Convective derivatives and Reynolds transport in curvilinear time-dependent coordinates

A fully covariant formulation of kinematics and dynamics of fluid flows and heat transfer is developed in time-dependent curvilinear coordinate systems. These moving and deformable reference frames have the same properties of a fluid motion and they can be successfully applied to a variety of problems ranging from numerical analysis to theoretical physics. The classical Reynolds transport theorem, the Euler formula and the acceleration addition theorem are extended to these general types of coordinates through a generalization of the convected differentiation concept originally introduced by Oldroyd (1950 Proc. R. Soc. A 200 523-41). A rigorous formulation of dynamical equations and conservation laws in curvilinear time-dependent coordinates could be the key for the construction of variational principles based on the method of constrained variations.