Analysis of Self-Similar Solutions of Euler System: SIB, Desingulatization and Impasse Points

This paper shows that the singularity induced bifurcation (SIB) phenomenon and desingularization tool from nonlinear differential-algebraic equations (DAEs) are intrinsic parts of the compressible model of Euler flow in the self-similar framework. The nonlinear DAEs in such a setting include the crossings of sonic (singularity) points by a smooth trajectory. The linearization of DAEs around the sonic points is characterized by the divergence of eigenvalues through infinity and a presence of folded nodes. Also, the singularity manifold includes mostly the impasse points, where trajectory collapses at or originates from. The desingularization tool from DAEs results in a singularity-free model, that is a system of ordinary differential equations (ODEs), having regular nodes as its equilibria. Partial reversal of the direction of the independent self-similar variable results in a transformation of the regular nodes into folded ones for the original Euler DAE system. The Euler DAE system is a system of DAEs obtained after the similarity variable
$x=t/r^{\lambda }$
is applied in the Euler systems of three partial differential equations for conservation of mass, balance of momenum, and balance of energy. The smooth trajectory connecting two equilibria of the Euler DAE system passes through two sonic points, traversing through the super- and subsonic areas for the independent self-similar variable
$-\infty < x < \infty $
. The point
$x=0$
is also of an interesting nature, with an infinite number of possible smooth trajectories passing through it. The analysis is based on a combination of both analytic and numeric approaches. Overall, the paper links the topics from the areas of fluid flows (in the Eulerian framework), self-similar solutions and DAEs.