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Multiquadric RBF Method and Asymptotic Analysis to Study the Influence of Orientation on the Reaction Fronts Propagation

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Abstract

In this work, we study the influence of orientation on the stability conditions of the reaction front where the monomer is solid and the polymer is liquid. The mathematical model includes the heat equation, the concentration equation and the Navier–Stokes equation under the Boussinesq approximation. We use the method proposed by Zeldovich and Frank-Kamenetskii to perform asymptotic analysis. We then perform a stability analysis. The linearized problem is solved numerically using a multiquadric radial basis function method (MQ-RBF) to find the stability boundary. This will allow us to deduce the influence of each control parameter of the problem on this stability, in particular the angle of inclination of the experimental tube.

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REFERENCES

  1. P. M. Goldfeder, V. A. Volpert, V. M. Ilyashenko, A. M. Khan, J. A. Pojman, and S. E. Solovyov, “Mathematical modeling of free-radical polymerization fronts,” J. Phys. Chem. B 101, 3474–3482 (1997). https://doi.org/10.1021/jp962150v

    Article  CAS  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, 21 (1962) [in Russian].

    Google Scholar 

  3. A. P. Aldushin, and S. G. Kasparyan, “Thermodiffusional instability of a combustion front,” Sov. Phys. Dokl. 24, 29 (1979).

    ADS  Google Scholar 

  4. 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, 127–165 (1985). https://doi.org/10.1080/00102208508947001

    Article  CAS  Google Scholar 

  5. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1987).

    Google Scholar 

  6. Y. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions (Consultants Bureau, New York, 1985).

    Book  Google Scholar 

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

    Article  Google Scholar 

  8. H. Rouah, L. Salhi and A. Taik, “Influence of some critical parameters on the stability of reaction fronts in liquid medium,” Int. J. Model. Identif. Control 36 (), 42–56 (2020). https://doi.org/10.1504/IJMIC.2020.115394

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

    Article  MathSciNet  Google Scholar 

  10. B. J. Matkowsky and G. I. Sivashinsky, “Acceleration effects on the stability of flame propagation,” SIAM J. Appl. Math. 37, 669–685 (1979). https://doi.org/10.1137/0137050

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. M. Bazile Jr., H. A. Nichols, J. A. Pojman, and V. Volpert, “Effect of orientation on thermoset frontal polymerization,” J. Polym. Sci. Part A: Polym. Chem. 40, 3504–3508 (2002). https://doi.org/10.1002/pola.10447

    Article  ADS  CAS  Google Scholar 

  13. 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. 10, 36 (1997).

    CAS  Google Scholar 

  14. V. A. Volpert, V. A. Volpert, V. M. Ilyashenko, and J.A. Pojman, “Frontal polymerization in a porous medium,” Chem. Eng. Sci. 53, 1655–1665 (1998).

    Article  CAS  Google Scholar 

  15. M. A. Mujeebu, M. Z. Abdullah, M. A. Bakar, A. A. Mohamad, R. M. N. Muhad, and M. K. Abdullah, “Combustion in porous media and its applications—A comprehensive survey,” J. Environ. Manage. 90, 2287–2312 (2009). https://doi.org/10.1016/j.jenvman.2008.10.009

    Article  CAS  PubMed  Google Scholar 

  16. L. D. Su, Z. W. Jiang and T. S. Jiang, “Numerical solution for a kind of nonlinear telegraph equations using radial basis functions,” Commun. Comput. Inf. Sci. 391, 140–149 (2013).

    Google Scholar 

  17. R. L. Hardy, “Multiquadric equations of topography and other irregular surfaces,” J. Geophys. Res. 76, 1905– 1915 (1971).

    Article  ADS  Google Scholar 

  18. R. Franke, A Critical Comparison of Some Methods for the Interpolation of Scattered Data. Technical Report (Naval Postgraduate School, New York, 1979).

    Book  Google Scholar 

  19. G. E. Fasshauer and J. Zhang, “On Choosing Optimal Shape Parameters for RBF Approximation,” Numer. Algorithms 45, 345–368 (2007). https://doi.org/10.1007/s11075-007-9072-8

    Article  ADS  MathSciNet  Google Scholar 

  20. M. D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge Univ. Press, Cambridge, 2003).

    Book  Google Scholar 

  21. G. E. Fasshauer, Meshfree Approximation Methods with MATLAB (World Scientific, Singapore, 2007).

    Book  Google Scholar 

  22. Ya. B. Zeldovich and D. A. Frank-Kamenetskii, “A theory of thermal propagation of flame,” Acta Physicochim. USSR 9, 341–350 (1938).

    CAS  Google Scholar 

  23. D.A. Schult, “Matched asymptotic expansions and the closure problem for combustion waves,” SIAM J. Appl. Math. 60, 136–155 (2000). https://doi.org/10.1137/S0036139998337152

    Article  MathSciNet  Google Scholar 

  24. A. Halassi, D. Ouazar, and A. Taik, “RBF methods for solving laterally averaged Saint-Venant equations: Application to eutrophication prevention through aeration,” Int. J. Comput. Fluid Dyn. 29, 464–477 (2015). https://doi.org/10.1080/10618562.2015.1129056

    Article  MathSciNet  Google Scholar 

  25. F. Benkhaldoun, A. Halassi, D. Ouazar, M. Seaid, and A. Taik, “Slope limiters for radial basis functions applied to conservation laws with discontinuous flux function,” Eng. Anal. Boundary Elem. 66, 49–65 (2016).

    Article  MathSciNet  Google Scholar 

  26. Y. Alhuri, F. Benkhaldoun, D. Ouazar, M. Seaid, and A. Taik, “A meshless method for numerical simulation of depth–averaged turbulence flows using a \( k \)– \( \varepsilon \) model,” Int. J. Numer. Methods Fluids 80, 3–22 (2016). https://doi.org/10.1002/fld.4067

    Article  ADS  MathSciNet  CAS  Google Scholar 

  27. F. Benkhaldoun, A. Halassi, D. Ouazar, M. Seaid, and A. Taik, “A stabilized meshless method for time-dependent convection-dominated flow problems,” Math. Comput. Simul. 137, 159–176 (2017). https://doi.org/10.1016/j.matcom.2016.11.003

    Article  MathSciNet  Google Scholar 

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Polydisciplinary Faculty of Sidi Bennour, Chouaib Doukkali University, El Jadida, Morocco

    Y. Joundy

  2. University Hassan II, FST-Mohammadia, Casablanca, Morocco

    H. Rouah & A. Taik

Authors
  1. Y. Joundy
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  2. H. Rouah
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  3. A. Taik
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Correspondence to H. Rouah.

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CONFLICT OF INTEREST. The authors of this work declare that they have no conflicts of interest.

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Joundy, Y., Rouah, H. & Taik, A. Multiquadric RBF Method and Asymptotic Analysis to Study the Influence of Orientation on the Reaction Fronts Propagation. J. Appl. Ind. Math. 17, 943–955 (2023). https://doi.org/10.1134/S1990478923040208

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