Linear stability analysis of reaction fronts in liquids
Authors:
M. Garbey, A. Taik and V. Volpert
Journal:
Quart. Appl. Math. 54 (1996), 225-247
MSC:
Primary 76E15; Secondary 35K57, 35R35, 76U05, 80A32
DOI:
https://doi.org/10.1090/qam/1388014
MathSciNet review:
MR1388014
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Abstract: The propagation of a reaction front for liquid-to-solid reaction is studied. The model includes the heat equation, an equation for the concentration of the liquid reactant, and the equations of liquid motion under the Boussinesq approximation. The linear stability of the reaction front is studied, and conditions for cellular and oscillatory instability are determined.
A. P. Aldushin and S. G. Kasparyan, Thermo-diffusion instability of combustion front, Doklady Akademii Nauk SSSR (5) 244, 67–70 (1979) (Russian)
A. S. Babadzhanyan, Vit. A. Volpert, Vl. A. Volpert, S. P. Davtyan, and I. N. Megrabova, Frontal regimes of an exothermal reaction with radially symmetric injection of reagents, Combustion, Explosion and Shock Waves (6) 24, 711–719 (1988)
A. S. Babadzhanyan, Vit. A. Volpert, Vl. A. Volpert, S. P. Davtyan, and I. N. Megrabova, Stability of frontal regimes for the occurrence of an exothermic reaction with radially symmetric supply of the reagents, Combustion, Explosion and Shock Waves (1) 25, 23–31 (1989)
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) (Russian)
A. Bayliss and B. J. Matkowsky, Nonlinear dynamics of cellular flames, SIAM J. Applied Math. (2) 52, 396–415 (1992)
G. Bazsa and I. R. Epstein, Traveling waves in the nitric acid-iron (II) reaction, J. Phys. Chem. (14) 89, 3050–3053 (1985)
V. P. Begishev, V. A. Volpert, S. P. Davtyan, and A. Y. Malkin, Some features of the anionic activated $\varepsilon$-caprolactam polymerization under wave propagation conditions, Dokl. Phys. Chemistry (4) 29, 1075–1077 (1984)
S. A. Bostandzhiyan, V. I. Boyarchenko, P. V. Zhirkov, and Zh. A. Zinenko, Low-temperature polymerization conditions in a flow-through reactor, J. Appl. Mech. Tech. Phys. (1) 20, 99–104 (1979)
S. A. Bostandzhiyan, M. V. Shulikovskaya, and S. P. Davtyan, 6th International School on Modeling Heat and Mass Transfer Processes in Chemical and Biochemical Reactors, Varna, 1989, p. 231
A. A. Butakov, E. I. Maksimov, and K. G. Shkadinsky, Theory of chemical displacement reactors, Combustion, Explosion and Shock Waves (1) 14, 48–54 (1978)
A. A. Butakov and E. A. Stessel, Influence of convection on occurrence of the exothermic reaction in tubular reactors, Doklady Akademii Nauk SSSR (6) 237, 1422–1425 (1977) (Russian)
A. A. Butakov and A. M. Zanin, Effects of viscosity variation on an exothermic reaction in a flow system, Combustion, Explosion and Shock Waves (5) 14, 628–631 (1978)
N. M. Chechilo, R. Ya. Khvilivitsky, and N. S. Enikolopyan, The phenomenon of propagation of the polymerization reaction, Dokl. Phys. Chemistry (4-6) 204, 512–513 (1972)
N. M. Chechilo and N. S. Enikolopyan, Structure of the polymerization wave front and propagation mechanism of the polymerization reaction, Dokl. Phys. Chemistry (4-6) 214, 174–176 (1974)
N. M. Chechilo and N. S. Enikolopyan, Effect of the concentration and nature of the initiators on the propagation process in polymerization, Dokl. Phys. Chemistry (4–6) 221, 391–394 (1975)
N. M. Chechilo and N. S. Enikolopyan, Effect of pressure and initial temperature of the reaction mixture during propagation of a polymerization reaction, Dokl. Phys. Chemistry (1–3) 230, 840–843 (1976)
S. P. Davtyan, P. V. Zhirkov, and S. A. Vol’fson, Problems of non-isothermal character in polymerization processes, Russ. Chemical Reviews (2) 53, 150–163 (1984)
D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, 1969
