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Elastico-viscous squeeze films. part I
19
Journal of Non-Newtonian Fluid Mechanics, 1 (1976) 19-37 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ELASTICO-VISCOUS
SQUEEZE FILMS. PART I
G. BRINDLEY, J.M. DAVIES and K. WALTERS Department
of Applied
Mathematics,
University
College of Wales, Aberystwyth
(Gt. Britain)
(Received July 2, 1975)
Summary This paper contains theoretical and experimental work on the squeeze-film situation for Newtonian and non-Newtonian liquids. To facilitate the interpretation of experimental results, a theoretical analysis is developed for an inelastic liquid having an arbitrary viscosity/shear rate relationship. In the process, it is found necessary to relax the common assumption that material planes which are initially horizontal remain so during the subsequent deformation. It is also shown how the inertia of the moving plate can be accommodated in the analysis. The experimental work contains a description of how a commercial rheometer can be adapted with ease to perform in a squeeze-film mode. Experimental data on polymer solutions indicate that under light-loading conditions, the behaviour of the liquids is predictable from a knowledge of the shear dependent viscosity only. However, under conditions of heavy loading (i.e. high Deborah number), viscoelastic effects are in evidence. Under these conditions, the liquids behave as better lubricants than one would predict from viscosity considerations.
1. Introduction In this paper, we shall be concerned with the squeeze-film flow described schematically in Fig. 1. The test fluid is contained between two horizontal circular flat plates which are at rest for times t < 0. At the instant t = 0, the top plate is released and falls under the influence of the normal load F. Facilities are available for determining the displacement h between the plates as a function of the time (see Fig. 2). In the present paper, we shall be mainly concerned with non-Newtoni~ elastic liquids and in particular with the influence of the elastic properties of the test liquids on the squeezing-flow characteristics. The squeeze-film situation is of interest for a number of reasons (cf. ref. [l] ). It is the basic geometry of the popular “plastometer” method of determining
20
Force
0
Ill LoadF
time
I
Fig. 1. Schematic diagram of squeeze-film situation. Fig. 2. Experimental conditions in a squeeze-film experiment.
the material properties of highly-viscous materials. It is also clearly of relevance within a lubrication context. Furthermore, it has attractions from a basic continuum mechanics standpoint, since the flow consists of an interesting combination of shear and extension in various parts of the flow field, the flow being dominated by shear near the plates and by extension in the middle of the gap. The only other study known to us which is comparable to that undertaken in the present project is the recent work of Leider and Bird [l] who concluded from their experiments that under light-loading conditions, an inelastic analysis based solely on viscosity considerations was sufficient to describe the flow, but that for heavy loading, the inelastic theory underestimated the spacing between the plates, often by an order of magnitude, which essentially implies that the elastico-viscous test fluids were behaving as better lubricants than one would predict from viscosity considerations. The present work is in broad general agreement with these interesting conclusions. In addition, we develop additional theory which is required in the interpretation of the experimental results and which has not been available hitherto. We also indicate how an existing commercial rotary rheometer (the Weissenberg rheogoniometer) can be easily adapted to perform in a squeeze-film mode. We find it convenient to define a “half-time” tllz as suggested by Leider and Bird [l]. tr,z is simply the time required for the plates to move from a distance apart ho to ho/2. We also find it convenient to define a Deborah number D, through the equation:
(1) where h is a characteristic relaxation time of the fluid. Low values of D, correspond to slightly elastic liquids and/or light loading and high values of D, correspond to highly elastic liquids and/or heavy loading.
21
2. Theoretical
considerations
We consider cylindrical polar coordinates (r,B,z), the z axis being along the normal axis to the plates, such that the lower and upper plates are given by z = 0 and z = h(t), respectively. Both plates are assumed to be of radius a. The physical components of the velocity vector in the r, 0 and z directions are denoted by u(,), qo) and u(,), respectively. Since h is usually very much smaller than a (h Q a), it is customary to apply the “lubrication approximation” to squeeze-film flow, which essentially involves the neglect of the inertia terms in the stress equations of motion and the application of an ordering process to the remainder of the terms in the governing equations. For example, we have U(r) =
O(l),
u(z) =
W/a), (2)
$ =O(l), c&=O(a/h). In the analysis, we can take utej = 0 from symmetry considerations sider u(,.) and u(,) to be functions of r, z and t but not 8. 2.1 Inelastic non-Newtonian We consider by an arbitrary given by *: Pik
=
-Pgik
’
and con-
liqzids
first the theory for a non-Newtonian inelastic liquid characterized viscosity/shear rate function 7)(q) so that the stress tensor pik is
(3)
P:k,
where (for incompressible fluids) p is an arbitrary isotropic pressure, ei:’ is the first rate-of-strain tensor and IZ is a suitable form of the second invariant of eji’. Within the lubrication approximation, we can take
If we neglect inertial effects,
ap -= ar
a&
- ar
ap _ a& -__+-+a.2 ar
+
1 F
@iIT,-
Pit-z,
r
the relevant
stress equations
P;oe,) + &P;IzJ>
aPizzj,
a.2
* We use standard tensor notation; covariant suffixes suffixes above and the usual summation convention Brackets placed around suffixes are used to denote
of motion
are:
(6)
(7) are written below and contravariant for repeated suffixes is assumed. “physical components”.
22
and, the equation of continuity is
-F_ $I(,)) +fg =0. Within the lubrication approximation, eqns. (6) and (7) become, on using eqn. (4):
(9)
ap,o az ’
(10)
where for convenience we have written T for the shear stress p(*,.), i.e. T = q(q)q. From eqns. (9) and (lo), we have:
T(q) =
zg+
(11)
A(r,t),
where A(r,t) is an arbitrary function of r and t. Since q = 0 on z = h/2 from symmetry, we have 7(q)=
(2-g 1c$f.
(12)
In order to solve eqn. (12) subject to the boundary conditions: U(r) =
v(,)
U(r)= 0,
=
0 on 2 = 0,
U(z)= k(t)
on z = h(t),
(13) I
(where the dot refers to differentiation with respect to t), we integrate the equation of continuity, eqn. (S), with respect to z from 0 to h, to give
ha
+3s shau’“‘dz
0
az
0
a,(ru~rj)dz
= 0.
