Tangential tensor fields on deformable surfaces—how to derive consistent <i>L</i>2-gradient flows

Nitschke, Ingo; Sadik, Souhayl; Voigt, Axel

doi:10.1093/imamat/hxae006

IMA Journal of Applied Mathematics

2023

DOI: 10.1093/imamat/hxae006

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Tangential tensor fields on deformable surfaces—how to derive consistent L2-gradient flows

Ingo Nitschke,

Souhayl Sadik,

Axel Voigt

Abstract: We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen cons… Show more

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Cited by 3 publications

References 59 publications

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Derivation and simulation of a two-phase fluid deformable surface model

Bachini,

Krause,

Nitschke

et al. 2023

J. Fluid Mech.

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To explore the impact of surface viscosity on coexisting fluid domains in biomembranes we consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modelled by incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces. We derive this model using the Lagrange–d’Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor–Hood element for the surface Navier–Stokes part, a semi-implicit approach for the Cahn–Hilliard part, higher-order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified subproblems.

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Derivation and simulation of a two-phase fluid deformable surface model

Bachini,

Krause,

Nitschke

et al. 2023

J. Fluid Mech.

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The interplay of geometry and coarsening in multicomponent lipid vesicles under the influence of hydrodynamics

2023

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We consider the impact of surface hydrodynamics on the interplay between membrane curvature and lipid composition in coarsening processes on model systems for biomembranes. This includes the influence on scaling laws and equilibrium configurations, which are investigated by computational studies of a surface two-phase flow problem with additional phase-dependent bending terms. These additional terms geometrically favor specific configurations. We find that the effect of hydrodynamics strongly depends on the composition. In situations where the composition allows a realization of a geometrically favored configuration, hydrodynamics enhances the evolution toward this configuration. We restrict our model and numerics to stationary surfaces of varying curvature and validate the numerical approach with various benchmark problems and convergence studies.

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Wrinkling of fluid deformable surfaces

Krause,

Voigt

2024

J. R. Soc. Interface.

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Wrinkling instabilities of thin elastic sheets can be used to generate periodic structures over a wide range of length scales. Viscosity of the thin elastic sheet or its surrounding medium has been shown to be responsible for dynamic processes. We here consider wrinkling of fluid deformable surfaces. In contrast with thin elastic sheets, with in-plane and out-of-plane elasticity, these surfaces are characterized by in-plane viscous flow and out-of-plane elasticity and have been established as model systems for biomembranes and cellular sheets. We use this hydrodynamic theory and numerically explore the formation of wrinkles and their coarsening, either by a continuous reduction of the enclosed volume or by the continuous increase of the surface area. Both lead to almost identical results for wrinkle formation and the coarsening process, for which a scaling law for the wavenumber is obtained for a broad range of surface viscosity and rate of change of volume or area. However, for large Reynolds numbers and small changes in volume or area, wrinkling can be suppressed and surface hydrodynamics allows for global shape changes following the minimal energy configurations of the Helfrich energy for corresponding reduced volumes.

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Tangential tensor fields on deformable surfaces—how to derive consistent L2-gradient flows

Cited by 3 publications

References 59 publications

Derivation and simulation of a two-phase fluid deformable surface model

Derivation and simulation of a two-phase fluid deformable surface model

The interplay of geometry and coarsening in multicomponent lipid vesicles under the influence of hydrodynamics

Wrinkling of fluid deformable surfaces

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