Bhatnagar, P. L. ; Prasad, Phoolan (1971) Study of selfsimilar and steady flows near singularities. II. A case of multiple characteristic velocity Proceedings of the Royal Society of London Series A: Mathematical, Physical & Engineering Sciences, 322 (1548). pp. 4562. ISSN 09628444

PDF
 Publisher Version
1MB 
Official URL: http://rspa.royalsocietypublishing.org/content/322...
Related URL: http://dx.doi.org/10.1098/rspa.1971.0053
Abstract
We consider here a system of firstorder 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 firstorder quasilinear partial differential equation. The simplicity of their method lies in the fact that all the results can be deduced from the phaseplane 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 selfsimilar 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 quasilinear equations and we have to consider the s + 1 dimensional phasespace of the steady equations. In the end we have also discussed two illustrative examples.
Item Type:  Article 

Source:  Copyright of this article belongs to Royal Society Publishing. 
ID Code:  4974 
Deposited On:  18 Oct 2010 07:54 
Last Modified:  16 May 2016 15:33 
Repository Staff Only: item control page