ICNPAA 2010: Mathematical Problems in
Engineering, Aerospace and Sciences

The presentation identifies a unified mathematical approach to description of a number of thermal instabilities typically observed in fire behaviour.

Consideration is mostly focused on compartment fires. The two key instabilities that are most important in fire behaviour are ignition and flashover.

Proposed mathematical framework for consideration of these instabilities relies on the concept of conjugate thermal explosion, proposed and developed recently by the author. This concept generalises the classical thermal explosion theory, and has been found very useful in interpreting fire instabilities, as well as some other combustion phenomena.

In the present talk, relevant mathematical models are developed, and their dynamics is investigated. Generally, the proposed formulation leads to non-linear integro-differential equations. As a specific example, dynamical model of fire flashover is investigated in more detail. For such a model, we study qualitatively the set of corresponding integro-differential equations, and derive critical conditions for flashover.

Finally, other potential application of conjugate thermal explosion theory are briefly discussed.