Karl Weissenberg  The 80th Birthday
Celebration Essays
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The Isolation of, and the Initial Measurements of the Weissenberg Effect
DR. R. J. RUSSELL
B.P. Research Centre, SunburyonThames, Middlesex. U.K.
1. INTRODUCTION
In the mid 1940’s it became necessary to establish design criteria for equipment to handle a class of nonNewtonian fluids which showed elastic aftereffects on discontinuing application of shear. The range of conditions for which these design criteria were required covered a very wide range of rates of shear under steady flow conditions and also embraced unsteady state flow conditions as exemplified by pumping through nozzles. These latter conditions received considerable attention since the pressure drops encountered were much higher than anticipated from the conventional nozzle discharge formulae, and a detailed rheological study was undertaken to establish the causes of the discrepancy. The progress of this study, which is the subject of this paper, lead to the identification of the Weissenberg effect in the flow of viscoelastic fluids.
2. CHARACTERISTICS OF FLUIDS TESTED
All the materials exhibited strong nonNewtonian flow characteristics, their viscosities, as determined in coaxial cylinder viscometers, decreasing very rapidly with increase of rate of shear. When an attempt was made to extend the upper limit of rate of shear by using capillary viscometers, large entrance and exit effects were found.
The variation of pressure with distance along the capillary showed that the end effects extended much farther into the capillary than would normally be expected. In addition, visual observation showed that the jet from the capillary exit always expanded to some extent.
There therefore appeared to be good reason to expect that the unusual flow effects encountered were a result of some form of elastic storage of energy. If this were so it was unreasonable to suppose that all the rheological behaviour that had been observed could be correlated in terms of a shearing stress/rate of shear relationship measured under steady conditions. Attention was therefore directed towards the quantification of the elastic phenomena observed and to ways of relating the elasticity effects with the behaviour observed in steady state flow.
3. PREVIOUS WORK
Experimentally, it had been found possible ^{ 1, 2, 3} to measure the relative moduli of elasticity of sols and gels using vibrational test apparatus. The determination of absolute values had proved difficult owing to geometrical effects in the apparatus. Theoretical studies due to Hencky ^{4} and Weissenberg ^{5,6} had indicated that the presence of elastic strains in liquids might result in significant decreases of apparent viscosity with increase in rate of shear. It therefore appeared that, provided reliable methods of measuring the elastic properties of the fluids under study could be developed, it should be possible to:
(i) check the theoretical deductions due to Weissenberg,
(ii) develop a more complete mechanical characterisation of the fluids than would otherwise have been possible.
4. EXPERIMENTAL PROGRAMME
The programme of work fell into two main sections:
(i) the measurement of rheological constants to characterise the mechanical behaviour of the fluids,
(ii) a detailed study of the phenomena exhibited by the fluids when studied under laminar flow conditions.
5. MEASUREMENT OF RHEOLOGICAL PROPERTIES
5. 1. Selection of Testing Procedures and Methods of Analysis
Weissenberg’s theoretical analysis indicated that under vibrational test conditions the vibrational frequency (expressed as angular velocity in radians/second) was equivalent to the rate of shear (expressed as reciprocal seconds) in stationary laminar flow. The analysis indicated that provided the imposed amplitude of the vibrational shear remained sufficiently small to ensure that the stress and strain tensors were essentially parallel to each other, the actual amplitude used in vibrational tests should not affect the frequency response of the liquid.
Alternatively, it was possible that the response of the liquid could be described in terms of the rate of shear (as for stationary laminar flow). In this case, since the displacement is given by the equation
Displacement=A f(r) sin wt
and hence the instantaneous rate of shear
where A = Amplitude
r = radial distance
f(r) = shape factor describing the change of amplitude with r
ω = angular velocity.
