slight curvature due to the influence of the curvature of 0,. 4. Towards a Two Zone Model It has been shown in the previous two sections that the Two Fluid Model is mathematically consistent. A modification can be made to the physical conception of the problem by considering the flow field to consist of only one fluid, but which exists in two distinct deformation regions or zones. Part of the fluid undergoes an irrotational deformation, the other part a shearing deformation, as in the 2 FM. The one fluid model is termed a Two Zone Model (2ZM). Of course, if the flow field actually contains a second distinct phase as in forced lubrication, wall slip, or the situation of a relatively cool cone being lubricated by a hot shear layer, then a 2ZM is inappropriate. Since there is only one fluid present, the 2ZM requires only one constitutive equation to be specified. If the method of Sections 2 and 3 is to be applicable then the constitutive equation must predict a high degree of shear-thinning in viscometric flow and exhibit high normal stresses in extensional flows. One such equation is a form of the so-called naive rheological model [8,27] and is defined by (4.1)-(4.5). The term naive is intended to indicate that the model contains only a small number of arbitrary parameters.
242 In an integral representation the stress ~- is given by
= 1 f,
M(t - t')(I-
EE "r) dt',
(4.1)
a.l_oo
where E is a strain tensor which is the solution to ~E/~t = A E ,
E( t',t') = l ,
(4.2)
with A = -a-+- l~ ( V u ) + - a- 2 1
(XTu) "r.
(4.3)
The non-affine parameter a is such that a = 1 corresponds to the Upper Convected and a = 0 to the Corotational formalism. The definition, so far, is similar to that in a n u m b e r of recent papers [28-30] except that in these publications an error has been made, in defining ~', by an omission of the factor 1/a in (4.1). This leads to the curious, but false result, that in simple shear flow, all stress components vanish identically when a = 0. The M e m o r y Function, M, is taken to have the form M(t-
t ' ) = Y'. 7/I, e x p [ - ( t - t')/~,k]
(4.4)
and the parameters r/k, Xk are related by the empirical equations r/k = r/0Xk/)-'~Xj, Xk = X/k~, k = 1,2, 3 . . . . .
(4.5)
J
which were first given in Spriggs [31]. Thus the total n u m b e r of u n k n o w n parameters (r/0, ~, a, a) is four. To a first approximation, the interface between the two zones is treated as a streamline. The upstream behaviour of the converging flow field will be similar to that of the 2 F M and can be determined by perturbing around the Newtonian solution. The aim of the analysis in this section is to determine the asymptotic downstream behaviour, i.e. close to the origin. Flow in the shear zone The naive rheological model possesses a power-law region at high shear rates. The shear stress is determined from the asymptotic expansion for large /3~5' and is given for both cone and wedge flows (provided the coordinate system is suitably identified) by
r/°~r(B~')l/~-~lS"°ll/" z,o-- - 2 a Z ( a ) sin((a + 1)~r/2a)
(4.6)
243 with p=(1-~2)“2andZ(cw)=
E
l/k*.
k-l
A similar expression
can be obtained for the first normal-stress-difference for simple shear flow valid at high shear rates
7
rr
-
ret? =-
T( pX)“a-llyrel”a-- px
?joh .qa) i
(4.7)
2 1*
cwsin( 7r/2a)
on the shear rate. The Clearly rre and r,, - 7ee have the same dependence order of magnitude inequalities (2.26) and (3.18) are satisfied, ensuring that the shear-stress terms dominate the normal-stress-difference terms in the shear layer equations. The Ellis model and the naive rheological model have the same highshear-rate, power-law region if
1
u/fa-
‘=jZ
[ 2 Z(a)
sin((l+
1)7r/2a)
1)
(4.8)
’
(4.9)
P=vo*
With these assignations the shear layer parameters x, T,, Q/K, are given by equations identical to the corresponding equations in .sections 2 and 3 for large l/s. For a complete evaluation of the shear layer it only remains to determine Ap* from the core zone. Flow in a core zone-wedge In wedge flow the core zone is a line sink with the kinematics which the rate-of-strain tensor defined in (4.3) is in 2-D
of (2.7), for
(4.10)
A=$(;
-:)*
The solution
of (4.2) for the strain tensor E is
E,, = (1 + 2Ks/r2)“‘, E,, = (1 + 2Ks/r2)-a/2, with all other components is given as an integral by r~~=~~mM(~)[l 7ee 7 f /omM(s)[
s = t - t’, of E being zero. The stress tensor defined
(4.11) by (4.1)
- (1 +2Ks/r2)o]ds, 1 - (1 + 2Ks/r’))“]ds.
(4.12)
244 All other components of r vanish identically. The stress at small values of r depends strongly on the value of a, and so therefore do Ap and Ap*. (At this stage the analysis could be continued in the style of a 2FM by leaving M(S) unspecified as there should be enough information in (4.12) to determine the asymptotic dependence of Ap for small r.) With M given by (4.4) the non-zero stress tensor components become
,
(4.13)
in which the qi and Xi are given by (4.5) and & = r2/2hiK. is obtained from (4.13) as Ap=
-
-i$r(rrr--ro,)=
=
%,KA.p*
Ap
(4.14)
r3
-dc~[G(,,E,)-G(-a,Si)], I
The quantity
(4.15)
’
where G(a,
&) = [(2a - l)&u-
2&-“]e~~I’(l
Unfortunately, it does not (4.15) in terms of a finite be evaluated numerically. of Ap* the summation is shown that as [ + 0 Ap* - (2 - l/a)<‘-”
+ a,&).
(4.16)
seem possible to express the infinite summation in number of known functions so Ap would have to To obtain an estimate of the asymptotic behaviour truncated to include only the first term. It can be
a * l/2,0
- 21Y(3/2)<3’2
a=
l/2
- 2[1n[
a = 0,
(4.17)
in which 5 = r 2/2X K. For the rubber compounds used here, 1 > a > l/2 and so the first of the asymptotes applies [8] (should values of a = l/2 arise, then Ap* < 0 which would necessitate a change in the analysis in the shear layer). Calculations can be carried out for the cone and for different rheological models but it is probably more important at this stage to investigate how higher terms in the approximation method for the 2FM and 2ZM might be generated.