(14)
On using the boundary conditions (13), eqn. (14) can be simplified to yield
Integrating eqn. (15) with respect to rand using the condition, vC;,I 0 on r = 0,
23
we have
At this point, we note a tendency have the functional form:
to conclude
from eqn. (16) that u(,) must
and by implication:
*te = -B(z,
t)
from the equation of continuity (where the dash refers to differentiation with respect to z). Under these conditions, material planes which are initially horizontal remain so during the subsequent deformation. The conclusion that eqn. (16) implies eqns. (17) and (18) is not generally valid,. In the appendix, we give a simple counterexample and also prove the result that one is justified in writing utr) and tuft) in the simplified forms given by eqns. (17) and (18) only when the viscosity function is representable in “power-law” form. This means that for the arbitrary viscosity function we are considering in this section, material planes which are initially horizontal can expect to experience “buckling” during the subsequent deformation. We are not justified therefore in writing eqns. (17) and (18), To proceed, we integrate eqn. (16) by parts to yield:
where 4 is given by eqn. (5). From eqn. (12) and a knowledge of the shear stress/ shear rate relationship, we can determine Q for any z and ap/ar, i.e. we can write:
and assume that q is known if r(q) is known. In the present work, the function 7(q) is obtained from independent experimental data. In order to invert eqn. (12) to yield q, we have found it convenient to use cubic-spline interpolation after an initial smoothing of the experimental data and to determine q numerically using Newton’s formula. Substituting eqn. (20).into eqn. (lS), we obtain
24
If we neglect the inertia of the moving plate, the total normal force on the plate due to the liquid is balanced by the applied load F, so that
Within the lubrication approximation, we can write: (23)
P(zz)l*=h= -Plt=h and integrating (22) by parts, we have F=-?rj-r a 2w%dr,
(24)
0
where we have used the boundary condition (25)
= 0.
&=h
The basic equations in the present study are eqns. (21) and (24). When the velocity of the moving plate is given (i.e. h is known) and the load F is required, eqn. (21) can be used to yield ap/ar as a function of r. The applied load F follows directly from eqn. (24). However, in our case, F is known and the velocity of the plate is required. The analysis is now far less straightforward. One method of proceeding is as follows. We introduce the following non-dimensional variables: r = ar*
q=
($ 1
4*
z = hz”
where q. is a characteristic viscosity, e.g. the limiting viscosity at very small rates of shear. Equations (12) and (19) now become:
q*($q*)q*= (&$?E;
(27)
and r* -= 2
3
s 0
z*q*dz*.
(28)
The total force is given by
(r*)2 $$d,..
(29)
We see from eqns. (27) and (28) that provided ah/h’ is fixed, a unique solution for ilp*/ar* as a function of r* can be obtained. We now proceed as follows. For a specific value of h (h, say) we ma*ke an initial guess of h (h, say) for a given load F2. f- ~o~esponding to PI and h, there is a load F1, which differs from Fa unless the initial guess of hI is exact. Examination of eqns. (27)-(29) now reveals’ that for fixed Q, no and ah/h2, a load F2 may be associated with a new h and h (h, and ha, say) with
(31) We see therefore that by varying h, , we can determine in a simple way a range of values of h, and ita corresponding to a load F2. (It is not difficult to deduce the most appropriate values of h, to provide the required range for h,.) The half-time t,,, is given by:
t,,2 = -
hodh, f $312
(32)
v, h2
which can be evaluated using Simpson’s rule and the computed values of h, and hs. Under conditions of severe loading, we have found it necessary to include the acceleration of the plate in the theoretical analysis. We write the equation of motion of the moving plate in the form: (33)
M%=--Mg+F,
where M is the mass of the plate and F (which is now the force on the plate due to the liquid rather than the applied load) is given by eqn. (24). Since F is a function of h and h, we essentially have to solve a second-order differential equation for h(t) subject to h(0) = ho,
h(0)
= 0.
(34)
The equation may be solved nume~ca~y using a standard fourth-order RungeKutta method to yield h as a function of time. In the section dealing with the ? AI may be conveniently obtained for the closest power-law representation of the viscosity function. The Scott equation (see eqn. 37) is available to facilitate this operation.
26
interpre~t~on of expe~men~ results, we show that the “‘plate-inertia” correction has an important influence under conditions of heavy luading. The foregoing analysis has been relatively complicated and basically numericaf on account of our desire to handle an arbitrary viscosity function q(q). For special forms of this function, an analytic solution to the problem can sometimes be obtained. For example, the relation between the load F and the displacement h in the case of a Newtonian liquid of viscosity no is given by the simple “Stefan” equation [ 23 : -3nQjk4 FEZ..----2h3
’
while for a power-Iaw fluid with a viscosity function given by:
where m and n are constants, we have the following equation due to Scott [3] * F I ___ 27mI(2n +-- 1)” s) ..-.. (+“a’“” nn/$Zn+l+ _t 3) -I”
(37)
The corresponding expression for the half-time t,,, is 133)
2.2 Ekastico-viscous liquids There have been a limited number of attempts to solve the squeeze-film flow problem for various elastico-viscous models (cf. refs. [4-81). The analyses of Tanner [ 6] and Kramer [ 7 ] predict that elastico-viscous liquids are poorer lubricants than one would anticipate on the basis of viscosity considerations. In this section, we outline briefly the solution to the problem for a second-order fluid model [cf- ref. [3] )? which is arguably a suitable model for use in the case of slightly elastic Iiquids or for liquids of moderate elasticity under conditions of light loading (i.e. low plate-acceleration}. The equations of state for the second-order fluid model can be written in the form of eqn. f3] and:
where e@ is the second rate-of&r&r tensor [9], a1 is a constant viscosity coefficient and cyzand a3 are material constants which are zero for a Newtonian liquid. In the present analysis, we assume that cy2and cy3are small enough for squares and higher powers to be neglected. * When m = na and n = 1, eqn. (31) reduces ta eqn. (351 as expected.
27
The appropriate after some routine
velocity distribution is given by eqns. (17) and (18) and, mathematics, the equation for B(z,t) can be shown to be:
where E is an arbitrary function of t. The solution of eqn. (40) subject to the appropriate boundary conditions (obtained from eqn. 13) is given by (41) From eqn. (41) it can be shown that the relevant pCzz) is given by:
normal stress component
(42) where we have imposed
the boundary
ptzz) = 0 when
z = h.
Substituting
~-=a,
condition
* (43)
eqn. (42) into eqn. (22). we obtain (44)
Our particular concern in the present paper is with the h(t) behaviour for a constant load F. If we write h1 for the dependence of h on t for a visco-inelastic liquid (which in this case corresponds to writing a2 = e3 = 0 in eqn. (39) so that h1 is essentially given by the “Newtonian” expression), it can be shown from eqn. (44) that (45) It is not difficult to deduce from eqn. (45) that e2 - 3&s/2 must be less than zero for the elastic liquid to be a better lubricant than a Newtonian liquid of the same viscosity. In terms of the first normal stress difference v1 and second normal stress difference v2, this condition is equivalent to: 2v, + iv2 < 0
(46)
for improved lubrication. On the basis of available evidence this is a most un_..._ * The need for, and choice of, a boundary condition of this sort is an important consideration in the general elastico-viscous squeeze-film problem. It will be considered in detail in a later paper.
