# American Institute of Mathematical Sciences

September  2014, 19(7): 1815-1835. doi: 10.3934/dcdsb.2014.19.1815

## Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour

 1 Dipartimento di Matematica, Università di Pisa, Pisa, Italy, Italy 2 Dipartimento di Scienze di Base e Applicate, per l'Ingegneria - Sezione Matematica, Sapienza Università di Roma, Rome, Italy 3 School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  March 2013 Revised  March 2014 Published  August 2014

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 to 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.
Citation: Giovambattista Amendola, Sandra Carillo, John Murrough Golden, Adele Manes. Viscoelastic fluids: Free energies, differential problems and asymptotic behaviour. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 1815-1835. doi: 10.3934/dcdsb.2014.19.1815
##### References:
 [1] G. Amendola, The minimum free energy for incompressible viscoelastic fluids, Math. Methods Appl. Sci., 29 (2006), 2201-2223. doi: 10.1002/mma.769. [2] G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. of Mech. and Appl. Math., 57 (2004), 429-446. doi: 10.1093/qjmam/57.3.429. [3] G. Amendola and M. Fabrizio, Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model, Diff. Int. Eq., 20 (2007), 445-466. [4] G. Amendola, M. Fabrizio, J. M. Golden and B. Lazzari, Free energies and asymptotic behaviour for incompressible viscoelastic fluids, Appl. Anal., 88 (2009), 789-805. doi: 10.1080/00036810903042117. [5] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0. [6] S. Breuer and E. T. Onat, On the determination of free energy in viscoelastic solids, Z. angew. Math. Phys., 15 (1964), 184-191. doi: 10.1007/BF01602660. [7] S. Breuer and E. T. Onat, On recoverable work in viscoelasticity, Z. Angew. Math. Phys., 15 (1981), 13-21. [8] S. Carillo, Existence, Uniqueness and Exponential Decay: An Evolution Problem in Heat Conduction with Memory, Quarterly of Appl. Math., 69 (2011), 635-649. S 0033-569X(2011)01223-1, Article Electronically published on July 7, (2011). doi: 10.1090/S0033-569X-2011-01223-1. [9] S. Carillo, An evolution problem in materials with fading memory: solution's existence and uniqueness, Complex Variables and Elliptic Equations An International Journal, 56 (2011), 481-492. doi: 10.1080/17476931003786667. [10] S. Carillo, Materials with Memory: Free energies & solutions' exponential decay, Communications on Pure And Applied Analysis, 9 (2010), 1235-1248. doi: 10.3934/cpaa.2010.9.1235. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] W. A. Day, Some results on the least work needed to produce a given strain in a given time in a viscoelastic material and a uniqueness theorem for dynamic viscoelasticity, Quart. J. Mech. Appl. Math., 23 (1970), 469-479. doi: 10.1093/qjmam/23.4.469. [13] G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity, 43 (1996), 247-278. doi: 10.1007/BF00042503. [14] L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity, 54 (1999), 141-185. doi: 10.1023/A:1007646017347. [15] L. Deseri, M. Fabrizio and J. M. Golden, The concept of minimal state in viscoelasticity: New free energies and applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96. doi: 10.1007/s00205-005-0406-1. [16] M. Fabrizio, G. Gentili and J. M. Golden, The minimum free energy for a class of compressible viscoelastic fluids, Advances Diff. Eq., 7 (2002), 319-342. [17] M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373. doi: 10.1007/BF00375062. [18] M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., 60 (2002), 341-381. [19] M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Int. Eq., 6 (1993), 491-505. [20] M. Fabrizio and A. Morro, Reversible processes in thermodynamics of continuous media, J. Nonequil. Thermodyn., 16 (1991), 1-12. doi: 10.1515/jnet.1991.16.1.1. [21] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970807. [22] G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math., 60 (2002), 153-182. [23] J. M. Golden, Free energy in the frequency domain: The scalar case, Quart. Appl. Math., 58 (2000), 127-150. [24] D. Graffi and M. Fabrizio, On the notion of state for viscoelastic materials of "rate'' type, Atti. Accad. Naz. Lincei, 83 (1990), 201-208. [25] D. Graffi and M. Fabrizio, Nonuniqueness of free energy for viscoelastic materials, Atti Accad. Naz. Lincei, 83 (1990), 209-214. [26] W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [28] M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal., 62 (1976), 303-321.