245 5. Conclusions An approximation method, the Two Fluid or Two Zone Model, for the solution of flows in converging wedges and cones has been put forward. The Model is mathematically consistent and a number of solutions have been obtained. Dynamical consistency for a variety of polymers was established using experimental results. There are a number of points in relation to the Model which require further investigation. One of these which was not discussed in other sections is how the downstream boundary conditions may be satisfied. The manner in which this could be done is described in Pearson and Trottnow [32] for die-entry flow into a short capillary. Their entry-flow is that of a sink ( ug = u,+= 0) which undergoes a sudden velocity rearrangement at the capillary entrance. The extension of this idea to the 2FM is straightforward. The 2FM and 2ZM are applicable to a wide variety of possible die-entry flows, including situations where lubricating agents are present or where a thermal boundary layer arises. The Models can easily be extended to flow in a converging annulus. The philosophy of the’Models should also be suitable for other flows in which high extensions occur; as in calendering between two large rolls, for example. Although recirculating vortices arise in the Model shear layer next to the walls and extend throughout the length of the die-entry, they are not of the “thickness” observed in the experimental flows of certain polymers such as LDPE, when 0 is not small. Indeed the assumptions of the Model rely on the shear layer being thin. Velocity measurements of the recirculating regions indicate that the deformation rates are small except in the neighbourhood of the core [lo]. So, most of the vortex could be modelled as a Newtonian fluid. An improvement to the Model for fluids which exhibit large entrance vortices could be achieved by splitting the flow field into three regions: a core and a shear layer as presently and thirdly a Newtonian vortex. The shear layer would act as a thin transition layer from the core to the vortex. Finally, a comment about the physical stability of the 2FM. It is well known that a number of polymers exhibit a critical shear rate above which the flow may become unstable (see for example [ 1,211) though there is some dependence on the flow geometry. In particular, Vinogradov et al. [33] showed experimentally that the instabilities in entrance flow occur at points of highest stress concentration; for example, at the join between an entrance region and a parallel duct such as a capillary. The shearing deformation in the 2FM occurs in a thin region and not across the whole field and is therefore concentrated there. Thus if a critical shear rate is exceeded, it will occur at the thinnest point of the shear layer which happens to be at the downstream exit. It could easily arise that a critical shear rate is exceeded in
246 the entrance region but simultaneously (and after substantial stress relaxation) not be attained far downstream in a capillary or other parallel-sided duct, Acknowledgements The authors are indebted to Dr. S.M. Richardson for the time he has spent in many fruitful discussions of this topic and above all for his patience and encouragement to one of us (AMH). A.M. Hull expresses his gratitude to the Science Research Council for their support during his Ph.D. work, part of which formed the basis for this paper. References 1 A.M. Hull, J. Non-Newtonian Fluid Mech., 8 (1981) 327-336. 2 R.B. Bird, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, John Wiley & Sons, 1977, Vol. 1. 3 J.R. Black and M.M. Denn, J. Non-Newtonian Fluid Mech., 1 (1976) 83-92. 4 K. Strauss, Acta Mechanica, 20 (1972) 233-246. 5 J.R. Black, Ph.D. Dissertation, University of Delaware, Newark, 1974. 6 P. Schtimmer, Rheol. Acta, 7 (1968) 271-277. 7 M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, 1973, p. 41. 8. A.M. Hull, Ph.D. Thesis, London University, 1981. 9 A.B. Metzner, Rheol. Acta, 10 (1971) 434-444. 10 T.J.F. Pickup, Ph.D. Thesis, Cambridge University, 1970. 11 J.R.A. Pearson and T.J.F. Pickup, Polymer, 14 (1973) 209-214. 12 J.L. White and A. Kondo, J. Non-Newtonian Fluid Mech., 3 (1977) 41-64. 13 H.M. Laun and H. Miinstedt, Rheol. Acta 17 (1978) 415-425. 14 Y. Ide and J.L. White, J. Appl. Polym. Sci., 22 (1978) 1061-1079. 15 A.M. Hull, J. Non-Newtonian Fluid Mechanics (in press). 16 A.C. Pipkin and D.R. Owen, Phys. Fluids, 10 (1967) 836-843. 17 D.C. Pountney and K. Walters, Phys. Fluids, 21 (1978) 1482-1484. 18 D.C. Pountney and K. Walters, Phys. Fluids, 22 (1979) 1007. 19 C.D. Han, K.U. Kim, N. Siskovic and C.R. Huang, J. Appl. Polym. Sci., 17 (1973) 95-103. 20 Y. Ide and J.L. White, J. Non-Newtonian Fluid Mech., 2 (1977) 281-298. 21 D.M. Turner and M.D. Moore, Plastics and Rubber: Processing, 5 (1980) 81-84. 22 E. Uhland, Rheol. Acta, 18 (1979) l-24. 23 H. Winter, C. Macosko and K. Bennett, Rheol. Acta, 18 (1979) 323-334. 24 R.T. Fenner, Principles of Polymer Processing, Macmillan Press Ltd., 1979, pp. 147-148. 25 M. Abramowitz and LA. Stegun (Eds.), Handbook of Mathematical Functions, Applied Maths Series No. 55, U.S. Nat. Bureau of Standards, Washington, D.C., 1964. 26 C.D. Han, Rheology in Polymer Processing, Academic Press, 3rd ed., 1976, p. 55. 27 C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 2 (1977) 221-253. 28 0. Hassager and S. Pedersen, J. Non-Newtonian Fluid Mech., 4 (1978) 261-268. 29 M.W. Johnson, Jr. and D. Segalman, J. Non-Newtonian Fluid Mech., 2 (1977) 255-270. 30 H.C. Lau and W.R. Schowalter, J. Rheol., 24 (1980) 507-516.
247 31 T.W. Sprigs, Chem. Eng. Sci., 20 (1965) 931-940. 32 J.R.A. Pearson and R. Trottnow, J. Non-Newtonian Fluid Mech., 4 (1978) 195-215. 33 G.V. Vinogradov, N.I. Insarova, B.B. Boiko and E.K. Borisenkova, Polym. Eng. Sci., 12 (1972) 323-334.
ON THE CONVERGING FLOW OF VISCOELASTIC THROUGH CONES AND WEDGES +
219
FLUIDS
A.M. HULL * and J.R.A. PEARSON ** Department of Chemical Engineering and Chemical Technology, Imperial College of Science and Technology, Prince Consort Road, London (Gt. Britain) (Received March 15, 1983)
Summary A novel approximation method for obtaining solutions to the slow flow of viscoelastic fluids through converging cones and wedges is presented. The method involves considering two distinct regions in the flow field divided by a streamline: an inner irrotational sink flow occupying most of the flow field and an outer highly sheared self-lubricating layer adjacent to the walls. This procedure greatly simplifies the governing equations and allows analytic expressions to be obtained. Since the two flow regions are separated by a streamline, different constitutive equations can be used for the inner and outer flow fields. This enables constitutive models to be used which model polymer melt rheology reasonably accurately and still retain analytic solutions. The method is applied to both wedges and cones; extensions to converging annuli are straightforward. Finally, an example flow field is considered in relation to a typical die-entry flow in polymer processing.
1. Introduction The situation to be considered is that of the slow flow of viscoelastic fluids through converging wedges and cones (see Fig. 1). Part of the justification for this is to arrive at a better understanding of die-entry flows + Dedicated to the memory of Professor J.G. Oldroyd * Present address: Du Pont de Nemours International S.A., Polymer Products Department, 28 Rue Alexandre Gavard, 1227 Carouge, Geneva (Switzerland). ** Present address: Schlumberger Cambridge Research, PO Box 153, Cambridge CB2 3BE (Gt. Britain) 0377-0257/84/$03.00
0 1984 Elsevier Science Publishers B.V.
220 in polymer processing. By assuming that the die-entry has a large contraction ratio occuring over a short distance or that the flow rates are high, the entrance geometry is idealised to one of infinite extent upstream with a point or line efflux downstream.
Fig. 1. Geometry
and coordinate
system of the flow problem.
