Abstract
In this work we propose a novel generalised form of the Phan–Thien and Tanner (PTT) model by considering a new functional form of the nonlinear expression characterizing the destruction of physical network junctions and entanglements. This new function of the trace of the stress tensor is given by the generalized Mittag–Leffler function, and contains the familiar exponential form of the original Phan–Thien and Tanner model as a limiting case, but affords additional fitting flexibility through the inclusion of one or two additional fitting constants. We perform fits to experimental data in shear and extension and show that this generalized expression allows a better description of the rheological responses for a range of complex materials such as polymer melts and semidilute polymer solutions. By using an appropriate information criterion, we also demonstrate that the resulting generelized model remains parsimonious but provides improved fits of real world data.
Highlights
In 1977 Nhan Phan–Thien and Roger I
We were not initially expecting a better fit with the new model for weak shearing flows, it turns out that the ratedependent variation of the rate of destruction of network junctions can still be felt in simple shear flows (by considering high α and ε values which changes the dominant balance of the terms in the equations for η(γ ) and Ψ1(γ ))
In this subsection we investigate the influence of the Mittag–Leffler function on the velocity and stress profiles, studying the influence of α and β on the velocity profile for channel and pipe flows and comparing these results with the exponential proposed a new constitutive equation (PTT) model, whose analytical solution is given in the work of Oliveira and Pinho [13]
Summary
In 1977 Nhan Phan–Thien and Roger I. The values of resulting the argument (t/T ≪ 1) fractional viscoelastic constitutive models allow a better fit to rheological data and at the same time make use of a smaller number of parameters, when compared, for example, with multi-mode approaches [4] Based on these considerations, the Mittag–Leffler function seems to be a good candidate to better describe the rate of destruction of junctions and the behavior of complex materials. A way to verify that the new model (with extra fitting parameters) provides a better description of real experimental data is to use an appropriate statistical measure of the fitting quality that penalizes the addition of extra parameters This is possible using, for example, the Akaike information criterion (AIC), proposed by Hirotugu Akaike in 1973 [7,8]. We consider the possibility of constitutive instabilities in the system dynamics due to non-monotonic variations in the steady flow curve for certain ranges of model parameters Section 5
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