Compressive flow between parallel disks: II. oscillatory behavior of viscoelastic materials under a constant load

Journal of Non -Newtonian Fluid Mechanics, 14 ( 1984) 30 l-325 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

301

COMPRESSIVE FLOW BETWEEN PARALLEL DISKS: II. OSCILLATORY BEHAVIOR OF VISCOELASTIC MATERIALS UNDER A CONSTANT LOAD $

S.J. LEE *, M.M. DENN **, M.J. CROCHET +, A.B. METZNER and G.J. RIGGINS ++

Center for Composite Materials, University of Delaware, Newark, DE 19711 (U.S.A.)

(Received January 31, 1983)

Summary

Compressive flow of viscoelastic materials between parallel disks under a constant load is studied analytically, numerically and experimentally. The key features of the unsteady deformation of viscoelastic materials are determined analytically using linear approximations to both the momentum and constitutive equations. In place of the monotonic “squeezing” found when Newtonian fluids are used, one finds in this case that oscillations arise when a critical value of a dimensionless group representing the ratio of elastic to inertial forces is exceeded.

In order to study the process in detail, finite-element numerical calculations are used with the full equations for quantitative calculation of the oscillatory behavior of fluids described by contravariant convected Maxwell models; it is found that this calculation is in surprisingly close agreement with the linear approximation. Experimental measurements, utilizing three fluids of widely different properties, support the major predictions of the analysis.

An important analytical conclusion arising from this study is that inertial terms can quite generally not be neglected, even for slow flows of viscous materials, in deformation processes starting from rest with a previously-undeformed fluid. This observation is derived from the fact that in viscoelastic

* Dedicated to the memory of Professor J.G. Oldroyd * Present address:, Dept. of Applied Mathematics, University College of Wales, Aberyst-

wyth, U.K. ** Present address: Dept. of Chemical Engineering, University of California, Berkeley, CA

94720, U.S.A. + Permanent address: Universite Catholique de Louvain, Louvain-la-neuve, Belgium

++ Present Address: Dept. of Bioengineering, Pennsylvania State University, University Park, , PA 16802, U.S.A.

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materials the resistance to deformation arising out of the rheological properties may be either small or zero at zero + time, hence small for short times of deformation, and the inertial resistance to motion may consequently dominate. Thus, such motions of viscoelastic materials differ appreciably from those of their Newtonian counterparts, for which the “zero Reynolds number” approximation may be quite useful.

1. Introduction

The problem to be considered here is that of the compressive ffow of viscoelastic fluids between two horizontal circular flat disks, shown sche- matically in Fig. 1. The test fluid is contained between two disks which are at rest for times t < 0; at t = 0 the upper disk is released and falls under a constant load m. The spacing between the disks is measured as a function of time.

This compressive flow between disks has been of interest for many reasons.

(1) It is encountered in certain polymer processing operations such as compression molding and stamping. The polymeric charges in these processes are frequently filled with fibers, whose orientation will determine the material properties of final products, and the orientation of the fibers is believed to be determined by the flow behavior of the polymeric medium.

(2) It arises in lubrication systems, and there has been a controversy as to whether or not viscoelastic lubricants will perform better than Newtonian lubricants, since modern motor oils often contain polymeric additives which render them viscoelastic.

(3) It is encountered in the popular plustometer method for determining the material properties of highly-viscous materials.

111

Load m

2H 1

Fig. 1. Schematic diagram of the compressive flow experiment.

303

(4) Most of all, this flow is of particular interest to rheologists since both shearing and extensional deformations are present under transient conditions, the flow being dominated by shear near the wall and by extension in the middle of the gap. Therefore, this compressive flow is a good candidate to be used in evaluating proposed constitutive equations, especially their transient responses, and further improving them.

Compressive flows of Newtonian and power-law fluids were analyzed by Stefan [l] and Scott [2], respectively. The Stefan and Scott equations have been verified experimentally by many researchers [3-71; they apply when R/H, the ratio of radius to height of the region being deformed, is large. Deviations arise because of edge effects when R/H < 10, and these have been described [7].

Various researchers have considered the case of viscoelastic fluids in compressive flow. All except Metzner [8] and Tichy and Winer [9] have predicted that viscoelastic fluids squeeze out faster than the corresponding inelastic fluids; this prediction is the opposite of many available experimental results.

Tanner [lo] analyzed the flow for a contravariant convected Maxwell fluid with a power-law viscosity and a constant relaxation time. He argued that the normal stress effects are small compared to the shear stress effects and predicted that viscoelastic fluids would be squeezed more rapidly than would the corresponding power-law fluids.