It has proved to be rather difficult to obtain solutions for flows in converging sections. In fact, for a viscoelastic fluid involving at least one objective time derivative of the stress tensor (in the differential formulation of the constitutive equation) only one exact solution has been discovered [ 11. This solution is for a 180” wedge (half-angle 90’) and is for the General Linear Upper convected Viscoelastic Fluid defined in [2] as
(14 in which 7 is the stress tensor, yIol the strain tensor, both tensors in the upper convected formalism, and M is the memory function. expansion solutions Otherwise, inverse coordinate (l/r) or “indirect” have been obtained using various constitutive equations. These are constructed either as perturbation or series expansions [3-61. They are only valid in regions far from the apex of the cone or wedge. This is as expected; however it should be mentioned that the expansions result in asymptotic diverging series [7,8]. It would seem to be obvious that for solutions valid in the neighbourhood of the apex, direct expansions in r are required. However, as shown in [8] there is a significant difficulty in doing this due to the effect of the upstream history of deformation. That is, the direct expansion or downstream flow involves the history. of the flow in a non-trivial way and it does not seem possible to resolve this with a matched solution. For these reasons a different approach is explained in this paper. The Extensional Primary Field approximation (EPF), which was originally pro-
221 posed independently by Metzner [9] and Pickup and Pearson [ 10,111, is used as a starting point. The EPF results from diagonalising the stress tensor, 7
7 =
i
711
[ol
M
[ol
722
PI
[ol LOI 733
I
(l-2)
by neglecting the shear stress components T,~, T,~, 723. Of course, 7 must be defined relative to some non-rotating coordinate system. The EPF cannot satisfy the no-slip boundary condition so it can only apply to some core region not adjacent to the walls (see Fig. 2). The object here is to develop and extend the EPF showing how to treat the remainder of the flow field and, in particular, satisfying the no-slip boundary condition. The kinematic justification for the EPF is from experimental observation of die-entry flows, surveys of which are given in [8,12]. Provided that the flow remains slow, a condition satisfied except for high fluxes of polymer solutions, most of the flow field in the converging region has almost radial streamlines, i.e. v0 -=xv, almost everywhere. More important, most of the outflow comes from the central portion of the flow whether recirculating vortices are formed or not. This means that the velocity field is proportional to l/r or l/r2 for a wedge or cone, in which the coordinate systems are cylindrical or spherical polar, respectively. Furthermore, there is considerable evidence that over most of a central core region there is little variation in the magnitude of the velocity across the core region. Consider now the whole flow field which is illustrated schematically in Fig. 3. Part a shows flow towards the apex of a cone or wedge with half angle 0. If the radial velocity component is smooth as in part b (i), then
core
of extent of EPF
Fig. 2. Die-entry
showing
streamlines
for core region.
222 (ii)
e=B / *
ll (iI
‘“r
0
e-,
0
Fig. 3. (a) Geometry of converging wedge or cone of half angle 8 with polar coordinates (r, 0). A -core region; B-self-lubricating region. 8, (r) refers to position of interface between lubricating and core regions. (b) Radial velocity component profiles as functions of B (at some given r): (i) smooth profile with no lubrication; (ii) kinked profile with sharp change in shear rate (l/r)au,/M at t9 = t?,(r), indicating lubrication effect.
r- ‘Clv,/SJ is of the same order of magnitude as av,/& over most of the flow region, and so neither simple shear flow nor irrotational flow are suitable approximations. If, however, a cone region where there is little radial variation in v, is assumed to arise so that av,./ib B r-‘av,/M, then there must also be a strongly sheared self-lubricating layer close to the wall with r-‘av,/ahJ z=- av,./ib, as in part b (ii). Thus the flow may be approximated by a simple shear flow within 8, < ]0] < 0, and an irrotational, sink flow within 0 G 181~ 8,. (The velocity profile in part b (ii) should not be taken as invariable, as it is possible that recirculation occurs in the wall layer implying a positive radial velocity in the neighbourhood of the wall.) In this manner, the no-slip boundary condition at the wall can be satisfied. Metzner [9] has proposed, as a criterion for the existence of an EPF,
N,,=J$L sh
where NTr is the Trouton ratio and the simple shear viscosity The assumed kinematics for tion for dynamic consistency, particular strain rate be much of magnitude greater than the
(1.3)
between the uniaxial extensional viscosity Q qsh. the extended model require a stronger condinamely that the extensional stresses at a greater than the shear rate, possibly an order strain rate. In crude terms this is
223 N,
>> l/(0
- 6,) z+ 2/a.
(1.4)
Recent experimental data on the stress in uniaxial extension have been reported for various polymer systems including low-density polyethylene (LDPE) [ 131, polystyrene (PS) [ 141 and two fully compounded rubbers, poly(isobutene-co-isoprene) (Butyl rubber) and poly(chloroprene) (Neoprene) [15]. Although extremely high strains were recorded, no steady state was reached, which means that a true viscosity in uniaxial extension could not be measured by filament extension. However, a measure of the stress likely in extensional flow is given by the maximum stress achieved in the experiment. In turn, this will be taken as a measure of the stresses achieved in the sink flows occurring in the core region. For the shear layer, the stress involved either in the start-up of simple shear or at steady values, can be taken as representative. From the experimental data shown in Hull [ 151, it can be seen that the extensional stresses are one or two orders of magnitude greater than the shear stresses at comparable deformation rates. This represents sufficient dynamical justification. To a first approximation the interface 8, separating the flow field will be treated as a streamline, i.e. there is no interchange between fluid undergoing simple shear and fluid undergoing sink flow. In sections 2 and 3 explicit calculations are presented in which the fluid in the core is described by a UCMF (Upper Convected Maxwell Fluid) and the fluid in the shear layer by the Ellis Viscosity Equation. Such a model where the two regions are not linked by the same constitutive equation will be termed a Two-Fluid Model (2FM). In section 4, the same calculations are done for a certain “naive” rheological model. Here the two regions are linked by the same constitutive equation. Such a model will be termed a Two-Zone Model (2ZM). The 2ZM does not rely upon 8, being a streamline and so should open up possibilities of calculating higher-order terms in the approximation method. The term 2FM is only meant to imply that different rheological aspects of what may well be one single fluid are being considered. A higher-order approximation scheme would have to be based on a matching of an almost viscometric flow field to an almost extensional flow field. Pipkin and Owen [ 161 have considered the form taken by a general constitutive equation in the first case and Pountney and Walters [ 17,181 in the second. 2. Wedge flow The equations of continuity and stress equilibrium with respect to a cylindrical polar coordinate system for steady, planar flow in a wedge ( r,B),
224 with corresponding
velocity
components
(u, U) are
(2-l) (2.2) (2.3) The boundary
conditions
for no slip at the wail, i.e. at 0 = + 0, require
u(r,f
0) = u(r,+ 0) = 0. (24 In the steady state, the flux per unit width 2@of fluid crossing every section of the wedge must be the same, namely
Q=
(2.5)
fLde
The convention chosen is that the flux in the converging direction is negative. The flow is now separated by an interface defined at 8 = e,(y). At this stage of the approximation 0, is considered to be constant locally and at most slowly varying with radial distance: thus it is required that r
ae,/ib e
1.
(2.6)
The flow within the region 181G 8, is assumed to be a sink flow, locally, with a velocity field satisfying the continuity equation (2.1) given by ti=
-K/r,ti=O
K a constant.
(2.7)
Quantities designated by “ A ” refer to the core but will only be used where possible confusion with the shear layer may arise. The formal boundary conditions at the interface are the continuity of velocity and of the normal and tangential tractions. Neglecting the possible slight curvature of 8,, these are Li = u, (6 = U), (2.8) B + +e*=p + ree,
(2.9) (2.10)
&I = rre 7
all evaluated at 0 = 8,. Because the core is a sink flow, +,.* will vanish identically. However, it is to be expected that rre f 0, so to the first approximation (2.10) will not be satisfied. However, a measure of the accuracy of the method is provided by the ratio of the tractions in (2.9) and (2.10). Thus it will be desirable for
brel c=
IP+%el +B,=
I%4 IP+ reel e=8, K
I.
