Shear viscosity mathematical aspects

The viscosity at rest, i.e., zero shear rate, is clinically referred to as zero shear viscosity (unit Pas = Pascal seconds = 1000 cp = 1000 centipoise). This value can not be accurately determined by a rheometer, but is rather extrapolated mathematically, using various computation methods. Viscosity at rest is occasionally employed as the single measure for the assessment and classification of behavioral properties of viscoelastic substances. However, viscosity at rest merely reflects one aspect of the varied biophysical characteristics of viscoelastic substances which merit differentiation. 

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

Several authors have extended the calculations of EiksteiK to non-spherical parti-cies The models that have been used are oblate or prolate ellipsoids of revolution long cylinders or a stiff row of spheres K The calculations are more difficult than in the case of spherical particles, not only in the mathematical sense, but also on account of two new physical aspects of the problem. It will be evident that the contribution to the viscosity of a long or flat particle depends upon its orientation. This orientation is constantly modified by the toppling over of the particles in the field of shear and by the rotational BROWNian motion of the particles. The viscosity is therefore dependent upon the rate of shear. At low rates of shear, the BROWNian motion prevails and the orientation of the particles is completely at random. At high shear, however, or for large particles, the BROWNian motion is negligible and the orientation completely determined by hydrodynamics. 

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