Karl Weissenberg - The 80th Birthday
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A Study of Weissenberg’s Holistic Approach to Biorheology
DR. G. W. SCOTT-BLAIR
Grist Cottage, Iffley Oxford, England
1. WEISSENBERG’S PHILOSOPHY
I hope that my old friend Karl Weissenberg will not mind my describing his scientific philosophy as “holistic” without first consulting him.
I do not imply, of course, that Weissenberg is necessarily a disciple of General Smuts, who coined this term (though Smuts had one of the most acute minds that I have found in anyone with whom I have discussed physics). Still less need we suppose that he is a follower of the Gestalt School, which, unknown to Smuts, was developing such similar ideas as his own in far-away Germany.
Yet I cannot refrain from quoting one passage from a lecture by Max Wertheimer,1 given before the Kant Society in Berlin in 1924 (is it even possible that Weissenberg was present?), in which he said: (The italics are in the original) “Is it necessary that all mathematics be established upon a piece-wise basis? What sort of a mathematical system would it be in which this were not the case? There have been attempts to answer the latter question but almost always they have fallen back in the end upon old procedures…..It is not enough, and certainly does not constitute a solution of the principal problem if one shows that the axioms of mathematics are both piece-meal and at the same time evince something of the opposite character. The problem has been scientifically grasped only when an attack specifically designed to yield positive results has been launched. Just how this attack is to be made seems to many mathematicians a colossal problem, but perhaps the quantum theory will force the mathematicians to attack it.”
The term “holistic” is not a pun: it comes directly from a Greek word meaning “a whole” and Weissenberg is one of the first rheologists to view rheology as a part of the “whole” of modern physics, including Relativity and the Quantum Theory. (I believe that Umstätter2 introduced Relativity into his theory of “Stukturmechanik” but I cannot find the exact reference and, in fact, I never understood his work.)
Weissenberg, however, has the gift of extreme clarity of exposition and a few quite short quotations from his papers will show how he has always viewed rheology and biorheology within the context of the whole of modern physics.
He told me once that he had asked Einstein for advice in advancing his career as a mathematical physicist; and Einstein had said “design and make experimental apparatus”.
Weissenberg was most successful in taking this advice. The apparatus he invented in another branch of physics before he left Germany may perhaps be described elsewhere in this book. In rheology, his “rheogoniometer”, now existing in many modified forms, opened a new chapter in experimentation. I remember wondering, when the name first appeared, what it had to do with “goniometry” - the measurement of angles. The point is, of course, that it measures components of stress, strain and strain-rate “at all angles”. These components can be resolved in the form of tensors, thus making possible the identification of invariants.
Of course these tensors were well-known before the time of Weissenberg. (In early times, I proposed that the motto for the British Society of Rheology might well be Omne in Amore. We were so often told “It’s all in Love!”3).*1 But it was Weissenberg and that other doyen of rheology (whose eightieth birthday we celebrated six years ago), Markus Reiner, who introduced tensor theory to many young, and some not so young, rheologists.
It is in a passage in his article on the rheology of blood in the book published in honour of Markus Reiner,4 that Weissenberg perhaps most clearly explains his “holistic” approach. This passage is worth quoting at some length.
“Once the instrumentation has been established one can make quantitative observations, and then face the second task. This consists in establishment of ‘law and order’ by incorporating the observations into the mathematical framework of a theory…….One can no longer accept the classical point of view according to which the system under consideration is observed undisturbed and in isolation, the observed data having the significance of exact and absolute characteristics. Instead, one has to realize that every quantitative observation is the result of an interaction between the observed system and the observing one at the very moment when the interaction takes place…….The inevitable interaction of the observed and the observing systems and the attendant uncertainty and relativity of the data have already been fully recognized as basic concepts in all modern theories of physics and it is important that this recognition should be extended throughout all natural sciences including chemistry, biology, and last but not least, haemorheology……” *2
“This research tool [the mathematics of quantum theory and relativity] dealt separately with uncertainty and relativity, using the quantum statistical theory of probability waves for the former and the group theory of transformation for the latter……Among the invariants so constructed, the quantities of matter and energy play a prominent part, so that the development of a useful theoretical scheme for circulatory problems in the living body requires tracing of the movement, through space and time, of each parties of matter and each portion of energy for the intake of food, drink and air to the discharge of excretions and waste products. Another invariant known as the ‘geodetic’ then serves as a guide to the path through space and time taken by the matter and energy, however complicated this path may be because of the presence of mechancal, electromagnetic and osmotic forces.”