(2.11)
225 Since 8, is a streamline there is an extra kinematic condition on the flux which is sufficient to determine a unique solution. If the flux in the core is 20 and the flux in the shear layers is 2Q, then
e=i,+Q,
(2.12)
0 = - K8,, constant,
(2.13)
Q = 4”“~ dr, constant.
(2.14)
I
The effect of the core flow on the shear layer will be shown to be through a( p + fee)/& determined at the interface from (2.9), and also through the radial velocity of the interface, -K/r. The expression a($ + feo)/i3r for the UCMF given by (2.15) can be calculated, by first evaluating kinematics of (2.7): (2K.s/r2
P[o]=
0
,O
0
0'
(1 +ZKs/r”)-’
0
0
O/
and then integrating
the strain
tensor
for the line sink
ys=t-t’
(2.16)
(2.15) to give
7TT= -P./h5,
+ee = (P/M( 1 - &%W),
(2.17)
in which all other stress tensor components vanish, E, is the exponential integral [25,p. 2281 and 5 = r*/(2AK) is a dimensionless local Deborah Number. The required expression is: (2.18) where (2.19)
Ap* = 1 - 5 - 25* + (3 + 2<)c2eEE,(<).
A graph of Ap* as a function oft is shown in Fig. 4 having asymptotes AP* - 2.0, 1.0 for large, small 5. Since Ap* does not vary substantially with 5, a suitable order of magnitude for Ap defined in (2.18) is (2.20) The shear layer equations
may be found
from the continuity
and stress
226 2 -
AP*
I--/
0
0
2
4
6
8
4
10
Fig. 4. Graph of Ap* defined in (2.19) as a function of 5.
equilibrium equations (2.1)-(2.3) once a new variable p = 0 - 8 (which measures the angle from the wall to any given point in the shear layer) is introduced. The transformed equations are (2.21)
(2.22) (2.23) Let x = 0 - 13,, so that TX is the thickness of the shear layer and assume the usual order of magnitude requirements for lubrication theory, i.e. r-u/K= O(l),
Y-’ a/@? = 0( l/rx)
and a/h
= 0( l/r).
According to the usual theory it is expected that 2) = 0( Kx/r). If v is larger than this, then retaining the dominant term in the continuity equation (2.21) gives au/@3 = 0 and the boundary conditions at the wall cannot be satisfied except trivially for 0 = 0 (in which case au/i!@ is not the dominant term). The largest term in the strain rate tensor 9 is (2.24) It is now appropriate to specify a constitutive equation for the stress r. Assume a power-law type dependence (which is usual for polymers at high shear rates) with 7rl3= -M&In,
(2.25)
r,, - ree = -Pl?J”
p, q, m, n being constants.
Using (2.24) to give an order
of magnitude
for (2.25), the order
of the
227 terms in eqns. (2.22)-(2.23)
in order of appearance are {eqn. (2.22))
and
4 -K
i xr2
r
n -j.x
1,
{eqn. (2.23))
xr3
The shear-stress term dominates the normal-stress term in (2.22) provided
-.&
K
4
1.
XP i xr2 i
(2.26)
For the Butyl rubber examined n = m and q/p = 0( 1). This is also the case for a number of melts including HDPE, PP and PS [ 191. In addition n = m at high shear rates is a feature of the White-Metzner Model which was developed specifically to model the shear stress and the normal stress difference in simple shear flow [20]. Their model is confirmed by data for LDPE and HDPE. In any case, provided m - n < 1 it will always be possible to choose x small enough so that (2.26) is satisfied in some region not including the origin. As the origin is approached, the normal stress difference may have to be taken into consideration. However, in what follows it will be assumed that the shear stress terms dominate. Written in terms of (r,6), (2.22) becomes upon retaining the largest terms: (2.27) Of the terms in (2.23) only the normal stress term is significant when n -C 1. Therefore in the variables ( r,O), (2.28) and p + roe does not vary across the shear layer but is determined at the interface. For consistency the terms in (2.27) must have the same order of magnitude. Thus x
=
o((;)( ;)‘-n)“~‘+n~.
(2.29)
If n = 1 and writting q = p, eqn. (2.28) is obtained only if p/P=
1
(2.30)
and the consistency relation (2.29) is x = 0( ~/fi)‘/2 -=x 1, a stronger require-
228 ment than (2.30). The 2FM with a Newtonian shear layer is a possible model for wedge flow of a melt in at least two particular cases. (i) The melt contains a low molecular weight phase which is sufficiently mobile to migrate to the surface at the wall to form a lubricating layer. Rubber compounds exhibiting wall slip, particularly EPDM and Nitrile compounds [21] and polymers such as HDPE [22] and PVC are examples of such materials. (ii) The die-entry is lubricated by an external agent of low viscosity which is pumped into the region adjacent to the wall. Winter et al. [23] have used this technique in an experiment intended to achieve pure extensional motions in the entry region, although Caswell [unpublished, reported at 1981 workshop on extensional flow, Ross Priory] disputes the possibility of achieving such motions for 0 /K 1. Otherwise if the melt is the same material in both regions then the zero-shear-rate viscosities will be identical, p = fi. However as discussed in [24] temperature rises may occur at sufficiently high flux in the die-entry. Most of the temperature variation occurs in a thin layer either adjacent or very close to the wall. In such cases, it should be possible to contain the temperature layer within the shear layer, which would lower the effective viscosity p. For Butyl rubber a temperature rise of about 15°C causes the viscosity p to be halved with a corresponding decrease in Q [8]. So to this level of approximation temperature rises increase the suitability of the 2FM. This conclusion is valid for all values of n. Rather than using a Power-Law model, a solution of (2.27) is obtained for the more realistic Ellis model given by [2] for the viscosity n (2.31)
/J/11 = 1 + IQ/.$-‘,
where s and (Yare constants; (Yis equivalent to l/n in eqn. (2.25). This model has both a Newtonian constant-viscosity region at low shear rates and a power-law region at high shear rates. An expression for the shear rate y,./,Bis obtained from (2.24) and (2.31) (2.32) The shear stress results from an integration 7re = -rBAp
- K,(r),
where K,(r) Let
is a function
71
=
-t-Ape,
-
K,
of (2.27) using (2.18) (2.33)
satisfying
the boundary
conditions.
(2.34)
229 be the shear stress at the interface r, = -rApO
and (2.35)
- K,
be the shear stress at the wall, Then since Ap < 0, (2.36)
7i<7W,7i
the velocity
U, it is necessary
to distinguish
two possibili-
Case I Backflow does not occur in the shear layer, in which case 7W< 0 and rre < 0 everywhere within the shear layer. Case II Backflow does occur, in which case rW> 0 and Q = 0 for some 8 E (8,,0]. For Case I, substitution $=i(rApe+K,)(l Integrating UC
of rre in (2.33) into (2.32) gives
+ [rApB+Kl/s]LI-‘).
to obtain
u satisfying
(2.37)
the boundary
condition
(2.4) gives
&(t[(rAp@+K,)2-(rApB+K,)2]
-
1 +( (Y+ l)s”-1
EcrAp0
+ K,)a+’
- (rAp8 + IL,)“‘]).