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##### References:
 [1] G. Amendola, The minimum free energy for incompressible viscoelastic fluids, Math. Methods Appl. Sci., 29 (2006), 2201-2223. doi: 10.1002/mma.769. [2] G. Amendola and S. Carillo, Thermal work and minimum free energy in a heat conductor with memory, Quart. J. of Mech. and Appl. Math., 57 (2004), 429-446. doi: 10.1093/qjmam/57.3.429. [3] G. Amendola and M. Fabrizio, Maximum recoverable work for incompressible viscoelastic fluids and application to a discrete spectrum model, Diff. Int. Eq., 20 (2007), 445-466. [4] G. Amendola, M. Fabrizio, J. M. Golden and B. Lazzari, Free energies and asymptotic behaviour for incompressible viscoelastic fluids, Appl. Anal., 88 (2009), 789-805. doi: 10.1080/00036810903042117. [5] G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials with Memory, Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0. [6] S. Breuer and E. T. Onat, On the determination of free energy in viscoelastic solids, Z. angew. Math. Phys., 15 (1964), 184-191. doi: 10.1007/BF01602660. [7] S. Breuer and E. T. Onat, On recoverable work in viscoelasticity, Z. Angew. Math. Phys., 15 (1981), 13-21. [8] S. Carillo, Existence, Uniqueness and Exponential Decay: An Evolution Problem in Heat Conduction with Memory, Quarterly of Appl. Math., 69 (2011), 635-649. S 0033-569X(2011)01223-1, Article Electronically published on July 7, (2011). doi: 10.1090/S0033-569X-2011-01223-1. [9] S. Carillo, An evolution problem in materials with fading memory: solution's existence and uniqueness, Complex Variables and Elliptic Equations An International Journal, 56 (2011), 481-492. doi: 10.1080/17476931003786667. [10] S. Carillo, Materials with Memory: Free energies & solutions' exponential decay, Communications on Pure And Applied Analysis, 9 (2010), 1235-1248. doi: 10.3934/cpaa.2010.9.1235. [11] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308. [12] W. A. Day, Some results on the least work needed to produce a given strain in a given time in a viscoelastic material and a uniqueness theorem for dynamic viscoelasticity, Quart. J. Mech. Appl. Math., 23 (1970), 469-479. doi: 10.1093/qjmam/23.4.469. [13] G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity, 43 (1996), 247-278. doi: 10.1007/BF00042503. [14] L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity, 54 (1999), 141-185. doi: 10.1023/A:1007646017347. [15] L. Deseri, M. Fabrizio and J. M. Golden, The concept of minimal state in viscoelasticity: New free energies and applications to PDEs, Arch. Rational Mech. Anal., 181 (2006), 43-96. doi: 10.1007/s00205-005-0406-1. [16] M. Fabrizio, G. Gentili and J. M. Golden, The minimum free energy for a class of compressible viscoelastic fluids, Advances Diff. Eq., 7 (2002), 319-342. [17] M. Fabrizio, C. Giorgi and A. Morro, Free energies and dissipation properties for systems with memory, Arch. Rational Mech. Anal., 125 (1994), 341-373. doi: 10.1007/BF00375062. [18] M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., 60 (2002), 341-381. [19] M. Fabrizio and B. Lazzari, On asymptotic stability for linear viscoelastic fluids, Diff. Int. Eq., 6 (1993), 491-505. [20] M. Fabrizio and A. Morro, Reversible processes in thermodynamics of continuous media, J. Nonequil. Thermodyn., 16 (1991), 1-12. doi: 10.1515/jnet.1991.16.1.1. [21] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992. doi: 10.1137/1.9781611970807. [22] G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quart. Appl. Math., 60 (2002), 153-182. [23] J. M. Golden, Free energy in the frequency domain: The scalar case, Quart. Appl. Math., 58 (2000), 127-150. [24] D. Graffi and M. Fabrizio, On the notion of state for viscoelastic materials of "rate'' type, Atti. Accad. Naz. Lincei, 83 (1990), 201-208. [25] D. Graffi and M. Fabrizio, Nonuniqueness of free energy for viscoelastic materials, Atti Accad. Naz. Lincei, 83 (1990), 209-214. [26] W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [28] M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal., 62 (1976), 303-321.
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