M. Garbey, H. G. Kaper, G. K. Leaf, and B. J. Matkowsky, Nonlinear analysis of condensed phase surface combustion, European J. Applied Math. 1, 73–89 (1990)
M. Garbey, H. G. Kaper, G. K. Leaf, and B. J. Matkowsky, Linear stability analysis of cylindrical flames, Quart. Appl. Math. XLVII, 691–704 (1989)
M. Garbey, H. G. Kaper, G. K. Leaf, and B. J. Matkowsky, Using Maple for the analysis of bifurcation phenomena in condensed phase surface combustion, J. Symbolic Computation 12, 89–113 (1991)
G. Z. Gershuni and E. M. Zhuhovitsky, Convective Instabilities of Incompressible Liquid, Moscow, Nauka, 1972
S. I. Khudyaev, On the asymptotic theory of the stationary combustion wave, Khimicheskaya Physica (5) 6, 681–691 (1987) (Russian)
A. Ya. Malkin, Yu. B. Navochnik, and V. P. Begishev, Polymer Process Engrg. 1, 71 (1983)
S. B. Margolis, An asymptotic theory of condensed two-phase flame propagation, SIAM J. Appl. Math. 43, 351–369 (1983)
S. B. Margolis and R. C. Armstrong, Two asymptotic models for solid propellant combustion, Combust. Sci. Technol. 47, 1–38 (1986)
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, Combustion Sci. and Tech. 43, 127–165 (1985)
B. J. Matkowsky and G. I. Sivashinsky, Propagation of a pulsating front in solid fuel combustion, SIAM J. Appl. Math. 35, 465–478 (1978)
A. G. Merzhanov, SHS-process: Combustion theory and practice, Archivum Combustionis (1–2) 1, 23–48 (1981)
A. G. Merzhanov, A. K. Filonenko, and I. P. Borovinskaya, New phenomena in combustion of condensed systems, Dokl. Phys. Chem. 208, 122–125 (1973)
A. G. Merzhanov and B. I. Khaikin, Theory of combustion waves in homogeneous media, Progr. Ener. Combustion Sci. 14, 1–98 (1988)
H. Miike, S. C. Muller, and B. Hess, Oscillatory deformation of chemical waves induced by surface flow, Physical Review Letters (18) 61, 2109–2112 (1988)
H. Miike, S. C. Muller, and B. Hess, Oscillatory hydrodynamic flow induced by chemical waves, Chem. Physics Letters (5, 6) 144, 515–520 (1988)
I. Nagypal, G. Bazsa, and I. R. Epstein, Gravity-induced anisotropies in chemical waves, J. Amer. Chem. Soc. (13) 108, 3635–3640 (1986)
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)
T. Plesser, H. Wilke, and K. H. Winters, Interaction of chemical waves with convective flows induced by density gradients. A comparison between experiments and computer simulations, Chem. Physics Letters (1, 2) 200, 158–162 (1992)
J. A. Pojman, R. Graven, A. Khan, and W. West, Convective instabilities in traveling fronts of addition polymerization, J. Physical Chemistry 96, 7466–7472 (1992)
J. A. Pojman and I. R. Epstein, Convective effects on chemical waves. 1. Mechanisms and stability criteria, J. Phys. Chem. (12) 94, 4966–4972 (1990)
J. A. Pojman, I. R. Epstein, T. J. McManus, and K. Scowalter, Convective effects on chemical waves. 2. Simple convection in the iodate-arsenous acid system, J. Phys. Chem. (3) 95, 1299–1306 (1991)
J. A. Pojman, I. P. Nagy, and I. R. Epstein, Convective effects in chemical waves. 3. Multicomponent convection in the iron (II)-nitric acid system, J. Phys. Chem. (3) 95, 1306–1311 (1991)
D. S. Riley and S. H. Davis, Long-wave interactions in morphological and convective instabilities, IMA J. Applied Math. 45, 267–285 (1990)
S. E. Solovyov, V. A. Volpert, and S. P. Davtyan, Radially symmetric flow of reacting liquid with changing viscosity, SIAM J. Appl. Math. 53, 907–914 (1993)
M. V. Shulikovskaya, S. A. Bostandzhiyan, and S. P. Davtyan, Frontal radical polymerization in spherical plug-flow reactors, Teor. osnovy khim. tehnologii (3) 23, 340–345 (1989) (Russian)
A. G. Strunina, A. V. Dvoryankin, and A. G. Merzhanov, Unstable regimes of thermite system combustion, Combustion, Explosion and Shock Waves 19, 158–163 (1983)
D. A. Vaganov, Zh. Prikl. Mekh. Tekh. Fiz. (1), 114–122 (1977) (Russian)
D. A. Vaganov and S. I. Khudyaev, On a stationary problem of combustion theory, Combustion, Explosion and Shock Waves 5, 167–172 (1969)
