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**
Weissenberg’s Contributions
to Rheology**

**
PROFESSOR W. PHILIPPOFF**

*
Newark*
*College* *of* *Engineering,* *Newark,* *N.* *J.,*
*U.S.A.*

Karl Weissenberg’s ideas have had a profound influence on the development of rheology, as the author, who was associated with him in the 1930’s can testify. He was mostly vastly ahead of the accepted science of the time and arrived at his conclusions by an extraordinary intuition of the physical reasons of the phenomena, even if this involved the development of completely new concepts. However, his work is for the most part not widely known, as he published it in journals of limited circulation and further did not bother to put his conclusions into directly provable formulas. An outline of his work in Rheology will be attempted here.

In the first paper^{1} he introduced
the concept of elasticity in the steady flow of liquids to account for the
non-Newtonian viscosity of polymer solutions and used an energy balance
(triangular diagram) for the flow of “general” liquids: the viscous loss, the
potential (elastic) energy and the kinetic energy. The reasoning, as continued
through his whole work, was abstract and in terms of continuum mechanics,
without discussing how or why elasticity occurs at all in molecular concepts.

In the next papers^{2,3 }the flow of
cellulose acetate in dioxane was described and analyzed. The experimental proof
of the laminar flow of this material was given and a constant viscosity at low
rates of shear was found, both facts contradicting the ideas current at the
time. As a side derivation the “Mooney-Rabinowitsch” correction was developed,
which was apparently not considered to be important by him at the time, but has
been used ever since for the comparison, of capillary and rotational instrument
data.

In another paper^{4} the general theory
of the mechanics of deformable bodies was developed, in which the concept of
“mechanical equation of state” was introduced, now called “constitutive
equation”. He used the generalized Maxwell body (S-body) and Voigt body (P-body)
as well as their combinations (SP-body). A series of time derivatives of stress
and strain was used, taken up some 20 years later by Oldroyd ref.^{20}.
A conclusion of the presence of an elasticity in the liquid was the prediction
of the dependence of the viscosity measured in shear vibrations on the
frequency. This was subsequently found by Eisenschitz^{5} and Philippoff^{6}.
Now pursuant to the extensive work of J. Ferry and his school the elasticity in
vibrational tests is generally accepted, much more so than the one in steady
flow.

In another paper^{7} the general theory
was further elaborated, using to some extent extensions of the tensor calculus,
without bothering to give a proof. Especially the non-Newtonian viscosity was
shown to be the result of a finite angle between the principal stress and
principal axis of the deformation velocity. Here also the “Goniometry” was
emphasized, the distribution of stresses at a point in space.

The Weissenberg Effect: During the war years
the so called “Weissenberg-Effect” (the climbing of a visco-elastic liquid up a
rotating rod), was probably first empirically observed; it was his intuition
that connected it with elasticity and “normal stresses” in flow.
Phenomenologically this was described in papers ^{10,11,12} but the
detailed explanation was given in^{9} and especially in the more widely
known paper^{12}. In this last paper he also introduced the “recoverable
shear” in steady flow as an explanation for the effects. Much later this was
experimentally proved by the author.

The Weissenberg Postulate: In^{9} and^{12}
the now often quoted Weissenberg assumption or postulate P_{22} = P_{33}
was developed for any material in plane stationary laminar flow from completely
general symmetry conditions of motion: where there are equal velocity gradients
(say zero) there are equal stresses. This is the direct result of his completely
general discussion of stresses and strains at a point or “Goniometry”.

The Rheogoniometer: Parallel to the theory, S.
M. Freeman^{9 12 }and especially J. E. Roberts^{14} developed
the “Weissenberg Rheogoniometer” to a practically usable instrument, with which
Russell^{8} and Jobling^{15} together with Roberts conducted
extensive tests which proved the logarithmic distribution of pressures along the
radius, the validity of the formula for the total force and (Roberts) the
equality P_{22} = P_{33}.14 This proof, though presented as a
result in 1953^{16} has unfortunately never been accurately described in
a wide circulation publication.

A summary of Weissenberg’s work was presented
at a seminar at Columbia University in 1963^{17} but has also a limited
distribution. In it his general principles of coordinate transformations,
similarity and symmetry are discussed at length, also there is a detailed
discussion of the “postulate”. However, this publication does not include all
the former developments.

