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A non‐conventional discontinuous Lagrangian for viscous flow

Although many attempts for finding a variational formulation of Navier‐Stokes equations have been made, a Lagrangian for viscous flow has not been established yet. An auspicious suggestion was made by Scholle [1] by extending Seliger and Whitham's Lagrangian [2] with additional terms ending up with some partial success: on the one hand, the phenomenon ‘viscosity’ occurs in a qualitatively correct manner, on the other hand the equations of motion resulting from the variation of Hamilton's principle differ from Navier‐Stokes equations and therefore, their solutions reveal noticeable quantitative differences to those of Navier‐Stokes equations. In this paper the Lagrangian [1] is modified by applying an innovative idea by Anthony [3], motivated by the reformulation of the Lagrangian in terms of complex fields, which can also be understood as the inversion of Madelungs idea [4] of reformulating the complex Schrödinger's equation into a hydrodynamic form. The prize one has to pay is that the resulting Lagrangian is discontinuous and therefore the mathematical treatment of the related variational problem challenging. Furthermore, an additional parameter, ω0, has to be introduced. However, it is demonstrated that Navier‐Stokes equations are recovered by the limit ω0 → ∞, whereas the case of finite ω0 can be interpreted as a generalization towards non‐equilibrium thermodynamics [3]. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)