Arbitrary LagrangianEulerian finite element method for curved and deforming surfaces. I. General theory and application to fluid interfaces
Abstract
An arbitrary LagrangianEulerian (ALE) finite element method for arbitrarily curved and deforming twodimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose inplane velocity need not depend on the inplane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with inplane incompressible flows. Numerical infsup instabilities associated with inplane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to nonaxisymmetric perturbations, and unstable with respect to longwavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.
 Publication:

Journal of Computational Physics
 Pub Date:
 April 2020
 DOI:
 10.1016/j.jcp.2020.109253
 arXiv:
 arXiv:1812.05086
 Bibcode:
 2020JCoPh.40709253S
 Keywords:

 Interfacial flows;
 Fluid film;
 Arbitrary LagrangianEulerian;
 Finite element method;
 Differential geometry;
 Lipid membrane;
 Physics  Computational Physics;
 Condensed Matter  Soft Condensed Matter;
 Physics  Biological Physics;
 Physics  Fluid Dynamics
 EPrint:
 59 pages, 16 figures