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Description
A new arbitrary Lagrangian-Eulerian (ALE) formulation for area-incompressible Navier-Stokes flow on evolving surfaces is presented. The new formulation extends the surface ALE formulation of [1] to more general surface motions. It is based on a new 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 pressure, fluid velocity and surface motion, where the latter two share the same normal velocity. The new theory is implemented in the nonlinear finite element framework of [2] using the pressure stabilization scheme of [3] and the mesh stabilization scheme of [4], which are all adapted here to the new ALE frame. The implementation is verified through several manufactured steady and transient solutions, obtaining optimal convergence rates in all cases. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.
[1] A. Sahu, Y.A.D. Omar, R.A. Sauer and K.K. Mandadapu (2020), Arbitrary Lagrangian-Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces, J. Comput. Phys., 407:109253
[2] R.A. Sauer, T.X. Duong and C.J. Corbett (2014), A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements, Comput. Methods Appl. Mech. Engrg., 271:48-68
[3] C.R. Dohrmann and P.B. Bochev (2004), A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Int. J. Numer. Methods Fluids, 46:183-201
[4] R.A. Sauer (2014), Stabilized finite element formulations for liquid membranes and their application to droplet contact, Int. J. Numer. Meth. Fluids, 75(7):519-545