Metzner [8] appears to be the first to have recognized the possible importance of the extensional flow as well as the shearing flow in this problem. He mentioned that extraordinarily high stresses are predicted for rapid extensional deformations of viscoelastic fluids, and he predicted slower squeezing of viscoelastic fluids (contravariant convected Maxwell fluids) based upon the “extensional primary field” approximation [ 111. Williams and Tanner [ 121 also considered a combination of shear and extensional effects but concluded that extensional effects are small compared to the shearing effects.

Kramer’s analysis [ 131 is unique in that the particle path equations are solved numerically using a convected coordinate system without neglecting any of the normal stresses. He used the integral constitutive equation of Lodge’s rubberlike liquid [14] with a single exponential memory function, which is identical to the contravariant convected Maxwell fluid. By assuming parallel squeezing and negligible inertia, he predicted an initial instantaneous drop when the load is applied to the fluid, and consequently the more rapid squeezing of viscoelastic fluids. The initial drop implies that the material has no resistance at the instant t = 0 except that of its own inertia; no resistance of the material implies a zero apparent viscosity, hence infinite Reynolds number. Thus, there is an internal inconsistency in this analysis which does

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not appear to have been noted heretofore: inertia must be taken into account in compressive flow of viscoelastic fluids under a constant load, even though the creeping flow approximation (negligible inertia) is very useful in the inelastic cases. This has been done previously only in the Tichy- Winer [9] analysis, which is not a constant load analysis as are all the others considered here.

Leider and Bird [ 151 have suggested that the use of a rheological model which can describe stress overshoot in simple shear flow is imperative to explain the slower squeezing of viscoelastic fluids. Note that the models used in the four previous analyses do predict the presence of a first normal stress difference, but do not predict stress overshoot or a second normal stress difference in shear flows. Leider and Bird proposed an empirical equation with an overshoot correction factor to be used in squeeze film problems.

Brindley et al. [5] analyzed the flow for the second-order fluid and again predicted faster squeezing of viscoelastic fluids. The second-order fluid shows first and second normal stress differences in shear flows, but no stress overshoot. Brindley and coworkers also presented some very interesting experimental results which show solid-like bouncing behavior under some severe loading conditions, but they did not provide a theoretical explanation of this bouncing behavior.

Experimental studies on viscoelastic fluids include those of Parlato [3], Leider [4], Brindley et al. [5], Grimm [6], and Tichy and Winer [9]. In general, with slow squeezing (low Deborah number), the squeezing behavior of viscoelastic fluids is close to that of the corresponding inelastic case. With fast squeezing (high Deborah number) most of experimental results show that viscoelastic materials squeeze out more slowly than the corresponding inelastic fluids, and under very severe loading conditions some materials even appear to “bounce back” after some amount of squeezing.

In this paper, we consider the compressive flow of fluids described by the contravariant convected Maxwell fluid under a constant load, with the inertia of the fluid and of the load included. First, we analyze a linearized problem to illustrate how the inertia and the elasticity may interact with each other to generate an oscillatory motion of the entire system. A finite-element numerical solution of the complete non-linear problem is then shown and compared to the solution of the linearized equations. Finally, experimental results are presented and compared to the predictions.

2. Problem formulation

Let us consider that the fluid is compressed between two parallel circular flat disks under a constant load m (see Fig. 2) and assume that the rheological properties of the fluid are described by the contravariant con-

305

+r 0 R

Fig. 2. The domain used in the analysis of compressive flow of viscoelastic materials.

vetted Maxwell model. Since there is no rotation, U, = 0, and a/&9 = 0. The velocity field in cylindrical coordinates then has the form

n= Ct.+, u8, 0,) = [u,(r, z, t), 0, r+D,(r, z, t>] (la)

and the deformation rate tensor d is given by

d=4(Vu+ VU=)= 0

I

av,/ar 0 f(aO,/aZ + av,/ar)

fur o >

f(avr/az + av,/ar) 0 a vz/aZ I

(lb)

showing the presence of both shearing and extensional deformation rate terms. The flow is then governed by the continuity equation,

f+uJ+$o, and the r- and z-components of the momentum equation,

Du,=_ ap ia a7

’ Dt ~+;@%)+yg+ Pg.