Thus the instantaneous rate of shear (and hence the average rate of shear) in vibrational testing is proportional to the product of frequency and amplitude. Hence studying the response of the liquid to variations of frequency and amplitude was used to test the applicability of the Weissenberg approach. With all the materials examined the stress developed in vibrational tests varied linearly with imposed amplitude but the response to frequency was far more complex. This was in accord with Weissenberg’s ideas, but not with the alternative hypothesis outlined above.
The complexity of the frequency response showed that several material constants would be required to fully describe the flow properties of the fluids. Therefore a general linear model of the form
was postulated where P,P’ etc., are the stress tensor and its time derivatives and S, Ś etc., are the strain tensor and its time derivatives.
Substituting (iw) for differentiation with respect to time in (1).
which enables the frequency response of the liquid to be defined in terms of a complex compliance r(=S/P) which is a function of frequency alone. Since the materials being tested were true liquids, b_{o}=0 and at low frequencies only terms up to the first derivative would be expected to be of importance. Under these conditions equation (1) reduces to
α_{o}/β_{1} P+α_{1}/β_{1} P'=Ś
i.e. the Maxwell equation of state, with the elasticity modulus (E) equal to b_{1}/a_{1} and the viscosity (h)= b_{1}a_{0}. At higher frequencies this simple description will be expected to fail and much recent work has been aimed to get descriptions for the higher frequency behaviour of viscoelastic fluids which are parsimonious in their requirements for material constants. Thus, the experimental validation of the Weissenberg hypothesis was best undertaken under low frequency conditions. This factor considerably influenced the development of the apparatus for the vibrational tests.
5. 2. Test Apparatus
The characteristics of the fluid and the desire to avoid the problem of analysing complicated shear motions dictated that the test apparatus should in principle consist of a cylindrical sample holder, which was vibrated about its axis. The elimination of end effects was achieved by arranging the cylinder with its axis vertical. The upper surface of the fluid was thus open to atmosphere and hence negligible forces were transmitted to it while end effects at the lower end were eliminated by floating the fluid on an inviscid liquid (water or mercury).
For this geometry of test apparatus the dependence of displacement on radial distance can be calculated in terms of frequency, fluid density and the complex compliance. Appendix 1 develops the relationship. Evaluating it for various ratios of the real to the imaginary part of the complex compliance shows that provided the imaginary part is not too large, the dependence of amplitude on radial distance gives a standing wave pattern in which blurred “nodes” are easily observable (see Figure 8, which shows that at a dimensionless radius of about 3.8 the amplitude is zero in the purely elastic case but increases with increase of the imaginary (viscous) component. By measurement of the amplitude and radius of the first antinode and the first node, it is possible to calculate both the real and the imaginary part of the compliance modulus. In the experimental apparatus the amplitude/radius dependence was found by placing a very thin line of fine aluminium powder on the surface of the fluid. Typical traces observed are shown in Figures 1 and 2. The range of the apparatus was extended to very low frequencies (where the radius of the first node exceeded the cylinder dimension) by adoption of a conicylindrical apparatus with a sufficiently small gap to reduce the inertia effects in the liquid to be negligible values and then measuring the relative amplitude and phase of the inner and outer cylinders as a function of frequency. Using these two apparatus allowed the dependence of the complex compliance on frequency to be followed over the theoretically important range.
All the fluids tested showed very similar variations of the compliance with frequency. Figure 3 shows typical curves for the real and imaginary components. The main features were:
(i) At low frequencies the real part of the compliance became independent of frequency while the imaginary part varied inversely with the frequency.
(ii) At high frequencies the real part decreased somewhat and then became constant. Over this frequency range the imaginary part ceased decreasing and became constant.
Thus the behaviour at low frequencies tended to that of a Maxwell body. Evaluation of the Maxwell viscosity was found to give the same value as calculated from very low rate of shear laminar flow tests. Weissenberg ^{5} had calculated the expected performance of a Maxwell body under laminar flow and these calculations had indicated that significant decreases of apparent viscosity should occur when the dimensionless group
hŚ 
E 
was of the order of unity. This prediction was confirmed for all the materials tested, h and E being derived from the vibrational tests while Ś was taken as the shear rate at which the first significant decrease of apparent viscosity occurred. At substantially higher rates of shear the Weissenberg prediction of the apparent viscosity were in general too low but at the equivalent vibrational frequencies the energy dissipated was higher than would have been obtained with the equivalent Maxwell body.
