Conjugate flows and amplitude bounds for internal solitary waves

Amplitude bounds imposed by the conservation of mass, momentum and energy for strongly nonlinear waves in stratified fluid are considered. We discuss the theoretical scheme which allows to determine broadening limits for solitary waves in the terms of a given upstream density profile. Attention is focused on the continuously stratified flows having multiple broadening limits. The role of the mean density profile and the influence of fine-scale stratification are analyzed.

Conjugate-flow theory for heterogeneous compressible fluids, with application to non-uniform suspensions of gas bubbles in liquids

Conjugate flows have been defined generally as flows uniform in the direction of streaming that separately satisfy the relevant hydrodynamical equations, so allowing a transition from one flow to its conjugate to be consistent with mass and energy conservation. In previous studies of various examples, certain general principles have been found to apply to conjugate flows: in particular, one in a pair of such flows is subcritical (subsonic) and the other supercritical (supersonic), the former having greater flow force (i.e. momentum flux plus pressure force). In this paper these principles are confirmed in another field of application, for which the theory of conjugate flows takes a novel course.The theoretical model defined in § 2 consists of a straight duct of arbitrary cross-section filled with a perfect fluid whose constitutive properties vary with cross-sectional position, and whose primary, prescribed flow is axial with a velocity distribution that may be non-uniform. In § 3 the possibility of a conjugate flow in the same duct is investigated, and its principal properties relative to those of the primary flow are deduced from certain simple inequalities between integrals over the cross-section. A Lagrangian description of the conjugate flow is essential, but the properties in question are established without the necessity of determining this flow explicitly. At the end of § 3, a modification of the model is discussed accounting for dissipative, flow-force conserving transitions (shocks). The application of the theory to flows of non-uniform suspensions of gas bubbles is considered in § 4.

A unified theory of conjugate flows

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.

On Benjamin's theory of conjugate vortex flows

Benjamin (1962) introduced the idea that a given ‘primary’ swirling flow, with cylindrical stream surfaces, may have associated with it ‘conjugate flows’, also swirling and cylindrical, which in a certain sense are equivalent to the primary one. He deduced that, in cases where such conjugate flows exist and where the primary flow cannot support standing waves of small amplitude, the conjugate flow nearest the primary one (a) can support such waves, and (b) has a ‘flow force’ greater than that of the primary flow. In the present paper these two results are proved rigorously by a method which differs from Benjamin's.