“It seems likely that a step in the right direction would be made by describing the flow in terms of probability waves (as in wave mechanics) and using the group theory of transformation to find the invariants in the maze of movements and to trace the geodetic pathways in the flow patterns.”
A similar line of thought was developed in Weissenberg’s contribution to the 4th International Congress of Rheology.5 In this article, which I shall not quote here verbatim, the analogy, not only with quantum theory, but with relativity is more strongly stressed. After all, the whole purpose of Einsten’s work, particularly in the “Special Theory”, lay in the requirements that there should be no “privileged” observers. Under extreme conditions, this meant modifying the classical laws of dynamics.
Yet, by analogy with the quantum theory, we see that in biological research, even for above the quantum level, measurements cannot be made without changing the system which is being studied, thus making the observer an essential feature of the measurement. Perhaps it was this paradox that made Einstein doubt, even until the end of his life, whether the current philosophy of the quantum theory could be final. The observer must be included in the observation.
2. RAYS TO RHEOLOGY
Historically, it is interesting to see how Weissenberg developed his ideas, through his work on X-ray analysis, to thermodynamics and, during the years when he had the privilege of having him with us in this country, to rheology and biorheology.
One of his first “linking papers” was written from Germany.6 “On the thermal, mechanical and X-ray analysis of swelling.” (By chance, the very next paper following this is Freundlich’s classical article “On Thixotropy”.) His concern with rheology followed quickly after he came to England. Older rheologists well remember his early lectures on the strange behaviour of many complex materials. This was illustrated by quite simple experiments, often demonstrated by his admirable partner, S. M. Freeman.
The experiments could be appreciated by any intelligent schoolboy - but the theory, especially of the normal stress effect which was afterwards given his name, involved considerable mathematical analysis and debate among many of the leading rheologists, notably Rivlin, Reiner and Pryce-Jones. (This last always called the rising of a visco-elastic material up a rotating rod, “the Mae West Effect”.)
In the early stages of his rheological career, Weissenberg stressed the importance of numerics which would describe the balance between the viscous and elastic components of strain in a viscoelastic system. (In a personal letter, Weissenberg quoted to me quite a large number of papers in which these numerics were discussed.)
Metzner, White and Denn7 have claimed that the most important numerics in rheology are (a) Reynolds’ Number, which gives the balance between viscous and inertia forces in flow, (b) Reiner’s “Deborah Number”, which compares the natural time-scale of a material with an experimental time and (c) Weissenberg’s Number, which gives the balance of elastic and viscous forces. (In all these cases, one is dealing in general with a whole of series of “numbers” - not just one.)
Weissenberg never forgot his link with Einstein. There is, of course, a formal parallelism between the method of making it possible to add a function of time to a series of lengths (by multiplying by the greatest possible speed of communication and introducing “i”) and that by which viscosities may be converted to a form which may be added to a “real” elastic modulus by multiplying by a frequency and again introducing “I” *3. But the real connection between Einstein’s philosophy and Weissenberg’s rheology is more fundamental. The reader is recommended to study the whole section on “Modern Ideas and Concepts on Research.” in his contribution to the 4th International Congress on Rheology: only the final paragraph will be quoted here: “According to Einstein, the development of methods of research which would lead to the best fit between theory and experiment should be the responsibility of the mathematical physicist because his training in both mathematics and physics should enable him to coordinate the requirements of the theoretician with those of the experimentalist in the formulation of the principles, design of instrumentation, selection of the variables to be measured, choice of the calibration laws of the scale divisions of the measuring instruments, and last but not least in critically reviewing the evidence offered by the data obtained from quantitative observations.”
3. ON TO BIORHEOLOGY
As Weissenberg pointed out in this article, the application of rheology to biorheological systems is by no means new. Nevertheless, it is only in recent times that extensive work has been done in this field. It is perhaps not surprising that by far the greater part of this work has been in the field of haemorheology. Blood is the most conspicuously “flowing” material in the body, its conditions of flow are extremely diverse, its coagulation in vivo constitutes the most frequent cause of death in men in our country and, though fortunately much less frequently, its failure to coagulate in wounds can also prove fatal. The current interest in the subject may well be illustrated by the fact that three books have recently appeared on haemorheology 8, 9, 10 and that in one of them10 there are no less than sixty-six pages of references. There are also Proceedings of various Congresses and Conferences dealing with the subject published recently.
