A Geometric Look at Momentum Flux and Stress in Fluid Mechanics

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-α equations, magneto-hydrodynamics 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.