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Constructing Poisson and Dissipative Brackets of Mixtures by using Lagrangian-to-Eulerian Transformation

Published online by Cambridge University Press:  05 May 2011

K.-C. Chen
K.-C. Chen*
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
* Professor, corresponding author
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Abstract

This paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian- to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets.

Keywords

Noncanonical poisson bracketDissipative bracketLagrangian-to-Eulerian transformationMicromorphic fluid
Type
Articles
Information
Journal of Mechanics , Volume 26 , Issue 2 , June 2010 , pp. 219 - 228
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2010

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References

1.Zhu, W. Q., “Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation,” Applied Mechanics Reviews, 59, pp. 230–248 (2006).CrossRefGoogle Scholar
2.Goldstein, H., Classical Mechanics, Addison-Wesley, Reading, MA (1980).Google Scholar
3.Arnold, V. I., “Sur La Geometrie Differentielle Des Groups De Lie De Dimension Infinie Et Ses Applications a L'Hydrodynamique Des Fluids Parfaits,” Annales de L'Institut Fourier, 16, pp. 319–361 (1966).Google Scholar
4.Marsden, J. E. and Weinstein, A., “The Hamiltonian Structure of the Maxwell-Vlasov Equation,” Physica D, 4, pp. 394–406 (1982).Google Scholar
5.Kaufman, A. N., “Dissipative Hamiltonian Systems: A Unifying Principle,” Physics Letters A, 100, pp. 419–422(1984).CrossRefGoogle Scholar
6.Morrison, P. J., “Bracket Formulation for Irreversible Classical Fields,” Physics Letters A, 100, pp. 423–427 (1984).CrossRefGoogle Scholar
7.Grmela, M., “Bracket Formulation of Dissipative Fluid Mechanics Equations,” Physics Letters A, 102, pp. 355–358(1984).Google Scholar
8.Edwards, B. J., “An Analysis of Single and Double Generator Thermodynamics Formulations for the Macroscopic Description of Complex Fluids,” Journal of Non-Equilibrium Thermodynamics, 23, pp. 301–333 (1998).Google Scholar
9.Grmela, M., Elafif, A. and Lebon, G., “Isothermal Nonstandard Diffusion in a Two-Component Fluid Mixture: A Hamiltonian Approach,” Journal of Non-Equilibrium Thermodynamics, 23, pp. 376–390 (1998).CrossRefGoogle Scholar
10.Öttinger, H. C., “Nonequilibrium Thermodynamics - A Tool for Applied Rheologists,” Applied Rheology, 9, pp. 17–26 (1999).Google Scholar
11.Beris, A. N., “Bracket Formulation as a Source for the Development of Dynamic Equation in Continuum Mechanics,” Journal of Non-Newtonian Fluid Mechanics, 96, pp. 119–136(2001).CrossRefGoogle Scholar
12.Mariano, P. M., “Mecahnics of Quasi-Periodic Alloys,” Journal of Nonlinear Science, 16, pp. 45–77 (2006).CrossRefGoogle Scholar
13.Liu, C. S., “Five Different Formulations of the FiniteStrain Perfectly Plastic Equations,” Computer Modeling in Engineering and Sciences, CMES, 17, pp. 73–94 (2007).Google Scholar
14.Abarbanel, H. D. I., Brown, R. and Yang, Y. M., “Hamiltonian Formulation of Inviscid Flows with Free Boundaries,” Physics of Fluids, 31, pp. 2802–2809 (1988).CrossRefGoogle Scholar
15.Marsden, J. E., Lectrues on Mechanics, Cambridge University, Cambridge (1992).Google Scholar
16.Marsden, J. E., Misiolek, G., Ortega, J. P., Perlmutter, M., and Ratiu, T. S., Hamiltonian Reduction by Stages, Springer-Verlag, New York (2007).Google Scholar
17.Edwards, B. J. and Beris, A. N., “Non-Canonical Poisson Bracket for Nonlinear Elasticity with Extensions to Viscoelasticity,” Journal of Physics A: Mathematical and General, 24, pp. 2461–2480 (1991).Google Scholar
18.Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, Springer, New York (1994).Google Scholar
19.Beris, A. N. and Edwards, B. J., Thermodynamics of Flowing Systems, Oxford University Press, New York (1994).Google Scholar
20.Edwards, B. J. and Beris, A. N., “Rotational Motion and Poisson Bracket Structures in Rigid Particle Systems and Anisotropic Fluid Theory,” Open Systems and Information Dynamics, 5, pp. 333–368 (1998).Google Scholar
21.Chen, K. C., “Noncanonical Poisson Brackets for Elastic and Micromorphic Solids,” International Journal of Solids and Structures, 44, pp. 7715–7730 (2007).CrossRefGoogle Scholar
22.Grmela, M., Lebon, G. and Lhuillier, D., “A Comparative Study of the Coupling of Flow with Non-Fickean Thermodiffusion. Part II: GENERIC,” Journal of Non-Equilibrium Thermodynamics, 28, pp. 23–50 (2003).Google Scholar
23.Elafif, A., Grmela, M. and Lebon, G., “Rheology and Diffusion in Simple and Complex Fluids,” Journal of Non-Newtonian Fluid Mechanics, 86, pp. 253–275 (1999).CrossRefGoogle Scholar
24.Hutter, K. and Jöhnk, K., Continuum Methods of Physical Modeling, Springer-Verlag, Berlin (2004)Google Scholar
25.Eringen, A. C., Microcontinuum Field Theories I. Foundations and Solids, Springer-Verlag, New York (1999).Google Scholar
26.Maugin, G. A., “The Method of Virtual Power in Continuum Mechanics: Applications to Coupled Fields,” ActaMechanica, 35, pp. 1–70 (1980).Google Scholar
27.Chen, K. C., “Microcontinuum Balance Equations Revisited: The Mesoscopic Approach,” Journal of Non-Equilibrium Thermodynamics, 32, pp. 435–458 (2007).CrossRefGoogle Scholar
28.Kotowski, R. and Radzikowska, E., “Variational Principle for Dissipative Processes in Continuum Physics,” International Journal of Engineering Science, 38, pp. 1129–1149(2000).Google Scholar
29.Fried, E., Gurtin, M. E. and Hutter, K., “A Void-Based Description of Compaction and Segregation in Flowing Granular Materials,” Continuum Mechanics and Thermodynamics, 16, pp. 199–219(2004).Google Scholar
30.Chen, K. C. and Tai, Y. C., “Volume-Weighted Mixture Theory for Granular Materials,” Continuum Mechanics and Thermodynamics, 19, pp. 457–474 (2008).CrossRefGoogle Scholar
31.Grmela, M. and Teichmann, J., “Lagrangian Formulation of Maxwell-Cattaneo Hydrodynamics,” International Journal of Engineering Science, 21, pp. 297–313 (1983).Google Scholar