28
realistic condition so that our analysis predicts that fluid elasticity gives rise to poorer lubrication in that the plates come together faster for non-zero values of aa and es. This is in general agreement with the conclusions of Tanner and Kramer. Since existing analyses seem to be at variance with the experimental work of Leider and also that presented in the next section, there is clearly need for more detailed theoretical studies involving more general equations of state and experimental conditions. We shall return to this subject in a later paper. 3. Experiment 3.1 Conuersion of the rheoguniometer
to a squeeze-film
apparatus
By making several simple modifications to a Weissenberg Rheogoniometer Rl6 (manufactured by Sangamo Controls Ltd.) we have been able to convert the instrument into a squeeze-film device. To accomplish this, the whole of the torquemeasuring system was removed. The torsion head transducer, normally used to measure the horizontal movement of its armature, was mounted on a right-angled bracket so that it was perpendicular to its usual position. A new transducer armature was made which was mounted vertically downwards and attached to the air-bearing rotor by a rigid radius arm E and clamp G (see Fig. 3). In this way, it was possible to measure small vertical movements of the air-bearing rotor and hence of the top plate.
Fig. 3. Schematic diagram of converted rheogoniometer.
29
A silver steel rod L, to which a weight pan W could be attached, was welded onto a cylindrical piece of metal which just fitted inside the hollow shaft of the air-bearing rotor. By tightening the clamp G, the silver steel rod and pan became rigidly fixed to the rotor, as did the torsion head displacement armature via the radius arm. The rod L passed through a hole in a Perspex block P, which fitted into the clamp normally used to grip the top of the torque-measuring bar. The clearance between the rod and the Perspex was of the order of 0.005 cm. Cut into the silver steel rod was a notch N, whose sides were parallel to each other and perpendicul~ to the axis of the rod. The notch was cut to a depth of half the diameter of the rod. The whole of the upper assembly could be supported by means of a parallelsided wedge M placed in this notch and held between the perspex block and the upper side of the notch. Rapid removal of this wedge caused a step function force of the type shown in Fig. 2 to be applied to the top plate, In the experiments, the air-bearing rotor was kept perpendicul~ during descent by means of the air-pressure system of the Weissenberg Rheogoniometer, and rotation in a horizontal plane was prevented by confining the radius arm to vertical movement only, using an improvised control. Since the transducer used to measure h had a new armature, it needed recalibrating. This was achieved by replacing the wedge by a number of feeler gauges in turn, and noting the displacement shown on the transducer meter. The linearity was found to be good up to displacements of the order of 0.2 cm. Both plates were adjusted to be parallel to each other and perpendicular to the vertical using standard rheogoniometer techniques. The maximum angle between the plates was estimated to be of the order of 0.0001 radians.
With the wedge in position, the air-bearing rotor and assembly were lowered towards the bottom plate. The gap between the plates was then set using feeler gauges to measure the distance apart of the plates, The gap-setting transducer was then set to zero, thereby allowing the top plate to be returned to the correct position after loading of the sample. The top plate was raised and the amount of fluid needed to fill the required gap was placed on the bottom plate using a syringe. The top plate was then carefully lowered to its correct position. Before activating the wedge and instigating the flow, the stresses developed in the test material during luading were allowed to relax. The signal from the displacement transducer was fed into a U.V.recorder so that a permanent record of the distance apart of the plates as a function of the time could be obtained after release of the activating mechanism. 3.3 Experiments
on Newtonian liquids
In order to estimate the effectiveness of the experimental techmque, silicone fluids (believed to be Newtonian under the operating conditions encountered in the experiments) were tested. The relevant operating formula is eqn. (35) which can be solved for the case of a constant load F to yield
30
Fig. 4. Squeeze-film results for a 140 poise silicone fluid. F = 423,800 on eqn. (47). Circles - experimental points obtained from U.V. traces.
dynes.
Full line - based
Fig. 5. Squeeze-film results for a 140 poise silicone fluid. Full line -based on eqn. (47). Circles - experimental points obtained from U.V. traces: full circles correspond to F = 423,800 dynes; open circles to F = 620,000 dynes.
h=
_L+4Ft
C h$j
3nq0a4
-
I
112
(471
Figures 4 and 5 contain typical results, the full lines representing eqn. (47) and the circles experimental points taken from the U.V.traces. In Fig. 4, the applied load is fixed and the initial spacing ho is varied and in Fig. 5, the load is varied for fixed ho. The agreement between theory and experiment in Figs. 4 and 5 was thought to be ~tisfa~tory, and we were led to believe that the modified rhe~goniome~r was functioning well in a squeeze-film mode.
In this paper, we report on experiments carried out on two p~lyisobutylen~ in-dekalin solutions. One was a 5% ~2OO/dekalin solution and the other a 1% BlOO/dekalin solution. Before examining these solutions in the squeeze-film apparatus, we characterized their simple-shear behaviour on the (unmodified) rheogoniometer. The relevant shear stress and first normal stress data are contained in Figs. 6 and 7, We observe that the solutions exhibit “power-law” behaviour over part of the shear-rate range but there are clearly regions which fall outside such a behaviour., The elastic liquids were tested in the squeeze-film apparatus for a wide range of applied loads and various values of ho: the loads varied between 640 and 5350 g and ho between 0.02 and 0.2 cm.
31
Pig. 6, Shear stress T and first normal stress difference trE data for the 5% I3200 in dekaIin solution obtained using mne-and-plate geeametry. Open symbols correspond to a 4O gap. Closed symbols correspond to a l$” gap.
.~~~~~_~.~ IO’
t
Fig. ?. Shear stress 7 and first normal sf;re~s difference ~1 data for the 1.5%13X00 in d&afin salution obtained using 1;’ gap in cone-and-plate geometry.
Figure 8 contitis a pfot af the observed half-times against those predicted by the Scott equation (38) for the 5% suhztion. A~~ern~nt is good for half-tomes up to 100 s, after which the half-times predicted by the Scott equation are always longer than those measured experimentally. The reason for this discrepancy can be traced to the fact that the reIevant shear-rates in the squeeze-film experiment were outside the power-law region and often in the ‘“first-Newtonian” region. The theoretical analysis was therefore overestimating the viscosity of the test solution. This conclusion is substantiated by the use of the analysis described in Section 2 for an arbitrary viscosity function, When the predicted half-times were obtained from this analysis, we see from Fig. 8 that the agreement between theory and experiment is exceffent. * We conclude Chat the squeeze-film behaviour of * The agreement between the predictions from the two anafyses for the fawer half-times is simply an indication that the dominant shear rates were essentially in the power-law region in the relevant experiments.