The flow of blood in the body follows many different patterns. There is much current debate about the incidence of turbulence. It used to be thought that this occurred only in the very large vessels, such as the aorta, and even that “heart murmurs” were caused by it. It is now known that this is not the case. However, blood vessels are not cylindrical and little is known about the values of Reynold’s numbers at which turbulence starts in vessels of elliptical or more complex cross-sections. Indeed the whole question of the relationship between Reynold’s numbers and the onset of turbulence is at present subject to controversy.
It is indeed not unlikely that at points of junction between vessels there is local turbulence (see ref. 10). In many vessels, streamline flow occurs, at least approximately. But the presence of the so-called “plasmatic zone”, probably first observed by Malpighi in 1686 (see ref. 5) and definitely observed by Poiseuille in the eighteen thirties, was first studied quantitatively by Fahraeus and Lindquist11 in 1931. The region nearest the vessel wall is not altogether free of corpuscles but there is a definite tendency for them to migrate towards the centre of the vessel. This gives rise to an apparent fall in viscosity in very narrow vessels, a phenomenon first observed by Bingham and Green12 for paints and studied in detail by the present writer and his colleagues for clays and soils at about the same time as Fahraeus was working on blood. Schofield and Scott Blair13 called this “the sigma phenomenon” but could not find any adequate explanation for it. (Later, Dix and Scott Blair14 found an explanation for the sigma phenomenon in very thick soil pastes, but I have never felt that this should be applied to such systems as blood.)
The complexity of the problem is seen when it is remembered that, unlike the flow of a Newtonian liquid through a capillary viscometer, in the case of blood in vivo, we are dealing with a non-Newtonian suspension of non-spherical deformable particles flowing in a non-cylindrical, non-rigid permeable tube of uneven bore! This makes the “rules” about as difficult as those of Alice in Wonderland’s game of croquet!
Even with a model system of rigid spherical particles suspended in a liquid of the same density, Segrè and Silberberg15 found not only that particles near the wall of the tube moved in towards the centre, but that particles near the centre moved outwards towards the walls. This is known as “the pinch effect” (not to be confused with the same term used in nuclear physics).
Though not immediately confirmed by other workers, later experiments showed that this was because the effect occurs only within a limited range of Reynolds’ numbers.
Reasons were already known for the inward motion of non-spherical and / or deformable particles: some suggestions have been offered to account for this in the case of rigid spheres, but the reason for their inward motion is still under discussion.
4. THE RHEOLOGY OF BLOOD
The viscosity of blood and its continuous phase (plasma) is generally measured extracorporeally. The only practical method of measuring blood flow in vivo is by means of an electromagnetic flow-meter, originally designed by Kolin.16 The technique has, of course, been considerably improved since then. (See Wyatt17).
Many studies have been made on blood removed from the body; but here there is a real difficulty. When blood comes into contact with any material other than the interior surface of a blood vessel, it starts to coagulate and its rheological properties change. This makes work with “native”*4 blood difficult. Copley18 has proposed that the internal surfaces of the vessels are coated with a layer of fibrin-like material. The permeability of this is homeostatically controlled and when this control fails, the vessel may become either hardened (sclerosis) or permeable as the case may be. There is some visual (microscopic) evidence for such a layer and it seems that, if a glass capillary is lined with fibrin, the flow of anticoagulated blood is slightly accelerated.
Much work has been done on the elastic properties of blood vessels and on the complex conditions of “pulsatile flow” produced by the heart-beat. This is well summarized in an article by Bergel and Schultz.19 The addition of anticoagulants, especially the much-used heparin, produces very complex changes. Even such compounds as sodium citrate or oxalate, interfere with the ionic balance. One can never be sure, therefore, that the results of such extra vivum*5 tests really reflect conditions in the body.
(Although I had no opportunity to complete the experiments, preliminary tests showed that passing the blood through a suitable ion exchange resin gave by far the best results. I am indebted to Dr. S. G. Rainsford for suggesting this method.)
