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The paper studies the equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Using a continuum description of the membrane's motions based on the surface Navier–Stokes equations with bending forces, the paper derives differential equations governing the mechanical equilibrium. The equilibrium conditions are found to be independent of lateral viscosity and relate tension, pressure, and tangential velocity of the fluid. These conditions suggest that either the lateral fluid motion ceases or non-decaying stationary flow of mass can only be supported by surfaces with Killing vector fields, such as axisymmetric shapes. A shape equation is derived that extends the classical Helfrich model with an area constraint to membranes of non-negligible mass. Furthermore, the paper suggests a simple numerical method to compute solutions of the shape equation. Numerical experiments conducted reveal a diverse family of equilibrium configurations. The stability of equilibrium states involving lateral flow of mass remains an unresolved question. 

The paper considers a system of equations that models a lateral flow of a Boussinesq–Scriven fluid on a passively evolving surface embedded in [Formula: see text]. For the resulting Navier–Stokes type system, posed on a smooth closed time-dependent surface, we introduce a weak formulation in terms of functional spaces on a space-time manifold defined by the surface evolution. The weak formulation is shown to be well-posed for any finite final time and without smallness conditions on data. We further extend an unfitted finite element method, known as TraceFEM, to compute solutions to the fluid system. Convergence of the method is demonstrated numerically. In another series of experiments we visualize lateral flows induced by smooth deformations of a material surface. 

Abstract The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ 3 . The method employs parametric P k - P k −1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin--Helmholtz instability problem on the unit sphere. 

Abstract The paper discusses a reuse of matrix factorization as a building block in the Augmented Lagrangian (AL) and modified AL preconditioners for nonsymmetric saddle point linear algebraic systems. The strategy is applied to solve two‐dimensional incompressible fluid problems with efficiency rates independent of the Reynolds number. The solver is then tested to simulate motion of a surface fluid, an example of a two‐dimensional flow motivated by an interest in lateral fluidity of inextensible viscous membranes. Numerical examples include the Kelvin–Helmholtz instability problem posed on the sphere and on the torus. Some new eigenvalue estimates for the AL preconditioner are derived.