Second gradient viscoelastic fluids: dissipation principle and free energies

Abstract

We consider a generalization of the constitutive equation for an incompressible second order fluid, by including thermal and viscoelastic effects in the expression for the stress tensor. The presence of the histories of the strain rate tensor and its gradient yields a non-simple material, for which the laws of thermodynamics assume a appropriate modified form. These laws are expressed in terms of the internal mechanical power which is evaluated, using the dynamical equation for the fluid. Generalized thermodynamic constraints on the constitutive equation are presented. The required properties of free energy functionals are discussed. In particular, it is shown that they differ from the standard Graffi conditions. Various free energy functionals, which are well-known in relation to simple materials, are generalized so that they apply to this fluid. In particular, expressions for the minimum free energy and a more recently introduced explicit functional of the minimal state are proposed. Derivations of various formulae are abbreviated if closely analogous proofs already exist in the literature.

Appendix

Various notations used in the main paper are defined here.

The real axis is denoted by R, while R ⁺=[0,+∞) and R ⁻=(−∞,0]. Also, R ^− −=(−∞,0) and R ⁺⁺=(0,+∞).

The Fourier transform of any function f:→R→R ⁿ is defined by

(3.32)

where

(3.33)

The half-range Fourier cosine and sine transforms are given by

(3.34)

If f(u) vanishes as u→+∞, we have

(3.35)

If f′(0) is non-zero, then

(3.36)

by virtue of (4.4).

Finally, we define the following subsets of the complex z-plane C:

(3.37)