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A Physicists' View on Constitutive Equations

Wednesday, September 29, 2004 - 1:30pm - 2:20pm
Keller 3-180
Harald Pleiner (Max Planck Institute for Polymer Research)
Joint work with M. Liu (Inst. f. Theoret. Physik, Universitaet Tuebingen, D 72076 Tuebingen, Germany), and Helmut R. Brand (Theoret. Physik III, Universitaet Bayreuth, D 95440 Bayreuth, Germany)

Hydrodynamic equations for various kinds of complex fluids (simple liquids, binary mixtures, liquid crystals, superfluids, crystals, etc.) can be derived rigorously using general physical laws and principles. This hydrodynamic method is generalized to include slowly relaxing quantities, in particular those describing viscoelasticity. By this procedure it is guaranteed that the resulting equations are in agreement with basic physical laws and requirements.

We start with the nonlinear hydrodynamic equations for elastic media derived from basic physical principles. For the Eulerian strain tensor the lower convected time derivative is obtained, unambiguously [1-3]. Adding a relaxation term the permanent elasticity is transformed into viscoelasticity [1,2], where both, the short time and the long time limit, are given correctly. The dynamic equation for the strain tensor obtained that way still shows the lower convected derivative universally. It covers the usual non-Newtonian effects, like shear thinning, strain hardening, stress overshoot, normal stress differences and Weissenberg effect, non exponential stress relaxation, etc. [4]. When brought into the more familiar form of a dynamic equation for the stress tensor (constitutive equation), it comprises most of the well-known ad-hoc models (Maxwell, Oldroyd, Giesekus), and is more general in structure than those, but is in disagreement with some of them (Johnson-Segalman, Jeffries) [5]. It imposes some restrictions on, and reveals some interdependencies of, the various non-Newtonian contributions that are otherwise introduced heuristically. It is shown how these contributions originate from (nonlinear) elasticity, viscosity, strain relaxation and convection. The time derivative for the stress tensor is no longer of the lower convected type, but is material dependent. We also discuss the connection to those descriptions of viscoelasticity that utilize an orientational (or configurational) order parameter [6].

[1] H. Temmen, H. Pleiner, M. Liu, and H.R. Brand, Phys. Rev. Lett. 84 (2000) 3228; 86 (2001) 745.

[2] H. Pleiner, M. Liu, and H.R. Brand, Rheol. Acta 39 (2000) 560.

[3] M. Grmela, Phys. Lett. A, 296 (2002) 97.

[4] O. Mueller, M. Liu, H. Pleiner, and H.R. Brand, to be published.

[5] H. Pleiner, M. Liu, and H.R. Brand, Rheol. Acta, DOI: 10.1007/s00397-004-0365-8 (2004).

[6] H. Pleiner, M. Liu, and H.R. Brand, Rheol. Acta 41 (2002) 375.