At low shear-rates, blood is non-Newtonian and I showed some years ago, that Casson’s20 plot of the square roots of stress and sheat-rate gives good straight lines.21 This equation is now widely used and holds well except at low and high rates of shear. But it also appears that another much older equation, that proposed by Herschel and Bulkley22 holds just as well over this range23 and a possible, though simplified model has been proposed to explain this, such as does not exist for the Casson equation, especially if applied to bovine blood which does not form rod-like structures of red cells (“rouleaux”). There is evidence that at very low stresses, blood does not flow at all, i.e. it has a yield-value.
But, in the body, there are many other conditions of blood-flow. For example, there is the pulsatile flow of arterial blood. The equations of such flow are very complex and are to be regarded as belonging to haemodynaniics rather than to rheology.
In the microcirculation, the capillaries are often smaller than the normal diameter of the erythrocytes (red cells). These must therefore be deformed in order to pass along the capillaries.
In an ingenious set of experiments, Katchalsky et al24 showed that, comparing the production of haemolysis (rupture of cells) at different rates of change of ambient osmotic pressure, erythrocytes must behave like a “Kelvin model”; i.e. they will deform if compressed slowly but rupture if stressed suddenly.
A somewhat complex type of flow, known as “bolus flow” has also been studied. (For references see 8.) This is caused by the viscous drag of the fluid in the spaces between the erythrocytes, which flow along the vessel in a slightly curved form, lying across, and almost filling the vessel. Between them, there are eddies of plasma.
5. PATHOLOGICAL BLOOD CONDITIONS
From this very brief account of some of the problems of blood flow, it will be clear why Weissenberg adopts a holistic approach to the problem.
Before discussing coagulation, brief mention should be made of the pathological significance of blood viscosity. In what was perhaps the earliest book on this subject, Nesfield25 points out that the age-long habit of bleeding patients for almost every complaint, though it doubtless killed many people, may have been helpful in quite a few conditions. It is the case that in many pathological conditions, there is present what Dintenfass10 has called “the high-viscosity syndrome”. The most striking instance is Raynaud’s disease, in which very high viscosities sometimes occur. (The same has been claimed for multiple sclerosis, but my own experience has shown that, in the active phase, although viscosities are sometimes very high, in others, equally severe, they may be very low, though these latter cases may well be associated with anaemia.)
Although high blood viscosity is not the prime cause of cerebral infarcation or atherosclerosis, it is commonly associated with these conditions. A correlation has been found between hypertension and high blood viscosity.
Knisely,26 in a long series of papers, has claimed that “in perfect health” (sic) erythrocytes, since they bear similar electric charges, do not adhere to one another, whereas in any pathological condition (including normal pregnancy!) they tend to form clumps. It is true, of course, that at the point of death “sludging” of blood is a common phenomenon. But this “clumping” or “sludging” came to be badly confused with the formation of rouleaux and with the fact, independently discovered and studied by Fahraeus in Sweden, that a high “erythrocyte sedimentation rate” (ESR) generally indicates the presence of some fairly serious pathological condition.
It may seem paradoxical, that Harkness et al27 has found that very similar information may be obtained with less danger of subjective errors by measuring the viscosity of plasma in a capillary viscometer. A high viscosity is associated with a high ESR, the opposite of what one might at first sight suppose.
Since thrombosis is now responsible for the largest number of male deaths in Western Countries (especially coronary), the coagulation of blood is of great clinical importance. As already stated, when blood comes into contact with any foreign substance, a series of reactions is initiated. This ends in the formation of thrombin, which reacts with the fibrinogen in the blood to form fibrin. The fibrin then polymerizes into a “clot”.
Although a neat list of some dozen reactions has been proposed, recent work has shown that the whole process is vastly more complex and it is not yet fully understood.
It is true that an almost total absence of any one of the “clotting factors” will prevent anything like normal clotting, but serious deficiencies in all but two of these factors are fortunately either extremely rare or unknown. The two less rare deficiencies, haemophilia (factor VIII) and Christmas disease (factor IX) can now be controlled by giving the patient the appropriate factor. Both these conditions (but not all the still rarer deficiencies) are inherited by the male through the female, but of course the “chain” must be started by some chance mutation. The famous Hesse series, which included the case of the heir to the throne in Imperial Russia before the Revolution, probably started with Queen Victoria.