D. A. Vasquez, J. W. Wilder, and B. F. Edwards, Hydrodynamic instability of chemical waves, J. Chem. Phys. (3) 98, 2138–2143 (1993)
D. A. Vasquez, B. F. Edwards, and J. W. Wilder, Onset of convection for autocatalytic reaction fronts: Laterally bounded systems, Physical Review A 43, 6694–6699 (1991)
H. J. Viljoen, J. E. Gatica, and V. Hlavacek, Bifurcation analysis of chemically driven convection, Chem. Engrg. Sci. (2) 45, 503–517 (1990)
Vit. A. Volpert, VI. A. Volpert, S. P. Davtyan, I. N. Megrabova, and N. F. Surkov, Two-dimensional combustion modes in condensed flow, SIAM J. Appl. Math. 52, 368–383 (1992)
V. A. Volpert, A. I. Volpert, and A. G. Merzhanov, Application of bifurcation theory to investigation of spin waves of combustion, Dokl. Phys. Chem. 262, 55–58 (1982)
V. A. Volpert, A. I. Volpert, and A. G. Merzhanov, Analysis of nonunidimensional combustion regimes by bifurcation theory methods, Dokl. Phys. Chem. 263, 239–242 (1982)
V. A. Volpert and S. P. Davtyan, The problem of wave existence for the process of polymerization and crystallization, Dokl. Akademii Nauk SSSR (1) 268, 62–65 (1983)
V. A. Volpert, I. N. Megrabova, S. P. Davtyan, and V. P. Begishev, Propagation of caprolactam polymerization wave, Combustion, Explosion and Shock Waves (4) 21, 441–447 (1985)
Vit. Volpert and Vl. Volpert, Propagation of frontal polymerization-crystallization waves, European J. Appl. Math. 5, 201–215 (1994)
Vit. A. Volpert and Vl. A. Volpert, Propagation velocity estimation for condensed phase combustion, SIAM J. Applied Math. (4) 51, 1074–1089 (1991)
Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985
Ya. B. Zeldovich and D. A. Frank-Kamenetsky, The theory of thermal propagation of flames, Zh. Fiz. Khim. 12, 100–105 (1938) (Russian)
G. V. Zhizhin and A. S. Segal, Hydrodynamic stability of the spherical front of a reaction accompanied by a strong increase in viscosity, Fluid Dynamics (3) 23, 361–367 (1988)
G. V. Zhizhin and A. S. Segal, Hydrodynamic stability of a cylindrical reaction front associated with a strong increase of viscosity, J. Appl. Mech. Tech. Phys. (2) 29, 216–224 (1988)
A. P. Aldushin and S. G. Kasparyan, Thermo-diffusion instability of combustion front, Doklady Akademii Nauk SSSR (5) 244, 67–70 (1979) (Russian)
A. S. Babadzhanyan, Vit. A. Volpert, Vl. A. Volpert, S. P. Davtyan, and I. N. Megrabova, Frontal regimes of an exothermal reaction with radially symmetric injection of reagents, Combustion, Explosion and Shock Waves (6) 24, 711–719 (1988)
A. S. Babadzhanyan, Vit. A. Volpert, Vl. A. Volpert, S. P. Davtyan, and I. N. Megrabova, Stability of frontal regimes for the occurrence of an exothermic reaction with radially symmetric supply of the reagents, Combustion, Explosion and Shock Waves (1) 25, 23–31 (1989)
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) (Russian)
A. Bayliss and B. J. Matkowsky, Nonlinear dynamics of cellular flames, SIAM J. Applied Math. (2) 52, 396–415 (1992)
G. Bazsa and I. R. Epstein, Traveling waves in the nitric acid-iron (II) reaction, J. Phys. Chem. (14) 89, 3050–3053 (1985)
V. P. Begishev, V. A. Volpert, S. P. Davtyan, and A. Y. Malkin, Some features of the anionic activated $\varepsilon$-caprolactam polymerization under wave propagation conditions, Dokl. Phys. Chemistry (4) 29, 1075–1077 (1984)
S. A. Bostandzhiyan, V. I. Boyarchenko, P. V. Zhirkov, and Zh. A. Zinenko, Low-temperature polymerization conditions in a flow-through reactor, J. Appl. Mech. Tech. Phys. (1) 20, 99–104 (1979)
S. A. Bostandzhiyan, M. V. Shulikovskaya, and S. P. Davtyan, 6th International School on Modeling Heat and Mass Transfer Processes in Chemical and Biochemical Reactors, Varna, 1989, p. 231
