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

TL;DR

This paper critiques Rayleigh's classical stability analysis of Couette flow, identifies a fundamental mistake regarding energy density, and proposes an alternative approach using an action principle that aligns better with observations.

Contribution

It reveals a common mistake in the literature's interpretation of energy density in Navier-Stokes equations and introduces an action principle-based analysis for better stability predictions.

Findings

Rayleigh's analysis is fundamentally flawed due to misuse of energy density.

An alternative action principle approach yields results consistent with observations.

New experiments are proposed to test the revised stability criteria.

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 was duly noted. More successful approaches have been found, but little was done to discover the weak point of Rayleigh's argument, what is the reason that it fails. This paper identifies the mistake as one that is endemic in the literature. Since the physics of the problem remains poorly understood, a discussion of this paradox 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 here that %no energy density is compatible with the Navier-Stokes equation, in the context of Couette flow or in general, and that the use of any expression as 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. The action principle provides a kinetic potential; a new criterion for stability recognizes the profound effect of the surface adhesion and the tensile strength of water. It is in full agreement with observation. Several new experiments are suggested.