Ind. Eng. Chem. Res. 1994,33, 2434-2436
2434
Partial Wall Slip in Polymer Flow? Roger I. Tanner Department of Mechanical and Mechatronic Engineering, Uniuersity of Sydney, NSW ZOOS, Australia Recently a great deal of attention has been given to the slip of polymers on the walls of containing channels. The present paper discusses the question of where slip begins in a plane stick-alip flow when the pressure gradient far upstream is insufficient to cause complete slip. Some experimentally verifiable results are produced using the J-integral method. 1. Introduction
*s"-
The question of whether or not a fluid slips or sticks at solid walls boundine flow is an old one. For the classical Newtonian small-moleculefluids of engineering the question has long been answered fluids stick to the wall, and this boundawcondition is widelv believed. For Dolvmeric fluids the question is of greater interest; for exambliCohen and Metzner (1986) investigated the apparent slip of polymer solutions in caDillarv flow. In these cases it is . . clear t hat the solvent does not slip. but there is an apparent overall s l h due to local concentration variations near the wall. For molten polymers, however, real slip appears to 'occur at the wall; see, for example, Denn (1992) and Hatzikiriakos et al. (1993) for a discussion of earlier work. An interesting early discussion of the mechanism of slip is that of J. R. A. Pearson, who considered that the ratio of moleculesize tosurface roughness scale isdecisive: when the molecules are small compared with the surface roughness scale, then no slip occurs, but when large macromolecules are present, slip may well happen. The onset of slip occurs a t a certain wall shear stress T ~which , in the case of several polymers, for example, linear low density polyethylene, is of order 0.1 MPa (Hatzikiriakos etal., 1993). After thecriticalwallshear stress isexceeded, increasing slip sets in, and the slip speed a t the wall (us) is a function of 17. - rc. When the shear stress on the wall ( T ~ is ) uniform, the picture is clear, but when the shear stress varies along the wall, it is more difficult to understand. For example, Phan-Thien (1988) did a numerical simulation of extrudate swell including wall slip; he found reduced extrudate swelling when slip occurred near the exit lip and he found that slip actually began upstream of the exit. This is in accordance with a suggestion expressed by Luo and Tanner (1988). (The reader is reminded that near the exit plane in extrusions without wall slip there is a very rapid rise in shear stress near theexit,rising theoreticallytoan infmitelevel (Nickell et al., 1974; Tanner, 1988).) Thus one expects some slip near any die exit, which will, in turn, cause a change in the shear stress distribution a t the wall. Phan-Thien (1988) used a trial and error procedure to fix the extent of the region of slip near the exit. The present paper seeks to study this question for the stick-slip problem, which is quite similar in character to the extrusion problem. 2. The Stick-Slip Problem
Figure 1 shows the (incompressible Newtonian) stickslip problem: upstream there is a plane channel flow, downstream a uniform flow. This problem has been well studied (Georgiou et al., 1993, Tanner and Huang, 1993) and accurate results are known for the Newtonian and power-lawcases. Nearthesingularpoint S,theshearstress 'Dedicated to Professor Arthur B. Metzner. 0888-5885/94/2633-2434$04.50/0
LV;a.W? ................
0
.-
.-x 0-
d.hra
s"i,
* I
..
_ / .
It
aw
A
y-
-
.Ilp
Figure 1. Stick-slip problem. Newtonian m e .
