A geometric look at momentum flux and stress in fluid mechanics

Andrew D. Gilbert; Jacques Vanneste

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arXiv · 1911.06613

A geometric look at momentum flux and stress in fluid mechanics

Andrew D. Gilbert·Jacques Vanneste

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Abstract

We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form valued 2-forms, and their divergence as a covariant exterior derivative. We review the necessary tools of differential geometry and obtain the corresponding coordinate-free form of the equations of motion for a variety of inviscid fluid models -- compressible and incompressible Euler equations, Lagrangian-averaged Euler-$\alpha$ equations, magnetohydrodynamics and shallow-water models -- using a variational derivation which automatically yields a symmetric momentum flux. We also consider dissipative effects and discuss the geometric form of the Navier--Stokes equations for viscous fluids and of the Oldroyd-B model for visco-elastic fluids.

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Andrew D. Gilbert, Jacques Vanneste. 2019-11-15. A geometric look at momentum flux and stress in fluid mechanics. https://arxiv.org/abs/1911.06613

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