(2)

t3b)

D/Dt is the material time derivative defined by (a/at + v - v). r,,, T@~, T,,,

and rrZ are the non-zero components of the extra-stress tensor; for the kinematics given by eqn. (la), the stress components given by the Maxwell

306

constitutive equation are:

=q( $+$). (4d)

(4’4

(44

Equations (l)-(4) must be solved simultaneously, subject to the following boundary and initial conditions:

v, = 0 at z = 0, 2H) no-slip condition,

v, = 0 atz=O

v,= -V(t) atz=2H I impermeable boundary, (5)

v, = 0 atr=O

a2:=0 atr=O i

symmetry conditions, (6) tlr

S, the surface force vector, vanishes on the free surface for all (t).

p,r,v=o- at t = O-, i.e., just before

imposition of the load (7)

In eqn. (5), V(t) represents the downward velocity of the upper disk. The total force exerted by the fluid on the upper disk is calculated through

the following integration:

F = oR( -ozrr)r=2H27rrdr. J

a = -p + r,, is the axial component of the total stress. This force is bylanced by the dynamic force exerted by the load,

F=m(g-dV-,/dt), (9)

in which mdV/dt represents the inertia of the load. It is not possible to solve analytically eqns. (2), (3) and (4) under the

boundary and initial conditions (5)-(7). It is therefore necessary either to make some assumptions which reduce the problem to a simpler form, or to solve the problem numerically. The following sections will present both an analytical approach, based upon simplifying assumptions, and a rigorous complete solution using finite-element numerical methods.

307

3. Analytical solution (a linearized problem)

We shall introduce the following assumptions intended to reduce the problem into an analytically-solvable form, and yet retain the important features of the problem:

(a) The material can be described adequately by a linearized model equivalent to a Jeffreys or Oldroyd B fluid. This enables one to drop the higher-order terms of eqns. (4) but adds a retardation time term to the right-hand side of those equations. The importance of a retardation time is discussed more fully elsewhere [ 161, and this term can be dropped when such a constitutive feature is not of interest. It does enable some examination of the phenomenological importance of retardation times in rheological phenomena with no increase in analytic complexity.

(b) The nonlinear inertial terms in the momentum equation can be neglected (i.e., pDu/Dt = pi!lu/&); this is consistent with neglecting nonlin- earities in the constitutive equation, and implies also that v2/H -=x dv/dt.

(c) The velocity field is assumed to be that of a Newtonian fluid, i.e.

0, =f(z, t), (lOa)

v, = - :rf’(z, t), (lob)

where

f(z, t) = V4[(z/H)’ - 3(z/H)‘] (11)

and the prime denotes differentiation with respect to z. (d) R/H x- 1. (e) The boundary condition at the free surface is necessary to identify the

pressure, and it is approximated by the following integration [7]:

J 02H( -p + 7rr)r=Rdz = 0.

The stress components of the linearized Jeffreys model are written as

(12)

(134

(13b)

(13c)

rrr+x,g= -77 ff+A,z ) i

af’ 1 a?M r&+X,-= at -v f’+h,,I 7 ( af' 1

a7 r,, + A,> ( aff

at =27l f'+A,x 7 1 a7,

i afff

rrr+X1,,= -+qr f"+X,x , i (134

308

from which each component of the extra-stress tensor 7 is obtained as

7 = rz f” + X2$&) exp( $)dt’.

1

(14b)

( 144

The linearized momentum equation is then readily integrated, paralleling the development in [7], to obtain the following expression for the force:

Equation (15) has been simplified by use of the approximation R/H B 1. Equating F in eqn. (15) with m( g - dV/dt) leads to an equation for I/, when this is multiplied by exp(t/X,) and differentiated with respect to t the following second-order ordinary differential equation is obtained:

d2V P*h ~

311x2 dV

dt2 +p* 1+----

I 1 2p*H2 dt+

Here, p* is an effective density inertia:

8mH p*=p 1+- I 1 rR4p .

377 __ v= p*g. 2H2

(16)

which accounts for both fluid and load

(17)

V2/H has been neglected relative to dV/dt in passing from eqns. (15)-( 16); equivalently, H has been taken to be approximately constant for the small deformation admissible in the linear theory.

Equation (16) contains the essential physics of this analysis. It is a linear, second-order, ordinary differential equation with constant coefficients (as long as changes in H are small and H is taken as a constant). The system is of second order because of the possible interaction between the inertia terms in the momentum balance and the viscoelastic relaxation terms in the constitutive equation. In terms of the displacement variable X,V (the distance travelled in one relaxation time), Equation (16) is equivalent to the equation for the displacement of a linear massspring-dashpot system, with the “ viscous damping” contributed by the term (p*/X,)(l + 3qh2/2p*H2). The sole contribution of the retardation time is therefore to increase damping. The solution will be oscillatory when the following condition is satisfied:

P *2 I I 1 [ m2 2 6P*h <O

2p*H2 H2 ’

or, equivalently,

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(18)

(19)

where the modified elasticity number is defined

El* = qX,/p*H2. (20)

The elasticity number, which is the ratio of fluid relaxation time to a diffusion time, and is independent of kinematics, was employed by Denn and Porteous [ 171 to identify conditions under which the elasticity can be expected to be important in viscoelastic fluid flow. Astarita and Marrucci [ 181 note its significance in inertially-dominated quasi-viscometric flows. It was first introduced by Tordella [ 191 to predict the onset of melt fracture, and later by Boger and Binnington [20] to correlate data on entry pressure losses in a capillary rheometer.