32
lo5 -
-g 10Lz! ; 1 103h 5 b 2 102.M b $ lo-
Measured
Half -T/me Isecl
Fig. 8. Predicted and observed half-times tained using eqn. (38). Crosses obtained viscosity function.
the 5% solution is predictable effects are apparently absent. this solution were carried out, (i.e. higher Deborah numbers) region.
for the 5% B200 in dekalin solution. Circles obusing analysis given in Section 2 for an arbitrary
from viscosity considerations only and viscoelastic Due to experimental limitations when the tests on it was not possible to consider shorter half-times to see if viscoelastic effects were in evidence in this
10’ I
3Q lo*.!!? g &
lo-‘-
Q $
-2
20b F! Q G3
-
8 .g
k-3 . Measure~2Ha~f _ Tz-i
Fig. 9. Predicted and observed tained using eqn. (38). Crosses
lsecl loo
half-times obtained
*’
for the 1% BlOO in dekalin solution. after applying inertial correction.
Circles
ob-
-4
h
ld ---“--.
‘.----____
34
corresponding theoretical curves (with and without the acceleration correction). The effect of the acceleration correction is clearly discernible, as is the marked difference between both theoretical curves and the experimental traces . In their work on squeeze-films, Leider and Bird [l] reached the same broad general conclusions as our own. Leider and Bird were also able to correlate all their results on one master curve using suitably defined nondimensional variables. For completeness, we attempted to apply their ideas to the present experiments and test fluids but without notable success [lo]. 4. Conclusions (i) The Weissenberg rheogoniometer can be modified without undue experimental difficulty to act as a squeeze-film apparatus. (ii) For a wide range of conditions (associated with light loading) the behaviour of elastic liquids in a squeeze-film situation is predictable from a knowledge of the shear-dependent viscosity only. This conclusion is based on a new theoretical analysis for an arbitrary viscosity function developed in Section 2. (iii) Viscoelastic effects are in evidence in squeeze-film situations when dilute polymer solutions are exposed to conditions of heavy loading (high Deborah number). The solutions behave as better lubricants than would be expected from viscosity considerations. (iv) Under some severe loading conditions, solid-like bouncing behaviour is possible. Acknowledgements We have benefitted from stimulating discussions with Professor R.B. Bird, Professor R.I. Tanner and Mr. J.F. Hutton, and helpful assistance from Mr. J. Astin and Dr. J.M. Broadbent. The experimental work was carried out under a contract from the Department of Trade and Industry and the paper is published by permission of the Director of the Department’s National Engineering Laboratory References 1 P.J. Leider and R.B. Bird, Univ. of Wisconsin Rheology Research Center Rep. No. 22, 1973, Ind. Eng. Chem. Fund., 13 (1974) 336,342. 2 J. Stefan and K. Sitzgber, Akad. Wiss. Math. Natur. Wien, 69 (1874) 713. 3 J.R. Scott, Trans. Inst. Rubber Ind., 7 (1931) 169. 4 P. Parlato, M.Ch.E. Thesis, Univ. of Delaware, 1969. 5 A.B. Metzner, Trans ASME, 90F (1968) 531. 6 R.I. Tanner, ASLE Trans., 8 (1965) 179. 7 J.M. Kramer, Univ. of Wisconsin Rheology Research Center Rep. No. 15, 1972, Appl. Sci. Res., 30 (1974) 1. 8 J.M. Davies, Ph.D. thesis, Univ. of Wales, 1974. 9 J.G. Oldroyd, Proc. R. Sot., A200 (1950) 523. 10 G. Brindley, Ph.D. thesis, Univ. of Wales, 1974.
Appendix In Section 2, we noted a tendency to conclude from eqn. (16) that UC,)must
35
have the functional form:
and (by implication) U(Z)= -B(2, t).
t-42)
Such a conclusion, if valid, leads to a significant simp~~fi~atio~ in any theoretical analysis, since one is essentially left with the solution of ordinary rather than partiut differential equations. Unfo~~~ately, this conclusion is not generally valid as the following counter example illustrates. UC,)= $3’(r, t) f zfz - h)(2z - A)C(r, t)
CA31
satisfies eqn. (16) and the boundary conditions (13) but not eqn. (Al), so that one is not justified in writing eqns. (Al) and (A2) in general. However, the significant simplification embodied in eqns. (Al) and (A2) makes it worthwhile to consider under what conditions these simple forms are justified. We are therefore led to seek the constraints on the viscosity function (for inelastic liquids) which are compatible with eqns. (Al) and (AZ). We begin by differentiating eqn. (12) with respect to r to yield * B”(z, t) dr m-c 2 dq Differentiating eqn. (12) with respect to z yields
From eqns. fA4) and (A5), we can easily see that
W6) The left-hand side of this equation is a function of z and t only and the righthand side is a function of r and t only. Since r and z are independent coordinates, we can equate both sides of eqn. (A6) to a function of the time k(t), so that
* We are grateful to Mr. J. Astin for suggesting this approach to the problem.
Inte~ating
eqn. (A7), we obtain = i In (z - ih) + G(t),
lnB”(z,t)
where G is an arbitrary
function
(A91
of t, so that
=a[B”(2,t)]k,
z-yl
where 01is a function
(A101 of t. Similarly,
integration
of eqn. (A8) yields:
ap= prkt ar where fl is a function we obtain 7(q) =
(All) of t. Substituting
eqns. (AlO) and (All)
cY/3[rB”(z,t)]“.
into eqn. (12),
(AW
But
a+, g3”(*,t),
Q= a2 =
so that eqn. (A12) can be written
dq) = wk,
in the form: (A13)
where m = 2krrj3. Equation (A13) is the familiar power-law representation for the shear stress. We conclude that the velocity components u(,.) and u(,) can be written in the simplified forms given by eqns. (Al) and (A2) only when the shear stress/shear rate relationship is expressible in power-law form.
Fig. 12. A schematic representation of the buckling of (initially horizontal) material planes at some time after the release of the top plate. The effect is exaggerated for illustration purposes.
37
When eqns. (Al) and (A2) are not appropriate, the analysis developed in Section 2 has to be used. This allows for material planes, which are initially horizontal, to warp during the squeeze-film deformation. Figure 12 contains a schematic representation of this behaviour based on the results for the polymer solution considered in Fig. 8.
