Theory
Basic problems of detonation theory are:
1) To determine the steadystate detonation solutions of the reactive Euler equations for a given reactive medium, be it a gaseous mixture or a condensed material.
2) To understand stability properties of the steadystate solutions, in particular to linear perturbations.
3) To be able to predict nonlinear dynamics of detonations, including such processes as ignition and failure, pulsating dynamics, and cell formation.
Many open problems still remain in all of the three areas above. My interest is in the development of simplified rational models of various problems of detonation dynamics that would afford insight into physical mechanism of the phenomena.
Some of my previous work on these problems is described below.
D. S. Stewart and A. R. Kasimov, Theory of detonation with an embedded sonic locus, SIAM Journal of Applied Mathematics, 66, No. 2, 384407, 2005. (PDF) In this paper we introduce a general definition of a selfsustained detonation wave as a structure composed of a lead shock front and a following unsteady transonic flow. A sonic locus is defined as a limiting characteristic surface and it is shown that this definition is consistent with steadystate as well as linearly unsteady sonic conditions. 
A. R. Kasimov and D. S. Stewart, Asymptotic theory of evolution and failure of selfsustained detonations, Journal of Fluid Mechanics, 525, 161192, 2005 (PDF). Based on the general definition of a sonic locus as a characteristic surface and introducing two asymptotic approximations, of small leadshock curvature and slow time variation, we derived a fully nonlinear theory of multidimensional detonation. The principal result of the theory is an evolution equation relating detonation acceleration, speed, and shock curvature. It was shown that when specalized to onedimensional radially expanding detonations, the evolution equation can describe blast initiation of detonation and predicts critical conditions of initiation in close agreement with experiment. 
A. R. Kasimov and D. S. Stewart, Theory of detonation initiation and comparison with experiment, Report #1035, Theoretical & Applied Mechanics, University of Illinois at UrbanaChampaign, 2004.Champaign, 2004 (PDF). This is a detailed description of how our theory of weaklycurved slowlyvarying detonations can be used to predict critical conditions for direct initiation of detonation in various gaseous mixtures. A. R. Kasimov and D. S. Stewart, Spinning instability of gaseous detonations, Journal of Fluid Mechanics, 466, 179203, 2002 (PDF). A linear stability analysis of gaseous detonation propagating in a cylindrical tube is performed. D. S. Stewart and A. R. Kasimov, State of Detonation Stability Theory and Its Application to Propulsion, Journal of Propulsion and Power, 22, No. 6, 12301244, 2006 (PDF). The paper reviews the state of detonation stability theory and its possible implications for propulsion applications.
