Beris-Edwards models on evolving surfaces: A Lagrange-D'Alembert approach

Using the Lagrange-d’Alembert principle, we develop several thermodynamically consistent surface Beris-Edwards models for thin, fluidic, deformable, liquid crystal films. These models consider the simultaneous, coupled evolution of the thin film along with the $Q$-tensor field on it by taking into account the inextensible viscous hydrodynamics of the film and a Landau-de Gennes-Helfrich energy. We present a general model with three-dimensional $Q$-tensor dynamics on the evolving film and several reduced models, constrained with Lagrange multipliers for additional modeling assumptions such as uniaxiality of the $Q$-tensor, conformity of eigenframes of the $Q$-tensor field to the surface, stationarity of the surface and several others. In addition, we introduce different time derivatives to account for different physical interpretations of how the surface deformations and liquid-crystal orientations interact. We relate the derived models to established models in simplified situations and mention potential applications in biology.