Constitutive equations Cross model

Ishii (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase regimes. AML Report ANL-77-47 Ide H, Matsumura H, Tanaka Y, Fukano T (1997) Flow patterns and frictional pressure drop in gas-liquid two-phase flow in vertical capUlary channels with rectangular cross section, Trans JSME Ser B 63 452-160... 

There are numerous other GNF models, such as the Casson model (used in food rheology), the Ellis, the Powell-Eyring model, and the Reiner-Pillippoff model. These are reviewed in the literature. In Appendix A we list the parameters of the Power Law, the Carreau, and the Cross constitutive equations for common polymers evaluated using oscillatory and capillary flow viscometry. 

In this chapter we have developed the general constitutive equation for a viscous liquid. We found that by using the rate of deformation or strain rate tensor 2D, we can write Newton s viscosity law properly in three dimensions. By making the coefficient of 2D dependent on invariants of 2D, we can derive models like the power law. Cross, and Carreau. We also showed how to introduce a three-dimensional yield stress to describe plastic materials with models like those Bingham and Casson. We saw two ways to describe the temperature dependence of viscosity and the importance of shear heating. 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

The feed is sent to the residue side of the first stage with a membrane area of 1.5 X 102 cm . The value of 0 is determined to satisfy the summation equations, and the product rates and compositions are calculated. The residue from the first stage is sent to the residue side of the second stage which also has a membrane area of 1.5 x I02 cm . The residue from the second stage is the residue product of the cross-flow model, and the combined stream of the permeates from the two stages constitutes the permeate product of the model. The results are tabulated as follows ... 

Sperling has studied theoretical conditions for the formation of domains in sequential IPNs using cross-linking degree for each network, as well as thermodynamics of mixing and interfacial tension for sequential IPNs, where separation occurs by the nucleation mechanism. The derivation of the basic equation for IPN domain diameters is based on a physical model of sequential IPNs, according to which polymer II, which is formed in a swollen network I, constitutes a spherical core and is in a contracted (deformed) state, while polymer I surrounds the core and is in an expanded (deformed) state. 

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