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

TL;DR

The paper develops a stabilized finite element framework for Navier–Stokes flow on self-evolving, curvilinear surfaces using an ALE description. A four-field model (fluid velocity, surface tension, mesh velocity, and surface position) is discretized monolithically, with the unknown in-plane mesh motion driven by in-plane membrane elasticity to stabilize the mesh without altering physical behavior. Stability is achieved via the Dohrmann–Bochev pressure stabilization, integrated over the current surface, and the time integration employs a second-order implicit trapezoidal scheme. The approach is validated on a suite of fixed and deforming-surface problems, including shear flows on evolving surfaces, octahedral vortex configurations, and soap-bubble inflation, with optimal convergence observed across cases and clear advantages when using C1-continuous discretizations such as NURBS. The results demonstrate robust handling of large deformations, dynamic surface evolution, and complex boundary conditions, reinforcing the method’s potential for accurate surface FSI analyses in membranous and soap-film contexts.

Abstract

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 (2025). 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 C1-continuous surface discretizations, for example based on NURBS.

Figures (30)

• Figure 1: ALE surface description: The fluid surface $\mathcal{S}$ is described by the ALE mapping $\boldsymbol{x}=\boldsymbol{x}(\zeta^\alpha,t)$ from the ALE parameter domain $\mathcal{P}$. This mapping is then used for surface integration and discretization. It is generally different from the Lagrangian mapping $\boldsymbol{x}=\hat{\boldsymbol{x}}(\xi^\alpha,t)$ from the material parameter domain $\hat{\mathcal{P}}$. From the two mappings follow the surface velocities $\boldsymbol{v}_\mathrm{m} := \partial\boldsymbol{x}/\partial t|_{\zeta^\alpha}$ and $\boldsymbol{v} := \partial\hat{\boldsymbol{x}}/\partial t|_{\xi^\alpha}$, and their in-plane difference $\boldsymbol{u} := \boldsymbol{v} - \boldsymbol{v}_\mathrm{m}$.
• Figure 2: Simple shear flow on a rigid sphere: (a) Flow field $\boldsymbol{v}$ and (b) convergence behavior for ALE case a and load case 1. Optimal convergence rates are obtained for all fields: At least $O(h^3)=O(n^{-1.5}_\mathrm{el})$ for $\boldsymbol{v}$, and $O(h^2)=O(n^{-1}_\mathrm{el})$ for $q$, $p$ and $\omega$.
• Figure 3: Simple shear flow on a rigid sphere: (a) Surface tension $q$, (b) surface vorticity $\omega$ and (c) lateral surface pressure $p$ for ALE case a and load case 1. The plots are normalized by $q_0 = \rho r^2\omega_0^2$, $w_0$ and $p_0 = \rho r\omega_0^2$.
• Figure 4: Simple shear flow on a rigid sphere: (a) Flow field $\boldsymbol{v}$ and (b) convergence behavior for the prescribed constant mesh velocity $\boldsymbol{v}_\mathrm{m}=\boldsymbol{c}_0$ given in \ref{['e:vm0']} (ALE case b, load case 1). In this case, $\boldsymbol{v}_\mathrm{m}$, and hence also $\boldsymbol{v}$, have out-of-plane components. As in Fig. \ref{['f:shearflow1']}, optimal convergence rates are obtained for all fields. The dashed line in (b) is the line from Fig. \ref{['f:shearflow1']}b.
• Figure 5: Simple shear flow on a rigid sphere: (a) Flow field $\boldsymbol{v}$ and (b) convergence behavior due to the prescribed fixed mesh distortion $\boldsymbol{x}_\mathrm{m}$ given in \ref{['e:xm1']} & \ref{['e:tm0']} (ALE case c, load case 1). Again, optimal convergence rates are obtained for all fields. The dashed lines in (b) are the lines from Fig. \ref{['f:shearflow1']}b.

Theorems & Definitions (10)

• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Remark 5.1
• Remark 5.2
• Remark B.1
• Remark B.2