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Using the Lagrange-D'Alembert principle we develop thermodynamicallyconsistent surface Beris-Edwards models. These models couple viscousinextensible surface flow with a Landau-de Gennes-Helfrich energy and considerthe simultaneous relaxation of the surface Q-tensor field and the surface, bytaking hydrodynamics of the surface into account. We consider differentformulations, a general model with three-dimensional surface Q-tensor dynamicsand possible constraints incorporated by Lagrange multipliers and a surfaceconforming model with tangential anchoring of the surface Q-tensor field andpossible additional constraints. In addition to different treatments of thesurface Q-tensor, which introduces different coupling mechanisms with thegeometric properties of the surface, we also consider different timederivatives to account for different physical interpretations of surfacenematics. We relate the derived models to established models in simplifiedsituations, compare the different formulations with respect to numericalrealizations and mention potential applications in biology.

Beris-Edwards Models on Evolving Surfaces: A Lagrange-D'Alembert Approach