A unified theory of conjugate flows

Abstract

Various examples of flow systems are known in which the study of conjugate flows (i.e. flows uniform in the direction of streaming which separately satisfy the hydrodynamical equations) is crucial to the understanding of observed wave phenomena. Open-channel flows are the best-known example, with which remarkable qualitative similarities have been revealed in studies of other systems: for instance, it has appeared in general that any pair of conjugate flows is transcritical (i.e. if one flow is supercritical according to a generalized definition, then the other is subcritical). So far the common ground among theoretical treatments has been defined only by intuitive analogies, and the aim of this paper is to give unity to the whole subject by identifying the elements that are intrinsically responsible for universal properties. The problem is accordingly considered in the form of an abstract (nonlinear) operator equation, whose solution representing a conjugate flow is a vector in a linear space of finite or infinite dimensions: all known examples are reducible to this form and other applications may be anticipated. The generalized treatment on these lines must have recourse to new methods, however, of a more powerful kind than would suffice for the ad hoc treatment of particular examples. A resume of the required mathematical material is presented in §2. The main substance of the paper is in § 3. In § 3.1 the supercritical and subcritical classification of flows is explained generally, being shown to depend on the eigenvalues of the Frechet derivative of the nonlinear operator presented by the hydrodynamical problem. In §3.3 fixed-point principles are used to define general conditions under which the existence of conjugate flows in a proposed category is guaranteed, and also in this subsection a special argument is given to exemplify the transcritical property of conjugate flows. Several aspects are covered in §3.4 by means of index theory, in particular the problem of classifying a multiplicity of conjugate flows possible in a given system and the question of what conditions ensure uniqueness. In §3.5 variational methods are used to account for the differences in flow force that appear to be an essential attribute of frictionless conjugate flows (flow force is a scalar property which is generally stationary to small variations about a solution of the hydrodynamical equations). The last three sections of the paper present treatments of specific examples illustrating the unified viewpoint given by the theory. Proofs of two topological theorems used in §3.4 are presented in appendix 1, and in appendix 2 the reasons for the variational significance of flow force are examined.