Journal of Non-Newtonian Fluid Mechanics, 1 (1976) 19-37 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
ELASTICO-VISCOUS
SQUEEZE FILMS. PART I
G. BRINDLEY, J.M. DAVIES and K. WALTERS Department
of Applied
Mathematics,
University
College of Wales, Aberystwyth
(Gt. Britain)
(Received July 2, 1975)
Summary This paper contains theoretical and experimental work on the squeeze-film situation for Newtonian and non-Newtonian liquids. To facilitate the interpretation of experimental results, a theoretical analysis is developed for an inelastic liquid having an arbitrary viscosity/shear rate relationship. In the process, it is found necessary to relax the common assumption that material planes which are initially horizontal remain so during the subsequent deformation. It is also shown how the inertia of the moving plate can be accommodated in the analysis. The experimental work contains a description of how a commercial rheometer can be adapted with ease to perform in a squeeze-film mode. Experimental data on polymer solutions indicate that under light-loading conditions, the behaviour of the liquids is predictable from a knowledge of the shear dependent viscosity only. However, under conditions of heavy loading (i.e. high Deborah number), viscoelastic effects are in evidence. Under these conditions, the liquids behave as better lubricants than one would predict from viscosity considerations.
1. Introduction In this paper, we shall be concerned with the squeeze-film flow described schematically in Fig. 1. The test fluid is contained between two horizontal circular flat plates which are at rest for times t < 0. At the instant t = 0, the top plate is released and falls under the influence of the normal load F. Facilities are available for determining the displacement h between the plates as a function of the time (see Fig. 2). In the present paper, we shall be mainly concerned with non-Newtoni~ elastic liquids and in particular with the influence of the elastic properties of the test liquids on the squeezing-flow characteristics. The squeeze-film situation is of interest for a number of reasons (cf. ref. [l] ). It is the basic geometry of the popular “plastometer” method of determining
20
Force
0
Ill LoadF
time
I
Fig. 1. Schematic diagram of squeeze-film situation. Fig. 2. Experimental conditions in a squeeze-film experiment.
the material properties of highly-viscous materials. It is also clearly of relevance within a lubrication context. Furthermore, it has attractions from a basic continuum mechanics standpoint, since the flow consists of an interesting combination of shear and extension in various parts of the flow field, the flow being dominated by shear near the plates and by extension in the middle of the gap. The only other study known to us which is comparable to that undertaken in the present project is the recent work of Leider and Bird [l] who concluded from their experiments that under light-loading conditions, an inelastic analysis based solely on viscosity considerations was sufficient to describe the flow, but that for heavy loading, the inelastic theory underestimated the spacing between the plates, often by an order of magnitude, which essentially implies that the elastico-viscous test fluids were behaving as better lubricants than one would predict from viscosity considerations. The present work is in broad general agreement with these interesting conclusions. In addition, we develop additional theory which is required in the interpretation of the experimental results and which has not been available hitherto. We also indicate how an existing commercial rotary rheometer (the Weissenberg rheogoniometer) can be easily adapted to perform in a squeeze-film mode. We find it convenient to define a “half-time” tllz as suggested by Leider and Bird [l]. tr,z is simply the time required for the plates to move from a distance apart ho to ho/2. We also find it convenient to define a Deborah number D, through the equation:
(1) where h is a characteristic relaxation time of the fluid. Low values of D, correspond to slightly elastic liquids and/or light loading and high values of D, correspond to highly elastic liquids and/or heavy loading.
21
2. Theoretical
considerations
We consider cylindrical polar coordinates (r,B,z), the z axis being along the normal axis to the plates, such that the lower and upper plates are given by z = 0 and z = h(t), respectively. Both plates are assumed to be of radius a. The physical components of the velocity vector in the r, 0 and z directions are denoted by u(,), qo) and u(,), respectively. Since h is usually very much smaller than a (h Q a), it is customary to apply the “lubrication approximation” to squeeze-film flow, which essentially involves the neglect of the inertia terms in the stress equations of motion and the application of an ordering process to the remainder of the terms in the governing equations. For example, we have U(r) =
O(l),
u(z) =
W/a), (2)
$ =O(l), c&=O(a/h). In the analysis, we can take utej = 0 from symmetry considerations sider u(,.) and u(,) to be functions of r, z and t but not 8. 2.1 Inelastic non-Newtonian We consider by an arbitrary given by *: Pik
=
-Pgik
’
and con-
liqzids
first the theory for a non-Newtonian inelastic liquid characterized viscosity/shear rate function 7)(q) so that the stress tensor pik is
(3)
P:k,
where (for incompressible fluids) p is an arbitrary isotropic pressure, ei:’ is the first rate-of-strain tensor and IZ is a suitable form of the second invariant of eji’. Within the lubrication approximation, we can take
If we neglect inertial effects,
ap -= ar
a&
- ar
ap _ a& -__+-+a.2 ar
+
1 F
@iIT,-
Pit-z,
r
the relevant
stress equations
P;oe,) + &P;IzJ>
aPizzj,
a.2
* We use standard tensor notation; covariant suffixes suffixes above and the usual summation convention Brackets placed around suffixes are used to denote
of motion
are:
(6)
(7) are written below and contravariant for repeated suffixes is assumed. “physical components”.
22
and, the equation of continuity is
-F_ $I(,)) +fg =0. Within the lubrication approximation, eqns. (6) and (7) become, on using eqn. (4):
(9)
ap,o az ’
(10)
where for convenience we have written T for the shear stress p(*,.), i.e. T = q(q)q. From eqns. (9) and (lo), we have:
T(q) =
zg+
(11)
A(r,t),
where A(r,t) is an arbitrary function of r and t. Since q = 0 on z = h/2 from symmetry, we have 7(q)=
(2-g 1c$f.
(12)
In order to solve eqn. (12) subject to the boundary conditions: U(r) =
v(,)
U(r)= 0,
=
0 on 2 = 0,
U(z)= k(t)
on z = h(t),
(13) I
(where the dot refers to differentiation with respect to t), we integrate the equation of continuity, eqn. (S), with respect to z from 0 to h, to give
ha
+3s shau’“‘dz
0
az
0
a,(ru~rj)dz
= 0.
