Abstract :
Time-independent swirling flows in rotationally symmetric pipes of constant and
varying diameter are constructed using variational techniques. In Part I, by E. van
Groesen, B. W. van de Fliert, and E. Fledderus J. Math. Anal. Appl., 192 1995.,
764]788.. critical flows in pipes of uniform cross-section were found by extremizing
the cross-sectional energy at constrained value of the cross-sectional helicity and
the axial flow rate. In this paper we add the cross-sectional angular momentum to
the constraints and find an unfolding of the family of flows obtained in Part I. The
flows are steady Beltrami-type flows, independent of the axial coordinate, and show
the phenomenon of flow-reversal at the central axis of the pipe. Closely related to
the unfolding is the introduction of a complexity parameter; instead of taking only
discrete values for the flows studied in Part I, now the complexity can change in a
continuous way. An approximation for swirling flows in a pipe with varying circular
cross-section is constructed in a quasi-homogeneous way as a succession in the
axial direction of critical flows in uniform pipes. The evolution of the constraint
values in the axial direction is determined by the conservation laws for the energy,
angular momentum, and the axial flux. A clear presentation of the mechanism
behind the evolution of parameters in the base-flow is so obtained. The consistency
conditions that follow from the conservation laws are shown to be related to the
solvability conditions of the linearized equation.