(2.38)
Let {=r,+,-ri=
-rApx
(2.39)
and rAp0
k=
+ K,
- rA px
z---z TV T1- 7w
It can easily be shown satisfies KAPP
r
=
xl’
r
(2.40)
that the boundary
-k-i+&
k
condition
ka+l
The flux Q is calculated
i&Y=
-‘w>o. s (2.8) requires
- (k + l)‘+‘]j.
that k
(2.41)
from (2.14) as
1
1
5 a-’
-xc2-z-6+-s a+1 0
[
ka+l
+
1
-k lx+2
i
u+2 - (k + 1)-+2)],
(2.42)
230 and the interface
shear stress pi is
7i = (1 + k)rApx.
(2.43)
In principle, it is possible to obtain x and k from (2.41), (2.42) once Q and Q (i.e. K) are given. The corresponding results for Case II can be shown to be
KAW-
Xl2
i
r
QAPP -=
__-#
r
2a+1s i
-s a-’
1
j-L+-
0
.ol+1
(J
- (1 -i,“‘)),
(2.44
;-~+_L(~)“-’ 1 .a+2 + (1 _jy+2 (Y+2 ( J
j( ja+Li
)
(2.45)
N
7I = (1 -j)rApx,
(2.46)
where j is defined
as (2.47)
The asymptotic behaviour of x and k and therefore 7, for large and small values of l/s is readily obtained. For large Z/S, corresponding either to small 5 or a Power-Law fluid, eliminating x and l/s from (2.41) (2.42) and (2&l), (2.45) kU+‘+(ka+2-(k+
Q
--_ K
1)a+2)/(a+2)
CaseI 3
k a+’ _ (k + l)“+’ --
J
.a+1
_
( J ‘u+2 + (1 +“+‘)/(a .a+1 _ (1 _j)a+l J
+ 2) Case II,
(2.48)
(2.49)
The critical value of Q/K signifying the transition between Case I and II is given when k = -j = 0; thus (Q/K),,i, = - l/(a + 2). A graph of k as a function of Q/K, for various values of (Yis shown in Fig. 5. The asymptote as Q/K + m is clearly k = -j - - l/2, and by using the binomial theorem it can be shown that the asymptote as k + 00 is Q/K - - l/2. (When Q/K < - l/2 no meaningful solution is obtained since it requires x < 0). The important conclusion to be drawn as far as the question of uniqueness is concerned is that k is a single-valued function of
Q/K-
231
k
no reverse
_
Fig. 5. Graph of k as a function of Q/K as given in (2.48) and (2.49).
In terms of k, x and 71 are given by
(-ku+’ + (1 + k)a+l)-’ ( -’ .a+1 + (1 -j,,C’)-’
Case I,
Case II,
(2.50)
(2.51)
1
(2.52)
I__
PK r2
4(a + 1)Ap”;
X (((1 -j)“+l
-ja+1)-1)“0+2(1
-j)
Case II,
(2.53)
where the asymptote of Ap as 5 + 0, which corresponds to l/s + 00, has been used in place of Ap itself.
232 For small l/s, corresponding either to 5 + cc or to a Newtonian fluid, in the shear layer the equations for Case I and II are identical. So from (2.41) (2.47)
Q k’2 + ‘I6
--
K
k+
as
l/s
+ 03
l/2
with the explicit solution k=
-(Q/K+
This expression X3
r* -
-j-&Q/K+ -
(2.54)
for k
1/3)/(2Q/K+
1).
(2.55)
for k is used to obtain x and 7r:
I>,
E(48Ap**/~/fi)“~(Q/K+ r*
(2.56) 2/3)(1
+ 2Q/K)-2’3
as[/s-,O.
(2.57)
Because it is required that rr be finite, thus eliminating the possibility of 1 + 2Q/K + 0, the thickness of the shear layer being small depends essentially on p/fi being small as suggested previously using order of magnitude arguments. For reasonable internal consistency of the approximation method the expression in (2.11) is evaluated. The normal traction p + roe at the interface is calculated from (2.18) by a straightforward integration with Ap* replaced by its asymptotic value as S/s -+ 0. Clearly p + roe - fiK/r* in this limit, thus the internal consistency parameter C-
(/&‘3(48)“3(Ap*)2’3(Q/K+
as Z/s -+ 0.
2/3)(1
- 2Q/K)-2’3 (2.58)
The approximation is therefore reasonably accurate or internally consistent either for a Newtonian fluid in the shear layer or for an Ellis fluid far from the origin when p/ii -=x 1. This is precisely the condition for x -GZ1. (That Q/K should approach - 2/3 is inadmissable since x would then be negative). From the expression (2.58) for x and the graph of Ap* in Fig. 4, it can be seen that if shear thinning does not occur, the relative thickness of the shear layer, x/O, increases as Ap* decreases and that the maximum variation x max/X,in = (0.5) - “3 - 1.3. It has been shown that the 2FM is a reasonable method when a true lubricating agent exists between the main body of the melt and the wall, but when this is not the case i.e. p = ,ii the 2FM breaks down upstream (r + 00). This means that it is not possible to determine Q and consequently x and k
233 from the upstream conditions. So far in the analysis Q, the flux in the shear layer, has been treated as an arbitrary constant, but in view of the above remarks it might be desirable to involve some principle, preferably physical, which enables Q to be determined from the downstream flow. Returning to the original argument for the 2FM, it was stated that the elongational stress should be much greater than the shear stress, so that the interface boundary condition on the shear stress (2.10) is insignificant compared to that for the normal traction (2.9). This can now be made stronger by requiring that the value of Q should be such that the interface shear stress is a minimum. In terms of the accuracy parameter C, C should be a minimum with respect to k in the limit as l/s -+ cc. From (2.52) and (2.53) and the asymptotic value of p + ree calculated from (2.18) (1 +k) [(1 + k)a+’ _ ,k,,+l]‘Aa+2)
Ci
(2.59)
as S/s + cc
for Case I and II with kc[ - l/2, cc). It can be shown numerically that the minimum in C lies on the envelope of the set of curves in Fig. 5 as indicated. For all values of (Yin the range of interest, Q/K is negative (a net inflow in the shear layer) and the wall shear stress rW is positive indicating that backflow occurs with the velocity being positive adjacent to the wall. The upstream flow field can be calculated using an indirect (l/r) expansion by perturbing about Newtonian flow, the first term of which determines the flux Q. This is matched to the downstream 2FM through Q. Successive terms in the indirect expansion can only be matched once higher-order terms are calculated downstream. However there is sufficient information to map out the essential features of the whole flow field. Before doing this, further examination of (2.54), apart from the value of k, shows that 2
cc0
ST
(a-
l)Aa+2)
as{/s+
cc.
(2.60)
i PK i
Since (Y> 1 for a shear thinning fluid, (2.60) confirms the order of magnitude arguments which suggested that the shear stresses become negligible in their influence on the core flow as the origin is approached. 3. Cone flow The analysis for two-fluid flow in a cone follows that for flow in a wedge. For this reason the method will not be described in such detail as in Section 2 except where variations occur.
234 The equations of continuity and stress equilibrium with respect to a spherical coordinate system for steady axisymmetric flow (r, t?, $) and velocity components (u, 0, w) are
(3.4) The boundary u(r,
conditions
+O) = v(r,f
are
0) = 0,
(3.5)
and the flux Q is given by e=
2n/@r2u
dr.
(3.6)
0
For an extensional d = -K/r2,
flow field in the cone, the core velocity
is of the form
6 = $ = 0.