Besides the mentioned publications he has often influenced his collaborators, who published ensuing mathematical developments without expressly stating his authorship. Such cases are the mentioned Rabinowitsch-formula, Jobling’s expression for the total force in the cone-and-plate and also the exact formulation of the recoverable shear and normal-stresses. Only the “Effect”, the “Postulate” and the Rheogoniometer are generally associated with his name in Rheology.

**
CONCLUSION:**

An example of his intuition is that of elasticity: when he introduced it in 1928 nothing was known about its possible origin. Only much later, in the 1950’s, the theory of rubber elasticity was founded through the work of W. Kuhn, Treloar, Guth and James and others. This was more recently, nearly 30 years after the original proposal, adapted to flowing solutions by the author. Due to this unfortunate publication of his ideas and conclusions in not widely accessible periodicals, his work has not exerted its due influence on rheologists in the world. It is partly due to the difficulty in reading his papers that require an exceptional amount of thinking and meditation on the part of the reader: he does not write in an easily understood way.

His work must range with the classical papers
in deformational mechanics such as the, equally not widely known, works of E.
and F. Cosserat^{18} and J. Finger,^{19} that to some extent
have been reformulated in more recent developments.

It is to be hoped that now his life’s work should be summarized and published in a widely understood form to achieve its well deserved influence on modern Rheology.

**
REFERENCES**

1. Herzog, R. O. and K. Weissenberg,
*Kolloid-z.,* *46,* 277, 1928.

2. Weissenberg K., *Mitt-Staatl.*
*Material* *p.* *Asst.* *Sonderheft* *V,* 1929.

3. Eissenschitz R., B. Rabinowitsch and K. Weissenberg, ibid., IX.

4. Weissenberg K., *Abh.* *
Preuss.* *Akad.* *Wiss.* *Heft.* *2,* 1931.

5. Eisenschitz R. and W. Philippoff,
*Naturwiss,* *21,* *145,* 1933.

6. Philippoff W., *Physik-Z.,* *35,*
884, 900, 1934.

7. Weissenberg K., *Arcives* *Geneve*
*5,* *17,* 1, 1935.

Reiner M. and K. Weissenberg, *Rheology*
*Leaflet,* 1939 Nr. 10, 11.

8. Russell, R. T., *Thesis,* Imperial
College London, 1946.

9. Weissenberg K., *Conf.* *British*
*Rheologist’s* *Club,* 36 1946.

Freeman, S. M., ibid., 68, 1946.

10. Weissenberg K., *Nature,* *
159,* 310, 1947.

Weissenberg K., *Shirley* *
Inst.* *Memo.*
*XXI,* 203, 1947.

Chadwick G. E., S. E. Shorter and K. Weissenberg, ibid., 223, 1947.

11. Freeman S. M. and K. Weissenberg, *
Nature,* *161,* 324 and *162,* 320, 1948.

12. Weissenberg K., *Proc.* *1st* *
Rheology* *Congress* *Holland,* 1948.

Freeman S. M., ibid.

13. Weissenberg K., *Proc.* *Roy.* *
Soc.,* *200A,* 183, 1950.

Lodge A. and K., Weissenberg *Some* *
Recent* *Dev.* *Rheol.,* 129, 1950.

14. Roberts J. E., *Ministry* *of* *
Supply* *ADE* *Rep.,* 13, 1952.

15*.*
Jobling A., *Ph.D.* *Thesis,* Univ. Cambridge, 1956.

16. Roberts J. E., *Proc.* *2nd.* *
Congr.* *of* *Rheology,* Oxford, 91, 1953.

Weissenberg K., *Proc.* *Faraday* *
Soc.,* Oxford Meeting, 1959.

17. Weissenberg K., *Seminar* *at* *
Columbia* *University* *N.Y.,* 1963.

18. Cosserat E. and F., *Ann.* *fac.*
*Sci.* *Toulouse,* *10,* 1896.

19. Finger J., *Akad.* *Wiss.* *
Wien.* *Sitzber.* *Ila,* *103,* 1073, 1894.

**
Preface /
Acknowledgements /
Biographical Notes**

**
Weissenberg’s
Influence on Crystallography**

**
Karl Weissenberg
and the Development of X-Ray 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 Normal-Stress 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**