Skip to main content
SpringerLink
Log in

Influence of vibrations on convective instability of polymerization fronts

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

Propagation of polymerization fronts with liquid monomer and solid polymer is considered, and the influence of vibrations on critical conditions of convective instability is studied. The model includes the heat equation, the equation for the concentration and the Navier-Stokes equations considered under the Boussinesq approximation. Linear stability analysis of the problem is fulfilled, and the convective instability boundary is found depending on the amplitude and on the frequency of vibrations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A.P. Aldushin and S.G. Kasparyan, Thermodiffusional instability of a combustion front. Soviet Physics – Doklady 24 (1979) 29–31.

    Google Scholar 

  2. G. I. Barenblatt, Ya. B. Zeldovich and A. G. Istratov, Diffusive-thermal stability of a laminar flame. Zh. Prikl. Mekh. Tekh. Fiz. 4 (1962) 21 (in Russian).

    Google Scholar 

  3. S.B. Margolis, H.G. Kaper, G.K. Leaf and B.J. Matkowsky, Bifurcation of pulsating and spinning reaction fronts in condensed two-phase combustion. Combust. Sci. Technol. 43 (1985) 127–165.

    Google Scholar 

  4. B.J. Matkowsky and G.I. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion. SIAM J. Appl. Math. 35 (1978) 465–478.

    Google Scholar 

  5. K.G. Shkadinsky, B.I. Khaikin and A.G. Merzhanov, Propagation of a pulsating exothermic reaction front in the condensed phase. Combustion, Explosion, and Shock Waves 7 (1971) 15–22.

    Google Scholar 

  6. P. Clavin, Dynamic behavior of premixed flame fronts in laminar and turbulent flows. Progr. Energy Comb. Sci. 11 (1985) 1–59.

    Google Scholar 

  7. A.G. Istratov and V.B. Librovich, Effect of the transfer processes on stability of a planar flame front. J. Appl. Math. Mech. 30 (1966) 451–466 (in Russian).

    Google Scholar 

  8. L.D. Landau and E.M. Lifshitz, Fluid Mechanics. New York: Pergamon (1987) 683 pp.

    Google Scholar 

  9. M. Matalon and B.J. Matkowsky, Flames in fluids: Their interaction and stability. Combust. Sci. and Technol. 34 (1983) 295–316.

    Google Scholar 

  10. Ya.B. Zeldovich, G.I. Barenblatt, V.B. Librovich and G.M. Makhviladze, The Mathematical Theory of Combustion and Explosions. New York: Consultants Bureau (1985) 597 pp.

    Google Scholar 

  11. B.J. Matkowsky and G.I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation. SIAM J. Appl. Math. 37 (1979) 686–699.

    Google Scholar 

  12. J. Pojman, V. Volpert and T. Dumont et al. Bubble behavior and convection in frontal polymerization on the KC-135 aircraft. AIAA Conference, Reno, 2000.

  13. B. McCaughey, J.A. Pojman, C. Simmons and V.A. Volpert, The effect of convection on a propagating front with a liquid product: comparison of theory and experiments. Chaos 8 (1998) 520–529.

    Google Scholar 

  14. G. Bowden, M. Garbey, V. Ilyashenko, J. Pojman, S. Solovyov and V. Volpert, The effect of convection on a propagating front with a solid product: comparison of theory and experiments. J. Chem. Physics 101 (1997) 678–686. Influence of vibrations on convective instability of polymerization fronts 31

    Google Scholar 

  15. M. Garbey, A. Taik and V. Volpert, Influence of natural convection on stability of reaction fronts in liquids. Quart. Appl. Math. 53 (1998) 1–35.

    Google Scholar 

  16. M. Garbey, A. Taik and V. Volpert, Linear stability analysis of reaction fronts in liquids. Quart. Appl. Math. 54 (1996) 225–247.

    Google Scholar 

  17. P. Pelce and D. Rochweger, Parametric control of microstructures in directional solidifaction. Phys. Rev. A 37 (1992) 5042–5053.

    Google Scholar 

  18. Ya. B. Zel'dovich and D. A. Frank-Kamenetskii, A theory of thermal propagation of flame. Acta Physicochim. USSR 9 (1938) 341–350 (Russian).

    Google Scholar 

  19. L. Landau, On the theory of slow combustion. Acta Physicochimica URSS (1944) 77–85.

  20. G. Searby, Acoustic instability in premixed flames. Combust. Sci. Technol. 81 (1992) 221–231.

    Google Scholar 

  21. J. D. Alexander and C. A. Lundquist, Motions in fluids caused by microgravitational acceleration and their modification by relative rotation. AIAA Journal 261 (1988) 34–39.