Far commoner than the failure of blood to clot when it should, is its clotting within a vessel. There are various types of thrombus, but the commonest is started by the linking together of platelets. These, though the smallest of the blood corpuscles, are living organisms and they throw out “arms” (pseudo-podia) which link them together. Behind this barrage of platelets, comes a “white thrombus” of large leucocytes, and behind them again the erythrocytes collect, until the whole vessel is blocked.
When a patient recovers from a thrombosis, a repetition is discouraged either by giving anticoagulants (which must be done with great care to avoid bleeding) or by giving drugs which will encourage the enzymatic breakdown of the fibrin clot (“fibrinolysis”). This process, which occurs physiologically following the healing of wounds, also involves another complex and not fully understood series of reactions. It is greatly accelerated in certain pathological conditions, e.g. some diseases of the liver and prostate cancer.
During the later stages at least in the polymerization of the fibrin, the clot has quite simple rheological properties behaving approximately as a Maxwell model: a dashpot and a spring in series.28 The complex modulus of the coagulating blood has been measured by the present writer, both with a specially designed rheometer and with the well-known Hartert Thombelastograph. Many tests have been done over a period of some twelve years in both bovine and human blood, in the latter case including a variety of pathological conditions.
The same equation has invariably been found to hold within the limits of experimental error. Although it is a simple exponential function, it does not appear to have been used before. Many papers have been written in the course of this work: only the last need be quoted.29 The argument, even in this paper, can be further simplified since the two postulated equations need not themselves be exponential.
6. OTHER BIOLOGICAL MATERIALS
Biorheology is, of course, not restricted to blood. Many other body fluids, as well as such structures as bone, skin, muscle have been studied. This volume is dedicated to Karl Weissenberg and he developed his interest in biorheology too soon before his retirement to have time to turn his attention to these other materials. It is quite natural that blood should have had his first consideration.
I shall therefore conclude with only a brief account of the practical significance of a few current investigations in other fields. Much work was done some thirty years ago on the properties of both bovine and human (uterine) cervical mucus. This mucus shows marked elastic properties at about the time of ovulation and simple instruments have been designed to measure these properties and so to determine the time of ovulation in women and of “heat” (which immediately preceeds ovulation) in the cow. The consistency of the mucus also increases in pregnancy and, since hormonal tests do not work for cows, a measure of this consistency gives the earliest indication of pregnancy.
These methods have not been widely used, partly because great care must be taken in extracting the sample from the cervix and, in the case of bovine pregnancy, because so many foetuses appear to be lost between the 28th day (when the test should operate) and about the 40th (by which time the foetus can be detected by rectal palpation) that the test has seemed hardly worth while in practice, except for research purposes. However, the prevalent use of contraceptive pills may well revive interest in this field. The viscosity of semen is also important in relation to male fertility.
Recent work on bronchial mucus, especially by Professor Lynne Reid and her colleagues at the Institute of Diseases of the Chest in London, has shown interesting relationships between the consistency of the mucus and the pathology of the patients. Vanous substances are used to make coughing more easy.
Another group of diseases that is very prevalent in this country comprises the rheumatic complaints. These are, in general, caused by unwanted friction at the joints. Nature’s lubricant, the synovial fluid, is not very efficient as lubricants go, but its replacement by artificial lubricants has not been altogether successful. In earlier times, no correlation was found betwen the rheological properties of synovial fluid and the type of rheumatic disease; however, with improved instrumentation, it appears that such correlations exist. But Wright and his colleagues at Leeds have also studied the friction between the surfaces of the bones at the joints. This research therefore falls within the range of “Tribology” (the study of friction and wear).
Many other body fluids have been studied, including sputum, intraocular fluid, cerebrospinal fluid etc. It is perhaps strange that very little appears to have been done on the progressively changing rheological properties of digesting foodstuffs in their course through the alimentary tract, as mentioned by Weissenberg, it is apparent that very wide fields await investigation within the range of biorheology.
Karl Weissenberg entered this field towards the end of his research career but, when the subject has been much more widely developed, I feel sure that his name will be remembered as a pioneer who linked this new branch of rheology with the rapidly developing fields of modern physical thought. May he live to be a hundred!