The solution to eqn. (16) for small displacements, in which H can be taken as constant, and for parameter values such that eqn. (19) is satisfied, is

v= 2p*gH2 311 +exp[ - (1 +t?El*)$](CsinF- 2p~2cos~],

(21) where the period of the oscillations is given by

T = 2?rh,/[6 El* - (1 + +( h,/h,) El*)2]‘/2. (22)

Equation (21) incorporates the condition that the displacement is initially from rest (V = 0); the constant C is determined from the initial condition on dV/dt, which lies between zero and a maximum value equal to the gravitational acceleration, g. The maximum value, which assumes no fluid bulk or viscous resistance to deformation, has been used in the calculations shown subsequently. For this case, the initial amplitude of the oscillation is given approximately by

amplitude = gT/vr , (23)

so that the amplitude will scale with the period. The criterion given by eqn. (19) is somewhat conservative for the actual

observation of oscillations. Oscillations will be observed experimentally only if the time scale for damping is larger than the period for oscillations. The time scale for damping follows from eqn. (21) as a[ 1 + (3/2)(X 2/X,) El*]/A,,

310

where a is rather arbitrary but between about 1 and 3; a = 1 corresponds to 63% decay in the exponential, a = 2 to 86%, and a = 3 to 95%. The criterion that the time scale for damping exceed the oscillation period is

El* > 4a2 + u2 6a2 /1+;?,*[. (24)

A choice of a = 2 corresponds to a coefficient on the right-hand side of the inequality of 1.8, rather than the value of 0.17 given in eqn. (19).

Equation (16) can also be solved numerically without the restriction of constant H, together with the relation d H/d t = - V/2, using Gear’s method [21]. Computed curves of height versus time are shown in Figs. 3 and 4 for various values of relaxation time (A, = 0.003, 0.01, 0.03 s) and retardation time (A, = O., 0.001, 0.003, 0.01 s), and are compared there to the corresponding Newtonian curves. In Fig. 3, the retardation time is set to zero and the relaxation time varied. In Fig. 4, the relaxation time is kept constant (X, = 0.03 s) and the retardation time varied. The onset of oscillations is in accordance with the criteria given by eqn. (24). The oscillation period is about the same as the one estimated from the approximate solution eqn. (22). A small increase of the retardation time damps the oscillation greatly, as also expected from the condition (24) and the fact that the coefficient of the viscous damping term in eqn. (19) is linear in X2 with a coefficient that is proportional to El *. Note that most of the vicoelastic curves remain below the corresponding Newtonian curves, which implies that the linear viscoelastic materials show faster squeezing that Newtonian fluids.

I .oo

0.25

o.oc I

1 = 100.

R,=2.

H, = 0.05

Ill = 5000.

00 0.02 0.04 0.06 0.08 0.10

Time

Fig. 3. H/H,, vs. time in the compressive flow of linear viscoelastic materials: the effect of the relaxation time.

311

0

h 0.50- I 7) = 100.

R,= 2. Ii,= 0.05

0.25 - m =5000. _

o.oo_ I I I I 0.00 0.02 0.04 0.06 0.08 0.10

Time

Fig. 4. H/H, vs. time in the compressive flow of linear viscoelastic materials: the effect of the

retardation time.

4. Numerical solution (finite-element method)

The numerical solution of the full system of the non-linear equations (2)-(4) with the boundary and initial conditions (5)-(7), will now be considered, using a finite-element method. The problem is complicated since we have the unsteady inertia and elasticity terms in the momentum and constitutive equations, respectively. The domain for numerical solution changes as time goes on, and the unsteady transient equations have to be solved between time steps. Thus, we will pose the problem in a more natural way, following the pathlines of each material point in Lagrangian coordinates. Each nodal point in the grid moves along its own pathline as time goes on, and we use the material time derivative directly, instead of using partial time derivatives in the equations.

The time derivatives are treated by an implicit three-point-recurrence finite-difference scheme with variable time steps, which requires at least two previous solutions. To solve for the solution at t = t,+, based on two previous solutions (at t, and t,_ ,), the material time derivates are discretized on the time axis by

DA t-1 Dt n+l = at.at,_,(:t, + *t,_,j [At,-I( WI +Atn--l)An+l

- (At, + At,_,)*A, + (At,)&_,], (25)

312

where At, = t,, , - t,. All other terms in the system of equations involve the unknown variables at t = t, + ,. The Newton-Raphson method is used to treat the nonlinear terms, and the predictor-corrector method is used to improve the shape of the grid for each time step.

