A non-conventional discontinuous Lagrangian for viscous flow

• Main Authors: M. Scholle, F. Marner
• Format: Article
• Language: English
• Published: The Royal Society 2017-01-01
• Series: Royal Society Open Science
• Subjects: potential fields; variational calculus; navier–stokes equations; non-equilibrium thermodynamics; analogies; first integrals
• Online Access: https://royalsocietypublishing.org/doi/pdf/10.1098/rsos.160447

Drawing an analogy with quantum mechanics, a new Lagrangian is proposed for a variational formulation of the Navier–Stokes equations which to-date has remained elusive. A key feature is that the resulting Lagrangian is discontinuous in nature, posing additional challenges apropos the mathematical treatment of the related variational problem, all of which are resolvable. In addition to extending Lagrange's formalism to problems involving discontinuous behaviour, it is demonstrated that the associated equations of motion can self-consistently be interpreted within the framework of thermodynamics beyond local equilibrium, with the limiting case recovering the classical Navier–Stokes equations. Perspectives for applying the new formalism to discontinuous physical phenomena such as phase and grain boundaries, shock waves and flame fronts are provided.