Differential constitutive equations

Equation 10.3, with G t) given by a single exponential function as in Eq. 4.12, can be written in the form of a differential equation as shown below. 

Note that for a step shear strain of % at t = 0, the resulting shear stress is  

And for steady simple shear, the long-time, limiting stress is  

To generalize Eq. 10.18 to describe flows having any kinematics, we replace the shear stress and shear rate by the corresponding tensorial quantities to obtain the generalized Maxwell 

And this simple, two-constant model can be further generalized to accommodate a discrete spectrum of relaxation times by writing Eq. 10.15 for each relaxation mode and summing the stresses resulting from solving each equation. 

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The Oldroyd-type differential constitutive equations for incompressible viscoelastic fluids can in general can be written as (Oldroyd, 1950)... 

Some of the integral or differential constitutive equations presented in this and the previous section have an exact equivalent in the other group. There are, however, equations in both groups that have no equivalent in the other category. 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

A differential constitutive equation the Phan-Thien Tanner mod ... 

R.G.Larson, Convected derivatives for differential constitutive equations, J. of Non-Newt. Fluid Mech. 2i (1987), 331-342. 

M. Renaxdy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., 65 (1985), 449-451. 

Computation of 2D viscoelastic flows for a differential constitutive equation... 

Severe difficulties have been encountered for several years in the numerical simulation of viscoelastic flow for differential constitutive equations. Let us now give a summary of the numerical problems previously presented. 

In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. 

In this Section, the numerical results compared to experimental data were obtained at CEMEF, with the differential constitutive equations already presented. 

This identity is useful for relating integral and differential constitutive equations, as we shall see in Section 3.4.4. A thorough discussion of this and other relationships among kinematic tensors can be found in Astarita and Marrucci (1974). 

Unlike simple differential constitutive equations as the one previously addressed, constitutive equations may present special types of derivatives such as the substantial derivative, or other types of derivatives in which a hypothetical frame of observation of the flow is allowed to translate, rotate, and/or deformate [33], The Criminale-Ericsen-Filbey (CEF) equation, written here as Equation 22.21, is an example of this type of equations. The CEF equation is relatively simple, and it is explicit in the stress tensor. The latter is a feature not shared by all rheological relationships belonging in the category of equations with special types of derivatives [35]. 

Verbeeten WMH, Peters GWM, Baaijens FPT (2001) Differential constitutive equations for polymer melts The extended Pom-Pom model. J Rheol 45 823-843 Verbeeten WMH, Peters GWM, Baaijens FPT (2002) Viscoelastic analysis of complex polymer melt flows using the extended Pom-Pom model. J Non-Newtonian Fluid Mech 108 301-326 Verleye V, Dupret F (1993) Prediction of fiber orientation in complex injection molded parts. 

The combination of Equations 4.30-4.32 and the elimination of the subscripts for the Maxwell and Kelvin models give the third-order linear differential constitutive equation ... 

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