Bracket formulation as a source for the development of dynamic equations in continuum mechanics

Journal of Non-Newtonian Fluid Mechanics

Volumn 96, Issue 1-2, 2001, Pages 119-136

  Beris, Antony N ^a,b  

^a University of Delaware   (United States)

^b UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

CONTINUUM MECHANICS; ENTROPY; HEAT LOSSES; MATHEMATICAL TRANSFORMATIONS; POISSON DISTRIBUTION; THERMODYNAMIC STABILITY;

LEGENDRE TRANSFORMS; NONEQUILIBRIUM THERMODYNAMICS;

NON NEWTONIAN FLOW;

DYNAMIC MODELING; MATHEMATICAL MODELING; THERMODYNAMICS;

EID: 0034745515     PISSN: 03770257     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0257(00)00131-2     Document Type: Article

References (61)

1. I. Newton, Philosophiae Naturalis Principia Mathematica, Streater, London, 1687.
2. C. Lanczos, The Variational Principles of Mechanics, 4th Edition, Dover Publications, New York, 1986.
3. R. Dugas, A History of Mechanics, Dover Publications, New York, 1988.
4. N.A. Doughty, Lagrangian Interaction, Addison-Wesley, Reading, MA, 1990.
5. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer, New York, 1994.
6. T. Frankel, The Geometry of Physics, Cambridge University Press, Cambridge, 1997.
7. S.-D. Poisson, Mémoire sur la variation des constantes arbitraires dans les questions de mécanique, J. École Polytech. 8 (1809) 266-344.
8. V.I. Arnold, Sur la géometrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, Grenoble 16 (1966) 319-361.
9. A. Weinstein, J.E. Marsden, The Hamiltonian structure of the Maxwell-Vlasov equations, Physica D 4 (1982) 394-406.
10. J.E. Marsden, A. Weinstein, Coadjoint orbits, vortices and Clebsch variables for incompressible fluids, Physica D 7 (1983) 305-323.
11. P.J. Morrison, The Maxwell-Vlasov equations as a continuous Hamiltonian system, Phys. Lett. A 80 (1980) 383-386.
12. P.J. Morrison, M. Tabor, Y.M. Treve (Eds.), Poisson brackets for fluids and plasmas, in: Proceedings of the AIP Conference on Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems, Vol. 88, 1982, pp. 13-46.
13. D. Lewis, J.E. Marsden, T.S. Ratiu, Stability and bifurcation of a rotating liquid drop, J. Math. Phys. 28 (1987) 2508-2515.
14. D. Lewis, J.C. Simo, Nonlinear stability of rotating pseudo-rigid bodies, Proc. Roy. Soc. London A 427 (1990) 281-319.
15. J.E. Marsden, OM. O'Reilly, F.J. Wicklin, B.W. Zombro, Symmetry, stability, geometric phases, and mechanical integrators, Nonlinear Science Today 1 (1991) 4-11; 14-21.
16. A.N. Kaufman, Dissipative Hamiltonian systems: a unifying principle, Phys. Lett. 100A (1984) 419-422.
17. P.J. Morrison, Bracket formulation for irreversible classical fields, Phys. Lett. 100A (1984) 423-427.
18. M. Grmela, Bracket formulation of dissipative fluid mechanics equations, Phys. Lett. 102A (1984) 355-358.
19. B.J. Edwards, H.C. Öttinger, Time-structure invariance criteria for closure approximations, Phys. Rev. E 56 (1997) 4097-4103.
20. H.C. Öttinger, A.N. Beris, Thermodynamically consistent reptation model without independent alignment, J. Chem. Phys. 110 (1999) 6593-6596.
21. A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems, Oxford University Press, New York, 1994.
22. M. Grmela, H.C. Öttinger, Dynamics and Thermodynamics of complex fluids. I. Development of the GENERIC formalism, Phys. Rev. E 56 (1997) 6620-6632.
23. H.C. Öttinger, M. Grmela, Dynamics and Thermodynamics of complex fluids. II. Illustrations of the GENERIC formalism, Phys. Rev. E 56 (1997) 6633-6655.
24. B.J. Edwards, An analysis of single and double generator thermodynamic formalisms for the macroscopic description of complex fluids, J. Non-Equilib. Thermodyn. 23 (1998) 300-332.
25. B.J. Edwards, A.N. Beris, H.C. Öttinger, An analysis of single and double generator thermodynamic formalisms for complex fluids. II. The microscopic description, J. Non-Equilib. Thermodyn. 23 (1998) 334-350.
26. H.C. Öttinger, GENERIC formulation of Boltzmann's kinetic equation, J. Non-Equilib. Thermodyn. 22 (1997) 386-391.
27. M. Grmela, P.J. Carreau, Conformation tensor rheological models, J. Non-Newtonian Fluid Mech. 23 (1987) 271-294.
28. M. Grmela, Hamiltonian dynamics of incompressible elastic fluids, Phys. Lett. A130 (1988) 81-86.
29. M. Grmela, Hamiltonian mechanics of complex fluids, J. Phys. A: Math. Gen. 22 (1989) 4375-4394.
30. M. Grmela, Dependence of the stress tensor on the intramolecular viscosity, J. Rheol. 33 (1989) 207-231.
31. A.N. Beris, B.J. Edwards, Poisson bracket formulation of viscoelastic flow equations of differential type: a unified approach, J. Rheol. 34 (1990) 503-538.