    Google Scholar 

  22. K. M. Nigam and K. D. P. Nigam, An approach in modeling the tubular polymer reactor. J. Appl. Polymer Sci. 28 (1983) 887–900.

    Google Scholar 

  23. A. YA. Malkin, L. I. Sherysheva, S. G. Kulichikhin and P. V. Zhirkov, The flow of a polymerizing liquid in a cylindrical pipe. Polymer Eng. Sci. 23 (1983) 804–809.

    Google Scholar 

  24. C. N. Lekakou and S.M. Richardson, Simulation of reacting flow during filling in reaction injection molding (RIM). Polymer Eng. Sci. 26 (1986) 1264–1275.

    Google Scholar 

  25. L. T. Manzione, Simulation of cavity filling and curing in reaction injection molding. Polym. Sci. Technol. 9 (1982) 263–278.

    Google Scholar 

  26. A. I Fedoseyev and J. D. Alexander, An inverse finite element method for pure and binary solidification problems. J. Comp. Phys. 130 (1997) 243–255.

    Google Scholar 

  27. A. I. Fedoseyev and J. D. Alexander, Thermovibrational flow in Bridgman Melt Growth Configurations. American Institute of Aeronautics and Astronautics (1999).

  28. J. D. Alexander, J. Ouazzani and F. Rosenberger, Analysis of the low gravity tolerance of Bridgman-Stockbarger crystal growth. I. Steady and impulse accelerations. J. Crystal Growth 97 (1989) 285–302.

    Google Scholar 

  29. J. D. Alexander, J. Ouazzani and F. Rosenberger, Analysis of the low gravity tolerance of Bridgman-Stockbarger crystal growth. II. Transitient and periodic accelerations. J. Crystal Growth 113 (1991) 21–38.

    Google Scholar 

  30. J. A. Pojman, Frontal polymerization in microgravity. AIAA 98-08113, 36th Aerospace Sciences Meeting (1998)

  31. J. A. Pojman, A. M. Khan and L. J. Mathias, Frontal polymerization in microgravity: results from the Conquest I sounding rocket flight. Microgravity Sci. Technol. X (1997) 36–40.

    Google Scholar 

  32. J. A. Pojman, A. M. Khan and L. Mathias, Frontal polymerization in microgravity. Polym. Prepr. (Am Chem. Soc. Div. Polym. Chem.) 38 (1997) 717–718.

    Google Scholar 

  33. J. I. D. Alexander, Low-gravity experiment sensitivity to residual acceleration: A Review. Microgravity Sci. Technol. III (1990) 52–68.

    Google Scholar 

  34. J. I. D. Alexander, Residual gravity jitter effects on fluid Processes. Microgravity Sci. Technol. VII (1994) 131–136.

    Google Scholar 

  35. M. Ishikawa, S. Nakamura, S. Yoda, H. Samejima and T. Goshozono, Respective motion bubbles to periodic g-jitter. Microgravity Sci. Technol. VII/2 (1994) 164–168.

    Google Scholar 

  36. H. Tang, Z. H. Cao., F. Liu and W. R. Hu, Effects of g-jitter on the critical Marangoni number. Microgravity Sci. Technol. VII/1 (1994) 137–141.

    Google Scholar 

  37. G. H. Wolf, The Dynamic Stabilization of the Rayleigh-Taylor instability and the corresponding dynamic equilibrium. Z. Phys. 227 (1969) 291–300.

    Google Scholar 

  38. V. A. Briskman, Parametrical stabilization of fluid interface. Dokl. Akad. Nauk SSSR 226/5 (1976) 1041–1044.

    Google Scholar 

  39. B. V. Novozhilov, The rate of propagation of the front of an exothermic reaction in a condensed phase, J. Dokl. Phys. Chem. 141 (1961) 836–838.

    Google Scholar 

  40. D. A. Schult, Matched asymptotic expansions and the closure problem for combustion waves. SIAM J. Appl. Math. 60 (2000) 136–155.

    Google Scholar 

  41. L. G. Loitsyanskii. Mechanics of Liquids and Gases. New York: Pergamon Press (1966) 802 pp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UMR 5585 CNRS, Analyse Numérique, Université Lyon 1, 69622, Villeurbanne Cedex, France

    K. Allali & V. Volpert

  2. Department of Chemistry and Biochemistry, University of Southern Mississippi, Hattiesburg, MS, 39406, U.S.A.

    J.A. Pojman

Authors
  1. K. Allali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Volpert
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J.A. Pojman
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Allali, K., Volpert, V. & Pojman, J. Influence of vibrations on convective instability of polymerization fronts. Journal of Engineering Mathematics 41, 13–31 (2001). https://doi.org/10.1023/A:1011878929608

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011878929608

Search

Navigation