,
(.5xy
C-r,
in [his case) at the wall is given, for the ~
~
h .. "J
where the channel half-width is h and the mean channel speed is n;the viscosity is 8. The (dimensionless) distance (relative to h) from the point S is r. From (1). one sees that as r 0, the shear stress become infinitely large. This clearly indicates that the no-slip, Newtonian physics mustbreakdownnearthesingular point. (Notethatearly results for the Newtonian problem given by Richardson (1970) show a different coefficient of r'l2 which appears to be in error.) In the present paper we will postulate a slip law so that the slip velocity at any wall point is given by
-
us= fClrl-7.) sgn r us= 0
(171 5
(1.1> T J rc)
(2)
wheref(.) isan experimentally determined function. More complex laws, depending for example on normal force, are possible, but we will only consider eq 2 here. In Figure 1, we suppose that the segment ST slips, while T A sticks to the wall; 1 is an unknown distance. We now work in dimensionless parameter terms, effectively, setting Li = h = 7 = 1. Applying the Rice J-integral method (Rice, 1968; Tanner and Huang, 1993) to the contour r = ABCDST in Figure 1,it can be shown (Tanner and Huang, 1993) that the J-integral in the I-y plane is zero. Hence
where J is taken around any closed contour not containing a singular point and ds is an element of the contour r. We do not expect any singular points on the contour ABCDST when slip occurs, but J is singular when the no-slip condition is employed. In expression 3, the dissipation integral W is
W
= t r , , d(D,,)
where I,, is the deviatoric stress 0 1994 American Chemical Society
(4)
*
~
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2436 Tij
+
(5)
= aij p6ij
ai, is the total stress, p is the pressure, and 6ij is the unit tensor. Dij is the rate of deformation tensor defined as
where U i is the velocity component in the x i direction. For the Newtonian case, which we consider to begin with, we find
w = 1/2q;/z
(7)
where +,the shear rate, is defined in the general case as
+
= (2Di$lij)1/2
(8)
ti = aijnj
(9)
Let us now compute the J-integral (3) around the contour ADCBST. (i) On AB the velocity is fully developed, and so auildx is zero. The contribution to Jfrom the first term, assuming a Newtonian profile on AB, is, in dimensionless terms (a = h = q = l),just 3. (ii) On BC dy = 0 and the traction is normal to the velocity gradient, so there is no contribution to J. (iii) On CD there are no stresses, hence no contribution; DS is similar to (ii) and also produces no contribution. (iv) On the solid wall dy = 0, so there is no contribution to J from STA; also since duild, is zero on TA, this contributes nothing. On ST we are left with a contribution -2Jkti duilax dx. Hence we find the relation, since au/dx = 0 on ST,
where is the shear stress at location x . Now on the x-axis, on the segment ST, the connection between u and T is given by eq 2, since u = us there. Hence (10) becomes
and (11) enables us to find T(O), the shear stress at S. Then we can find u,(O) from (2); this quantity should be measurable in an experiment. This does not answer the question of where the "separation" point T occurs. Without a detailed assessment of the shear stress, we have not found an accurate estimate. To find a first approximation to the slip length, we let the shear stress on the wall be estimated by using the shear stress in the no-slip case. We find an approximation to T as (Newtonian case) = T~
0.5 1
1.5
2
2.5
3
3.5
4
4.5
5
X
The other quantity not yet defined in (3) is ti, the traction on a surface with a normal unit vector ni:
T
0
+ 1.382/(x/h)lI2
(12)
where T,.,o is the wall shear stress far upstream, and the second term is obtained from (1). This is a close approximation in the no-slip case; see Figure 2. An approximate value of 1 is then given by eq 13 (in dimensionless terms), setting T = T~ in (12), 1 = ~ / ~ ( I T , J - Two)
2
The minimum value of 1 occurs when
-
TWO
(13) 0, hence
Figure 2. Comparison of approximation (eq 12) with numerical solution of shear stress in no-slip case.
l,
-
6 / m c2
(14)
In dimensional terms, 1
-
6q2~2/?rh1~c - 7&12
(15)
or, in terms of the dimensional upstream wall shear stress (T& = 3qii/h) (15) becomes
-
For the case where T~ = 0.1 MPa and is half of this, (16) gives 1 0.212h which is a very substantial slip distance. The slipzone increases rapidly as ~dapproaches 70.