(3.7)
Again a Maxwell Fluid is taken for the inner region with a constitutive equation in the integral upper convected formalism as defined in (2.15). The strain tensor ytoI is calculated from the velocity field (3.7) to be /
\
0 A
Y[O]
0
=
0
2/3
0
\
I (3-g)
Evaluation
of the stress tensor components
yields (3.9) (3.10)
235 (3.11) in which I?( a, 5) is the incomplete gamma function [25, p. 2601 and = r3/(3M) is a dimensionless local Deborah Number or cube of radial 5 distance. In order to carry out a complete order of magnitude evaluation of the shear layer equations, the normal traction gradient is calculated to be
-$;[r’(i,,-i,,)]=
Ap=$(P+fos)= -413
+
4t-
l/3
_
4t
-w
e[lY(1/3,()
+ 2 +3++. (3.12)
Asymptotic
expansions
for large and small 5 are obtained (3.13)
and
Ap-
-
(3.14)
Thus Ap shows a character in conical flow different from that in wedge flow. A graph of Ap* is shown in Fig. 6. Although far from the origin the behaviour is similar to the wedge, i.e. it conforms to the Newtonian asymptote, at the origin it has a singularity. It is now shown, using an order-of-magnitude analysis for a power-law fluid similar to that in Section 2, how to obtain the shear layer equations. Introduce the variable /? = 0 - 8 and transform eqns. (3.1), (3.3) to (3.15)
“0
1
2
Fig. 6. Graph of Ap*([)
3
4
defined in (3.12).
k
5
236
%)I+& +%)+ la7
+7ee-%4
-J ap
re
cot(0 - p)7 rB r
o = 7
r
~~(r3r~~)-~~(p+r~e)+cot(q-~)r~S=0. Assume
the following
@-8,=x~
(3.16) (3.17)
orders of magnitude
1,
r-la/ap=o(i,,hx), a/ar=o(i/r), u=O(Kx/r2), Tre=O(q(K/Xr3)"),
r2u/K=
rrr
O(l),
?e and
%e - 7++ = o(
p( K/xr3)“).
The second normal stress difference, 7ee - Q+, is chosen to have the same order of magnitude as the first, 7rr - rse, on the basis of the limited experimental results available. Han [26] reports that for a wide range of polymer melts 7ee - T++,varies between -0.1 and - 0.7 of q, - res. The shear stress term r-l a( rrs)/M will dominate over the normal stress terms in (3.16) provided 4
K
xP i xr3 i
“-* >> 1 >
(3.18)
which is similar to the condition (2.26) and will be assumed to be valid, The normal stress term r- ’ a( p + ree)/M will dominate the other terms provided g
4
‘-?I
K
l-1 xr3
22~
1 asr+
cc
(3.19)
and g 47 i
K
4’3--nA,,3X-l+n
>>
asr+()
(3.20)
i
Thus, far from the origin where the flow field must be Newtonian, the 2FM is valid if p/p z+ 1, the same condition as for the wedge. At the other extreme, close to the origin the 2FM is valid at least for all values of n G 1, which is slightly different from the wedge result. So to first order the pressure does not vary across the shear layer. For consistency it is expected that x will have the order of magnitude (3.21)
231
(3.22) Retaining the largest terms the shear layer equations in terms of (r,B),
become,
when written
(3.23) The shear stress rre is to be determined shear rate yre given by
using the Ellis model (2.31) and a
1 au Y,e = ; ae . The boundary
(3.24) conditions
at the interface
8 = 8, are
ti= u,
(3.25)
B + +*~BB = P + 7003
(3.26)
and the fluxes &,Q in the cone and shear layer are Q = 2rr2/Oz4 Q=
(3.27)
de,
(3.28)
-2n:8,.
Noting the strong similarity between the equations for cone and wedge flow, the equations for the flux and the shear stress at the boundary of the shear layer are readily obtained for both Case I and II corresponding to negative and positive r,,,. They are given by QAPP ---=
2rr2
_xs2 i
ck”+2-(k+
where k is a solution KAPP r2
---=
l)a+2)])
CaseI,
(3.29)
CaseII,
(3.30)
of
XC2-k-1/2+--$
kU+‘-
(k + l)a+l])
Case I, (3.31)
a-1
j-1/2+&
0
S
[
J .a+’ - (1 -j)a+‘])
Case II,
(3.32)
238 and r, = (1 + k)rApx
Case I,
(3.33)
= (1 -j)rApx
Case II,
(3.34)
in which ri, l, k, j have the same meaning as in section 2. The asymptote for small c/s is identical to the wedge but with Q/K in (2.54)-(2.57) replaced by Q/(2rK) (once Ap* has been defined for the core). However the asymptote may correspond either to 5 --, cc, or to a Newtonian fluid throughout the shear layer (S + 00). As for the wedge, as 5 + cc, the 2FM is valid only when p/p z+ 1. But for s + cc and close to the origin Ap* has a singularity according to 5-i. Using the asymptote of Ap (3.14) and the definition of Ap* (3.12), the equations for x and ri corresponding to (2.56) and (2.57) are x3
71
_
-
4P(l/3>(3XK)“’ 3r -
(3.35)
~[~P(l,3)(3hK)“3]2’3(48;)“3(&+2/3)(l
+$)-2’3 (3.36)
as l/s + 0 though in such a manner that s + cc and l+ cc. The interface angle 6, approaches 0 as r + 0. The accuracy parameter C can be evaluated once p + ree is calculated from (3.14). Close to the origin P + ree -
2!J(l/3)(3XK)4’3 9hr4
as r ~ o
(3.37)
and C=0(r413)
asr+O.
(3.38)
This confirms the validity of the 2FM for a Newtonian shear layer sufficiently close to the origin. The asymptote for large S/S, corresponding to a Power-Law fluid, is similar to the wedge provided the correct asymptotic value of Ap* is used. Figure 5 with Q/K replaced by Q/~TK is the graph of the parameter k. Again the value of Q/(2rK) minimising the interface shear stressr, lies on the envelope in Fig. 5. The asymptotic results corresponding to (2.50)-(2.53) for x and r1 are ((1 + k)a+’
- Ikla+‘)-’
Case I & II,
(3.39)
239 l/la+Z) 71
--
-@Y
r3 (3.40)
Case I &II, with AP* - 4lY( 1/3)(3XK)“3/9r and the accuracy
or internal
as Z/S + cc, consistency
parameter
a-1
C = ( u3/$K)x(
r3/3hK)30.
(3.41)
By way of an example of the flow field expected for the 2FM consider a standard injection nozzle which is used for testing moulding machine performance. Referring to Fig. 7, the nozzle has the dimensions [24] D = 2.38 mm;
D, = 15.9 mm;
L=25.4mm;
20 = 90”.
The parameters for the Ellis fluid modelling the Butyl and Chloroprene rubber compounds are given in Table 1. Two choices of flow rate are made which are intended to correspond roughly to flows typical in injection moulding and extrusion. Defining a characteristic velocity U as in Table 2, choose (i) U = 10 ms- ’ for injection moulding and (ii) U = 0.1 ms-’ for extrusion. The parameter k is chosen such that C is a minimum with respect to k in the limit as c/s --) 00. The asymptotic forms of x, (S/S)“-’ and C are calculated for the mean velocity U at the exit of the converging region and are given in Table 2 for the Butyl rubber compound. The flow in a wedge and a cone are not completely similar; the large l/s asymptote depends on the fluid relaxation time X in the case of cone flow, whereas the wedge flow is independent of A. In the calculations for Table 2, A is taken as 12.5 s, a value derived from experimen-
Fig. 7. Typical cylindrical die geometry.
