Parametric transition from deflagration to detonation in stellar medium.

The nature of thermonuclear explosions of white-dwarf stars is a fundamental astrophysical issue, the first principle interpretation of which is still commonly regarded as an unresolved problem. There is a general consensus that stellar explosions are a manifestation of the deflagration-to-detonation transition of an outward propagating self-accelerating thermonuclear flame subjected to instability-induced corrugations. A similar problem arises in unconfined terrestrial flames where a positive feedback mechanism leading to the pressure runaway has been identified.

An elementary model for autoignition of laminar jets.

In this paper, we formulate and analyse an elementary model for autoignition of cylindrical laminar jets of fuel injected into an oxidizing ambient at rest. This study is motivated by renewed interest in analysis of hydrothermal flames for which such configuration is common. As a result of our analysis, we obtain a sharp characterization of the autoignition position in terms of the principal physical and geometrical parameters of the problem.

Gelfand-type problem for two-phase porous media.

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two-phase materials, where, in contrast to the classical Gelfand problem which uses a single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively owing to the absence of stationary temperature distribution.

Local accumulation times for source, diffusion, and degradation models in two and three dimensions.

We analyze the transient dynamics leading to the establishment of a steady state in reaction-diffusion problems that model several important processes in cell and developmental biology and account for the diffusion and degradation of locally produced chemical species. We derive expressions for the local accumulation time, a convenient characterization of the time scale of the transient at a given location, in two- and three-dimensional systems with first-order degradation kinetics, and analyze their dependence on the model parameters. We also study the relevance of the local accumulation time as a single measure of timing for the transient and demonstrate that, while it may be sufficient for describing the local concentration dynamics far from the source, a more delicate multi-scale description of the transient is needed near a tightly localized source in two and three dimensions.

Self-similar dynamics of morphogen gradients.

Morphogen gradients are concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues and play a fundamental role in various aspects of embryonic development. We discovered a family of self-similar solutions in a canonical class of nonlinear reaction-diffusion models describing the formation of morphogen gradients. These solutions are realized in the limit of infinitely high production rate at the tissue boundary and are given by the product of the steady state concentration profile and a function of the diffusion similarity variable.

Local kinetics of morphogen gradients.

Some aspects of pattern formation in developing embryos can be described by nonlinear reaction-diffusion equations. An important class of these models accounts for diffusion and degradation of a locally produced single chemical species. At long times, solutions of such models approach a steady state in which the concentration decays with distance from the source of production.