Elastic Fluid

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

Flow processes iaside the spinneret are governed by shear viscosity and shear rate. PET is a non-Newtonian elastic fluid. Spinning filament tension and molecular orientation depend on polymer temperature and viscosity, spinneret capillary diameter and length, spin speed, rate of filament cooling, inertia, and air drag (69,70). These variables combine to attenuate the fiber and orient and sometimes crystallize the molecular chains (71). 

The fluid mechanics origins of shock-compression science are reflected in the early literature, which builds upon fluid mechanics concepts and is more concerned with basic issues of wave propagation than solid state materials properties. Indeed, mechanical wave measurements, upon which much of shock-compression science is built, give no direct information on defects. This fluids bias has led to a situation in which there appears to be no published terse description of shock-compressed solids comparable to Kormer s for the perfect lattice. Davison and Graham described the situation as an elastic fluid approximation. A description of shock-compressed solids in terms of the benign shock paradigm might perhaps be stated as ... 

Visco-elastic fluids like pectin gels, behave like elastic solids and viscous liquids, and can only be clearly characterized by means of an oscillation test. In these tests the substance of interest is subjected to a harmonically oscillating shear deformation. This deformation y is given by a sine function, [ y = Yo sin ( t) ] by yo the deformation amplitude, and the angular velocity. The response of the system is an oscillating shear stress x with the same angular velocity . 

Several solids conveying models were developed by Campbell and his students at Clarkson University [19, 20]. These models will be referred to as either the Clarkson University models or the Campbell models. They proposed that the movement of the screw flight was pushing the polymer bed as the screw turns rather than the frictional force at the barrel moving the polymer pellets down the screw. For these models, they assumed that the solid bed behaved more like an elastic fluid rather than a solid and removed the torque balance constraint. Campbell and Dontula [20] reasoned that because the solid polymer pellets more closely resemble an elastic particulate fluid, no torque balance in the bed would be necessary. They further assumed that the force normal to the pushing flight was due to a combination of the force due to the pressure in the channel and a force proportional to the frictional force exerted at the barrel by the solid bed. The Campbell-Dontula model was first published as ... 

Campbell and Dontula [1] revisited the original solids conveying problem. They proposed that the flights of the screw cause the movement of the solids down the screw, and that the polymer pellets behaved more like an elastic fluid than a solid polymer plug. Therefore, a torque balance would not be applicable. The following is a list of the additional assumptions they made ... 

Under these circumstances [said Davy] a vivid action was soon observed to take place. The potash began to fuse at both its points of electrization. There was a violent effervescence at the upper surface at the lower, or negative, surface, there was no liberation of elastic fluid but small globules having a high metallic lustre, and being precisely similar in visible characters to quicksilver, appeared, some of which burnt with explosion and bright flame, as soon as they were formed, and others remained, and were merely tarnished, and finally covered by a white film which formed on their surfaces. 

Lavoisier delivered two shocks to the Aristotelian elements. His experiments on water led him to conclude in 1783 that it is not a simple substance at all, not properly called an element, as had always been thought . And, concerning that other fluid element of antiquity, he announced that atmospheric air is composed of two elastic fluids of different and opposite qualities , which he called mephitic air and highly respirable air . Neither water nor air, in other words, is an element. 

Y 1770 THE CHEMISTRY OF AIR had become the focus of intense activity. The physical behavior of air had been well worked out in the previous century, but the ability of air to lose its characteristic identity as an elastic fluid and become fixed in the solid state was a difficult fact to comprehend. In France, two men, Jean-Baptiste Michel Bucquet (1746— 1780) and Antoine-Laurent Lavoisier (1743—1794), independently undertook systematic investigations of these phenomena. After a brief collaboration, the unfortunate Bucquet died in 1780, leaving to Lavoisier the completion of a chemical revolution he had anticipated in 1773d... 

If it were permitted me to indulge in conjectures, I should say, that some experiments, which are not sufficiently complete to submit to public inspection, induce me to believe that every elastic fluid results from the combination of some solid or fluid body with the inflammable principle, or perhaps even with the matter of pure fire, and that on this combination the state of elasticity depends. I should add that the substance fixed in metallic calces, and which augments their weight, would not be, properly speaking, on this hypothesis, an elastic fluid, but the fixed part of an elastic fluid, which has been deprived of its inflammable principle. The principal action of charcoal, and all other... 

Upon reconsidering this subject, it occurred to me that I had never contemplated the effect of dijference of size in the particles of elastic fluids. By size I mean the hard particle at the centre and the atmosphere of heat taken together. If, for instance, there be not exactly the same number of atoms of oxygen in a given volume of air, as of azote in the same volume, then the sizes of the particles of oxygen must be different from those of azote. And if the sizes be different, then on the supposition that the repulsive power is heat, no equilibrium can be established by particles of unequal sizes pressing against each other. 

