Wagner, integral equation

These constitutive equations differ in their mathematical form the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. 

In axisymmetric flow situations, the global pressure drop in a capillary rheometer is well described by the three constitutive equations. If one focuses on the entrance pressure drop, the numerical entrance pressure drop related to Bagley correction is foimd to be less important than the corresponding experimental data for the differential models for LDPE and LLDPE melts. For the Wagner integral constitutive equation, the computed entrance pressure drops are found to be lower for both fluids, but the computed values are closer to the experimental data for LLDPE than those related to the LDPE melt. This descrepancy, previously reported in the literature, needs further investigation. 

This initial-value type integral equation can be calculated numerically using the method presented by Wagner [39] and Acrivos and Chambre [40]. The time-derivative of c, can be approximated numerically by dcjdt = (cy+i - c tN. Substituting into Equation 33 and discretizing time into time steps / yields ... 

Wagner (45) also has derived the relationship given by Equation 14, where a is the coefficient of expansion of the metal the first term on the left is —Sff.p, i.e., the partial excess entropy at constant pressure (free surface conditions) and the first term on the right is the corresponding partial excess entropy under constant volume. By combining Equation 9 with Equation 14 and integrating, Oates and Flanagan (47) have obtained Equation 15, where the reference con-... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

An important form of the second formula, which is also denoted as the Wagner equation, can be obtained by integration ... 

Flow experiments were carried out at LRMP for axisymmetric contractions and at CEMEF for plane geometries. Numerical simulations were performed at Laboratoire de Rheologie, with Wagner memory-integral constitutive equations, with the stream-tube method (sub-section 5.1) and at CEMEF, where the finite-... 

It may be helpful at this point to ejcplain why the formidable "central problem" discussed above doesn t show up in the traditional theory of mixed conduction as developed by Wagner. He only treats two situations ion blocking electrode conditons and open circuit conditions. When the electrodes are ion blocking, l vanishes in Equation, 32 (for each ionic species) in which case Equation 32 is easy to integrate. 

Gerber N, Wagner J (1972). Explanation of dose-dependent decline of diphenylhydantoin plasma levels by fitting to the integrated form of the Michaelis-Menten equation. Res Commun Chem Pathol Pharmacol 3 455. 

The mechanism is explained by Wagner [25]. Both oxygen anions and the metal cations are the rate-controlling species. The parabolic rate law has a theoretical basis and is very often encountered in high-temperature corrosion. Integration of Pick s law results to parabolic rate law equation when fep = CDAc. 

Wagner, M.H., Prediction of primary normal stress difference from shear viscosity data using a single integral constitutive equation. RheoL Acta, 16,43-50 (1977). 

Wagner [14,15] has provided a method for the prediction of normal stress difference from shear viscosity using a strain-dependent single-integral constitutive equation of the (Berstein-Kearsley-Zapas) 6KZ type. In an sq>propriately modified form, it can be written [13] as follows ... 

Wagner, M. H., Prediction of primary stress difference from shear viscosity data using a single integral constitutive equation, Rheol. Acta, 16, 43-50 (1977). Huppler, J. D., Ashare, E., and Holmes, L. A., Rheological properties of three solutions. Part I. Non-Newtonian viscosity, normal stresses, and complex viscosity, nans. Soc. Rheol., 11,159-179 (1967). 

Another strain measure that is closely related to the one defined by Eq. 10.14 was proposed by Wagner etal. [ 14]. This tensor involves a new scalar, which they call the molecular stress function. When used in an integral constitutive equation it was found to be able to describe the behavior of a high-density polyethylene in shear flow and several types of extensional deformation. 

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