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Irreversible Port-Hamiltonian Formulations for 1-Dimensional fluid systems

Ahlam Ouardi, Arijit Sarkar, Hector Ramirez, Yann Le Gorrec

Abstract

The Irreversible Port-Hamiltonian Systems (IPHS) framework is extended to the modelling of non-isentropic fluids with viscous dissipation in the Eulerian description. Building on earlier IPHS formulations for diffusion-driven and non-convective distributed systems, it is shown that convective transport can be consistently encompassed by the framework by modifying the underlying differential operators. After revisiting the constitutive relations of non-isentropic fluids in both Eulerian and Lagrangian coordinates, it is demonstrate how these systems fit within an extended IPHS formulation. Furthermore, an extended parametrisation of the boundary port variables which ensures that the first and second laws of Thermodynamics are fulfilled allows to define a general class of boundary controlled IPHS.

Irreversible Port-Hamiltonian Formulations for 1-Dimensional fluid systems

Abstract

The Irreversible Port-Hamiltonian Systems (IPHS) framework is extended to the modelling of non-isentropic fluids with viscous dissipation in the Eulerian description. Building on earlier IPHS formulations for diffusion-driven and non-convective distributed systems, it is shown that convective transport can be consistently encompassed by the framework by modifying the underlying differential operators. After revisiting the constitutive relations of non-isentropic fluids in both Eulerian and Lagrangian coordinates, it is demonstrate how these systems fit within an extended IPHS formulation. Furthermore, an extended parametrisation of the boundary port variables which ensures that the first and second laws of Thermodynamics are fulfilled allows to define a general class of boundary controlled IPHS.
Paper Structure (9 sections, 7 theorems, 64 equations, 2 figures)

This paper contains 9 sections, 7 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-isentropic compressible fluids
  3. Governing equations : Eulerian coordinate
  4. Governing equations: Lagrangian coordinate
  5. IPHS formulations of non isentropic compressible fluids
  6. IPHS fromulation in Eulerian coordinates
  7. IPHS formulation in Lagrangian coordinates
  8. Boundary controlled IPHS
  9. Conclusion

Key Result

Lemma 1

$\mathcal{A}$ is the formal adjoint of $\mathcal{A}$, i.e. $\mathcal{B}=\mathcal{A}^\ast$.

Figures (2)

  • Figure 1: Eulerian and Lagragian coordinates.
  • Figure 2: Small fluid element in spatial and material coordinates.

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 4.1
  • Definition 4.2
  • Lemma 4: First law of Thermodynamics
  • Lemma 5: Second law of Thermodynamics
  • ...and 5 more