Asymptotic analysis for strongly temperature-dependent rates

Asymptotic analysis for strongly temperature-dependent rates 

Although the asymptotic analysis was developed first on the basis of a formulation in the variables appearing in equation (42) [9], it is more instructive to begin with equation (46). For later convenience in analysis, the combinations a = a7(l + a ) (not to be confused with the a of Chapter 2) 

From equation (66) it is seen that as p becomes large the reaction term in equation (46) becomes very small ( exponentially small, since p appears inside the exponential) unless t is near unity. Hence for 1 — t of order unity, there is a zone in which the reaction rate is negligible and in which the convective and diffusive terms in equation (46) must be in balance. In describing this conyectiye-diffusive zone, an outer expansion of the form T = Tq(0 + H (p)zi( ) + H2(P)t2(0 + may be introduced, following the formalism of matched asymptotic expansions [35]. Here 

Substitution of this expansion into equation (46), passing to the limit p oo, subtracting the result from equation (46), dividing by Hi(p), again passing to the limit jS oo, and so on, produces the sequence of outer equations 

The downstream boundary conditions, y 0 as rj oo, may be applied to the inner equations. The first integral of equation (70) then is 

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