Derivation and transformation of variational principles with emphasis on inverse and hybrid problems in fluid mechanics: a systematic approach

A systematic approach to the derivation of variational principles (VPs) from the partial differential equations of fluid mechanics is suggested herein, consisting essentially of two major lines: (1) establishing a first VP via reversed deduction followed by extending it successively to a Family of subgeneralized VPs via a series of transformations. and (2) vice versa. Full advantage is taken of four powerful means - the functional variation with variable domain, the natural boundary/initial condition (BC/IC), the Lagrange multiplier, and the artificial interface. The occurrence of three kinds of variational crisis is demonstrated and methods for their removal are suggested. This approach has been used with great success in establishing VP-families in fluid mechanics with special attention to inverse and hybrid problems of flow in a rotating system.