Study of self-similar and steady flows near singularities. II. A case of multiple characteristic velocity

Abstract

We consider here a system of first-order quasilinear partial differential equations in two independent variables: t, time and x, spatial coordinate. In many physically realistic problems in fluid mechanics, a singularity of the system of ordinary differential equations representing the steady solutions represents a critical state where one of the characteristic velocities vanishes (e.g. sonic point in fluid mechanics). Kulikovskii & Slobodkina (1967) have shown that the stability of all the steady solutions near a singularity can be studied with the help of a simple first-order quasi-linear partial differential equation. The simplicity of their method lies in the fact that all the results can be deduced from the phase-plane of the steady equations. The analysis of Kulikovskii & Slobodkina is valid for any system of equations, totally hyperbolic or mixed type with the only assumption that the characteristic velocity under consideration is real and not multiple. We have earlier (1970, to be referred to as part I) extended their treatment to self-similar flows. In this paper we have shown that in the case of a characteristic velocity of multiplicity s (s > 1), it is still possible to approximate the system provided there exists exactly s linearly independent eigenvectors corresponding to this characteristic velocity. The approximate system consists of s quasi-linear equations and we have to consider the s + 1 dimensional phase-space of the steady equations. In the end we have also discussed two illustrative examples.