Fig. 1. Trace for liquid showing almost perfect elasticity.
Fig. 2. Trace for liquid showing significant viscous damping of elastic wave.
Fig. 3. Real and imaginary parts of complex compliance.
The close agreement between the observed laminar flow behaviour of the fluids tested and that predicted by Weissenberg’s theoretical work suggested that a study to determine if the basis of Weissenberg hypothesis. the rotation of the stress tensor towards the direction of the streamlines of flow as the rate of shear was increased, could be demonstrated. Any such rotation would cause the presence of at least one normal stress component when the measurement axes were defined as being orthogonal to each other with one axis being along the streamlines. The study therefore had as its main objective the development of a technique to first isolate and then measure the magnitude of any such normal component.
6. NORMAL STRESS MEASUREMENT TECHNIQUES
The main experimental problem involved in determining whether normal components existed was the likelihood that, if the measurement was attempted by any form of pressure gauge, the isotropic hydrostatic pressure component would change so that it exactly compensated the normal stress component developed due to shear. Although in principle such a change could be measured since it would cause a volume change of the liquid, in practical terms the volume change was expected to be so small as to be undetectable. Indirect methods of measurement were therefore sought and the most promising method appeared to be the study of the effects caused by curvature of the streamlines of flow. It appeared that a normal component of stress acting along the streamlines of flow (the P_{11} component) should act as a hoop stress and hence cause a buildup of pressure towards the centre of curvature.
7. TEST APPARATUS
The simplest technique to explore the postulated effect was to shear the fluid between two parallel discs, one stationary, the other rotating, as shown in Figure 4. The first apparatus built for the purpose is shown in Figure 5, which shows the two discs being forced apart by the hydrostatic pressure developed between them. This apparatus gave the first demonstration of the Weissenberg effect. It proved conclusively the existence of a normal stress component along the streamlines of flow and thus demonstrated the essential validity of the Weissenberg hypothesis.
Fig. 4. Arrangement to measure normal component acting along steamline (11Component),
Fig. 5. Apparatus giving first demonstration of Weissenberg effect.
To determine whether normal stress components at right angles to the streamlines (P_{22}) existed, the radial distribution of stress on the plate was studied. To facilitate interpretation of the results, the simple parallel plate apparatus was replaced by a cone and plate system, shown in Figure 6, in which the pressure distribution was measured by a set of manometers. The pressure distribution was described by the relationship:

Pr 
= 
K log R/r 
where 
Pr 
= 
Pressure at radius r 

R 
= 
outside radius 

K 
= 
constant 
This distribution is the one to be expected to arise from the action of a P_{11} component of stress with a zero P_{22} component. No direct method for measuring the third component of stress (P_{33}) was developed but no phenomena were observed which suggested that this component was other than zero. From a general view point it appears unlikely that the third normal stress component (P33) should be other than zero when the second component is equal to zero.
Fig. 6. Measurement of pressure distribution due to normal stress.
It was therefore concluded that the stress tensor developed during the shearing of viscoelastic fluid had the form:
P_{11} 
P_{12} 
0 
P_{12} 
0 
0 
0 
0 
0 
Breaking the tensor into a deviatory and an isotropic part, i.e.
⅔ P_{11} 
P_{12} 
0 

1 
0 
0 
P_{12} 
⅓ P_{11} 
0 
+P_{11}/3 
0 
1 
0 
0 
0 
⅓ P_{11} 

0 
0 
1 
suggests that the action of shearing increases the isotropic pressure. Hence the isotropic pressure should increase at points of high shear. This deduction is in line with observations made during vibrational flow measurements in the cylindrical apparatus where it was observed that at the nodes (the points of highest shear) the equilibrium liquid surface was slightly lower than at the antinodes. It also goes some way to explain the high nozzle pressure drops since the high shear rates within the nozzle would develop a hydrostatic pressure opposing flow.
