Influence of natural convection on stability of an inclined front propagation

Abstract

This research work may be considered as a continuation of a series of investigations concerning the influence of natural convection on stability of reaction fronts propagation. We consider an inclined propagating polymerization front. The governing equations consist of the heat equation, the equation for the depth of conversion for one-step chemical reaction and of the Navier-Stokes equations under the Boussinesq approximation. We first perform a formal asymptotic analysis in the limit of a large activation energy to get an approximate interface problem. Then, we fulfill the linear stability analysis of the stationary solution and find the perturbation equations. A meshless collocation method based on multiquadric radial basis functions has been applied for numerical simulations. The conditions of convective instabilities obtained are in good agreement with some previous studies. This shows that the proposed approach is accurate and that it helps in describing the influence of the propagation direction on stability of polymerization fronts.

Keywords

• Polymerization front,natural convection,asymptotic analysis,linear stability analysis,multiquadric radial basis functions method

References

1. K. Allali, A. Ducrot, A. Taik, and V. Volpert. In‡uence of vibrations on convective instability of polymerization fronts. Jour. Eng. Math., 41:13–31, 2001.
2. M. Bazile, H.A. Nichols, J.A. Pojman, and V. Volpert. E¤ect of orientation on thermoset frontal polymerization. Journal of Polymer Science Part A: Polymer Chemistry 40 (20), 3504-3508, 2002.
3. M. Belk, K.G. Kostarev, V. Volpert, and Yudina T.M. Frontal photopolymerization with convection. The Journal of Physical Chemistry B 107 (37), 10292-10298, 2003.
4. W. Cheney and W. Light. A course in approximation theory. William Allan, New York, 1999.
5. M. Garbey, A. Taik, and V. Volpert. Linear stability analysis of reaction fronts in liquids. Quart. Appl. Math., 1996.
6. R.L. Hardy. Multiquadric equations of topography and other irregular surfaces. Journal of Geophysical Research, 1971.
7. S.B. Margolis. An asymptotic theory of condensed two-phase ‡ame propagation. SIAM J. Applied Math., 43:351–369, 1983.
8. C. A. Micchelli. Interpolation of scattered data: distance matrices and conditionally positive de…nite functions. Constr. Approx., 2, 1986.

9. A. H. Nayfeh. Perturbation methods. Wiley, New York, 1973.
10. B.V. Novozhilov. The rate of propagation of the front of an exothermic reaction in a condensed phase. Proc. Academy Sci. USSR, Phys. Chem. Sect., 141:836–838, 1961.
11. S.A. Sarra and J.E. Kansa. A Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Di¤ erential Equations. Marshall University and University of California, Davis, 2009.
12. A. Volpert, Vit. Volpert, and Vl. Volpert. Travelling wave solutions of parabolic systems. AMS Providence, 1994.
13. Ya.B. Zeldovich, G.I. Barenblatt, V.B. Librovich, and G.M. Makhviladze. The mathematical theory of combustion and explosions. translated from the Russian by Donald McNeill, 1985.