Further Study on the Structure of Critical Points in Fluid Systems

Effects of Disturbance on the Bifurcation Point

Abstract

Matsuda (1983) has shown that critical points appearing in steady problems of fluid systems are classified mainly into two types (i.e. bifurcation point and snap point) in connection with the symmetry in the velocity field. According to the results of this study, a bifurcation point can appear only for the system whose external conditions have some elements of symmetry in the strict sense. For the system whose external conditions have no symmetry, only a snap point can appear as a critical point.
In an actually existing fluid system, even when the external conditions appear to have the symmetry, its symmetric conditions are, more or less, perturbed by unexpected or uncontrollable disturbances. In this study we wish to examine the structure of critical points appearing in such real fluid systems. For this purpose, we consider a system having the symmetric external conditions slightly disturbed; this slight deviation from the symmetric conditions is assumed to be characterized by a single parameter μ. With the spectral equations derived in Matsuda (1983), we examine how a bifurcation point appearing for the system with complete symmetric external conditions (i.e. for μ=0) is modified in the system without them (i.e. μ≠0). It is found that instead of a bifuracation point a snap point appears for, μ≠0. However, in the phase space the snap point and a steady solution existing separately from the snap point constitute a global structure analogous to a bifurcation point. This global structure is continuously modified to a bifurcation point as μ tends to zero.
In connection with the present problem we discuss the validity of idealized models as a tool of examining a real fluid system with disturbance.