32. B.J. Edwards, A.N. Beris, M. Grmela, Generalized constitutive equation for polymeric liquid crystals. Part 1. Model formulation using the Hamiltonian (Poisson bracket) formulation, J. Non-Newtonian Fluid Mech. 35 (1990) 51-72.
33. B.J. Edwards, A.N. Beris, M. Grmela, R.G. Larson, Generalized constitutive equation for polymeric liquid crystals. Part 2. Non-homogeneous systems, J. Non-Newtonian Fluid Mech. 36 (1990) 243-254.
34. B.J. Edwards, A.N. Beris, M. Grmela, The dynamical behavior of liquid crystals: a continuum description through generalized brackets, Mol. Cryst. Liq. Cryst. 201 (1991) 51-86.
35. B.J. Edwards, A.N. Beris, The dynamics of a thermotropic liquid crystal, Eur. J. Mech. B/Fluids 11 (1992) 121-142.
36. F. Dupret, J.M. Marchal, Loss of evolution in the flow of viscoelastic fluids, J. Non-Newtonian Fluid Mech. 20 (1986) 143-171.
37. G. Marrucci, G. Titomanlio, G.C. Sarti, Testing of a constitutive equation for entangled networks by elongational and shear data of polymer melts, Rheol. Acta 12 (1973) 269-275.
38. D. Acierno, F.P. La Mantia, G. Marrucci, G. Rizzo, G. Titomanlio, A nonlinear viscoelastic model with structure-dependent relaxation times. I. Basic formulation, J. Non-Newtonian Fluid Mech. 1 (1976) 125-146.
39. M. Doi, Molecular dynamics and rheological properties of concentrated solutions of rod-like polymers in isotropic and liquid crystalline phases, J. Polym. Sci.: Polym. Phys. Ed. 19 (1981) 229-243.
40. R.G. Larson, Arrested tumbling in shearing flows of liquid crystal polymers, Macromolecules 23 (1990) 3983-3992.
41. G. Marrucci, P.L. Maffettone, Nematic phase of rodlike polymers. I. Prediction of transient behavior at high shear rates, J. Rheol. 34 (1990) 1217-1230.
42. N.C. Andrews, B.J. Edwards, A.J. McHugh, Continuum dynamic behavior of homogeneous liquid-crystalline polymers under the imposition of shear and magnetic fields, J. Rheol. 39 (1995) 1161-1181.
43. N.C. Andrews, A.J. McHugh, B.J. Edwards, Out-of-plane orientational dynamics of polymer liquid crystals under flow, J. Rheol. 40 (1996) 459-469.
44. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, 2nd Edition, Wiley-Interscience, New York, 1987.
45. L. Onsager, Reciprocal relations in irreversible processes. Part 1, Phys. Rev. 37 (1931) 405-426.
46. L. Onsager, Reciprocal relations in irreversible processes. Part 2, Phys. Rev. 38 (1931) 2265-2279.
47. H.B.G. Casimir, On Onsager's principle of microscopic reversibility, Rev. Mod. Phys. 17 (1945) 343-350.
48. C. Truesdell, Rational Thermodynamics, 2nd Edition, Springer, New York, 1984.
49. B.J. Edwards, H.C. Öttinger, R.J.J. Jongschaap, On the relationships between thermodynamic formalisms for complex fluids, J. Non-Equilib. Thermodyn. 22 (1997) 356-373.
50. B.J. Edwards, H.C. Öttinger, Time-structure invariance criteria for closure approximations, Phys. Rev. E 56 (1997) 4097-4103.
51. H.C. Öttinger, A thermodynamically admissible reptation model for fast flows of entangled polymers, J. Rheol. 43 (1999) 1461-1493.
52. H.C. Öttinger, Thermodynamically admissible reptation models with anisotropic tube cross sections and convective constraint release, J. Non-Newtonian Fluid Mech. 89 (2000) 165-185.
53. A.N. Beris, V.G. Mavrantzas, On the compatibility between various macroscopic formalisms for the concentration and flow of dilute polymer solutions, J. Rheol. 38 (1994) 1235-1250.
54. M.D. Graham, Stress/concentration/slip instabilities in couette flow of polymer solutions, in: Paper Presented in the Session 128 at the 1999 National AIChE Meeting on Fundamental Research in Fluid Mechanics: Non-Newtonian Flows, Dallas, November 1999.
55. B.J. Edwards, A.N. Beris, V.G. Mavrantzas, A model with two coupled Maxwell modes, J. Rheol. 40 (1996) 917-942.
56. V.G. Mavrantzas, A.N. Beris, A hierarchical model for surface effects on chain conformation and rheology of polymer solutions. I. General formulation, J. Chem. Phys. 110 (1999) 616-627.
57. V.G. Mavrantzas, A.N. Beris, A hierarchical model for surface effects on chain conformation and rheology of polymer solutions. II. Application to a neutral surface, J. Chem. Phys. 110 (1999) 628-638.
58. P. Hain, G. Lielens, R. Keunings, V. Legat, The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations, J. Non-Newtonian Fluid Mech. 79 (1998) 387-403.
59. R. Keunings, Advances in the computer modeling of the flow of polymeric liquids, Comp. Fluid Dynamics, 1999, in press.
60. B.J. Edwards, A.N. Beris, Noncanonical Poisson bracket for nonlinear elasticity with extensions to viscoelasticity, J. Phys. A: Math. Gen. 24 (1991) 2461-2480.
61. B.J. Edwards, A.N. Beris, Rotational motion and Poisson bracket structures in rigid particle dynamics and anisotropic fluid theory, Open Syst. & Information Dyn. 5 (1998) 333-368.