(14)
On using the boundary conditions (13), eqn. (14) can be simplified to yield
Integrating eqn. (15) with respect to rand using the condition, vC;,I 0 on r = 0,
23
we have
At this point, we note a tendency have the functional form:
to conclude
from eqn. (16) that u(,) must
and by implication:
*te = -B(z,
t)
from the equation of continuity (where the dash refers to differentiation with respect to z). Under these conditions, material planes which are initially horizontal remain so during the subsequent deformation. The conclusion that eqn. (16) implies eqns. (17) and (18) is not generally valid,. In the appendix, we give a simple counterexample and also prove the result that one is justified in writing utr) and tuft) in the simplified forms given by eqns. (17) and (18) only when the viscosity function is representable in “power-law” form. This means that for the arbitrary viscosity function we are considering in this section, material planes which are initially horizontal can expect to experience “buckling” during the subsequent deformation. We are not justified therefore in writing eqns. (17) and (18), To proceed, we integrate eqn. (16) by parts to yield:
where 4 is given by eqn. (5). From eqn. (12) and a knowledge of the shear stress/ shear rate relationship, we can determine Q for any z and ap/ar, i.e. we can write:
and assume that q is known if r(q) is known. In the present work, the function 7(q) is obtained from independent experimental data. In order to invert eqn. (12) to yield q, we have found it convenient to use cubic-spline interpolation after an initial smoothing of the experimental data and to determine q numerically using Newton’s formula. Substituting eqn. (20).into eqn. (lS), we obtain
24
If we neglect the inertia of the moving plate, the total normal force on the plate due to the liquid is balanced by the applied load F, so that
Within the lubrication approximation, we can write: (23)
P(zz)l*=h= -Plt=h and integrating (22) by parts, we have F=-?rj-r a 2w%dr,
(24)
0
where we have used the boundary condition (25)
= 0.
&=h
The basic equations in the present study are eqns. (21) and (24). When the velocity of the moving plate is given (i.e. h is known) and the load F is required, eqn. (21) can be used to yield ap/ar as a function of r. The applied load F follows directly from eqn. (24). However, in our case, F is known and the velocity of the plate is required. The analysis is now far less straightforward. One method of proceeding is as follows. We introduce the following non-dimensional variables: r = ar*
q=
($ 1
4*
z = hz”
where q. is a characteristic viscosity, e.g. the limiting viscosity at very small rates of shear. Equations (12) and (19) now become:
q*($q*)q*= (&$?E;
(27)
and r* -= 2
3
s 0
z*q*dz*.
(28)
The total force is given by
(r*)2 $$d,..
(29)
We see from eqns. (27) and (28) that provided ah/h’ is fixed, a unique solution for ilp*/ar* as a function of r* can be obtained. We now proceed as follows. For a specific value of h (h, say) we ma*ke an initial guess of h (h, say) for a given load F2. f- ~o~esponding to PI and h, there is a load F1, which differs from Fa unless the initial guess of hI is exact. Examination of eqns. (27)-(29) now reveals’ that for fixed Q, no and ah/h2, a load F2 may be associated with a new h and h (h, and ha, say) with
(31) We see therefore that by varying h, , we can determine in a simple way a range of values of h, and ita corresponding to a load F2. (It is not difficult to deduce the most appropriate values of h, to provide the required range for h,.) The half-time t,,, is given by:
t,,2 = -
hodh, f $312
(32)
v, h2
which can be evaluated using Simpson’s rule and the computed values of h, and hs. Under conditions of severe loading, we have found it necessary to include the acceleration of the plate in the theoretical analysis. We write the equation of motion of the moving plate in the form: (33)
M%=--Mg+F,
where M is the mass of the plate and F (which is now the force on the plate due to the liquid rather than the applied load) is given by eqn. (24). Since F is a function of h and h, we essentially have to solve a second-order differential equation for h(t) subject to h(0) = ho,
h(0)
= 0.
(34)
The equation may be solved nume~ca~y using a standard fourth-order RungeKutta method to yield h as a function of time. In the section dealing with the ? AI may be conveniently obtained for the closest power-law representation of the viscosity function. The Scott equation (see eqn. 37) is available to facilitate this operation.
26
interpre~t~on of expe~men~ results, we show that the “‘plate-inertia” correction has an important influence under conditions of heavy luading. The foregoing analysis has been relatively complicated and basically numericaf on account of our desire to handle an arbitrary viscosity function q(q). For special forms of this function, an analytic solution to the problem can sometimes be obtained. For example, the relation between the load F and the displacement h in the case of a Newtonian liquid of viscosity no is given by the simple “Stefan” equation [ 23 : -3nQjk4 FEZ..----2h3
’
while for a power-Iaw fluid with a viscosity function given by:
where m and n are constants, we have the following equation due to Scott [3] * F I ___ 27mI(2n +-- 1)” s) ..-.. (+“a’“” nn/$Zn+l+ _t 3) -I”
(37)
The corresponding expression for the half-time t,,, is 133)
2.2 Ekastico-viscous liquids There have been a limited number of attempts to solve the squeeze-film flow problem for various elastico-viscous models (cf. refs. [4-81). The analyses of Tanner [ 6] and Kramer [ 7 ] predict that elastico-viscous liquids are poorer lubricants than one would anticipate on the basis of viscosity considerations. In this section, we outline briefly the solution to the problem for a second-order fluid model [cf- ref. [3] )? which is arguably a suitable model for use in the case of slightly elastic Iiquids or for liquids of moderate elasticity under conditions of light loading (i.e. low plate-acceleration}. The equations of state for the second-order fluid model can be written in the form of eqn. f3] and:
where e@ is the second rate-of&r&r tensor [9], a1 is a constant viscosity coefficient and cyzand a3 are material constants which are zero for a Newtonian liquid. In the present analysis, we assume that cy2and cy3are small enough for squares and higher powers to be neglected. * When m = na and n = 1, eqn. (31) reduces ta eqn. (351 as expected.
27
The appropriate after some routine
velocity distribution is given by eqns. (17) and (18) and, mathematics, the equation for B(z,t) can be shown to be:
where E is an arbitrary function of t. The solution of eqn. (40) subject to the appropriate boundary conditions (obtained from eqn. 13) is given by (41) From eqn. (41) it can be shown that the relevant pCzz) is given by:
normal stress component
(42) where we have imposed
the boundary
ptzz) = 0 when
z = h.
Substituting
~-=a,
condition
* (43)
eqn. (42) into eqn. (22). we obtain (44)
Our particular concern in the present paper is with the h(t) behaviour for a constant load F. If we write h1 for the dependence of h on t for a visco-inelastic liquid (which in this case corresponds to writing a2 = e3 = 0 in eqn. (39) so that h1 is essentially given by the “Newtonian” expression), it can be shown from eqn. (44) that (45) It is not difficult to deduce from eqn. (45) that e2 - 3&s/2 must be less than zero for the elastic liquid to be a better lubricant than a Newtonian liquid of the same viscosity. In terms of the first normal stress difference v1 and second normal stress difference v2, this condition is equivalent to: 2v, + iv2 < 0
(46)
for improved lubrication. On the basis of available evidence this is a most un_..._ * The need for, and choice of, a boundary condition of this sort is an important consideration in the general elastico-viscous squeeze-film problem. It will be considered in detail in a later paper.
