Codeformational

In codeformational equations, the basic kinematic quantities are the displacement functions. This generally means using the respective Cauchy and Finger tensors deformation gradient tensor X /9 Xfi ] by the following equations... 

The most efficient mechanism of drop breakup involves its deformation into a fiber followed by the thread disintegration under the influence of capillary forces. Fibrillation occurs in both steady state shear and uniaxial extension. In shear (= rotation + extension) the process is less efficient and limited to low-X region, e.g. X < 2. In irrotatlonal uniaxial extension (in absence of the interphase slip) the phases codeform into threadlike structures. 

Table 2.4. Binary metal/metal systems observed to exhibit solid-state amorphiz-ation by interdiffusion or other type of driven mixing (e.g. mechanical codeformation). The table lists the type of experiment, the typical reaction temperature, TR (for the case of interdiffusion reaction) and references. (B thin-film bilayer diffusion couples, M thin-film multilayer diffusion couple, S interdiffusion of polycrystalline layer of one component with single crystal of another component, MA mechanical alloying of the metals, MAT thermal reaction of a mechanically deformed composite...

With the homogeneous flow assumption, whereby V(RR) = 0, the definition of (RR)(i) as given by Eq. (6.25) is the same as the codeformational time derivative or convected time derivative shown in Appendix 6.A. 

Both yjo] and X[oj are codeformational rheological tensors. They can describe the deformation and the stress response to a deformation without interference from the rigid-body rotation (see Chapter 5). In other words, these tensors and their time derivatives, integrations, and combinations... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

At finite deformations, equation 59 can be shown to be incorrect because it is not objective i.e., it predicts results which erroneously depend on the orientation of the sample with respect to laboratory coordinates. This error can be eliminated by replacing j/j in equation 59 by the components of a corotational rate-of-strain tensor or the components of one of several possible codeformational rate-of-strain tensors either of these replacements ensures that the unwanted dependence of cy on the instantaneous orientation of a fluid particle in space is removed. If the stress-strain relations are linear within the changing coordinate frame, equation 59 is modified only be replacing y,-j with a different strain rate tensor whose definition is complicated and beyond the scope of this discussion. The corresponding corotational model is that of Goddard and Miller and the codeformational models correspond to those of Lodge or Oldroyd, Walters, and Fredrickson. ... 

Here, b/br is called the convected derivative due to Oldroyd (1950), and it is the fixed coordinate equivalent of the material derivative of a second-order tensor referred to in convected coordinates. The physical interpretation of the right-hand side of Eq. (2.104) may be given as follows. The first two terms represent the derivative of tensor a j with time, with the fixed coordinate held constant (i.e., Da /Dr), which may be considered as the time rate of change as seen by an observer in a fixed coordinate system. The third and fourth terms represent the stretching and rotational motions of a material element referred to in a fixed coordinate system. This is because the velocity gradient dv fdx (or the velocity gradient tensor L defined by Eq. (2.59)) may be considered as a sum of the rate of pure stretching and the material derivative of the finite rotation. For this reason, the convected derivative is sometimes referred to as the codeformational derivative (Bird et al. 1987). 

By replacing the partial time derivative with a nonlinear time derivative, which is based on a codeforming and translating coordinate system, the constitutive equation given in Eq. 3.1 now becomes (Bird et al., 1987a) ... 

A model consisting of the codeformational MaxweU constitutive equation coupled to a kinetic equation for breaking and re-formation of micelles is presented to reproduce most of the nonlinear viscoelastic properties of wormlike micelles. This simple model is also able to predict shear banding in steady shear and pipe flows as well as the long transients and oscillations that accompany this phenomenon. Even though the model requires six parameters, all of them can be evaluated from single and independent rheological experiments, and then they can be used to predict other flow situations. The predictions of our model are compared with experimental data for aqueous micellar solutions of cetyltrimethylammonium tosilate (CTAT). 

The model consists of the codeformational Maxwell equation coupled to a kinetic equation to account for the breaking and re-formation of micelles [15,24]. For simple shear flow, the model simplifies to the following system of ordinary differential equations ... 

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