240 TABLE
1
Ellis fluid parameters
for rubber compounds
Ellis fluid parameters
p(Nsm-‘)
s (Nm-*)
(Y
Butyl Chloroprene
1.05 x 106 2 x106
4.4x 104 7 x104
7.7 5.0
TABLE 2 2FM parameters for Butyl rubber for flow in a nozzle at given mean velocities; in metres, r0 = 1.7 X 10e3 m.
(9 U(m s-‘) K=U(r,cos45°)2(m3s-‘) k X (s/s)“-’ C Q
(m3 s-‘)
8 (m3 s-l)
TABLE
c (UsY-
(ii) 0.1
10.0 1.4x 10-s -0.41 1.9X lo3 r2.9 1.2X 10K3 r-l4 4.4X lo3 r2.9 -3.5x1o-6
1.4x -0.41 1.5 X 1.0x 3.6 X - 3.5 x
lo5 r2.9 102’ r-14 IO5 r2.9 10-s
-6.9x
-6.9x
lo-’
1O-5
10-7
3
Values of the 2FM parameters
X
r is measured
at core extrance,
r =
r,
(9
(ii)
0.23” 9.2 x 1O-3 3.2x 1O24
18” 0.75 2.6 x lOI
tal data [8]. The values of r, and r, as defined
in Fig. 3 are
r0 = 1.7 X 10m3 m, r, = 1 .l X 10m2 m Using the expressions in Table 2, the angle subtended by the shear layer and the values of C and ( {/s)~- ’ at r = r, are as in Table 3. From the results in Tables 2 and 3, it is apparent that the 2FM becomes increasingly consistent with increasing flow. rates. Also the large l/s asymptote is valid at these rates throughout the converging region and the shear layer extends back along the whole core. However, as the flow rate decreases, the shear stress at the interface becomes increasingly important at the beginning of the conic section. It may prove necessary to modify the model to accommodate this feature. The streamlines of the flow field are sketched in Fig. 8. The “known”
241
B
-
known
-----
hype
streamlines _
-__-__--_---_-
Fi’g. 8. Sketch of streamlines
in converging
region.
streamlines refer to the sections of the flow field which can be calculated using the 2FM in the conic section and the well-known solution in the upstream tube flow. The “hypothetical” streamlines refer to the region where some matching remains to be done. Since the flux through the shear layer is small compared to that through the core, the interface streamline, when traced back upstream, is very close to the wall. The core streamlines show a slight curvature due to the influence of the curvature of 0,. 4. Towards a Two Zone Model It has been shown in the previous two sections that the Two Fluid Model is mathematically consistent. A modification can be made to the physical conception of the problem by considering the flow field to consist of only one fluid, but which exists in two distinct deformation regions or zones. Part of the fluid undergoes an irrotational deformation, the other part a shearing deformation, as in the 2 FM. The one fluid model is termed a Two Zone Model (2ZM). Of course, if the flow field actually contains a second distinct phase as in forced lubrication, wall slip, or the situation of a relatively cool cone being lubricated by a hot shear layer, then a 2ZM is inappropriate. Since there is only one fluid present, the 2ZM requires only one constitutive equation to be specified. If the method of Sections 2 and 3 is to be applicable then the constitutive equation must predict a high degree of shear-thinning in viscometric flow and exhibit high normal stresses in extensional flows. One such equation is a form of the so-called naive rheological model [8,27] and is defined by (4.1)-(4.5). The term naive is intended to indicate that the model contains only a small number of arbitrary parameters.
242 In an integral representation the stress ~- is given by
= 1 f,
M(t - t')(I-
EE "r) dt',
(4.1)
a.l_oo
where E is a strain tensor which is the solution to ~E/~t = A E ,
E( t',t') = l ,
(4.2)
with A = -a-+- l~ ( V u ) + - a- 2 1
(XTu) "r.
(4.3)
The non-affine parameter a is such that a = 1 corresponds to the Upper Convected and a = 0 to the Corotational formalism. The definition, so far, is similar to that in a n u m b e r of recent papers [28-30] except that in these publications an error has been made, in defining ~', by an omission of the factor 1/a in (4.1). This leads to the curious, but false result, that in simple shear flow, all stress components vanish identically when a = 0. The M e m o r y Function, M, is taken to have the form M(t-
t ' ) = Y'. 7/I, e x p [ - ( t - t')/~,k]
(4.4)
and the parameters r/k, Xk are related by the empirical equations r/k = r/0Xk/)-'~Xj, Xk = X/k~, k = 1,2, 3 . . . . .
(4.5)
J
which were first given in Spriggs [31]. Thus the total n u m b e r of u n k n o w n parameters (r/0, ~, a, a) is four. To a first approximation, the interface between the two zones is treated as a streamline. The upstream behaviour of the converging flow field will be similar to that of the 2 F M and can be determined by perturbing around the Newtonian solution. The aim of the analysis in this section is to determine the asymptotic downstream behaviour, i.e. close to the origin. Flow in the shear zone The naive rheological model possesses a power-law region at high shear rates. The shear stress is determined from the asymptotic expansion for large /3~5' and is given for both cone and wedge flows (provided the coordinate system is suitably identified) by
r/°~r(B~')l/~-~lS"°ll/" z,o-- - 2 a Z ( a ) sin((a + 1)~r/2a)
(4.6)
243 with p=(1-~2)“2andZ(cw)=
E
l/k*.
k-l
A similar expression
can be obtained for the first normal-stress-difference for simple shear flow valid at high shear rates
7
rr
-
ret? =-
T( pX)“a-llyrel”a-- px
?joh .qa) i
(4.7)
2 1*
cwsin( 7r/2a)
on the shear rate. The Clearly rre and r,, - 7ee have the same dependence order of magnitude inequalities (2.26) and (3.18) are satisfied, ensuring that the shear-stress terms dominate the normal-stress-difference terms in the shear layer equations. The Ellis model and the naive rheological model have the same highshear-rate, power-law region if
1
u/fa-
‘=jZ
[ 2 Z(a)
sin((l+
1)7r/2a)
1)
(4.8)
’
(4.9)
P=vo*
With these assignations the shear layer parameters x, T,, Q/K, are given by equations identical to the corresponding equations in .sections 2 and 3 for large l/s. For a complete evaluation of the shear layer it only remains to determine Ap* from the core zone. Flow in a core zone-wedge In wedge flow the core zone is a line sink with the kinematics which the rate-of-strain tensor defined in (4.3) is in 2-D
of (2.7), for
(4.10)
A=$(;
-:)*
The solution
of (4.2) for the strain tensor E is
E,, = (1 + 2Ks/r2)“‘, E,, = (1 + 2Ks/r2)-a/2, with all other components is given as an integral by r~~=~~mM(~)[l 7ee 7 f /omM(s)[
s = t - t’, of E being zero. The stress tensor defined
(4.11) by (4.1)
- (1 +2Ks/r2)o]ds, 1 - (1 + 2Ks/r’))“]ds.
