Safe storage and transportation of combustible and potentially explosive materials (fuel, propellants, food grains), the efficient and reliable operation of practical devices, and the necessity of limiting pollution, all require an improved understanding of the combustion process. Central to combustion is the notion of energy conversion, from chemical to thermal and/or mechanical. Inevitably, the interplay between the liberation of energy and the mechanical and transport properties of the medium leads to wave motion. I am interested in obtaining a mathematical description of combustion waves; their genesis, propagation, response to disturbances, mutual interaction, and extinction. The effort is directed largely at unsteady processes, for which observations of actual events have marched ahead of quantitative, and on occasions even qualitative, description.
From a mathematical point of view, the problems under study are modeled by systems of nonlinear PDEs. The aim is the construction of explicit solutions, with strong emphasis on analytical techniques (especially asymptotics and singular perturbations), supplemented by a judicious use of numerics.
Development of instructional materials, involving web technology in particular, is a recent interest. I have collaborated with a group of colleagues who are developing a multimedia environment -- a calculus world -- that would help a student learn and acquire the fundamental concepts of calculus through exploration and discovery. I have also participated in the development of web-based modules in the area of mathematical methods under the Links project: an NSF-supported initiative on Mathematics Across The Curriculum.
"Asymptotic Treatment of Chemically Reactive Systems," a Volume in the Applicable Mathematics series, Pitman Publishing, 1983.
"Two-phase modelling of DDT: structure of the velocity relaxation zone", Physics of Fluids, 9, pp. 3885-3897, 1997 (with S. F. Son, J. B. Bdzil, R. Menikoff and D. S. Stewart).
"Blowup in semilinear parabolic equations with weak diffusion," Combustion Theory and Modelling, 2, pp. 283-292, 1998 (with M. Short).
"Two-phase modelling of deflagration-to-detonation transition in granular materials: a critical examination of modeling issues", Physics of Fluids, 11, pp. 378-402, 1999 (with J. B. Bdzil, R. Menikoff, S. F. Son and D.S. Stewart).DDT1.pdf
"The hydrodynamic mechanisms of pulsating detonation-wave instability," Phil. Trans. R. Soc. Lond. A, 357, pp.3621-3637, 1999 (with M. Short and J. J. Quirk).Pulse.pdf
"Two-phase modelling of DDT in granular materials: reduced equations", Physics of Fluids, 13, pp. 3002-3024, 2001 (with J. B. Bdzil, R. Menikoff, S. F. Son and D.S. Stewart).DDT3
"Plane strain deformation of an elastic material compressed in a rough rectangular cavity," International Journal of Engineering Science, 40, pp. 991-1010, 2002 (with Arwen Warlock, Hsu-Kuang Ching and R. C. Batra).
"Lean combustion in near-critical swirling flows," Combustion Theory and Modelling, 6, pp. 625-645, 2002 (with Z. Rusak and J. J. Choi).Swirl-1.pdf
"Mechanisms of detonation formation due to a temperature gradient," Combustion Theory and Modelling, 6, pp. 553-594, 2002 (with D. Schwendeman, J. Quirk and T. Hawa).Gradient.pdf
"Nondiffusive hot spot in a confined, narrow domain," Journal of Engineering Mathematics, 45, pp. 335-366, 2003 (with M. Short).Hot spot.pdf
"Effect of thermal nonhomogeneity on explosion or detonation in an annular cookoff," Proceedings of the 12th Symposium (International) on Detonation, 2002 (with D. Schwendeman).Annulus.pdf
"Detonation initiation: modelling, computation and mechanisms." CHT04C2.pdf
"The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow," Journal of Computational Physics, 212, pp. 490-526, 2006 (with C. W. Wahle and D. W. Schwendeman).mpJCP.pdf
"Effect of near-critical swirl on the Burke-Schumann reaction zone," Journal of Engineering Mathematics, 54, pp. 181-196, 2006 (with Kang-Ho Sohn and Zvi Rusak).
"On the structure and accuracy of programmed burn," Combustion Theory and Modelling, 10, 289-321, 2006 (with. J. B. Bdzil and D. S. Stewart). PB.pdf
"Homogeneous ignition: chain-branched reactions," Journal of Engineering Mathematics, 56, pp. 105-128, 2006 (with P. A. Blythe and M. Short).
"A study of detonation diffraction in the ignition-and-growth model," (with D. W. Schwendeman, J. B. Bdzil and W. D. Henshaw). Combustion Theory and Modelling, 11, 781-822, 2007. I&G.pdf
"Numerical simulations of premixed reaction with swirl," Combustion Theory and Modelling, 11, 863-887, 2007 (with J. J. Choi and Zvi Rusak).
"A study of detonation evolution and structure for a model of compressible two-phase reactive flow," Combustion Theory and Modelling, 12,159-204, 2007 (with C. W. Wahle and D. W. Schwendeman).
"A high-resolution Godunov method for compressible multi-material flow on overlapping grids," Journal of Computational Physics, 223, pp. 262-297, 2007 (with J. W. Banks and D. W. Schwendeman).
"A study of detonation propagation and diffraction with compliant confinement," Combustion Theory and Modelling, 12, 769-808, 2008 (with J. W. Banks, D. W. Schwendeman and W. D. Henshaw).
"Shock-induced chain-branched ignition," Proceedings of the Combustion Institute, 32, 2371-2377, 2009 (with P. A. Blythe and Mark Short).
"A study of detonation diffraction and failure for a model of compressible two-phase reactive flow," Combustion Theory and Modelling, 14, 331-366, 2010 (with D. W. Schwendeman and W. D. Henshaw). mprxn2d.pdf
"A numerical study of shock-induced cavity collapse," Shock Waves, 22 (2012), 89-117 (with M. Ozlem, D. W. Schwendeman and W. D. Henshaw).
"A comparative study of two macro-scale models of condensed-phase explosive," IMA J. Applied Math., 77 (2012), 2-17 (with D. W. Schwendeman and W. D. Henshaw).
"A Hybrid Two-Phase Mixture Model of Detonation Diffraction with Compliant Confinement," Comptes Rendus Mecanique, submitted, (2012) (with D. W. Schwendeman and W. D. Henshaw).
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