28
realistic condition so that our analysis predicts that fluid elasticity gives rise to poorer lubrication in that the plates come together faster for non-zero values of aa and es. This is in general agreement with the conclusions of Tanner and Kramer. Since existing analyses seem to be at variance with the experimental work of Leider and also that presented in the next section, there is clearly need for more detailed theoretical studies involving more general equations of state and experimental conditions. We shall return to this subject in a later paper. 3. Experiment 3.1 Conuersion of the rheoguniometer
to a squeeze-film
apparatus
By making several simple modifications to a Weissenberg Rheogoniometer Rl6 (manufactured by Sangamo Controls Ltd.) we have been able to convert the instrument into a squeeze-film device. To accomplish this, the whole of the torquemeasuring system was removed. The torsion head transducer, normally used to measure the horizontal movement of its armature, was mounted on a right-angled bracket so that it was perpendicular to its usual position. A new transducer armature was made which was mounted vertically downwards and attached to the air-bearing rotor by a rigid radius arm E and clamp G (see Fig. 3). In this way, it was possible to measure small vertical movements of the air-bearing rotor and hence of the top plate.
Fig. 3. Schematic diagram of converted rheogoniometer.
29
A silver steel rod L, to which a weight pan W could be attached, was welded onto a cylindrical piece of metal which just fitted inside the hollow shaft of the air-bearing rotor. By tightening the clamp G, the silver steel rod and pan became rigidly fixed to the rotor, as did the torsion head displacement armature via the radius arm. The rod L passed through a hole in a Perspex block P, which fitted into the clamp normally used to grip the top of the torque-measuring bar. The clearance between the rod and the Perspex was of the order of 0.005 cm. Cut into the silver steel rod was a notch N, whose sides were parallel to each other and perpendicul~ to the axis of the rod. The notch was cut to a depth of half the diameter of the rod. The whole of the upper assembly could be supported by means of a parallelsided wedge M placed in this notch and held between the perspex block and the upper side of the notch. Rapid removal of this wedge caused a step function force of the type shown in Fig. 2 to be applied to the top plate, In the experiments, the air-bearing rotor was kept perpendicul~ during descent by means of the air-pressure system of the Weissenberg Rheogoniometer, and rotation in a horizontal plane was prevented by confining the radius arm to vertical movement only, using an improvised control. Since the transducer used to measure h had a new armature, it needed recalibrating. This was achieved by replacing the wedge by a number of feeler gauges in turn, and noting the displacement shown on the transducer meter. The linearity was found to be good up to displacements of the order of 0.2 cm. Both plates were adjusted to be parallel to each other and perpendicular to the vertical using standard rheogoniometer techniques. The maximum angle between the plates was estimated to be of the order of 0.0001 radians.
With the wedge in position, the air-bearing rotor and assembly were lowered towards the bottom plate. The gap between the plates was then set using feeler gauges to measure the distance apart of the plates, The gap-setting transducer was then set to zero, thereby allowing the top plate to be returned to the correct position after loading of the sample. The top plate was raised and the amount of fluid needed to fill the required gap was placed on the bottom plate using a syringe. The top plate was then carefully lowered to its correct position. Before activating the wedge and instigating the flow, the stresses developed in the test material during luading were allowed to relax. The signal from the displacement transducer was fed into a U.V.recorder so that a permanent record of the distance apart of the plates as a function of the time could be obtained after release of the activating mechanism. 3.3 Experiments
on Newtonian liquids
In order to estimate the effectiveness of the experimental techmque, silicone fluids (believed to be Newtonian under the operating conditions encountered in the experiments) were tested. The relevant operating formula is eqn. (35) which can be solved for the case of a constant load F to yield
30
Fig. 4. Squeeze-film results for a 140 poise silicone fluid. F = 423,800 on eqn. (47). Circles - experimental points obtained from U.V. traces.
dynes.
Full line - based
Fig. 5. Squeeze-film results for a 140 poise silicone fluid. Full line -based on eqn. (47). Circles - experimental points obtained from U.V. traces: full circles correspond to F = 423,800 dynes; open circles to F = 620,000 dynes.
h=
_L+4Ft
C h$j
3nq0a4
-
I
112
(471
Figures 4 and 5 contain typical results, the full lines representing eqn. (47) and the circles experimental points taken from the U.V.traces. In Fig. 4, the applied load is fixed and the initial spacing ho is varied and in Fig. 5, the load is varied for fixed ho. The agreement between theory and experiment in Figs. 4 and 5 was thought to be ~tisfa~tory, and we were led to believe that the modified rhe~goniome~r was functioning well in a squeeze-film mode.
In this paper, we report on experiments carried out on two p~lyisobutylen~ in-dekalin solutions. One was a 5% ~2OO/dekalin solution and the other a 1% BlOO/dekalin solution. Before examining these solutions in the squeeze-film apparatus, we characterized their simple-shear behaviour on the (unmodified) rheogoniometer. The relevant shear stress and first normal stress data are contained in Figs. 6 and 7, We observe that the solutions exhibit “power-law” behaviour over part of the shear-rate range but there are clearly regions which fall outside such a behaviour., The elastic liquids were tested in the squeeze-film apparatus for a wide range of applied loads and various values of ho: the loads varied between 640 and 5350 g and ho between 0.02 and 0.2 cm.
31
Pig. 6, Shear stress T and first normal stress difference trE data for the 5% I3200 in dekaIin solution obtained using mne-and-plate geeametry. Open symbols correspond to a 4O gap. Closed symbols correspond to a l$” gap.
.~~~~~_~.~ IO’
t
Fig. ?. Shear stress 7 and first normal sf;re~s difference ~1 data for the 1.5%13X00 in d&afin salution obtained using 1;’ gap in cone-and-plate geometry.
Figure 8 contitis a pfot af the observed half-times against those predicted by the Scott equation (38) for the 5% suhztion. A~~ern~nt is good for half-tomes up to 100 s, after which the half-times predicted by the Scott equation are always longer than those measured experimentally. The reason for this discrepancy can be traced to the fact that the reIevant shear-rates in the squeeze-film experiment were outside the power-law region and often in the ‘“first-Newtonian” region. The theoretical analysis was therefore overestimating the viscosity of the test solution. This conclusion is substantiated by the use of the analysis described in Section 2 for an arbitrary viscosity function, When the predicted half-times were obtained from this analysis, we see from Fig. 8 that the agreement between theory and experiment is exceffent. * We conclude Chat the squeeze-film behaviour of * The agreement between the predictions from the two anafyses for the fawer half-times is simply an indication that the dominant shear rates were essentially in the power-law region in the relevant experiments.
