A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds -- General theory and analytical solutions

TL;DR

This paper introduces a novel ALE formulation for Navier-Stokes flows on self-evolving surfaces, enabling detailed analysis of fluidic membranes using surface meshes and analytical solutions for verification.

Contribution

It presents a general curvilinear surface ALE framework with analytical solutions, facilitating simulations on deforming surfaces and advancing computational methods for fluidic membranes.

Findings

Allows simulation of flows on deforming surfaces using surface meshes

Provides analytical solutions for validation of numerical methods

Enables detailed study of fluidic membranes like soap films and lipid bilayers

Abstract

A new arbitrary Lagrangian-Eulerian (ALE) formulation for Navier-Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterization that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier-Stokes equations, and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.