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Mathematical Physics Seminar

A classical mistake and what it tells us. How to do better with an action principle for Hydro and Thermo dynamics    

Christian Fronsdal  - University of California Los Angeles

Location:  Hill 705
Date & time: Thursday, 26 April 2018 at 12:00PM - 1:00PM

Abstract: Rayleigh’s stability analysis of cylindrical Couette flow, of 1889 and 1916, is in contradiction with observation. The analysis is repeated in many textbooks and reviews up to 2017, and its failure to agree with observation is duly noted; more successful approaches have been found, but little was done to discover the weak point of Rayleigh’s argument. This paper identifies the mistake as one that is endemic in the literature. Since the physics of the problem remains poorly understood, a discussion should prove useful. Briefly, the argument depends on the Navier-Stokes equation and on the assumption that a certain expression called “energy density” or “kinetic potential” can be interpreted and used as such. It is shown that the use of any expression for a kinetic potential is in conflict with the Navier-Stokes equation, in all but a very limited context. An alternative analysis of basic Couette flow, based on an Action Principle for compressible fluids, provides a Hamiltonian density as well as a kinetic potential. The two are not the same, even in the simplest cases. With the kinetic potential provided by the action principle, Rayleigh’s stability criterion is a necessary condition for stability, but the action principle applies to compressible flows and a new stability criterion achieves a better agreement with observation. The action is Galilei invariant. Several new experiments are suggested.        


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