Gauge freedom and objective rates in fluid deformable surfaces

Morphodynamic descriptions of fluid deformable surfaces are relevant for a range of biological and soft matter phenomena, spanning materials that can be passive or active, as well as ordered or topological. However, a principled, geometric formulation of the correct hydrodynamic equations has remained opaque, with objective rates proving a central, contentious issue. We argue that this is due to a conflation of several important notions that must be disambiguated when describing fluid deformable surfaces. These are the Eulerian and Lagrangian perspectives on fluid motion, and three different types of gauge freedom: in the ambient space; in the parameterisation of the surface; and in the choice of frame field on the surface. We clarify these ideas, and show that objective rates in fluid deformable surfaces are time derivatives that are invariant under the first of these gauge freedoms, and which also preserve the structure of the ambient metric. The latter condition reduces a potentially infinite number of possible objective rates to only two: the material derivative and the Jaumann rate. The material derivative is invariant under the Galilean group, and therefore applies to velocities, whose rate captures the conservation of momentum. The Jaumann derivative is invariant under all time-dependent isometries, and therefore applies to local order parameters, or symmetry-broken variables, such as the nematic $Q$ -tensor. We provide examples of material and Jaumann rates in two different frame fields that are pertinent to the current applications of the fluid mechanics of deformable surfaces.