  Using the Lagrange-D'Alembert principle we develop thermodynamically
consistent surface Beris-Edwards models. These models couple viscous
inextensible surface flow with a Landau-de Gennes-Helfrich energy and consider
the simultaneous relaxation of the surface Q-tensor field and the surface, by
taking hydrodynamics of the surface into account. We consider different
formulations, a general model with three-dimensional surface Q-tensor dynamics
and possible constraints incorporated by Lagrange multipliers and a surface
conforming model with tangential anchoring of the surface Q-tensor field and
possible additional constraints. In addition to different treatments of the
surface Q-tensor, which introduces different coupling mechanisms with the
geometric properties of the surface, we also consider different time
derivatives to account for different physical interpretations of surface
nematics. We relate the derived models to established models in simplified
situations, compare the different formulations with respect to numerical
realizations and mention potential applications in biology.
