The formulation of the Navier–Stokes equations on Riemannian manifolds

Chan, Chi Hin; Czubak, Magdalena; Disconzi, Marcelo M.

doi:10.1016/j.geomphys.2017.07.015

Cited by 44 publications

(52 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But the Killing fields on the sphere correspond to the rotating motion which is physically very natural. We think that this is one more argument in favor of L compared to ∆ H , in addition to the discussion in [2].…”

Section: For All Harmonic V}mentioning

confidence: 76%

“…It seems to us that the operator L is physically most natural candidate for the diffusion operator because it most naturally generalizes the constitutive laws which are used in the Euclidean spaces. Also in [2] the authors come to the conclusion that L is the best choice. However, in [6] it is argued that ∆ H should be used, at least in some situations.…”

Section: Model and The Diffusion Operatormentioning

confidence: 99%

“…Navier Stokes equations have been widely studied both form theoretical and applied points of view [12]. In recent years there has been a growing interest of this system on Riemannian manifolds [1,2,6,8,9,11] and the many references therein. There seems to be two different reasons for this interest.…”

Section: Introductionmentioning

confidence: 99%

“…There have been different choices for the diffusion operator for the system on the manifolds, and we discuss first some reasons why we think that the choice adopted below is the appropriate one. The same choice is advocated also in [2,11]. It turns out that this choice of diffusion operator has important consequences on the qualitative and asymptotic properties of the flow, and our choice implies that Killing vector fields are essential in the analysis.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Navier–Stokes equations on Riemannian manifolds

Samavaki

Tuomela

2020

Journal of Geometry and Physics

View full text Add to dashboard Cite

We study properties of the solutions to Navier-Stokes system on compact Riemannian manifolds. The motivation for such a formulation comes from atmospheric models as well as some thin film flows on curved surfaces. There are different choices of the diffusion operator which have been used in previous studies, and we make a few comments why the choice adopted below seems to us the correct one. This choice leads to the conclusion that Killing vector fields are essential in analyzing the qualitative properties of the flow. We give several results illustrating this and analyze also the linearized version of Navier-Stokes system which is interesting in numerical applications. Finally we consider the 2 dimensional case which has specific characteristics, and treat also the Coriolis effect which is essential in atmospheric flows.

show abstract

Section: For All Harmonic V}mentioning

confidence: 76%

Section: Model and The Diffusion Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Navier–Stokes equations on Riemannian manifolds

Samavaki

Tuomela

2020

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…The SDE's can be reexpressed by substituting the Stratonovich definition (20). The forward equations are expressed as…”

Section: Appendix A: Forward Sde In Ito Definitionmentioning

confidence: 99%

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

Koide

Kodama

2020

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, differently from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier-Stokes-Fourier (NSF) equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gaugeinvariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.

show abstract

A meshfree Lagrangian method for flow on manifolds

Suchde

2021

Numerical Methods in Fluids

View full text Add to dashboard Cite

In this article, we present a novel meshfree framework for fluid flow simulations on arbitrarily curved surfaces. First, we introduce a new meshfree Lagrangian framework to model flow on surfaces. Meshfree points or particles, which are used to discretize the domain, move in a Lagrangian sense along the given surface. This is done without discretizing the bulk around the surface, without parametrizing the surface, and without a background mesh. A key novelty that is introduced is the handling of flow with evolving free boundaries on a curved surface. The use of this framework to model flow on moving and deforming surfaces is also introduced. Then, we present the application of this framework to solve fluid flow problems defined on surfaces numerically. In combination with a meshfree generalized finite difference method (GFDM), we introduce a strong form meshfree collocation scheme to solve the Navier–Stokes equations posed on manifolds. Benchmark examples are proposed to validate the Lagrangian framework and the surface Navier–Stokes equations with the presence of free boundaries.

show abstract

The formulation of the Navier–Stokes equations on Riemannian manifolds

Cited by 44 publications

References 48 publications

Navier–Stokes equations on Riemannian manifolds

Navier–Stokes equations on Riemannian manifolds

Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

A meshfree Lagrangian method for flow on manifolds

Contact Info

Product

Resources

About