Mathematics

Elements of Structural Optimization by Zafer Gurdal, R. T. Haftka Raphael T. Haftka

By Zafer Gurdal, R. T. Haftka Raphael T. Haftka

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Although the ignition of a fuel drop has not been analyzed, similar problems have been studied in a fluid-mechanics-free context. Most present efforts in ignition are concerned with the fluid mechanics, particularly the compression waves that are generated, and the question of whether or not they can lead to detonation. Mathematically, the extinction problem is not as interesting as the ignition problem, but it is of physical interest and reveals some of the idiosyncrasies of the small Mach number equations.

Suppose that the initial data for J, p, X and Y correspond to some point on the steady response curve near E, but 2 has been reduced to a value less than 2£. With O reduced in this way the initial velocity, described by [105], will be different from the steady-state velocity. The initial response of the combustion field to this imbalance occurs rapidly, being described in terms of the fast time The significance of this scale is that time derivatives must then be retained in the flame-sheet description (dT/dt and tf-Tldx2 are both 0(0) within the reaction zone); it is the proper time scale to use in showing that the middle branch of the response is unstable [16].

Symmetric buoyancy-driven flow. coincident with the Jt-axis. It turns out that the temperature of the sheet is constant, so that rather than determine it by solving the equations for X and Y, we shall assign to it the value T*. It is then only necessary to solve the equations As always, the subscript /refers to the fresh mixture. The transport coefficients are assumed to be proportional to the temperature, so that this is not only more realistic than the adoption of constant values, but leads to simpler equations for a parabolic system in which the equation of state is Charles' law.

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