Integral Constitutive Equations

Single-integral constitutive equations for viscoelastic fluids... 

This full set of self-consistent equations is clearly very difficult to solve, even numerically. However, good approximations of closed integral type have been proposed. These essentially ignore the s-dependence of the survival and orientation functions, which makes them a physically appeaUng approach in the case of wormlike surfactants [71,72]. For ordinary monodisperse polymers the following approximate integral constitutive equation results ... 

H. J. Park, D. Kim, K.-J. Lee, and E. Mitsoulis, Numerical Simulation in Converging Channel Flow of the Fluid Ml Using an Integral Constitutive Equation, J. Non-Newt. Fluid Mech., 52, 69-89 (1994). 

A.C.Papanastasiou, L.E.Scriven, C.W.Macosko, An integral constitutive equation for mixed flows viscoelastic characterization, J. Rheol. 21 (1983), 387-410. 

It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the Phan-Thien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory munerical results were also obtained with multi-mode integral constitutive equations using a spectnun of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [19]. 

Various two- and three-dimensional flow situations can be considered by the method. Integral constitutive equations may be involved in 3D flows ... 

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-... 

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. 

Mitsonhs, E., Computational rheology with integral constitutive equations, Appl. Rheol, 9, 198-203 (1999). 

Laun, H.M. (1980) Stresses and recoverable strains of stretched polymer melts and their prediction by means of a single integral constitutive equation. Rheology, Vol.2, (ed. G. Astarita et ai). Plenum, New York, pp. 419-425. 

The general factorable single integral constitutive equation is an equation that describes well the viscoelastic properties of a large class of crosslinked rubbers > and the elasticoviscous properties of many polymer melt8 under various types of deformation An appropriate way to compare the stress-strain relation for cured elastomers and polymer melts therefore is to calculate the strain-dependent function contained in this constitutive equation from experimental results and to compare the strain measures so obtained. 

A general single integral constitutive equation results if the Boltzmann superposition principle is applied to a nonrspecified tensor functional, of the macroscopic strain, represented by the Finger... 

Lai, J. and Bakker, A. (1995) An integral constitutive equation for nonlinear plasto-viscoelastic behavior of high-density polyethylene. Polym. Eng. Sci., 35,1339. 

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). 

Derail et al. [10,11] used a non-linear integral constitutive equation of the KBKZ [20] type, which, for uniaxial extension, has the form... 

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 ... 

Before leaving the integral constitutive equations, we remark diat a class of equations has been proposed in wMch the functions 01 and 02 of eq 4.4.1S depend on the invariants of the strain rate tenscv D, rather than the strain tensor B (Bird et al., 1968). Many of the sinqiler examples of diese equations do not reduce to the equation of linear viscoelasticity at sniall strains (Gross and Maxwell, 1972 Astarita and Marrucci, 1974) and this class of equations has not been much favored lately. 

The K-BKZ and other integral constitutive equations discussed above can be regarded as generalizations of the Lodge integral, eq4.3.18. The upper-convected Maxwell (UCM) equation, which is the differential equivalent of the Lodge equation, can also be generalized to make possible more realistic predictions of nonlinear phenomena. 

Finally, more accurate differential and integral constitutive equations were presented, and their successes and failures in de-st bing experimental data, were discussed. No single nonlinear constitutive equation is best for all purposes, and thus one s choice of an appropriate constitutive equation must be guided by the problem at himd, the accuracy with which one wishes to solve the problem, and the effort one is willing to expend to solve it. Generally differential models of the Maxwell type are easier to implement numerically, and some are available in fluid mechanics codes. Also, some cmistitutive equations are better founded in molecular theory, as discussed in Chiqpter 11. 

As an example of a popular viscoelastic constitutive equation used in the past 25 years, vhich possesses enough degree of complexity so as to capture as accurately as possible the complex nature of polymeric liquids, we present here the K-BKZ integral constitutive equation with multiple relaxation times proposed by Papanastasiou et al. [27] and further modified by Luo and Tanner [28]. This is often referred to in the literature as K-BKZ/PSM model (from the initials of Papanastasiou, Scriven, Macosko) and is vritten as... 

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