The mixed finite-element method, which was first introduced by Kawahara and Takeuchi [22] and further studied by Crochet and Bezy [23] and Crochet and Keunings [24], has been adopted to solve the system of equations. We have chosen quadrilateral elements and the same shape functions as those of Crochet and Keunings: a linear shape function for the pressure and quadratic shape functions in the velocity and the stress components. Since the sudden change of the boundary conditions (from no-slip to free surface) causes a stress singularity at the edge of the disk, we use small elements around the edge of the disk to relax the singularity within a small neighborhood of the edge. The Galerkin finite element formulation of the equations is given elsewhere [23,25,26]. The resulting simultaneous linear algebraic system is

_L t=o. H/H,, = I.

t = 0.038 H/H, -0.756

I - t = 0.066 H/H, = 0.866

1

t =0.105 H/H,, =0.730 I

I’ t =0.133 H/H, =0.799

Fig. 5. The initial grid and the shapes of the boundaries at later times in the compressive flow of a contravariant convected Maxwell fluid.

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I? \ I

0.7 - R = 5. Ho= 0.5 Linear viscoelastic prediction

p = I. 0.6 - 7) = 1000.

x,=0.1 In = 5000.

0.5 I I I I 0.00 0.04 0.08 0. I 2 0.16 0.20

Time

Fig. 6. H/H, vs. time in the compressive flow of a contravariant convected Maxwell fluid, compared to the linear viscoelastic and Newtonian cases.

solved by the frontal elimination technique proposed by Irons [27] to reduce the use of the central core memory.

Typical values of the material properties ( p = 1, 77 = 1000, A, = O.l), the geometry (R = 5, H,, = 0.5), and the load (m = 5000) have been used in the calculation. These values correspond to p* = 11, so the inertia of the load dominates, and El* = 37. The initial grid used in the calculation, and the shapes of the boundaries at later times, are shown in Fig. 5; the calculated height as a function of time is shown in Fig. 6, together with the linearized and Newtonian cases. It is seen from this figure that the overall behaviour of the Maxwell material is not very different from that computed from the linearized equations. The Maxwell fluid shows a somewhat smaller amplitude of the oscillation and a slightly longer oscillation period. Note that both viscoelastic curves remain below the corresponding Newtonian curve.

The shear rate in this flow depends upon the position and the time. The maximum shear rate occurs at the edge of the disk and at the time when the closing speed reaches the highest value. In this calculation, this maximum value of shear rate, y,,,, is 210 s-l at time t = 0.022, and so the Weissenberg number ‘/,,, X has a value of 21. This is an unusually high value of the Weissenberg number for convergence of numerical calculations since the highest published value for which convergence has been obtained for steady flows of a Maxwell fluid with a singularity is only of order unity [ 16,28-301. However, the result may not be surprising if one notes that ours is a transient flow in which the stress levels are still low at the small value of the Deborah number at this instant (t/A, = 0.22). Calculations at higher values

314

of the Weissenberg number were attempted in order to use parameters more representative of those describing the experiments reported below, but the solution ceased to converge after several time steps when the Weissenberg number was increased to a magnitude of 36.

5. Experiments

Experiments on the compressive flow of Newtonian and viscoelastic materials under a constant load have been carried out to obtain the film thickness as a function of time.

5. I. Apparatus

The apparatus used in the experiments was composed of three parts: the squeezing equipment, the thickness measuring device, and the recording equipment.

Squeezing equipment: The squeezing equipment was composed of a rigid stationary part and a moving part. A central cylindrical rod, at the bottom of which a flat circular disk was attached, moved through two linear ball bushings which provided alignment. The test material was placed between a stationary bottom plate and this circular disk. At the instant t = 0, the load (all the moving parts) on the material is released. The upper plate moved down under the influence of the gravitational acceleration, and the test material was squeezed radially outward.

Thickness measuring device: Continuous measurement of the film thickness during squeezing was accomplished with an LVDT (linear variable differential transducer) manufactured by Schaevitz Engineering Co. The LVDT is composed of two parts, the body and the core. The body was fixed on the stationary part of the squeezing equipment and the core was attached to the moving cylindrical rod, so that the core moves with the rod.

Recording equipment: The continuous measurement from the LVDT was sent to an oscilloscope (Tektronix 531A) and appeared as a moving spot on the screen. A photograph of the trace of the moving spot was taken with a Tektronix C-13 camera and a Polaroid Land pack film camera back.