*1I cannot recall whether Love himself used tensor notation, but the principles were there.
*2A term introduced by Prof. A.L. Copley to include the rheology of blood, its components and vessels.
*3Compare the Lorentz equation: ds2=dx2+dy2 +dz2-c2 dt2 with the equation for a sinusoidally strained viscoelastic system: G*=G’+ i G”.
*4”Native” means untreated with anticoagulants.
*5 Better Latin than “ex vivo”.
1. Translated from the German in Ellis, W. D., “A Sourcebook of Gestalt Psychology”, Kegan Paul, London, 1938.
2. Umstatter, M., “Strukturmechanik” Steinkopf, Dresden and London,1948.
3. Love, A. E. H., “A Treatise on the Mathematical Theory of Elasticity”, Cambridge Univ. Press (2nd Edn.), 1906.
4. Abir, D. (Ed.), “Contributions to Mechanics”, Pergamon Press, Oxford,1969, p. 437.
5. Weissenberg, K., Proc. IVth Intern. Congr. Rheol. Part 4 (Ed. A. L. Copley), Interscience Publishers, New York, 1965, p. 19.
6. Herzog, R. O. and Weissenberg, K., Kolloid Zeir., 46, 277, 1928.
7. Metzner, A. B., White, J. L. and Denn, M. M., Amer. Inst. Chem. Engrs. 1., 12, 863, 1966.
8. Whitmore, R. L., “Rheology of the Circulation”, Pergamon Press, Oxford, 1968.
9. Larcan, A. and Stoltz, I. F., “Microcirculation et Hémorhéologie” Mason et Cie, Paris, 1970.
10. Dintenfass, L., “Blood Microrheology”, Butterworth, London, 1971.
11. Fahraeus, R. and Lindquist, T., Amer. J. Physiol. 96, 562. 1931.
12. Bingham, E. C. and Green, H., Proc. Amer. Soc. Test. Mater. 19, 640, 1919.
13. Schofield, R. K. and Scott Blair, G. W., J. Phys. Chem. 34, 248, 1930:35, 1212, 1931, etc.
14. Dix, J. F. and Scott Blair, G. W., J. Appl. Phys., 11, 574, 1940.
15. Segrè, G. and Silberberg, A., Nature, 189, 209, 1961: J. Fluid Mech.,14, 115, 136, 1962.
16. Kolin, A., Proc. Soc. Exp. Biol. and Med., 35, 53. 1936-7: 46, 235, 1941.
17. Wyatt, D. C., J. Sci. Instrum. (Ser. 2) 1, 1146, 1968: Phys. Med. Biol.,13, 529 1968.
18. Copley, A. L., Abstr. Comm. 19th Intern. Congr. Physiol. (Montreal), 1953.
19. Bergel, D. H. and Schultz, D. L., Prog. Biophys. Molec. Biol., 22, 3, 1971.
20. Casson, N., Brit. Soc. Rheol. Bull. No. 52, 5, 1957.
21. Scott Blair, G. W., Nature, 183, 613, 1959.
22. Herschel, W. H. and Bulkley, R., Kolloid Zeit., 39, 291, 1926.
23. Scott Blair, G. W., Rheol. Acta, 5, 184, 1966.
24. Katchalsky, A., Kedem, O., Klibansky, C. and de Vries, A., Chapter in “Flow Properties of Blood and other Biological Systems” (Ed. A. L. Copley and G. Stainsby), Pergamon Press, Oxford, 1960.
25. Nesfield, V.. “The Viscosity of Blood”. Cobden-Sanderson, 1938.
26. Kniseley, M. H., (his views are summarised in a Chapter in “Flow Properties of Blood and other Biological Systems”) Ed. A. L. Copley and C. Stainsby, Pergamon Press, Oxford, 1960.
27. Harkness, J., Houston, J. and Whittington, R. B., Brit. Hed. J., No. 4442, p. 268, 1946.
28. Scott Blair, G. W. and Burnett, J., Kolloid Zeit., 168, 98, 1960.
29. Scott Blair, G. W., Rheol. Acta, 10, 316, 1971.
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
The Effect of Molecular Weight and Concentration of Polymers in Solutions on the Normal Stress Coefficient
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
Dr. Karl Weissenberg, 1922-28
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