3. Non-Newtonian Case
If we model the fluid as a power law ( T = ~j.")(or any other variable-viscosity model), then the above type of analysis is still applicable. Let us consider a power-law model. Following Tanner and Huang (1993) we then replace (1) by V*
where K*is the dimensionless stress-intensity factor which has been tabulated for various values of n by Tanner and Huang (1993). The J-integral may be used as before, and one finds
w = p+"+l/(n+ 1)
(18)
The relation between J and K has also been tabulated by Tanner and Huang (1993). I t follows that, for the general inelastic case,
and K*(n+l)/n
I = 247, - Two1(n+l)/n
(20)
where all quantities are dimensionless. (In the power-law case, the stresses are made dimensionless by the quantity AiiIh)".) In dimensional terms, using the fact that the shear rate at the wall is
2436 Ind. Eng. Chem. Res., Vol. 33, No. 10,1994
we find the dimensional length of slip 1 to be
Acknowledgment It is a pleasure to acknowledgethe influence of Professor Arthur B. Metzner on my thinking, and I am especially grateful for his support and continued interest during my three extended visits to the University of Delaware in the period 1979-1992. Literature Cited
which reduces to (16) when n = 1. Taking n = 0.3, we find l/h = 0.342 when rc = 2rW,which is an increase over the Newtonian case; for n = 0.1 the result is llh = 0.138, which is a decrease over the Newtonian result. Thus generally lln is about 0.2 of the channel half-width for rc = 2 7 ~ 0 . Conclusion In this exploratory paper we have sought to understand how a flowing polymer separates from the die wall when the stress varies near an exit region. We have estimated the position of the onset of slip using the stress field found without wall slip. The action of slip will reduce stresses in the region, and hence our estimate of the slip region length is expected to be in excess of the actual slip length. Nevertheless, we expect that the order of magnitude of the slip region is about 10-30% of a channel half-width, and this conclusion is compatible with the computations of Phan-Thien (1988). The reduction of extrudate swelling due to the “lip slip” may also be understood in these terms. For the general case, computing will be necessary; here the relationship (19) can be used as a check on accuracy. The problem of incorporating the discontinuous boundary condition into the computer program may be difficult; an alternative possibility is to have a very rapid cutoff function when r approaches r cso that some slip always occurs, but it is exponentially small for T~ > 7 4 . Some of the above results will also be useful in understanding experimental arrangements designed to find the slip law, and this is currently under investigation.
Cohen, Y.; Metzner, A. B. An Analysis of Apparent Slip Flow of Polymer Solutions. Rheol. Acta 1986,25, 28. Denn, M. M. Surface-Induced Effects in Polymer Melt Flow. In Theoretical and Applied Rheology; Moldenaers, P., Keunings, R., Eds.; Elsevier: Amsterdam, 1992; Vol. 1, p 45. Georgiou, G. C.; Olson, L. G.; Schultz, W. W.; Sagen, S. A Singular Finite Element for Stokes Flow: The Stick-Slip Problem. Znt. J. Numer. Methods Fluids 1989,9, 1353. Hatzikiriakos, S. G.; Stewart, C. W.; Dealy, J. M. Effect of Surface Coatings on Wall Slip of LLDPE. Znt. Polym. Process. 1993,8, 30.
Luo, X. L.; Tanner, R. I. Finite Element Simulation of Long and Short Circular Die Extrusion Experiments Using Integral Models. Znt. J. Numer. Methods Eng. 1988,25, 9. Nickell, R. E.; Tanner, R. I.; Caswell, B. The Solution of Viscous Incompressible Jet and Free Surface Flows Using Finite Element Methods. J. Fluid Mech. 1974,65, 189. Phan-Thien, N. Influence of Wall Slip on Extrudate Swell: A Boundary Element Investigation. J.Non-Newtonian Fluid Mech. 1988, 26, 327.
Rice, J. R. A Path Independent Integral and the ApproximateAnalysis of Strain Concentration by Notches and Cracks. J. Appl. Mech. 1968,90,379.
Richardson, S . A Stick-slip Problem related to the Motion of a Free Jet at Low Reynolds Number. Proc. Cambridge Philos. SOC.1970, 67, 477.
Tanner, R. I. Engineering Rheology, revised ed.; Oxford Univ. Press: Oxford, 1988. Tanner, R. I.; Huang, X. Stress Singularities in Non-NewtonianStickSlip and Edge Flows. J. Non-Newtonian Fluid Mech. 1993,50, 135.
Received for review February 17, 1994 Revised manuscript received July 28, 1994 Accepted July 29, 1994’ Abstract published in Advance ACS Abstracts, September 1, 1994.