5.2. Materials

Two Newtonian fluids have been used as the standards to test the apparatus, and three different viscoelastic materials have been investigated, as follows:

Viscasil60000 (Newtonian): This is a viscous silicone fluid manufactured by General Electric Company, with a viscosity that is independent of

315

w+J + A

0 18°C k1,=1.6) A 23°C tar= 1.0)

LO5

n 35°C (a,= 0.44)

Fig. 7. Standard reduced-variable plot (from oscillatory shear measurements) for the silicone polymer.

deformation rate at a value of 60000 cs (at 25°C). The density is 0.97 g/cm3. Dow Corning 2OOj7uid, 12.500 (Newtonian): manufactured by Dow Corn-

ing Company, the viscosity is 12500 cs (at 25°C). The density is 0.975 g/cm3. Silicone polymer: This is a three-phase material (silicone resin, plasticizer,

and filler) manufactured by ICI in England. This material is known to have rheological properties close to those of a linear Maxwell fluid, (see for example fig. 19-2 of Denn 1311). Oscillatory shear data at three different temperatures have been obtained through the courtesy of K.F. Wissbrun of the Celanese Research Corporation. By taking 23°C as a reference temperature, ]q*]w are superposed on one mastercurve with the use of temperature- dependent horizontal shift factors ((or) (see Ferry [32] or Takaki and Bogue [33]) as shown in Fig. 7. The linear viscoelastic behavior of a Maxwell fluid can be written in the form

(26)

Equation (26) fits the data in Fig. 7, establishing that the linear viscoelastic properties are those of a Maxwell fluid over this range of the circular frequency. As shown in Fig. 8, the viscosity is quite independent of frequency or deformation rate; its value ranges from 1.1 X lo6 to 0.31 X lo6 poise at the temperatures of 18-35“C; the relaxation time varies from 0.18 to 0.05 s over the same temperature range: i.e., the modulus G( h, = n/G) is virtually independent of temperature.

316

IO6 + 0 18°C A A 23°C

m 35°C

+ A

Fig. 8. Product of viscosity and frequency (from oscillatory shear measurements) depicted to show independence of viscosity and frequency.

TLA-227: This fluid is a concentrate of a methacrylate copolymer in petroleum oil, manufactured by Texaco, Inc. Steady shear stress and first normal stress measurements in the low shear rate range were made on the Rheometrics mechanical spectrometer (of Celanese), and capillary measurements were made at high shear rates. The end correction (see Bagley [34]) in the capillary measurement was unnecessary in the long capillary tubes used (L/R = 160, 268). The results at various temperatures are shown in Fig. 9. The spectrometer and capillary data agree well at 27°C. The material shows slight shear thinning behavior (n = 0.86) at high shear rates.

PAA-water solution: This 3.3 wt.% aqueous solution of Separan AP-30 (a partially hydrolyzed polyacrylamide manufactured by Dow Chemical Com- pany) was characterized with a Weissenberg rheogoniometer. The shear stress and the first normal stress data are shown in Fig. 10. This material is highly shear thinning (n = 0.27) and highly elastic.

5.3. Experimental results and discussion

Experimental results on Newtonian fluids are shown in Fig. 11, in which H/H, is plotted against dimensionless time. The data agree well with the Stefan equation, and illustrate the quality of the apparatus and the tech- niques used.

For the viscoelastic fluids, experiments were carried out over wide ranges of experimental conditions in order to evaluate the generality of the oscil-

317

10’ 1 I I I I11111 I I I I11111 I III IO2 IO0 IO’ 102 103

Shear Rate (set?

Fig. 9. The shear stress and the first normal stress difference of TLA-227.

lo51

,022 IO0 IO’ IO2 IO3

Shear Rate (set-‘1

Fig. 10. The shear stress and the first normal stress difference of a solution of 3.3 wt.% of PPA in water.

318

TABLE 1

Experimental Variables and Conditions Studied

3.1 cm < 20.5 kg 0.51 <R -c 6.35 cm 0.074 < H,, < 0.146 cm

125 <?j < 660 Poise (Newtonian) 1.4 < q < 920000 Poise (viscoelastic) 5.6 iA, c 100 ms

latory behavior, and the importance of inertia even in very viscous systems. Table 1 summarizes the conditions used.

The data for the most viscous fluid used, the silicone polymer, are summarized in Table 2 and in Figs. 12- 14. Figure 12, a photograph of exp. 2, depicts the dramatically large values of the oscillations as recorded under the actual conditions of the experiments, which, at El* = 90, are well beyond the predicted onset of oscillatory behavior. Data taken from this curve are compared in Fig. 13 with the linear viscoelastic prediction and with Newto- nian behavior. The experimental period of oscillation, about 30 ms, agrees very well with the a priori prediction of 27 ms computed from eqn. (22). The linear viscoelastic predictions exhibit a much larger amplitude than the experiment. Reference to eqns. (20) and (24) shows that introduction of a retardation time only 4% as large as the relaxation time would serve to bring agreement between theory and experiment on the amplitude as well as the period of the oscillation. The retardation time of this fluid was not mea-

