Geometric and energy-aware decomposition of the Navier–Stokes equations: A port-Hamiltonian approach

Abstract

A port-Hamiltonian model for compressible Newtonian fluid dynamics is presented in entirely coordinate-independent geometric fashion. This is achieved by the use of tensor-valued differential forms that allow us to describe the interconnection of the power preserving structure which underlies the motion of perfect fluids to a dissipative port which encodes Newtonian constitutive relations of shear and bulk stresses. The relevant diffusion and the boundary terms characterizing the Navier–Stokes equations on a general Riemannian manifold arise naturally from the proposed construction.

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

“…In this manner, we avoid overloading this paper with all the technicalities involved in the derivation process. In a future sequel of this paper, we shall present the derivation of these equations from first principles in the port-Hamiltonian framework and highlight the underlying energetic structure, similar to our previous works on fluid mechanics [39,40,41,42].…”

A differential geometric description of thermodynamics in continuum mechanics with application to Fourier–Navier–Stokes fluids

“…In order to derive it, we calculate K using continuity and momentum equations ( 5) and (9). One way to perform the calculation is by considering the kinetic energy density a function of the velocity one-form ν = u ♭ and the mass density µ, i.e., K = 1 2 (ι ν # ν)µ, and computing the rate using the chain rule (see e.g., 23,24 ) as…”

“…Consequently, we have that ∇u ∧T p = −pdiv(u)V. Notice that this term is proportional to the divergence of the velocity vector field of the continuum, and appears with opposite sign in the kinetic and internal energy equations, representing a reversible exchange between these two energy fields due to work of compression or expansion. Using this result in (23) and the fact that DV Dt = div(u)V, we end up with the entropy equation…”

“…Whereas for the residual stress tensor T r , it is given by the viscous stress tensor, given by the sum of a bulk stress T λ and a shear stress T κ , defined by 11,23 T λ ∶= λ(div(u)V), (28)…”

Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

“…Since they are closed under interconnection (this feature stems from the properties of the Dirac structure [11]), pH systems have the potential to tackle complex multiphysical engineering applications. So far they were employed to model fluid-structure coupled phenomena [12], reactive flows [13], Euler and Navier-Stokes equations [14,15,16], thin mechanical and thermomecanical structures [17,18]. The interested reader may consult [19] for a comprehensive review on distributed pH systems.…”