A description of thermodynamics for continuum mechanical systems is presented in the coordinate-free language of exterior calculus. First, a careful description of the mathematical tools that are needed to formulate the relevant conservation laws is given. Second, following an axiomatic approach, the two thermodynamic principles will be described, leading to a consistent description of entropy creation mechanisms on manifolds. Third, a specialization to Fourier–Navier–Stokes fluids will be carried through.
References
1.
E.
Kanso
, M.
Arroyo
, Y.
Tong
, A.
Yavari
, J. G.
Marsden
, and M.
Desbrun
, “On the geometric character of stress in continuum mechanics
,” Z. Angew. Math. Phys.
58
, 843
–856
(2007
).2.
P.
Asinari
and E.
Chiavazzo
, “Overview of the entropy production of incompressible and compressible fluid dynamics
,” Meccanica
51
, 1245
–1255
(2016
).3.
R.
Abraham
, J. E.
Marsden
, and T.
Ratiu
, Manifolds, Tensor Analysis, and Applications
(Springer Science & Business Media
, 2012), Vol. 75.4.
T.
Frankel
, The Geometry of Physics: An Introduction
(Cambridge University Press
, 2011
).5.
J. E.
Marsden
and R. H.
Abraham
, “Hamiltonian mechanics on Lie groups and hydrodynamics
,” in Global Analysis (American Mathematical Society, 1970
).6.
J. E.
Marsden
, T.
Ratiu
, and A.
Weinstein
, “Reduction and Hamiltonian structures on duals of semidirect product Lie algebras
,” in Fluids and Plasmas: Geometry and Dynamics
, Contemporary Mathematics Vol. 28
(American Mathematical Society
, 1984
), pp. 55
–100
.7.
J. E.
Marsden
, T.
Ratiu
, and A.
Weinstein
, “Semidirect Products and Reduction in Mechanics
,” Trans. Am. Math. Soc.
281
, 147
(1984
).8.
P. J.
Morrison
, “Hamiltonian description of the ideal fluid
,” Rev. Mod. Phys.
70
, 467
–521
(1998
).9.
R.
Rashad
, F.
Califano
, F. P.
Schuller
, and S.
Stramigioli
, “Port-Hamiltonian modeling of ideal fluid flow: Part I. Foundations and kinetic energy
,” J. Geom. Phys.
164
, 104201
(2021
).10.
R.
Rashad
, F.
Califano
, F. P.
Schuller
, and S.
Stramigioli
, “Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow
,” J. Geom. Phys.
164
, 104199
(2021
).11.
A. D.
Gilbert
and J.
Vanneste
, “A geometric look at momentum flux and stress in fluid mechanics
,” arXiv:1911.06613 (2019
).12.
L. A.
Mora
, Y.
Le Gorrec
, D.
Matignon
, H.
Ramirez
, and J. I.
Yuz
, “On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids
,” Phys. Fluids
33
, 117117
(2021
).13.
A.
van der Schaft
and B.
Maschke
, “Geometry of thermodynamic processes
,” Entropy
20
, 925
(2018
).14.
M.
Grmela
, “Contact geometry of mesoscopic thermodynamics and dynamics
,” Entropy
16
, 1652
–1686
(2014
).15.
A. M.
Badlyan
, B.
Maschke
, C.
Beattie
, and V.
Mehrmann
, “Open physical systems: From GENERIC to port-Hamiltonian systems
,” arXiv:1804.04064 (2018
).16.
D. N.
Arnold
, R. S.
Falkt
, and R.
Whither
, “Finite element exterior calculus, homological techniques, and applications
,” Acta Numer.
15
, 1
–155
(2006
).17.
P. B.
Bochev
and J. M.
Hyman
, “Principles of mimetic discretizations of differential operators
,” in Compatible Spatial Discretizations
, edited by D. N.
Arnold
, P. B.
Bochev
, R. B.
Lehoucq
, R. A.
Nicolaides
, and M.
Shashkov
(Springer
, New York
, 2006
), pp. 89
–119
.18.
E. S.
Gawlik
and F.
Gay-Balmaz
, “A variational finite element discretization of compressible flow
,” in Foundations of Computational Mathematics
(Springer
, 2021
), Vol. 21, pp. 961
–1001
.19.
A. N.
Hirani
, “Discrete exterior calculus
,” Ph.D. thesis (California Institute of Technology
, 2003
.20.
F.
Califano
, R.
Rashad
, F. P.
Schuller
, and S.
Stramigioli
, “Energetic decomposition of distributed systems with moving material domains: The port-Hamiltonian model of fluid-structure interaction
,” J. Geom. Phys.
175
, 104477
(2022
).21.
V.
Arnold
, “Topological methods in hydrodynamics
,” Annu. Rev. Fluid Mech.
24
, 145
–166
(1992
).22.
B. F.
Schutz
, Geometrical Methods of Mathematical Physics
(Cambridge University Press
, 1980
).23.
F.
Califano
, R.
Rashad
, F. P.
Schuller
, and S.
Stramigioli
, “Geometric and energy-aware decomposition of the Navier–Stokes equations: A port-Hamiltonian approach
,” Phys. Fluids
33
, 047114
(2021
).24.
A. J.
van der Schaft
and B. M.
Maschke
, “Hamiltonian formulation of distributed-parameter systems with boundary energy flow
,” J. Geom. Phys.
42
, 166
–194
(2002
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.