It was found possible to obtain general relationships involving the normal and shear components of the stress tensor by plotting the dimensionless groups
P_{11}/E and P_{12}/E
against the group η Ś/E. As shown in Figure 7 the correlations were valid over a wide range, although at the higher values of η Ś/E the effects of the different higher frequency behaviours was becoming significant.
When η Ś/E was less than 1 the relationships tended to the values given by:
P_{12}/E  η Ś/E
P_{11}/E = [η Ś/E]^{2}
Thus at low rates of shear the P_{11} component became progressively less important and the behaviour tended more and more to that of a normal viscous liquid.
8. CONCLUSIONS
The work described is considered to provide experimental evidence on the validity of the Weissenberg hypothesis on the effect of elasticity in laminar flow. It indicates that Weissenberg’s early theoretical work ^{6}, which analysed the effects to be expected from the rotation of the stress tensor when large elastic strains were developed on shearing a viscoelastic fluid, required a minor extension to make it fit experimental observation. This was the inclusion of a normal stress component acting along the streamlines of flow.
Fig. 7. Dimensionless stress components vs Dimensionless rate of shear.
Fig. 8. Amplitude/Radius dependence for cylindrical apparatus.
9. ACKNOWLEDGEMENT
The author would like to express his thanks for the advice and encouragement of Dr. K. Weissenberg during the course of the work and also Dr. H. R. Fehling and P. O. Rosin for their support for the study.
10. REFERENCES
1. Freundlich, H. and Seifriz, W., Zeitsch. Physiol. Chem., 1922, p. 233.
2. Madelung E. and Flägge S., Ann. Physik, March 1935, p. 209.
3. Philippoff W., Physicalishe FS, 85, 1934, p. 883 and 900.
4. Hencky H., Ann. Physick, Ser, 5, 1929, p. 675.
5. Weissenberg K., Die. Mechanik deformberbarer Korpeör. Abhg. d Preuss Acad. d Wissenschafter 1931, Vol. 2.
6. Weissenberg K., Ach. des Sci. Phy. et Nat., Vol. 17, 5.
APPENDIX
CALCULATION OF AMPLITUDE/RADIUS DEPENDENCE FOR VIBRATION OF CYLINDRICAL VISCOELASTIC BODY
The differential equation controlling the motion is obtained by balancing the shearing forces due to viscoelasticity against the inertia forces developed by the vibrational motion. This gives the equation:
(1) (2)
The solution of (2) gives the displacement (D_{r}) at radius r as:
(3)
since the displacement is zero at r=o.
(4)
The maximum amplitude at radius r is given by:
(5)
Evaluation of F (rm, n/_{m}) gives the variation of amplitude with radius. Figure 8 gives the calculated relationship and shows that for small ratios of n/_{m} an easily characterisable standing wave form is developed.
LIST OF SYMBOLS
Preface / Acknowledgements / Biographical Notes
Weissenberg’s Influence on Crystallography
Karl Weissenberg and the Development of XRay Crystallography
The Isolation of, and the Initial Measurements of the Weissenberg Effect
The Role of Similitude in Continuum Mechanics
Elasticity in Incompressible Liquids
The Physical Meaning of Weissenberg's Hypothesis with Regard to the Second NormalStress Difference
A Study of Weissenberg's Holistic Approach to Biorheology
The Weissenberg Rheogoniometer Adapted for Biorheological Studies
Weissenberg’s Contributions to Rheology
The Early Development of the Rheogoniometer
Some of Weissenberg's More Important Contributions to Rheology: An Appreciation
Publications of Karl Weissenberg and Collaborators / List of Contributors
© Copyright John Harris