Constitutive differential

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. 

Outline of a decoupled scheme for the differential constitutive models... 

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. 

C. Guillope and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal., Th. Meth.. ppl., 15 (1990) 849-869. 

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

M. Renaxdy, Inflow boundary conditions for steady flows of viscoelcistic fluids with differential constitutive laws. Rocky Mount. J. Math., 18 (1988) 445-453, and 19 (1989) 561. 

IIOJ. Baranger and D. Sandri, Finite element method for the approximation of viscoelastic fluid flow with a differential constitutive law. First European Computational Fluid Dynamics Conference, Bruxelles, 1992, C. Hirsch (ed.), Elsevier, Amsterdam, 1993, 1021-1025. 

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). 

The methods utilized to measure the viscoelastic functions are often close to the stress patterns occurring in certain conditions of use of polymeric materials. Consequently, information of technological importance can be obtained from knowledge of these functions. Even the so-called ultimate properties imply molecular mechanisms that are closely related to those involved in viscoelastic behavior. Chapters 16 and 17 deal with the stress-strain multiaxial problems in viscoelasticity. Application of the boundary problems for engineering apphcations is made on the basis of the integral and differential constitutive stress-strain relationships. Several problems of the classical theory of elasticity are revisited as viscoelastic problems. Two special cases that are of special interest from the experimental point of view are studied viscoelastic beams in flexion and viscoelastic rods in torsion. 

Goetzl, E. J., Kong, Y., Voice, J. K. Cutting edge differential constitutive expression of functional receptors for lysophosphatidic acid by human blood lymphocytes. J Immunol 164 (2000) 4996-4999. 

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]. 

Consider the optimal design of a unit operation or a chemical plant or consider a problem of optimal process control. In these situations as in many other similar situations, the constraint equations (algebraic, differential, or algebraic-differential) constitute the most significant part of the overall problem. 

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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