(4.12)
244 All other components of r vanish identically. The stress at small values of r depends strongly on the value of a, and so therefore do Ap and Ap*. (At this stage the analysis could be continued in the style of a 2FM by leaving M(S) unspecified as there should be enough information in (4.12) to determine the asymptotic dependence of Ap for small r.) With M given by (4.4) the non-zero stress tensor components become
,
(4.13)
in which the qi and Xi are given by (4.5) and & = r2/2hiK. is obtained from (4.13) as Ap=
-
-i$r(rrr--ro,)=
=
%,KA.p*
Ap
(4.14)
r3
-dc~[G(,,E,)-G(-a,Si)], I
The quantity
(4.15)
’
where G(a,
&) = [(2a - l)&u-
2&-“]e~~I’(l
Unfortunately, it does not (4.15) in terms of a finite be evaluated numerically. of Ap* the summation is shown that as [ + 0 Ap* - (2 - l/a)<‘-”
+ a,&).
(4.16)
seem possible to express the infinite summation in number of known functions so Ap would have to To obtain an estimate of the asymptotic behaviour truncated to include only the first term. It can be
a * l/2,0
- 21Y(3/2)<3’2
a=
l/2
- 2[1n[
a = 0,
(4.17)
in which 5 = r 2/2X K. For the rubber compounds used here, 1 > a > l/2 and so the first of the asymptotes applies [8] (should values of a = l/2 arise, then Ap* < 0 which would necessitate a change in the analysis in the shear layer). Calculations can be carried out for the cone and for different rheological models but it is probably more important at this stage to investigate how higher terms in the approximation method for the 2FM and 2ZM might be generated.
245 5. Conclusions An approximation method, the Two Fluid or Two Zone Model, for the solution of flows in converging wedges and cones has been put forward. The Model is mathematically consistent and a number of solutions have been obtained. Dynamical consistency for a variety of polymers was established using experimental results. There are a number of points in relation to the Model which require further investigation. One of these which was not discussed in other sections is how the downstream boundary conditions may be satisfied. The manner in which this could be done is described in Pearson and Trottnow [32] for die-entry flow into a short capillary. Their entry-flow is that of a sink ( ug = u,+= 0) which undergoes a sudden velocity rearrangement at the capillary entrance. The extension of this idea to the 2FM is straightforward. The 2FM and 2ZM are applicable to a wide variety of possible die-entry flows, including situations where lubricating agents are present or where a thermal boundary layer arises. The Models can easily be extended to flow in a converging annulus. The philosophy of the’Models should also be suitable for other flows in which high extensions occur; as in calendering between two large rolls, for example. Although recirculating vortices arise in the Model shear layer next to the walls and extend throughout the length of the die-entry, they are not of the “thickness” observed in the experimental flows of certain polymers such as LDPE, when 0 is not small. Indeed the assumptions of the Model rely on the shear layer being thin. Velocity measurements of the recirculating regions indicate that the deformation rates are small except in the neighbourhood of the core [lo]. So, most of the vortex could be modelled as a Newtonian fluid. An improvement to the Model for fluids which exhibit large entrance vortices could be achieved by splitting the flow field into three regions: a core and a shear layer as presently and thirdly a Newtonian vortex. The shear layer would act as a thin transition layer from the core to the vortex. Finally, a comment about the physical stability of the 2FM. It is well known that a number of polymers exhibit a critical shear rate above which the flow may become unstable (see for example [ 1,211) though there is some dependence on the flow geometry. In particular, Vinogradov et al. [33] showed experimentally that the instabilities in entrance flow occur at points of highest stress concentration; for example, at the join between an entrance region and a parallel duct such as a capillary. The shearing deformation in the 2FM occurs in a thin region and not across the whole field and is therefore concentrated there. Thus if a critical shear rate is exceeded, it will occur at the thinnest point of the shear layer which happens to be at the downstream exit. It could easily arise that a critical shear rate is exceeded in
246 the entrance region but simultaneously (and after substantial stress relaxation) not be attained far downstream in a capillary or other parallel-sided duct, Acknowledgements The authors are indebted to Dr. S.M. Richardson for the time he has spent in many fruitful discussions of this topic and above all for his patience and encouragement to one of us (AMH). A.M. Hull expresses his gratitude to the Science Research Council for their support during his Ph.D. work, part of which formed the basis for this paper. References 1 A.M. Hull, J. Non-Newtonian Fluid Mech., 8 (1981) 327-336. 2 R.B. Bird, R.C. Armstrong and 0. Hassager, Dynamics of Polymeric Liquids, John Wiley & Sons, 1977, Vol. 1. 3 J.R. Black and M.M. Denn, J. Non-Newtonian Fluid Mech., 1 (1976) 83-92. 4 K. Strauss, Acta Mechanica, 20 (1972) 233-246. 5 J.R. Black, Ph.D. Dissertation, University of Delaware, Newark, 1974. 6 P. Schtimmer, Rheol. Acta, 7 (1968) 271-277. 7 M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, 1973, p. 41. 8. A.M. Hull, Ph.D. Thesis, London University, 1981. 9 A.B. Metzner, Rheol. Acta, 10 (1971) 434-444. 10 T.J.F. Pickup, Ph.D. Thesis, Cambridge University, 1970. 11 J.R.A. Pearson and T.J.F. Pickup, Polymer, 14 (1973) 209-214. 12 J.L. White and A. Kondo, J. Non-Newtonian Fluid Mech., 3 (1977) 41-64. 13 H.M. Laun and H. Miinstedt, Rheol. Acta 17 (1978) 415-425. 14 Y. Ide and J.L. White, J. Appl. Polym. Sci., 22 (1978) 1061-1079. 15 A.M. Hull, J. Non-Newtonian Fluid Mechanics (in press). 16 A.C. Pipkin and D.R. Owen, Phys. Fluids, 10 (1967) 836-843. 17 D.C. Pountney and K. Walters, Phys. Fluids, 21 (1978) 1482-1484. 18 D.C. Pountney and K. Walters, Phys. Fluids, 22 (1979) 1007. 19 C.D. Han, K.U. Kim, N. Siskovic and C.R. Huang, J. Appl. Polym. Sci., 17 (1973) 95-103. 20 Y. Ide and J.L. White, J. Non-Newtonian Fluid Mech., 2 (1977) 281-298. 21 D.M. Turner and M.D. Moore, Plastics and Rubber: Processing, 5 (1980) 81-84. 22 E. Uhland, Rheol. Acta, 18 (1979) l-24. 23 H. Winter, C. Macosko and K. Bennett, Rheol. Acta, 18 (1979) 323-334. 24 R.T. Fenner, Principles of Polymer Processing, Macmillan Press Ltd., 1979, pp. 147-148. 25 M. Abramowitz and LA. Stegun (Eds.), Handbook of Mathematical Functions, Applied Maths Series No. 55, U.S. Nat. Bureau of Standards, Washington, D.C., 1964. 26 C.D. Han, Rheology in Polymer Processing, Academic Press, 3rd ed., 1976, p. 55. 27 C.J.S. Petrie, J. Non-Newtonian Fluid Mech., 2 (1977) 221-253. 28 0. Hassager and S. Pedersen, J. Non-Newtonian Fluid Mech., 4 (1978) 261-268. 29 M.W. Johnson, Jr. and D. Segalman, J. Non-Newtonian Fluid Mech., 2 (1977) 255-270. 30 H.C. Lau and W.R. Schowalter, J. Rheol., 24 (1980) 507-516.
247 31 T.W. Sprigs, Chem. Eng. Sci., 20 (1965) 931-940. 32 J.R.A. Pearson and R. Trottnow, J. Non-Newtonian Fluid Mech., 4 (1978) 195-215. 33 G.V. Vinogradov, N.I. Insarova, B.B. Boiko and E.K. Borisenkova, Polym. Eng. Sci., 12 (1972) 323-334.