Stability analysis for cylindrical Couette flow of compressible fluids

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. In an inspired paper, Landau, L. ["Theory of the superfluidity of helium II," Phys. Rev. 60, 356-358 (1941)] introduced the idea of two independent flows, "phonons" and "rotons," with strong emphasis on the idea that there is only one kind of fluid. The dynamical variables include two flows but only one density variable. In this paper, two-flow dynamics is created by merging two actions, neither 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, kinetic potential, and non-zero angular momentum remain, and are manifest as liquid tension, as it is well known to exist through the observation of the meniscus and configurations with negative pressure. The immediate effect of merging the two versions of classical hydrodynamics in a unique theory based on an action principle is to provide a Hamiltonian and a kinetic potential for compressible fluids with rotational flow. 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 effect of introducing spin, which explains a most perplexing experimental discovery: the ability of frozen helium to remember its angular momentum (spin).