Energy-based, geometric, and compositional formulation of fluid and plasma models

TL;DR

This work develops and demonstrates the Exergetic Port-Hamiltonian Systems (EPHS) framework for energy-based, coordinate-invariant modeling of coupled fluid and plasma dynamics. By composing models from storage, reversible, and irreversible subsystems through a power-preserving graphical language, it builds from a 1D ideal barotropic fluid to a 3D NSF fluid, and further to Maxwell, EMHD, and MHD models, ensuring thermodynamic consistency via exergy-based storage and Dirac/Onsager structures. The approach guarantees energy conservation and non-negative entropy production, while supporting reuse and modular extension across physical domains. The geometric, exterior-calculus formulation facilitates coordinate-free modeling and lays groundwork for structure-preserving discretization and potential integration with scientific machine learning in multiphysics contexts.

Abstract

Fluid dynamics plays a crucial role in various multiphysics applications, including energy systems, electronics cooling, and biomedical engineering. Developing models for complex coupled systems can be challenging and time-consuming. In particular, ensuring the consistent integration of models from diverse physical domains requires meticulous attention. Considering the example of (electro-)magneto hydrodynamics (on a fixed spatial domain and with linear polarization and magnetization), this article demonstrates how relatively complex models can be composed from simpler parts by means of a formal language for multiphysics modeling. The Exergetic Port-Hamiltonian Systems (EPHS) modeling language features a simple graphical syntax for expressing the energy-based interconnection of subsystems. This reduces cognitive load and facilitates communication, especially in multidisciplinary environments. As the example demonstrates, existing models can be easily integrated as subsystems of new models. Specifically, an ideal fluid model is used as a subsystem of a Navier-Stokes-Fourier fluid model, which in turn is reused as a subsystem of an (electro-)magneto hydrodynamics model. The energy-based, compositional approach simplifies understanding complex models, and it makes it easy to encapsulate, reuse, and replace (parts of) models. Moreover, structural properties of EPHS models guarantee fundamental properties of thermodynamic systems, such as conservation of energy, non-negative entropy production, and Onsager reciprocal relations.

Figures (12)

• Figure 1: Interconnection pattern for an ideal barotropic fluid model. Box $\mathtt{ke}$ represents storage of kinetic energy, which is exchanged both in terms of (specific) momentum via port $\mathtt{ke.p_s}$ and mass via port $\mathtt{ke.m}$. Box $\mathtt{pps}$ represents the reversible transformation between two different representations of momentum, namely specific momentum (or velocity) exchanged via port $\mathtt{pps.p_s}$ and momentum density exchanged via port $\mathtt{pps.p}$. As the transformation depends on the mass density, the box has a state port $\mathtt{pps.m}$, drawn as a dashed line. Box $\mathtt{sa}$ represents self-advection of kinetic energy and the boundary port $\mathtt{b_k}$ accounts for advection across the boundary $\partial \mathcal{Z}$. Box $\mathtt{ie}$ represents storage of internal energy, which is exchanged in terms of mass via port $\mathtt{ie.m}$. Box $\mathtt{adv}$ represents advection of internal energy in terms of mass and the boundary port $\mathtt{b_m}$ accounts for advection across the boundary $\partial \mathcal{Z}$.
• Figure 2: Interconnection pattern for an ideal fluid model. Box $\mathtt{kin}$ represents the kinetic energy subsystem and box $\mathtt{int}$ represents the internal energy subsystem. The boundary ports $\mathtt{b_k}$, $\mathtt{b_m}$ and $\mathtt{b_s}$, which account for advection of kinetic and internal energy across $\partial \mathcal{Z}$, are not exposed, leading to a model with impermeable boundary.
• Figure 3: The pattern from \ref{['fig:if_expanded']} is shown in a simplified form, where multiple ports with a common prefix are drawn as a single line, called a multiport.
• Figure 4: Interconnection pattern for the canonical definition of the kinetic energy system. Box $\mathtt{ke}$ represents storage of kinetic energy, which is exchanged in terms of momentum via port $\mathtt{ke.p}$ and mass via port $\mathtt{ke.m}$. Box $\mathtt{sa}$ represents self-advection of kinetic energy and the boundary port $\mathtt{b_k}$ accounts for advection across $\partial \mathcal{Z}$. The outer port $\mathtt{f.p}$ allows for exchange of kinetic energy with other systems on the same domain $\mathcal{Z}$ and the outer state port $\mathtt{f.m}$ shares information about the fluid mass with other systems on $\mathcal{Z}$.
• Figure 5: Interconnection pattern of the kinetic energy system. In contrast to the pattern in \ref{['fig:kin']}, kinetic energy is expressed in terms of specific momentum (or velocity), rather than momentum density. To retain the same outer interface, the additional box $\mathtt{pps}$ represents the reversible transformation between the two alternative state variables.