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Polymer Processing
Published in Anil Kumar, Rakesh K. Gupta, Fundamentals of Polymer Engineering, 2018
The major effect of polymer cooling is that it retards stress relaxation and, as mentioned previously in the chapter, some of the stress remains frozen-in, even after the molding has completely solidified. This stress relaxation cannot be predicted using an inelastic constitutive equation (why?); the simplest equation that we can use for the purpose is the upper-convected Maxwell model. In the absence of flow, the use of this model yields () τzx(t,x)=τzx(0,x)exp(−∫0tdt′λ0)
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
The Oldroyd eight-parameter model illustrates how the number of constants becomes prohibitively large if only frame invariance considerations are used to formulate constitutive equations based purely on continuum mechanics considerations. It provides useful qualitative descriptions but is not quantitatively accurate. Care must be exercised in using the Oldroyd models in elongational flows since the elongational viscosity can become infinite if the parameters are not chosen carefully. Various empirical differential models have also been suggested. The White-Metzner model, for example, is obtained by making the relaxation time and the viscosity in the upper convected Maxwell model functions of the shear rate. It is useful in describing flows where the coupling of shear thinning and elasticity is significant. The Giesekus model can be obtained from molecular arguments and, unlike the Oldroyd model, is quadratic in stress. Although the model is considerably more difficult to use, it predicts decreasing viscosity and normal stress coefficients with increasing shear rate and a finite second normal stress coefficient. Differential constitutive equations are often the most convenient and practical way to determine the role of viscoelasticity in real-world systems. They are best suited to describing flows in which both shearing and extension are involved or shearing flows in which normal stresses are important.
Mathematical regression models for rheological behavior of interaction between polymer-surfactant binary mixtures and electrolytes
Published in Journal of Dispersion Science and Technology, 2022
Tandrima Banerjee, Abhijit Samanta, Ajay Mandal
The viscosity of polymer solutions is not constant at different flow conditions due to its non-Newtonian behavior. The viscosity of polymer solutions is depended on shear rate and shear rate history. Several models have been proposed to describe the diversity and complexity in rheological behavior of polymer solutions, e.g., Jeffreys, Carreau, Ellis, and power-law models that describe time independent rheology, Herschel–Bulkley model that includes yield stress and upper-convected Maxwell model and Oldroyd-B that include viscoelastic properties.[35–39] However, working with non-Newtonian fluids requires one to account for the shear rate as an additional variable with direct impact on viscosity, as expressed by the Herschel–Bulkley model, especially when the viscosity does not vary uniformly and linearly with shear rate. Empirical relationships for the viscosity as a function of the shear rate and the concentrations of alkali, surfactant, polymer, and salt are not generally available in the literature.