Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour

Abstract

Some expressions for the free energy in the case of incompressible viscoelasticfluids are given. These are derived from free energies already introduced for other viscoelastic materials, adapted to incompressible fluids.A new free energy is given in terms of the minimal state descriptor. The internaldissipations related to these different functionals are also derived. Two equivalent expressions for the minimum free energy are given, one in terms of the history of strain and the other in terms of the minimal state variable. This latter quantity is also used to prove a theorem of existence and uniqueness of solutions to initial boundary value problems for incompressible fluids. Finally, the evolution of the system is described interms of a strongly continuous semigroup of linear contraction operators ona suitable Hilbert space. Thus, a theorem of existence and uniqueness ofsolutions admitted by such an evolution problem is proved.