Navier–Stokes Equations on Riemannian Manifolds: Mathematical Formulation and Numerical Simulations

Fluid flows on curved manifolds arise naturally in a variety of applications, ranging from surface fluid dynamics and geophysical flows to relativistic hydrodynamics. This talk presents the mathematical formulation and numerical simulation of incompressible fluid flows on Riemannian manifolds from an intrinsic perspective.

The first part introduces the minimal knowledge of Riemannian geometry required to derive and understand the Navier–Stokes equations on manifolds. Starting from the classical equations in Euclidean space, the governing equations are extended to general Riemannian manifolds using metric tensors, covariant derivatives, and intrinsic differential operators. This provides a unified mathematical framework applicable to both two- and three-dimensional manifolds.

The second part focuses on finite element discretisations and numerical simulations. An intrinsic finite element method formulated directly in parameter space is presented and compared with surface finite element methods. The proposed discretisation satisfies a discrete energy estimate, providing a theoretical basis for its stability. Numerical experiments demonstrate the evolution of incompressible viscous flows towards steady states characterised by Killing vector fields on both two- and three-dimensional manifolds. Eigenvalue problems associated with the underlying geometric differential operators are also considered to further compare and validate the numerical methods. Different numerical approaches are compared in terms of accuracy, efficiency, and their ability to capture the intrinsic geometry of the manifold.

The talk concludes with a discussion of ongoing and future work in geometric fluid dynamics, including fluid–structure interaction, mean curvature flow, and relativistic hydrodynamics.