A. A. Butakov, E. I. Maksimov, and K. G. Shkadinsky, Theory of chemical displacement reactors, Combustion, Explosion and Shock Waves (1) 14, 48–54 (1978)
A. A. Butakov and E. A. Stessel, Influence of convection on occurrence of the exothermic reaction in tubular reactors, Doklady Akademii Nauk SSSR (6) 237, 1422–1425 (1977) (Russian)
A. A. Butakov and A. M. Zanin, Effects of viscosity variation on an exothermic reaction in a flow system, Combustion, Explosion and Shock Waves (5) 14, 628–631 (1978)
N. M. Chechilo, R. Ya. Khvilivitsky, and N. S. Enikolopyan, The phenomenon of propagation of the polymerization reaction, Dokl. Phys. Chemistry (4-6) 204, 512–513 (1972)
N. M. Chechilo and N. S. Enikolopyan, Structure of the polymerization wave front and propagation mechanism of the polymerization reaction, Dokl. Phys. Chemistry (4-6) 214, 174–176 (1974)
N. M. Chechilo and N. S. Enikolopyan, Effect of the concentration and nature of the initiators on the propagation process in polymerization, Dokl. Phys. Chemistry (4–6) 221, 391–394 (1975)
N. M. Chechilo and N. S. Enikolopyan, Effect of pressure and initial temperature of the reaction mixture during propagation of a polymerization reaction, Dokl. Phys. Chemistry (1–3) 230, 840–843 (1976)
S. P. Davtyan, P. V. Zhirkov, and S. A. Vol’fson, Problems of non-isothermal character in polymerization processes, Russ. Chemical Reviews (2) 53, 150–163 (1984)
D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Plenum Press, New York, 1969
M. Garbey, H. G. Kaper, G. K. Leaf, and B. J. Matkowsky, Nonlinear analysis of condensed phase surface combustion, European J. Applied Math. 1, 73–89 (1990)
M. Garbey, H. G. Kaper, G. K. Leaf, and B. J. Matkowsky, Linear stability analysis of cylindrical flames, Quart. Appl. Math. XLVII, 691–704 (1989)
M. Garbey, H. G. Kaper, G. K. Leaf, and B. J. Matkowsky, Using Maple for the analysis of bifurcation phenomena in condensed phase surface combustion, J. Symbolic Computation 12, 89–113 (1991)
G. Z. Gershuni and E. M. Zhuhovitsky, Convective Instabilities of Incompressible Liquid, Moscow, Nauka, 1972
S. I. Khudyaev, On the asymptotic theory of the stationary combustion wave, Khimicheskaya Physica (5) 6, 681–691 (1987) (Russian)
A. Ya. Malkin, Yu. B. Navochnik, and V. P. Begishev, Polymer Process Engrg. 1, 71 (1983)
S. B. Margolis, An asymptotic theory of condensed two-phase flame propagation, SIAM J. Appl. Math. 43, 351–369 (1983)
S. B. Margolis and R. C. Armstrong, Two asymptotic models for solid propellant combustion, Combust. Sci. Technol. 47, 1–38 (1986)
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, Combustion Sci. and Tech. 43, 127–165 (1985)
B. J. Matkowsky and G. I. Sivashinsky, Propagation of a pulsating front in solid fuel combustion, SIAM J. Appl. Math. 35, 465–478 (1978)
A. G. Merzhanov, SHS-process: Combustion theory and practice, Archivum Combustionis (1–2) 1, 23–48 (1981)
A. G. Merzhanov, A. K. Filonenko, and I. P. Borovinskaya, New phenomena in combustion of condensed systems, Dokl. Phys. Chem. 208, 122–125 (1973)
A. G. Merzhanov and B. I. Khaikin, Theory of combustion waves in homogeneous media, Progr. Ener. Combustion Sci. 14, 1–98 (1988)
H. Miike, S. C. Muller, and B. Hess, Oscillatory deformation of chemical waves induced by surface flow, Physical Review Letters (18) 61, 2109–2112 (1988)
H. Miike, S. C. Muller, and B. Hess, Oscillatory hydrodynamic flow induced by chemical waves, Chem. Physics Letters (5, 6) 144, 515–520 (1988)
I. Nagypal, G. Bazsa, and I. R. Epstein, Gravity-induced anisotropies in chemical waves, J. Amer. Chem. Soc. (13) 108, 3635–3640 (1986)
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)
T. Plesser, H. Wilke, and K. H. Winters, Interaction of chemical waves with convective flows induced by density gradients. A comparison between experiments and computer simulations, Chem. Physics Letters (1, 2) 200, 158–162 (1992)
