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I/V;. Since X, is continuous, this means that v^ is continuous along the
separatiix if it is bounded there, since the limits from either side must be
equal. We have



lim v^.






(v, \) • IV . \ )


- *(1 X,) exp




lim -—

(v. \) - (v.. \ ) X^


= k{l K) exp




lim -^

(V, \) - (\\. \ ) Xy



where C(k, i) is bounded, uniformly in k ^ (0, K^ax]' and in
z € (immi ^max)- Hencc v^ is bounded along the separatrix, and possesses a
removable discontinuity at the transonic critical point. This completes the
proof of Theorem 4.1.

Denote the x coordinate of the transonic critical point in Theorem 4.1
jc^(k, i). We expect from physical considerations that x^. ^ ^ as k - 0,
because the plane wave critical point is at infinity. We may use (3.3b) to
write Xc as

\ (k ^)

x,{k, i)^ k-^ J ^(^' K,i)exp(A7/c-(X,K,i)) JX

where v(X, k, i) and c(X, k, i) denote v and c along the stable separatrix
from X = to the critical point \^{k, z). In order for the critical point to be
correctly modelled in the r - ^ limit it must remain within the region of



51



asymptotic validity for the model, rhis will be so if the shock radius :
increases faster than x^.. We now show that this is so.

PROPOSITION 7.1

lim»cc^(K, z) - 0.

K -0

Proof: We have

\ (k. :)

jc^(k, i) < sup |/:~V(\, K, i)exp(A7/c-(\, K, r))| f -j -

S \ :S \ ^ 1 A.

< sup |A: - v(X, K, r)exp(Ayc-(X, k, r))|(-log(l - \,(k, i)).

s \ < 1

In the proof for Theorem 4.1 we found that the separatrix solution is
bounded, uniformly in k and i, and bounded away from the vacuum locus
c = 0, so the supremum is bounded. Since \,- is a smooth function of k and
z,we have

limKlogfl - \ (K,i)) = 0.
This proves the proposition.

8. Conclusion. In Theorem 4.1 we show that the vector field (3.7)
possesses the familiar saddle point structure characteristic of weak detona-
tions. The story does not end here, however. The degenerate double zero
eigenvalue bifurcation point present in the velocity-reaction progress plane
for the ZND plane wave is infinite codimensionaJ. That is. an infinite
number of parameters are required to produce all possible topological



52



equivalence classes which can be obtained by a smooth perturbation of the
system. Only one bifurcation parameter (1/z) arises in the present analysis.
It may be that a generalization of the model will reveal additional bifurcation"
parameters which will produce a more complete unfolding of the bifurcation.
Also, additional bifurcations of the sonic critical point may alter the phase
plane structure at values of the shock radius z which are large compared to
the reaction zone width but small with respect to the rather stringent limits
imposed in Theorem 4.1. A study of these bifurcations could shed light on
phenomena such as detonation failure and reinitiation.

Aknowledgements The results of Sections 1-3 were circulated in
manuscript form in the summer of 1985 at the Los Alamos National Labora-
tory and were presented in a public lecture there in January 1986. We thank
J. Bdzil for comments on these occasions. The author also wishes to thank
B. Bukiet for his efforts in the numerical validation and implementation of
the model. A critical reading of a preliminary version of the manuscript by
W. Fickett, R. Menikoff, B. Plohr, and D. H. Sharp provided valuable
insights and corrections. The author is indebted to J. Glimm for his con-
tinual guidance.



-53



REFERENCES

[1] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential
Equations, Springer Verlag, New York (1983).

[2] V.I. Arnold, Ordinary Differential Equations, M.I.T. Press, Cambridge
(1978).

[3] J. Bdzil and R. Engelke, "A study of the steady-state reaction-zone struc-
ture of a homogeneous and a heterogeneous explosive," Phys. Fluids 26,
pp. 1210-1221 (1983).

[4] J. Bdzil and D. S. Stewart, "The shock dynamics of stable multidimen-
sional detonation," U. 111. Preprint (1986).

[5] B. Bukiet, "The effect of curvature on detonation speed," NYU Preprint
(1986).

[6] B. Bukiet, A Study of Some Numerical Methods for Two Dimensional
Curved Detonation Problems, NYU Thesis (1986).

[7] E. Dabora, J. Nicholls, and R. Morrison, "The influence of a compressi-
ble boundary on the propagation of gaseous detonations," Tenth Sympo-
sium (International) on Combustion, pp. 817-830 (1965).

[8] J. Erpenbeck, "Steady-state analysis of quasi-one-dimensional reactive
flow," Phys. Fluids, 11, pp. 1352-1370 (1968).

[9] Eyring, Powell, EXiffey and Parlin, "The stability of detonation," Chem.
Revs. 45, pp. 69-181 (1949).

[10] W. Fickctt and W. Davis, Detonation, University of California Press,
Berkeley (1979).



- 54 -



[11] J. Guckenheimer and P. Holmes, Nonlinear Oscillations. Dynamical S\s-
tems, and Bifurcations of Vector Fields, Springer V'erlag, New York
(1983).

[12] H. Jones, "A theory of the dependence of the rate of detonation of soHd
explosives on the diameter of the charge," Proc. Roy. Soc. (London),
A 189, pp. 415-427 (1947).

[13] C. L. Mader, Numerical Modeling of Detonations, University of Califor-
nia Press, Berkeley (1979).

[14] L. Pismen, "Dynamics of lumped chemically reacting systems near singu-
lar bifurcation points," Chem. Eng. Sci. 39, pp. 1063-1077 (1984).

[15] M. Sichel, "A hydrodynamic theory for the propagation of gaseous deto-
nations," AIAA J. 4, pp. 264-272 (1966).

[16] W. Wood and J. Kirkwood, "Diameter effect in condensed explosives,
the relation between velocity and radius of curvature in a detonation
wave," 7. Chem. Phys. 22, pp. 1920-1924 (1954).



55-



Appendix. We summarize here the construction of the Poincare
transformations leading to equations (6.1). The augmented system may be
written in the form



{A. I)



Vy = — a^X - aiK - a^v\ + a4KV + a^Kk + a(,K^

+ Kpi(v, \, k) + X^i(v, X)

X.^ — bivX + biK-X + byK- — b^X' + Kpjiv, \, k) + Xq2{v, X)

K, -

where v = v- v^,X=X - 1,/?, 6 ^3 2 and


1 2 4

Online LibraryJames JonesAn asymptotic analysis of an expanding detonation → online text (page 4 of 4)