Incompressible fluid problems on embedded surfaces: modeling and variational formulations
type: Item
instance of: scholarly article
title: Incompressible fluid problems on embedded surfaces: modeling and variational formulations
zbMATH Open document ID: 1406.35224
DOI: 10.4171/IFB/405
author: Thomas Jankuhn
author: Maxim A. Olshanskii
author: Arnold Reusken
published in: Interfaces and Free Boundaries
publication date: 2019-01-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1702.02989
review text: Summary: Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in \(\mathbb R^n\). For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76A20
Mathematics Subject Classification ID: 37E35
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 74K15
zbMATH DE Number: 7002974
zbMATH Keywords: fluids on surfaces
zbMATH Keywords: viscous material interface
zbMATH Keywords: fluidic membrane
zbMATH Keywords: Navier-Stokes equations on manifolds
MaRDI profile type: Publication
arXiv ID: 1702.02989
cites work: Q5736805
cites work: Lagrangian Navier-Stokes diffusions on manifolds: variational principle and stability
cites work: Q4167627
cites work: A stable numerical method for the dynamics of fluidic membranes
cites work: Parametric FEM for geometric biomembranes
cites work: On the two-phase Navier-Stokes equations with Boussinesq-Scriven surface fluid
cites work: The Navier-Stokes equations on the rotating 2-D sphere: Gevrey regularity and asymptotic degrees of freedom
cites work: Q4485429
cites work: Finite element methods for surface PDEs
cites work: Groups of diffeomorphisms and the motion of an incompressible fluid
cites work: Theory and practice of finite elements.
cites work: Finite Element Methods for Navier-Stokes Equations
cites work: A Trace Finite Element Method for Vector-Laplacians on Surfaces
cites work: Numerical Methods for Two-phase Incompressible Flows
cites work: A continuum theory of elastic material surfaces
cites work: Analysis of finite element methods for vector Laplacians on surfaces
cites work: Energetic variational approaches for incompressible fluid systems on an evolving surface
cites work: Inequality of Korn's type on compact surfaces without boundary
cites work: Navier-Stokes equations on Lipschitz domains in Riemannian manifolds.
cites work: Symmetry considerations for material surfaces
cites work: A finite element approach to incompressible two-phase flow on manifolds
cites work: A Finite Element Method for Elliptic Equations on Surfaces
cites work: Numerical simulation of incompressible two‐phase flows with a Boussinesq–Scriven interface stress tensor
cites work: The Interplay of Curvature and Vortices in Flow on Curved Surfaces
cites work: A semi-implicit finite element method for viscous lipid membranes
cites work: The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds
cites work: Q4888132
cites work: Interfacial Transport Phenomena
cites work: Analysis on Morrey Spaces and Applications to Navier-Stokes and Other Evolution Equations
cites work: Infinite-dimensional dynamical systems in mechanics and physics
OpenAlex ID: W2597331459
description: scientific article