J. A. Pojman, R. Graven, A. Khan, and W. West, Convective instabilities in traveling fronts of addition polymerization, J. Physical Chemistry 96, 7466–7472 (1992)
J. A. Pojman and I. R. Epstein, Convective effects on chemical waves. 1. Mechanisms and stability criteria, J. Phys. Chem. (12) 94, 4966–4972 (1990)
J. A. Pojman, I. R. Epstein, T. J. McManus, and K. Scowalter, Convective effects on chemical waves. 2. Simple convection in the iodate-arsenous acid system, J. Phys. Chem. (3) 95, 1299–1306 (1991)
J. A. Pojman, I. P. Nagy, and I. R. Epstein, Convective effects in chemical waves. 3. Multicomponent convection in the iron (II)-nitric acid system, J. Phys. Chem. (3) 95, 1306–1311 (1991)
D. S. Riley and S. H. Davis, Long-wave interactions in morphological and convective instabilities, IMA J. Applied Math. 45, 267–285 (1990)
S. E. Solovyov, V. A. Volpert, and S. P. Davtyan, Radially symmetric flow of reacting liquid with changing viscosity, SIAM J. Appl. Math. 53, 907–914 (1993)
M. V. Shulikovskaya, S. A. Bostandzhiyan, and S. P. Davtyan, Frontal radical polymerization in spherical plug-flow reactors, Teor. osnovy khim. tehnologii (3) 23, 340–345 (1989) (Russian)
A. G. Strunina, A. V. Dvoryankin, and A. G. Merzhanov, Unstable regimes of thermite system combustion, Combustion, Explosion and Shock Waves 19, 158–163 (1983)
D. A. Vaganov, Zh. Prikl. Mekh. Tekh. Fiz. (1), 114–122 (1977) (Russian)
D. A. Vaganov and S. I. Khudyaev, On a stationary problem of combustion theory, Combustion, Explosion and Shock Waves 5, 167–172 (1969)
D. A. Vasquez, J. W. Wilder, and B. F. Edwards, Hydrodynamic instability of chemical waves, J. Chem. Phys. (3) 98, 2138–2143 (1993)
D. A. Vasquez, B. F. Edwards, and J. W. Wilder, Onset of convection for autocatalytic reaction fronts: Laterally bounded systems, Physical Review A 43, 6694–6699 (1991)
H. J. Viljoen, J. E. Gatica, and V. Hlavacek, Bifurcation analysis of chemically driven convection, Chem. Engrg. Sci. (2) 45, 503–517 (1990)
Vit. A. Volpert, VI. A. Volpert, S. P. Davtyan, I. N. Megrabova, and N. F. Surkov, Two-dimensional combustion modes in condensed flow, SIAM J. Appl. Math. 52, 368–383 (1992)
V. A. Volpert, A. I. Volpert, and A. G. Merzhanov, Application of bifurcation theory to investigation of spin waves of combustion, Dokl. Phys. Chem. 262, 55–58 (1982)
V. A. Volpert, A. I. Volpert, and A. G. Merzhanov, Analysis of nonunidimensional combustion regimes by bifurcation theory methods, Dokl. Phys. Chem. 263, 239–242 (1982)
V. A. Volpert and S. P. Davtyan, The problem of wave existence for the process of polymerization and crystallization, Dokl. Akademii Nauk SSSR (1) 268, 62–65 (1983)
V. A. Volpert, I. N. Megrabova, S. P. Davtyan, and V. P. Begishev, Propagation of caprolactam polymerization wave, Combustion, Explosion and Shock Waves (4) 21, 441–447 (1985)
Vit. Volpert and Vl. Volpert, Propagation of frontal polymerization-crystallization waves, European J. Appl. Math. 5, 201–215 (1994)
Vit. A. Volpert and Vl. A. Volpert, Propagation velocity estimation for condensed phase combustion, SIAM J. Applied Math. (4) 51, 1074–1089 (1991)
Ya. B. Zeldovich, G. I. Barenblatt, V. B. Librovich, and G. M. Makhviladze, The Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, 1985
Ya. B. Zeldovich and D. A. Frank-Kamenetsky, The theory of thermal propagation of flames, Zh. Fiz. Khim. 12, 100–105 (1938) (Russian)
G. V. Zhizhin and A. S. Segal, Hydrodynamic stability of the spherical front of a reaction accompanied by a strong increase in viscosity, Fluid Dynamics (3) 23, 361–367 (1988)
G. V. Zhizhin and A. S. Segal, Hydrodynamic stability of a cylindrical reaction front associated with a strong increase of viscosity, J. Appl. Mech. Tech. Phys. (2) 29, 216–224 (1988)
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