Exterior and vector calculus views of incompressible Navier-Stokes port-Hamiltonian models

Rashad, Ramy; Califano, Federico; Brugnoli, Andrea; Schüller, F.

doi:10.1016/j.ifacol.2021.11.074

Cited by 3 publications

(4 citation statements)

References 46 publications

(128 reference statements)

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“…The mathematical study of alternative variational methods as applied to the Navier–Stokes equations in particular remains an ongoing endeavour (Oseledets 1989; Vujanovic & Jones 1989; Doering & Gibbon 1995; Fukagawa & Fujitani 2012; Jones 2015; Gay-Balmaz & Yoshimura 2017; Hieber, Robinson & Shibata 2017; Hochgerner 2018; Gay-Balmaz & Yoshimura 2019 a , b ; Rashad et al. 2021; Gonzalez & Taha 2022; Taha & Gonzalez 2022; Sanders 2023 b ).…”

Section: Literature Reviewmentioning

confidence: 99%

“…Rashad et al. (2021) modelled the incompressible Navier–Stokes equations in so-called ‘port-Hamiltonian’ framework rather than the standard Hamiltonian framework. Their model used vector calculus instead of exterior calculus to minimize the number of operators.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Within the last 80 years, many alternative methods have been developed in an attempt to circumvent the non-self-adjointness of dissipative systems (Prigogine & Glansdorff 1965;Biot 1970;Finlayson 1972a;Lebon & Lambermont 1973;Tonti 1973;Magri 1974;Telega 1979;Tonti 1984;Filippov 1989;Sieniutycz 2000;Robinson 2001;Galley 2013;Kim, Dargush & Lee 2016;Mottaghi et al 2019;Taroco et al 2020;Bersani & Caressa 2021;Junker & Balzani 2021). The mathematical study of alternative variational methods as applied to the Navier-Stokes equations in particular remains an ongoing endeavour (Oseledets 1989;Vujanovic & Jones 1989;Doering & Gibbon 1995;Fukagawa & Fujitani 2012;Jones 2015;Gay-Balmaz & Yoshimura 2017;Hieber, Robinson & Shibata 2017;Hochgerner 2018;Gay-Balmaz & Yoshimura 2019a,b;Rashad et al 2021;Sanders 2023b). Oseledets (1989) attempted to express the Navier-Stokes equations using Hamiltonian formalism.…”

Section: Literature Reviewmentioning

confidence: 99%

“…The study was able to derive the Navier-Stokes equation from a stochastic Hamiltonian system but ignored the stress tensor, was unable to separate configuration and momentum variables and did not establish energy conservation or dissipation. Rashad et al (2021) modelled the incompressible Navier-Stokes equations in so-called 'port-Hamiltonian' framework rather than the standard Hamiltonian framework. Their model used vector calculus instead of exterior calculus to minimize the number of operators.…”

Section: Literature Reviewmentioning

confidence: 99%

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A canonical Hamiltonian formulation of the Navier–Stokes problem

Sanders,

DeVoria,

Washuta

et al. 2024

J. Fluid Mech.

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This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ( $u_{i}$ , $p$ ) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$ . Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$ . If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

See 2 more Smart Citations

A canonical Hamiltonian formulation of the Navier–Stokes problem

Sanders,

DeVoria,

Washuta

et al. 2024

J. Fluid Mech.

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Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation

et al. 2023

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In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.

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