TABLE 2

Comparison of Linear Viscoelastic Theory With Experiment, Silicone Polymer

Experimental Conditions. m (9) R (cm) H, (cm) P* (g/cd

9 (poise) A, (s)

exp. 1 exp. 2

20 500 8600 1.252 0.505 0.11 0.13 2340 43 800 665 000 665 000 0.1 0.1

Results. T experimental (ms) T (22) eqn. Initial amplitude (cm) El*

5-10 30 5.3 27 0.0045 0.02 2400 90

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0 A Viscasil 60000 l l Dow Coring 12 500

- Stefon equation

P I

0.4 -

0.2 -

0.0 I I I I 0. 2. ‘a 4. 6. 6. 10

16 F g+ _- ( > 3i& R.7

Fig. 11. Dimensionless spacing (H/H,,) vs. dimensionless time [(16/3XF/sR2XH,/R)2r/lJ for the squeezing of Newtonian fluids. Experimental results compared to the Stefan equation.

.i set lime

Fig. 12. Oscilloscope tracing showing the film thickness vs. time of squeezing. Silicone polymer; m = 8600 g, R = 0.505 cm, 9 = 665000 p (at 24.4”Q A, = 0.1 s.

320

I .o

0.9

0.0

e 1

0.7

0.6

0.5

R =0.505cm

- Ho=0.13cm Linear viscoelastic

rl= 665,000 p.(24.4T)

m= 8600 g. I I

I. 50. 100. 150.

Time, millisec.

Fig. 13. H/H, vs. time in the squeezing of silicone polymer, compared to the corresponding Newtonian and linear viscoelastic cases: short time response.

sured, nor are we aware of any experimental technique which would enable the accurate measurement of so small a value of this parameter. Indeed, the experiment depicted in Figs. 12 and 13 may be the only one presently available which could be adapted for this purpose.

As expected from the calculations, the experimental curve in Fig. 13 falls below the Newtonian one-the experiment is of too short a duration to enable development of the high stress levels characteristics of rapidly-extend- ing viscoelastic fluids, and, when this particular experiment is extended in time, the deformation rates fall to sufficiently low levels that high extensional stresses are again not expected to arise.

The behavior of this same fluid over a greater period of time is shown in Fig. 14. The initial response appears to be an instantaneous drop, since the time scale is too large to reveal the detailed oscillatory motion at very early times; we suspect that oscillations of the kind depicted in Fig. 12 may have been present in the work of previous authors but simply not discerned.

It is also of interest to monitor the gradual change from smooth, monotonic squeezing (when El* = 0) towards the increased propensity for oscillation as the elasticity number of the experiment is increased. While the first “onset” is difficult to discern, it always arose at elasticity numbers of the order of unity, with an initial temporary flattening of the H-t curves discernable at El* as low as 0.1. One such plot, at El* = 0.3, is shown in Fig. 15. It is typical of results obtained, both for TLA and PAA, in a total of five experiments [35].

Data for the other two fluids, under conditions of oscillatory behavior. are

321

0.126 - cm

3 E Y 0 ._

if

O.-

I 1.5 set

Time

Fig. 14. Squeezing of a silicone polymer, m = 17500 g, R = 0.505 cm, 1) = 920000 p (at 2O”C), X = 0.1 s: long time response.

0.292 -

I I 0. 0.2

Time (set)

Fig. 15. Squeezing of the PAA solution showing the incipient onset of an oscillation. R cm, m = 4750 g and El* = 0.3. (The lower two curves, which trace the H - t behavior at times, are not relevant recordings.)

= 6.35 longer

322

TABLE 3

Comparison of Linear Viscoelastic Theory with Experiment, Shear-Thinning Viscoelastic Fluids

TLA-227 PAA in water

Experimental Conditions m (g) R (cm) HO (cm) P* (g/cm3) i, W’) 9 (poise) A, (s)

3140 4750 1.252 6.35 0.074 0.146 242 2 516-63 1530-47 190-255 1.4-18 0.007-0.03 0.0056-0.07

Results T, eqn. (22) (ms) T experimental (ms) El*

20-30 33 25 30 1.0-6.0 0.18-30

summarized in Table 3. The elasticity number (eqn. (20)) is very sensitive to shear-thinning of the fluid since it contains, in effect, the viscosity squared (A, = n/G, where G denotes the modulus, a quantity frequently nearly independent of deformation rate). Complications arising out of this fact may, however, be eliminated in computing the period of oscillation from eqn. (22), since (with X,/h, = 0)

T = (2~*p*H*,‘3G)“~ (27)

when the identity X, = n/G is employed. Thus, for PAA in water, eqn. (27) leaves no ambiguity whatever concerning the predicted period of oscillation and that for TLA is also not very great. The experimentally-observed periods of oscillation are believed to be accurate to + lo%, but not better than this, since the values of the elasticity number are only moderate in these experiments, and the magnitude of the “bounce” is only small.

