IRENE: a fluId layeR finitE-elemeNt softwarE

We present a finite-element software library, IRENE, which allows to solve numerically the dynamics of a viscous fluid layer embedded in three-dimensional space. Unlike finite-element libraries present in the literature, IRENE can handle two-dimensional open surfaces with a wide range of boundary conditions, and inter-surface obstacles with any shapes, and is built upon the user-friendly and versatile finite element computational software (FEniCS). Also, the library can describe a wide range of physical regimes–both low-Reynolds-number and inertia-dominated ones–capturing the complex coupling between in-plane flows, out-of-plane deformations, surface tension, and elastic response. We validate IRENE against known analytical and numerical results, and demonstrate its capabilities through physical examples. Overall, IRENE provides a versatile and efficient tool for understanding fluid-layer dynamics on multiple physical scales, from flows of lipidic membranes on a microscopic level, to fluid flows on a macroscopic scale, to atmospheric air flows on a planetary level.

💡 Research Summary

The manuscript introduces IRENE, an open‑source finite‑element (FE) library designed to simulate the dynamics of a viscous fluid layer that lives on a two‑dimensional surface embedded in three‑dimensional space. The authors first motivate the need for such a tool by highlighting the ubiquity of thin fluid layers in biology (lipid membranes, cell cortices), soft‑matter physics (soap films, coatings) and geophysical flows (low‑altitude cloud layers). Existing numerical approaches—boundary‑integral, immersed‑boundary, spectral, Monte‑Carlo, surface‑FEM, and conventional FE methods—are either limited to closed manifolds, one‑dimensional manifolds, or to low‑Reynolds‑number, laminar regimes. Consequently, they cannot treat open surfaces with arbitrary obstacles, complex boundary conditions, or turbulent inertial flows.

IRENE addresses these gaps in four major ways. (1) It supports open, possibly multiply‑connected surfaces, allowing arbitrary Dirichlet, Neumann, or mixed boundary conditions on any part of the domain. (2) It permits the insertion of obstacles of any shape (e.g., trans‑membrane proteins, topographic protrusions) directly into the surface mesh. (3) It is capable of handling both low‑Reynolds‑number (viscous‑dominated) and high‑Reynolds‑number (inertia‑dominated) regimes, thereby covering a broad spectrum of physical phenomena. (4) It is built on the FEniCS platform, which provides a high‑level Python interface, automatic differentiation, and a large community of developers; the code is released publicly on GitHub.

The governing equations are derived from surface continuum mechanics. The unknown fields are the tangential velocity (v_i), the normal velocity (w), the surface tension (\sigma), and the height function (z(\mathbf{x})). The system consists of a surface incompressibility condition, a momentum balance that includes surface tension, viscous stresses, curvature‑induced forces, and an elastic bending term derived from the Helfrich energy, and a kinematic relation linking the normal velocity to the time derivative of the height. The equations involve the mean curvature (H), Gaussian curvature (K), the Laplace–Beltrami operator (\nabla_{LB}), and covariant derivatives on the manifold, making them intrinsically fourth‑order in (z).

Direct FE discretisation of the fourth‑order PDE would lead to numerical blow‑up because standard continuous Lagrange elements have discontinuous second derivatives. IRENE circumvents this by introducing auxiliary fields: (\omega_i = \nabla_i z) and (\mu = H(\omega)). This reformulation reduces the highest derivative order to second, allowing a mixed variational formulation where only first derivatives appear after integration by parts. The weak form is constructed with test functions for each field, and penalty terms are added to enforce the auxiliary relations and boundary conditions weakly. This approach yields a well‑posed linear system that can be assembled efficiently with FEniCS.

For time‑dependent problems, the authors combine an Incremental Pressure Correction Scheme (IPCS) with Crank–Nicolson temporal discretisation to solve the Navier–Stokes equations on a deformable surface. This hybrid scheme preserves second‑order accuracy while handling the pressure–velocity coupling and the geometric nonlinearity of the moving manifold.

The library is validated against analytical solutions and previously published numerical data in three representative cases. (i) A steady‑state lipid membrane with a trans‑membrane protein in a circular domain reproduces the known Helfrich shape equation solution. (ii) Poiseuille flow in a curved channel demonstrates the correct coupling between inertia, curvature, and surface tension. (iii) Fully dynamic Navier–Stokes simulations on a deformable surface show the emergence of flow‑induced shape changes and agree with benchmark simulations. In all cases, IRENE matches reference results quantitatively and exhibits good computational performance.

The discussion emphasizes the modularity of the code: alternative elastic models, active stresses, or coupling to surrounding bulk fluids can be incorporated with minimal changes. The authors outline future extensions toward multi‑scale applications ranging from sub‑cellular membrane dynamics to planetary‑scale atmospheric layers. By releasing the software openly, they invite community contributions and aim to establish IRENE as a standard tool for researchers tackling coupled fluid‑elastic problems on curved surfaces.