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Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys
Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$
1. | Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, 1050 Brussels, Belgium |
References:
[1] |
M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions,'' 10th edition, Dover, New York, 1972. |
[2] |
M. Z. Bazant and H. A. Stone, Asymptotics of reaction-diffusion fronts with one static and one diffusing reactant, Physica D, 147 (2000), 95-121.
doi: 10.1016/S0167-2789(00)00140-8. |
[3] |
E. Ben-Naim and S. Redner, Inhomogeneous 2-species annihilation in the steady-state, J. Phys. A, 25 (1992), L575-L583.
doi: 10.1088/0305-4470/25/9/012. |
[4] |
M. A. Benarde, W. B. Snow, V. P. Olivieri and B. Davidson, Kinetics and Mechanism of Bacterial Disinfection by Chlorine Dioxide, Appl. Microbiol., 15 (1967), 257-265. |
[5] |
B. Chopard, M. Droz, J. Magnin and Z. Rácz, Localization-delocalization transition of a reaction-diffusion front near a semipermeable wall, Phys. Rev. E, 56 (1997), 5343-5350.
doi: 10.1103/PhysRevE.56.5343. |
[6] |
S. Cornell, M. Droz and B. Chopard, Role of fluctuations for inhomogeneous reaction-diffusion phenomena, Phys. Rev. A, 44 (1991), 4826-4832.
doi: 10.1103/PhysRevA.44.4826. |
[7] |
S. Cornell, Z. Koza and M. Droz, Dynamic multiscaling of the reaction-diffusion front for $mA+nB\rightarrow C$, Phys. Rev. E, 52 (1995), 3500-3505.
doi: 10.1103/PhysRevE.52.3500. |
[8] |
S. Cornell and M. Droz, Exotic reaction fronts in the steady state, Physica D, 103 (1997), 348-356.
doi: 10.1016/S0167-2789(96)00267-9. |
[9] |
P. V. Danckwerts, Unsteady-state diffusion or heat-conduction with moving boundary, Trans. Faraday Soc., 46 (1950), 701-712.
doi: 10.1039/tf9504600701. |
[10] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[11] |
L. Gálfi and Z. Rácz, Properties of the reaction front in an A+B$\rightarrow$C type reaction-diffusion process, Phys. Rev. A, 38 (1988), 3151-3154.
doi: 10.1103/PhysRevA.38.3151. |
[12] |
M. R. Gray, "Upgrading Petroleum Residues and Heavy Oils,'' CRC Press, New York, 1994, see page 118. |
[13] |
R. A. Greinke and H. B. Mark, Jr., Conductometric analysis of binary amine mixtures reacting via apparent fractional order kinetics, Anal. Chem., 39 (1967), 1572-1576.
doi: 10.1021/ac50156a027. |
[14] |
Y-E.L. Koo and R. Kopelman, