Formulation through asymptotic methods

We now restrict our attention to premixed gaseous combustibles and address the question of the intrinsic stability of their deflagrative processes. The topic is a large one that has undergone extensive advancement in recent years. Here we shall only be able to touch on some of the highlights. The 

Here we shall be concerned with the stability of planar deflagrations to non-one-dimensional perturbations. We shall see that a richer variety 

At large values of the ZePdovich number, the chemical reaction is confined to a thin sheet in the flow. For all purposes except the analysis of the sheet structure, the sheet may be treated as a surface—for example, G(x, t) 0—which in terms of the Cartesian coordinates (x, y, z) may be written locally as x = F(y, z, t) if the x coordinate is not parallel to the sheet in its local orientation. When analyses are pursued in outer-scale variables, it is convenient to work in a coordinate system that moves with the sheet. For the undisturbed flow, let the x coordinate be normal to the planar flame, 

The coordinate transformation also enables a nondimensional version of the equation for momentum conservation to be written from equations (1-2) and (1-5). For this purpose, Prandtl numbers based on the first and a modified second coefficient of viscosity, respectively, may be defined as Pri = /iooCp, /A and Pf2 = + Koo)Cp,ooMoo- To allow for 

The equation for conservation of energy will be written under the assumptions that the Mach number is small and that the work done by body forces is negligible, both of which are accurate approximations for the deflagration problems considered. In addition, we shall neglect the Dufour effect (and later the Soret effect), so that by use of equation (1-6) in equation (3-72), in which h = Yj= i we obtain 

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