Stability analysis for cylindrical Couette flow of compressible fluids

TL;DR

This paper presents a novel stability analysis of cylindrical Couette flow in compressible fluids using an Action Principle, linking stability to tensile strength and metastable states, and confirming predictions with recent experiments.

Contribution

It introduces a new theoretical framework merging classical hydrodynamics with Hamiltonian mechanics to analyze flow stability and predicts localized instabilities near the bubble point.

Findings

Stability criterion based on tensile strength of water.

First instability localized near the bubble point.

Theoretical predictions confirmed by recent experiments.

Abstract

A new analysis of basic Couette flow, is based on an Action Principle for compressible fluids, with a Hamiltonian as well as a kinetic potential. An effective criterion for stability recognizes the tensile strength of water. This interpretation relates the problem to capillary action and to metastable configurations (Berthelot's negative pressure experiment of 1850). We calculate the pressure and density profiles and find that the first instability of basic Couette flow is localized near the bubble point. This theoretical prediction has been confirmed by recent experiments. The theory is the result of merging the two versions of classical hydrodynamics, as advocated by Landau for superfluid Helium II, but here applied to fluids in general, in accord with a widely held opinion concerning superfluidity. In this paper two-flow dynamics is created by merging two actions, not by choosing between them, nor by combining the two vector fields as in the Navier-Stokes equation . At rest, as contributions to the mass flow they cancel, but a non-zero kinetic energy and kinetic potential as well as non-zero angular momentum remain, manifest as liquid tension, as is well known to exist by observation of the meniscus and configurations with negative pressure. (Fronsdal 2020b in preparation). This theory gives a very satisfactory characterization of the limit of stability of the most basic Couette flow. The inclusion of a vector field that is not a gradient has the additional affect of introducing spin, which explains a most perplexing experimental discovery: the ability of frozen Helium to remember its angular momentum.