A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds – stabilized finite element formulation

This work presents a stabilized finite element formulation of the arbitrary Lagrangian-Eulerian (ALE) surface theory for Navier-Stokes flow on self-evolving manifolds developed in Sauer [1]. The formulation is physically frame-invariant, applicable to large deformations, and relevant to fluidic surfaces such as soap films, capillary menisci and lipid membranes, which are complex and inherently unstable physical systems. It is applied here to area-incompressible surface flows using a stabilized pressure-velocity (or surface tension-velocity) formulation based on quadratic finite elements and implicit time integration. The unknown ALE mesh motion is determined by membrane elasticity such that the in-plane mesh motion is stabilized without affecting the physical behavior of the system. The resulting three-field system is monolithically coupled, and fully linearized within the Newton-Rhapson solution method. The new formulation is demonstrated on several challenging examples including shear flow on self-evolving surfaces and inflating soap bubbles with partial inflow on evolving boundaries. Optimal convergence rates are obtained in all cases. Particularly advantageous are $C^1$-continuous surface discretizations, for example based on NURBS.