  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.
