Tensor conformation

Grmela, M., and Carreau, P. J., Conformation tensor rheological models, J. Non-Newtonian Fluid Mech., 23, 271-294 (1987). 

Other examples of macroscopic constitutive equations employed for describing polymer rheology and mechanics include conformation tensor models as well as the more recently proposed Pom-Pom and Rolie-Poly models. Most of these approaches have been inspired by simple mechanical models of polymers. 

First, a coarse-grained description of the polymer melt is invoked through the definition of the conformation tensor, c, which is a global descriptor of the long-length scale conformation of polymer chains. The conformation tensor c is defined as the second moment tensor of the end-to-end distance vector of a polymer chain reduced by one-third of the unperturbed end-to-end distance and averaged over all chains in the system ... 

Figures 7a and 7b show the time evolution of the diagonal components Cxx, Cyy, and c z of the conformation tensor for the C24 and C78 melts, respectively. For both systems, the initial value of c x is significantly higher than 1, whereas those of Cyy and Czz are a little less than 1, indicative of the oriented conformations induced by the imposed steady-state elongational structure of flow field a x- As time evolves, c x decreases whereas Cyy and Czz increase continuously, approaching the steady-state, field-free value of 1, indicative of fully equilibrated, isotropic structures in the absence of any deforming or orienting field.

For a FENE-type model with nonlinear elastic force of the dumbell, the restituting elastic force of a molecule can be represented in the form F = —where fo is th characteristic spring restitution coefficient per unit mass and Fq = L / L — trA) with L the maximum length of the polymer. The evolution equation for the elastic contribution is given in terms of the deformation or conformational tensor A = (RR), which is now related to the elastic part of the stress tensor by the formula The equation is... 

The average elastic stress p>er unit volume of the solution/cluster compound is given through the relation = — qFqA with A the conformation tensor accounting for the average... 

Abstract We numerically investigate nonlinear regimes of shear-induced phase separation in entangled polymer solutions. For the purpose a time-dependent Ginzburg-Landau model describing the two-fluid dynamics of polymer and solvent is used. A conformation tensor is introduced as a new dynamic variable to represent chain deformations. Its variations give rise to a large viscoelastic stress. Above the coexistence curve, a dynamical steady state is attained, where fluctuations are enhanced on various spatial... 

We briefly set up a dynamic model of an entangled polymer solution in the semidilute regime, (f) > = and critical volume fraction and N being the polymerization index. In terms of the polymer volume fraction (f> and the conformation tensor W = Wij, the free energy is given by [10]... 

In the simulation of viscoelastic flow, a significant numerical problem, the so-called High Weissenberg Number Problem (HWNP), often occurs with loss of convergence of numerical algorithms. In order to alleviate the problem, we have, as the first attempt, implemented the conformation tensor Positive Definiteness Preserving Scheme (PDFS) by Stewart et al. [28], and then adapted and implemented the Log-Conformation Representation (LCR) approach by Fattal and Kupferman [7] in the viscoelastic two-phase flow solver in FS3D. 

On the right-hand side of the constitutive equation, Eq. (1.3), a diffusion term has been added, as proposed by Sureshkumar and Beris [81], so that in turbulent simulations the high wavenumber contributions of the conformation tensor do not diverge during the numerical integration of this equation in time. This parallels the introduction of a numerical diffusion term in any scalar advection equation (e.g., a concentration equation with negligible molecular diffusion) that is solved along with the flow equations under turbulent conditions [82]. In Eq. (1.3), Dq is the dimensionless numerical diffusivity [54-56]. The issue of the numerical diffusivity is further discussed in Sections 1.3.2 and 1.4.3. 

The conformation tensor has a definite physical origin and interpretation, typically associated with the second moment of a suitably defined chain end-to-end distribution function [84]... 

Equation (1.16) is a Helmholtz equation with the unknown sum j + c This is first transformed in the spectral domain where, due to the separability of the Helmholtz equation, the equations for each pair of Fourier modes (r, k) are fully decoupled with the only coupling appearing among the Chebyshev modes. Thus, by solving the Helmholtz equation with a fast solver [86], the updated solution for the conformation tensor is obtained, from which, with the aid of Eq. (1.6), the extra stress tensor n+i i calculated. 

Equation (1.17) is solved with the same solver as the Helmholtz equation for the conformation tensor by using the influence matrix method [87, 89). 

Figure 1.7 Vorticity (with blue) and trace of conformation tensor (with yellow) isosurfaces, close to their maximum values, obtained in a FENE-P polymer simulation of turbulent channel flow. The flow is from back to front with the bottom and top surfaces representing the channel walls. The close-to-maximum vorticity...

Fattal, R. and Kupferman, R. (2004) Constitutive laws for the matrix-logarithm of the conformation tensor. 

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