A Geometric Look at Momentum Flux and Stress in Fluid Mechanics

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, their divergence covariant exterior derivative. review necessary tools differential geometry obtain corresponding coordinate-free form equations motion for variety inviscid models—compressible incompressible Euler equations, Lagrangian-averaged Euler- $$\alpha $$ ? magnetohydrodynamics shallow-water models—using variational derivation which automatically yields symmetric momentum flux. also consider dissipative effects discuss Navier–Stokes viscous fluids Oldroyd-B model visco-elastic fluids.