2.1 The Beginning of Experimental Fluid Rheology

Much of the previous chapter has dealt with problems arising from solid rheology, in particular from fibre mechanics. We now need to address the beginnings of serious experimental fluid rheology. Some of the relevant background in Newtonian flow has already been discussed in the previous chapter, but it is important for us to highlight the innovative experimental work of Hagen (1839), a civil engineer, and Poiseuille (1835, 1846), a physician, both of whom studied flow in small-diameter tubes for different reasons; Hagen was looking for basic hydraulic information and Poiseuille was interested in blood flow. The experimental technique of Poiseuille was exemplary, and is still worthy of study. The same can be said of the seminal work of Couette1

The work of Hagen, Poiseuille and Couette set the stage for the 20th century preoccupation with non-Newtonian liquids. At this time, non-classical behaviour was most clearly observed through the variation of viscosity with shear rate. The first experiments that unequivocally demonstrated a non-Newtonian viscosity varying with shear rate seem to be those of Theodore Schwedoff, who was Dean of Sciences in Odessa in the late 1880s. He used a thin-gap Couette-type apparatus, and was therefore able to infer shear stress and shear rate fairly accurately. Two of his papers appeared in 1889 and 1890 in the Journal de Physique. In the first, he described concepts of viscosity and springiness that he sought to measure; he also dealt with fluids that showed a yield point, such as weak gelatine gels. He found that the material relaxed after a step strain, and he tried to use Maxwell’s ideas to describe what he saw. In his second paper, Schwedoff (1890) concentrated on viscosity. He concluded: “(the viscosity) is not constant, as one usually supposes: it varies with the speed of shearing”. Since he was dealing with gel materials that had a yield stress, he also saw that the viscosity tended to an infinite value as the shear rate went to zero. He was able to show a constant viscosity for glycerine, but using a 1% gelatine solution, there was a seven-fold variation in viscosity as the shear rate varied by about 15:1.

To describe the results, Schwedoff invented the generalized Maxwell model for the shear stress σ :

(2.1)

Here σy is a shear yield stress. We note that J G Butcher (1877) invented the terms “elastico-viscous” and “elastico-plastico-viscous”, of which (2.1) is an example, to describe material behaviour; the latter term is, happily, now rare.

The 1890 paper of Schwedoff was the forerunner to a multitude of papers on variable viscosity effects in a plethora of materials which were to occupy much of the literature of the first half of the 20th century (e.g. Hatschek 1913). According to Scott Blair (1938), there was a tendency to label all anomalous behaviour as manifestations of ‘plasticity’, with no clear idea as to what that meant.

In 1922, Bingham published an important book entitled “Fluidity and Plasticity”. It contained a mammoth 82 page bibliography and provided a great deal of information on measurements for various systems, including gel-like materials with a yield stress. It also included an important discussion on ‘wall-slip’ in viscometers. Strangely, the book does not have a very lengthy discussion on variable-viscosity systems, which probably reflects the mood at the time of writing.

This mood had to change, and it did so as a result of rheometrical experiments on a multitude of different materials. So, by 1939, Scott Blair, in the March issue of ‘British Plastics’, was able to report that “Rheological methods are in practice applied to an enormous variety of materials. Among the more obvious applications are the following: metals (elastic and plastic properties), oils, paints and varnishes, bitumens, tars and pitch, gums, inks and starches, flours for bread and biscuit making, cheese, butter, cream and milk, tooth pastes, soaps and toilet creams, ceramic clays, dyestuffs and fibres of all kinds…” (see also Scott Blair 1938). In order to classify the plethora of possibilities in a simple way, the British Rheologists Club published a working chart on rheological behaviour (Nature 149, 1942, p702) and invited further comments. Figure 2.1 contains a response from L Bilmes (1942) in the form of a circular chart.

References to many of the more relevant contributions to the expanding literature are provided in the books by Houwink (1937), Scott Blair (1938) and Philippoff2 (1942). The latter is a remarkable book, first published by Steinkopff in Germany during the Second World War and reprinted two years later at Ann Arbor in the United States. The book is an important source of reference on the state of colloidal rheology in the early 1940s. We note that the often misunderstood term ‘thixotropy’, which we discuss in Chapter 7, was invented in this period.

With the increasing interest in fluid rheology came the need to describe nonlinear fluids in an appropriate manner and a large number of empirical models describing the observed shear stress-shear rate behaviour were invented. This was essentially a curve-fitting exercise, and, to be attractive, the models had to contain as few constants as possible.

Most notable amongst the growing list of available models were the Bingham (1922) model, the Ostwald (1925)-de Waele (1923) power-law model, the Herschel-Bulkley (1926) model and the Ellis and Williamson models (see Houwink 1937, Tanner 1988). These are still important today in giving a compact, manageable description of shearing flow. One could say that if variable viscosity and linear viscoelasticity were the sum total of rheology, then it had all been done by 1930.

The empirical shear stress-shear rate models were used to derive the relevant flow rate-pressure drop relationships for capillary flow, and these could then be compared with experiment. However, it was soon realized that a procedure was needed to solve the important inverse problem, i.e. given a set of flow rates Q and pressure drops Δp from a given capillary, what shear stress σ - shear rate curve does this imply (if one exists)? This question was successfully addressed by Weissenberg, the provenance of the solution being unexpected, since, at the time (1928), Weissenberg was working at the Berlin crystallographic laboratory of R O Herzog at the Kaiser-Wilhelm Institut für Faserstoffchemie Berlin-Dahlem, which was not primarily interested in fluid mechanics. Weissenberg’s interest in these matters can be traced to a paper with Herzog (1928) entitled “On the thermal, mechanical and x-ray analysis of swelling” which was published in Kolloid-Zeitschrift. It is a remarkably wide-ranging paper and sets out a programme of research on the relations of material and molecular structures to thermal and mechanical properties. Whilst this was clearly only a preliminary statement of a vast programme of work, it was recognized that material behaviour might be described as an interplay of kinetic, thermal and elastic energy, and a triangular diagram with vertices labelled with these three forms of energy was displayed. In a surprising switch from this very general scene to a particular problem, on their p281, the first general analysis of the inverse capillary flow begins.

In this analysis, it is assumed that the velocity gradient (dw/dr) is a function of the shear stress, i.e.

(2.2)

Then follows their equation (2), which states that the volume discharge rate Q, the capillary radius R, length L and pressure drop Δp are related by

(2.3)

where

(2.4)

We are unable to see how this clearly incorrect formula was derived.

The correct inversion formula was published twice in 1929. The first publication, submitted on March 14, 1929 was authored by Eisenschitz, Rabinowitsch and Weissenberg (1929), but before it appeared Rabinowitsch (1929) published again (August 1929) in the Zeitschrift für Physikalische Chemie, a more accessible journal. However, both Rabinowitsch, in his 1929 paper, and Eisenschitz, in a 1933 paper, acknowledge unequivocally that the inversion method was due to Weissenberg. This brilliant analysis will bear repeating.

If w(r) is the axial velocity distribution with respect to a cylindrical polar coordinate system, we have

(2.5)

Recognizing that the fully-developed wall shear stress σw is given by

(2.6)

(2.5) can be written, after an integration by parts,

(2.7)

A crucial step in the Weissenberg analysis is the assumption of no slip at the wall, so that

(2.8)

in which case the first term on the right hand side of (2.7) vanishes. (Later, Mooney (1931) generalized the...