Open AccessViscoelastic fluids: Free energies, differential problems and asymptotic behaviour
Author(s) -
Giovambattista Amendola,
Sandra Carillo,
John M. Golden,
Adele Manes
Publication year - 2014
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2014.19.1815
Subject(s) - uniqueness , compressibility , viscoelasticity , mathematics , mathematical analysis , semigroup , hilbert space , boundary value problem , uniqueness theorem for poisson's equation , state variable , contraction (grammar) , dimension (graph theory) , pure mathematics , physics , mechanics , thermodynamics , medicine
Some expressions for the free energy in the case of incompressible viscoelastic fluids 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 internal dissipations 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 of initial boundary value problems for incompressible fluids. Finally, the evolution of the system is described in terms of a strongly continuous semigroup of linear contraction operators on a suitable Hilbert space. Thus, a theorem of existence and uniqueness of solutions admitted by such an evolution problem is proved