This idea occurred to me in 1805. I soon found that the sizes of the particles of elastic fluids must be different. For a measure of azotic gas and one of oxygen, if chemically united, would make nearly two measures of nitrous gas, and those two could not have more atoms of nitrous gas than the one measure had of azote or oxygen. Hence the suggestion that all gases of different kinds have a difference in the size of their atoms and thus we arrive at the reason for that diffusion of every gas through every other gas, without calling in any other repulsive power than the well-known one of heat. ... 

Fig. 5 [in Daltons plate] represents two atoms of hydrogen drawn in due proportions to those of azote, and coming in contact with them it is obvious that the atoms of hydrogen can apply one to the other with facility, but can not apply to those of azote, by reason of the rays not meeting each other in like circumstances hence, the cause of the intestine motion which takes place on the mixture of elastic fluids, till the exterior particles come to press on something solid. 

Quicklime does not attract air when in its ordinary form, but is capable of being joined to one particular species only, which is decomposed through the atm., either in the shape of an exceedingly subtle powder, or more probably in that of an elastic fluid. To this I have given the named fixed air. 

Quicklime, therefore, does not attract air when in its most ordinary form, hut is capable of being joined to one particular species only, which is dispersed through the atmosphere, either in the shape of an exceedingly subtile powder, or more probably in that of an elastic fluid. To this I have given the name of fixed air, and perhaps very improperly but I thought it better to use a word already familiar in philosophy than to invent a new name, before we be more fully acquainted with the nature and properties of this substance, which will probably be the subject of my future inquiry. 4... 

In 1774 Torbern Bergman presented his treatise on the atmospheric acid (Luftsaure or Aerial acid ) the most complete and systematic discussion of the sources, preparation, properties and combinations of carbon dioxide and carbonic acid. He begins by explaining that about 1770 he had informed his foreign correspondents of his ideas of the nature and properties of that elastic fluid, and cites Dr. Priestley who mentioned his ideas in the Philosophical transaction for 1772 and in a new edition of his work on airs had confirmed them by several fine experiments. 

We now know that common air consists of three elastic fluids mixed together viz., 1st of the aerial acid in its disengaged state, but in so small quantity that it alone cannot impart a visible redness to tincture of turnsol 2nd of an air unfit for sustaining flame, or being subservient to... 

That in comparing these facts with those reported in the preceding chapter, it would appear proven, that there combines with the metals during their calcination, an elastic fluid which becomes fixed, and it is to this fixation that is due their augmentation in weight. 

That several circumstances would seem to tend to the belief that all of the air that we breathe is not fit to be fixed for entering into combination of metallic calxes, but that there exists in the atmosphere a particular elastic fluid which occurs mixed with the air, and that at the moment when the quantity of this fluid contained under the bell jar is exhausted, that the calcination can no longer take place, etc.28... 

SAN of 0.040 Jm-2. This small value of r(I) is probably related to the fact that separation of chains at low stress levels occurs in the most favourable sites, the polymer behaves like a visco-elastic fluid. As Fig. 12 shows, with increasing stress more such sites are activated and the scattering vector increases. On the other hand, annealing leads to a deactivation of such sites and to a coarser structure of the formed fibrils [62]. It must be concluded that in this regime no chain scission or forced reptation occur. 

A. S. Lodge, Elastic Fluids, Academic Press, London, 1964. 

Y. Iso, D. L. Kuch, and C. Cohen, Orientation in Simple Shear Flow of Semi-dilute Fiber Suspensions 1. Weakly Elastic Fluids, J. Non-Newt. Fluid Mech., 62, 115-134 (1996). 

The formulation of proper constitutive relations is a complex problem and is the basis of the science of rheology, which cannot be covered here. This section presents only four relatively simple constitutive relations that have proved to be practically useful to chemical engineers. Elastic fluid behavior is expressly excluded from consideration. The following equations are a listing of these constitutive relations many others are possible ... 

In the case of elastic fluids and for simple shear flow, the first normal stress difference is N[ =on — o22- When shearing a fluid between two plates (x, direction), the first normal stress difference N( forces the plates apart (x2 direction). The first normal stress difference N i is shown together with the measured shear stress x as a function of the shear rate in Fig. 3.9. In the range of shear rates investigated, the shear stress in the case of silicone oil is substantially greater than the normal stress difference and we see substantially greater normal stress differences for viscoelastic PEO solution than for viscous silicone oil. 

The elastic-melt extruder makes use of the Weissenberg or rod climbing effect—which is observed when an elastic fluid is sheared or rotated inside a container by a rod. Because of viscoelasticity, the fluid climbs the rod. 

FIG. 15.31 (A) Efflux of an elastic fluid into a narrow tube from a large reservoir. (B) Die swell at efflux of an... 

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