Quasi-homogeneous Critical Swirling Flows in Expanding Pipes, Part II

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.