  Fluid deformable surfaces are ubiquitous in cell and tissue biology,
including lipid bilayers, the actomyosin cortex, or epithelial cell sheets.
These interfaces exhibit a complex interplay between elasticity, low Reynolds
number interfacial hydrodynamics, chemistry, and geometry, and govern important
biological processes such as cellular traffic, division, migration, or tissue
morphogenesis. To address the modelling challenges posed by this class of
problems, in which interfacial phenomena tightly interact with the shape and
dynamics of the surface, we develop a general continuum mechanics and
computational framework for fluid deformable surfaces. The dual solid-fluid
nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian
descriptions of deforming bodies. Here, we extend the notion of Arbitrarily
Lagrangian-Eulerian (ALE) formulations, well-established for bulk media, to
deforming surfaces. To systematically develop models for fluid deformable
surfaces, which consistently treat all couplings between fields and geometry,
we follow a nonlinear Onsager formalism according to which the dynamics
minimize a Rayleighian functional where dissipation, power input and energy
release rate compete. Finally, we propose new computational methods, which
build on Onsager's formalism and our ALE formulation, to deal with the
resulting stiff system of higher-order of partial differential equations. We
apply our theoretical and computational methodology to classical models for
lipid bilayers and the cell cortex. The methods developed here allow us to
formulate/simulate these models for the first time in their full
three-dimensional generality, accounting for finite curvatures and finite shape
changes.
