An arbitrary Lagrangian-Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. A formalism is provided for determining the equations of motion of a two-dimensional material using an irreversible thermodynamic analysis of curved surfaces. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity is independent of the in-plane material velocity, and which can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf-sup instabilities associated with in-plane 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-including the lid-driven cavity problem. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are numerically and analytically found to be unstable with respect to long-wavelength perturbations when their length exceeds their circumference.In this paper, and the subsequent manuscript in the series ‡ [1], we develop an arbitrary Lagrangian-Eulerian (ALE) theory for arbitrarily curved and deforming two-dimensional interfaces with in-plane fluidity. The theory is based on a surface discretization which is independent of the in-plane material flow, such that the surface mesh need not convect with the material. Consequently, two-dimensional materials with large in-plane flows on arbitrarily deforming surfaces can be modeled. In Part I, we apply our theory and use standard numerical techniques to devise an isoparametric ALE finite element method for incompressible fluid films. We then implement the finite element formulation, model the deformations and flows of such materials over time, and provide several numerical results for both flat and cylindrical geometries. In Part II, we extend the finite element formulation to lipid membranes and study membrane behavior in several biologically relevant situations. As the equations governing single-and multi-component lipid membranes reduce to the fluid film equations in the limit where no elastic energy is stored in the membrane, such a separation is natural and allows us to present our results in a more accessible manner.Two-dimensional fluids have played an increasingly important role in many engineering applications, in which they often arise at phase boundaries in multiphase systems [2]. For example, under the influence of gravity and capillary forces, foams will drain over time until the constituent bubbles burst [3]. Foam lifetime plays a key role in their viability for engineering applications, and there have consequently been many efforts to improve foam stability [2]. Similar efforts have been made to stabilize emulsions and collo...

In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved configurations. Previous theoretical and computational studies to understand membrane morphologies resulting from locally induced curvature are often limited to axisymmetric shapes, which severely restricts the physically admissible morphologies. Under the restriction of axisymmetry, past efforts predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction of axisymmetry by employing recent theoretical and numerical advances to understand arbitrarily curved and deforming lipid bilayers. Our non-axisymmetric morphologies reveal membrane morphologies which agree well with axisymmetric studies-however only if the resting tension of the membrane is low. When the resting tension is moderate to high, we show that (i) axisymmetric invaginations are unstable; and (ii) non-axisymmetric ridge-shaped structures are energetically favorable. We further study the dynamical effects resulting from the interplay between intramembrane viscous flow and induced curvature, and find the rate at which the locally induced curvature increases is a key determinant in the formation of ridges. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, while non-axisymmetric ridges are favored when the curvature is slowly increased: The rate of change of induced curvature affects the intramembrane viscous flow of lipids, which can impede the membrane's ability to transition into ridges. We conclude that the appearance of non-axisymmetric ridges indicates that axisymmetry cannot be generally assumed when understanding processes involving locally induced curvature. Our results hold potentially relevant implications for biological processes such as endocytosis, and physical phenomena like phase separation in lipid bilayers.

In various biological processes such as endocytosis and caveolae formation, the cell membrane is locally deformed into curved configurations. Previous theoretical and computational studies to understand membrane morphologies resulting from locally induced curvature are often limited to axisymmetric shapes, which severely restricts the physically admissible morphologies. Under the restriction of axisymmetry, past efforts predict that the cell membrane buds at low resting tensions and stalls at a flat pit at high resting tensions. In this work, we lift the restriction of axisymmetry by employing recent theoretical and numerical advances to understand arbitrarily curved and deforming lipid bilayers. Our non-axisymmetric morphologies reveal membrane morphologies which agree well with axisymmetric studies-however only if the resting tension of the membrane is low. When the resting tension is moderate to high, we show that (i) axisymmetric invaginations are unstable; and (ii) non-axisymmetric ridge-shaped structures are energetically favorable. We further study the dynamical effects resulting from the interplay between intramembrane viscous flow and induced curvature, and find the rate at which the locally induced curvature increases is a key determinant in the formation of ridges. In particular, we show that axisymmetric buds are favored when the induced curvature is rapidly increased, while non-axisymmetric ridges are favored when the curvature is slowly increased: The rate of change of induced curvature affects the intramembrane viscous flow of lipids, which can impede the membrane's ability to transition into ridges. We conclude that the appearance of non-axisymmetric ridges indicates that axisymmetry cannot be generally assumed when understanding processes involving locally induced curvature. Our results hold potentially relevant implications for biological processes such as endocytosis, and physical phenomena like phase separation in lipid bilayers.

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