32
lo5 -
-g 10Lz! ; 1 103h 5 b 2 102.M b $ lo-
Measured
Half -T/me Isecl
Fig. 8. Predicted and observed half-times tained using eqn. (38). Crosses obtained viscosity function.
the 5% solution is predictable effects are apparently absent. this solution were carried out, (i.e. higher Deborah numbers) region.
for the 5% B200 in dekalin solution. Circles obusing analysis given in Section 2 for an arbitrary
from viscosity considerations only and viscoelastic Due to experimental limitations when the tests on it was not possible to consider shorter half-times to see if viscoelastic effects were in evidence in this
10’ I
3Q lo*.!!? g &
lo-‘-
Q $
-2
20b F! Q G3
-
8 .g
k-3 . Measure~2Ha~f _ Tz-i
Fig. 9. Predicted and observed tained using eqn. (38). Crosses
lsecl loo
half-times obtained
*’
for the 1% BlOO in dekalin solution. after applying inertial correction.
Circles
ob-
-4
h
ld ---“--.
‘.----____
34
corresponding theoretical curves (with and without the acceleration correction). The effect of the acceleration correction is clearly discernible, as is the marked difference between both theoretical curves and the experimental traces . In their work on squeeze-films, Leider and Bird [l] reached the same broad general conclusions as our own. Leider and Bird were also able to correlate all their results on one master curve using suitably defined nondimensional variables. For completeness, we attempted to apply their ideas to the present experiments and test fluids but without notable success [lo]. 4. Conclusions (i) The Weissenberg rheogoniometer can be modified without undue experimental difficulty to act as a squeeze-film apparatus. (ii) For a wide range of conditions (associated with light loading) the behaviour of elastic liquids in a squeeze-film situation is predictable from a knowledge of the shear-dependent viscosity only. This conclusion is based on a new theoretical analysis for an arbitrary viscosity function developed in Section 2. (iii) Viscoelastic effects are in evidence in squeeze-film situations when dilute polymer solutions are exposed to conditions of heavy loading (high Deborah number). The solutions behave as better lubricants than would be expected from viscosity considerations. (iv) Under some severe loading conditions, solid-like bouncing behaviour is possible. Acknowledgements We have benefitted from stimulating discussions with Professor R.B. Bird, Professor R.I. Tanner and Mr. J.F. Hutton, and helpful assistance from Mr. J. Astin and Dr. J.M. Broadbent. The experimental work was carried out under a contract from the Department of Trade and Industry and the paper is published by permission of the Director of the Department’s National Engineering Laboratory References 1 P.J. Leider and R.B. Bird, Univ. of Wisconsin Rheology Research Center Rep. No. 22, 1973, Ind. Eng. Chem. Fund., 13 (1974) 336,342. 2 J. Stefan and K. Sitzgber, Akad. Wiss. Math. Natur. Wien, 69 (1874) 713. 3 J.R. Scott, Trans. Inst. Rubber Ind., 7 (1931) 169. 4 P. Parlato, M.Ch.E. Thesis, Univ. of Delaware, 1969. 5 A.B. Metzner, Trans ASME, 90F (1968) 531. 6 R.I. Tanner, ASLE Trans., 8 (1965) 179. 7 J.M. Kramer, Univ. of Wisconsin Rheology Research Center Rep. No. 15, 1972, Appl. Sci. Res., 30 (1974) 1. 8 J.M. Davies, Ph.D. thesis, Univ. of Wales, 1974. 9 J.G. Oldroyd, Proc. R. Sot., A200 (1950) 523. 10 G. Brindley, Ph.D. thesis, Univ. of Wales, 1974.
Appendix In Section 2, we noted a tendency to conclude from eqn. (16) that UC,)must
35
have the functional form:
and (by implication) U(Z)= -B(2, t).
t-42)
Such a conclusion, if valid, leads to a significant simp~~fi~atio~ in any theoretical analysis, since one is essentially left with the solution of ordinary rather than partiut differential equations. Unfo~~~ately, this conclusion is not generally valid as the following counter example illustrates. UC,)= $3’(r, t) f zfz - h)(2z - A)C(r, t)
CA31
satisfies eqn. (16) and the boundary conditions (13) but not eqn. (Al), so that one is not justified in writing eqns. (Al) and (A2) in general. However, the significant simplification embodied in eqns. (Al) and (A2) makes it worthwhile to consider under what conditions these simple forms are justified. We are therefore led to seek the constraints on the viscosity function (for inelastic liquids) which are compatible with eqns. (Al) and (AZ). We begin by differentiating eqn. (12) with respect to r to yield * B”(z, t) dr m-c 2 dq Differentiating eqn. (12) with respect to z yields
From eqns. fA4) and (A5), we can easily see that
W6) The left-hand side of this equation is a function of z and t only and the righthand side is a function of r and t only. Since r and z are independent coordinates, we can equate both sides of eqn. (A6) to a function of the time k(t), so that
* We are grateful to Mr. J. Astin for suggesting this approach to the problem.
Inte~ating
eqn. (A7), we obtain = i In (z - ih) + G(t),
lnB”(z,t)
where G is an arbitrary
function
(A91
of t, so that
=a[B”(2,t)]k,
z-yl
where 01is a function
(A101 of t. Similarly,
integration
of eqn. (A8) yields:
ap= prkt ar where fl is a function we obtain 7(q) =
(All) of t. Substituting
eqns. (AlO) and (All)
cY/3[rB”(z,t)]“.
into eqn. (12),
(AW
But
a+, g3”(*,t),
Q= a2 =
so that eqn. (A12) can be written
dq) = wk,
in the form: (A13)
where m = 2krrj3. Equation (A13) is the familiar power-law representation for the shear stress. We conclude that the velocity components u(,.) and u(,) can be written in the simplified forms given by eqns. (Al) and (A2) only when the shear stress/shear rate relationship is expressible in power-law form.
Fig. 12. A schematic representation of the buckling of (initially horizontal) material planes at some time after the release of the top plate. The effect is exaggerated for illustration purposes.
37
When eqns. (Al) and (A2) are not appropriate, the analysis developed in Section 2 has to be used. This allows for material planes, which are initially horizontal, to warp during the squeeze-film deformation. Figure 12 contains a schematic representation of this behaviour based on the results for the polymer solution considered in Fig. 8.