Dissipative port-Hamiltonian Formulation of Maxwell Viscoelastic Fluids

Authors

Luis A. Mora, Yann Le Gorrec, Hector Ramirez, Juan Yuz, Bernhard Maschke

Abstract

In this paper we consider general port-Hamiltonian formulations of multidimensional Maxwell’s viscoelastic fluids. Two different cases are considered to describe the energy fluxes in isentropic compressible and incompressible fluids. In the compressible case, the viscoelastic effects of shear and dilatational strains on the stress tensor are described individually through the corresponding constitutive equations. In the incompressible case, an approach based on the bulk modulus definition is proposed in order to obtain an appropriate characterization, from the port-Hamiltonian point of view, of the pressure and nonlinear terms in the momentum equation, associated with both dynamic pressure and vorticity of the flow.

Keywords

Port-Hamiltonian systems; Non-Newtonian Fluids; Maxwell’s viscoelasticity

Citation

• Journal: IFAC-PapersOnLine
• Year: 2021
• Volume: 54
• Issue: 14
• Pages: 430–435
• Publisher: Elsevier BV
• DOI: 10.1016/j.ifacol.2021.10.392
• Note: 3rd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems MICNON 2021- Tokyo, Japan, 15-17 September 2021

BibTeX

@article{Mora_2021,
  title={{Dissipative port-Hamiltonian Formulation of Maxwell Viscoelastic Fluids}},
  volume={54},
  ISSN={2405-8963},
  DOI={10.1016/j.ifacol.2021.10.392},
  number={14},
  journal={IFAC-PapersOnLine},
  publisher={Elsevier BV},
  author={Mora, Luis A. and Gorrec, Yann Le and Ramirez, Hector and Yuz, Juan and Maschke, Bernhard},
  year={2021},
  pages={430--435}
}

Download the bib file

References

• Altmann, R. & Schulze, P. A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. Systems & Control Letters vol. 100 51–55 (2017) – 10.1016/j.sysconle.2016.12.005
• Bird, (2015)
• Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics (Springer Basel, 2014). doi:10.1007/978-3-0348-0822-4 – 10.1007/978-3-0348-0822-4
• Bollada, P. C. & Phillips, T. N. On the Mathematical Modelling of a Compressible Viscoelastic Fluid. Archive for Rational Mechanics and Analysis vol. 205 1–26 (2012) – 10.1007/s00205-012-0496-5
• Brugnoli, A., Alazard, D., Pommier-Budinger, V. & Matignon, D. Port-Hamiltonian formulation and symplectic discretization of plate models Part I: Mindlin model for thick plates. Applied Mathematical Modelling vol. 75 940–960 (2019) – 10.1016/j.apm.2019.04.035
• Edwards, B. J. & Beris, A. N. Remarks concerning compressible viscoelastic fluid models. Journal of Non-Newtonian Fluid Mechanics vol. 36 411–417 (1990) – 10.1016/0377-0257(90)85021-p
• Gresho, Incompressible Flow and the Finite Element Method. (1998)
• Huo, X. & Yong, W.-A. Structural stability of a 1D compressible viscoelastic fluid model. Journal of Differential Equations vol. 261 1264–1284 (2016) – 10.1016/j.jde.2016.03.041
• Hütter, M., Carrozza, M. A., Hulsen, M. A. & Anderson, P. D. Behavior of viscoelastic models with thermal fluctuations. The European Physical Journal E vol. 43 (2020) – 10.1140/epje/i2020-11948-9
• John, (2016)
• Joseph, (1990)
• Landau, (1987)
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Macchelli, A., Le Gorrec, Y. & Ramírez, H. Boundary Energy-Shaping Control of an Ideal Compressible Isentropic Fluid in 1-D. IFAC-PapersOnLine vol. 50 5598–5603 (2017) – 10.1016/j.ifacol.2017.08.1105
• Mackay, A. T. & Phillips, T. N. On the derivation of macroscopic models for compressible viscoelastic fluids using the generalized bracket framework. Journal of Non-Newtonian Fluid Mechanics vol. 266 59–71 (2019) – 10.1016/j.jnnfm.2019.02.006
• Massey, (2012)
• Matignon, D. & Hélie, T. A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems. European Journal of Control vol. 19 486–494 (2013) – 10.1016/j.ejcon.2013.10.003
• Mora, L. A., Gorrec, Y. L., Matignon, D., Ramirez, H. & Yuz, J. I. About Dissipative and Pseudo Port-Hamiltonian Formulations of Irreversible Newtonian Compressible Flows. IFAC-PapersOnLine vol. 53 11521–11526 (2020) – 10.1016/j.ifacol.2020.12.604
• Murdock, (1993)
• Öttinger, Modeling complex fluids with a tensor and a scalar as structural variables. Revista Mexicana de Fisica (2002)
• Trenchant, V., Fares, Y., Ramirez, H. & Le Gorrec, Y. A port-Hamiltonian formulation of a 2D boundary controlled acoustic system. IFAC-PapersOnLine vol. 48 235–240 (2015) – 10.1016/j.ifacol.2015.10.245
• Trenchant, V., Ramirez, H., Le Gorrec, Y. & Kotyczka, P. Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct. Journal of Computational Physics vol. 373 673–697 (2018) – 10.1016/j.jcp.2018.06.051
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• van der Schaft, (2014)
• Vu, Distributed and backstepping boundary controls to achieve IDA-PBC design. IFAC-PapersOnLine (2015)