TABLE 4

Agreement between the predicted and experimental periods of oscillation

T predicted (ms) T experimental (ms)

Silicone, exp. 1 5.3 5-10 exp. 2 27 30

TLA 20-30 25 PAA 33 30

323

At some risk of redundancy, Table 4 summarizes the overall agreement between the predicted and experimental periods of the oscillation for all 4 experiments of Tables 2 and 3. We believe this to be a remarkable validation of the effects of a coupling between the inertial and viscoelastic properties of the system in these experiments.

Finally, we may note (Table 2) that the experimentally-observed ampli- tudes do scale with the periods of oscillation as predicted by eqn. (23).

6. Concluding remarks

(1) The compressive motion of viscoelastic materials under a constant load may or may not be oscillatory, depending upon the material properties, the magnitude of the load, and the system geometry. While small oscillations have been observed before, their origin does not appear to have been identified previously.

(2) When oscillation occurs, it is due to the combined effects of the inertia of the system and the elasticity of the fluid. The oscillation period depends only on system geometry and inertia, and the elastic modulus of the fluid. The amplitude scales with the period.

(3) When starting an experiment from rest with virgin, undeformed viscoelastic fluids, inertial effects are of importance even when the motions are slow ones, since the viscous stresses are always small at sufficiently short times of deformation. Consequently, the instantaneous ratio of inertia to viscous stresses (an instantaneous Reynolds number) is always high, and “zero Reynolds number” approximations are inappropriate. While the time- dependence of the viscous stresses is well-known, the consequent importance of inertia appears to have been quite generally overlooked.

(4) A small retardation time can have very large effects on the unsteady- state behavior of viscoelastic materials and deserves to be considered care- fully. Oscillatory squeezing experiments, although difficult, may represent the most accurate method available for measuring this fluid physical prop- erty.

(5) Linear approximations to both the material and system behavior may enjoy a surprising range of validity, enable the discernment of physical phenomena stripped of ancillary complications, and deserve more use than they have received.

(6) Under the experimental conditions reported herein the silicone polymer always squeezed out more rapidly than would a corresponding Newto- nian material. The reasons for this are clear and are discussed. Contrary results have also been obtained in a portion of our study [25] and these will be described in a subsequent paper. Thus, both kinds of behavior appear to be obtainable, depending upon the conditions used.

324

Acknowledgements

Financial support for this work was provided by the industrial sponsors of the Center for Composite Materials (Director: Professor R.B. Pipes). Dr. K.F. Wissbrun provided the majority of the rheological measurements. F.N. Cogswell, L.H. Keon and N. Sarkar arranged donations of the viscoelastic silicone polymer, the TLA and the PAA, respectively. We are grateful for all of this interest and support.

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(1977). See also AIChEJ, 24 (1978) 427. 7 S.J. Lee, M.M. Denn, M.J. Crochet and A.B. Metzner, J. Non-Newtonian Fluid Mech., 10

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325

31 M.M. Denn, Process Fluid Mechanics, Prentice-Hall 1980. 32 J.D. Ferry, Viscoelastic Properties of Polymers, 2nd ed., Wiley, 1970. 33 T. Takaki and D.C. Bogue, J. Appl. Polym. Sci., 19 (1975) 419. 34 E.B. Bagley, J. Appl. Phys. , 28 (1957) 624. 35 G.J. Riggins, B.Ch.E. Thesis, University of Delaware, Newark, Delaware (1982).

Note added in proof

While we believe this paper provides a clear articulation of exactly why inertial forces must be included when considering the sudden deformation of fluids previously at rest we are not the first to note oscillatory behavior arising from the coupling of inertial and viscoelastic forces. The 1964 movie by H. Markovitz included experiments suggested by W. Philippoff which show “bouncing” when a steel ball is dropped into a container of viscoelastic fluid; Ring and Waters (J. Phys. D: App. Phys. 5 (1972) 141) provided a numerical analysis of the impulsive motion of a sphere in a fluid which depicted such behavior; we have earlier drawn attention to the work of Brindley, Davies and Walters [5] and Mashelkar and Marrucci (Rheol. Acta 19 (1980) 426) coupled inertial and viscoelastic forces when considering flow across a cylinder imbedded in the fluid.

We are grateful to Dr. R.A. Mashelkar for assimilating this series of references for us.

Documents

Compressive flow between parallel disks: II. oscillatory behavior of viscoelastic materials under a constant load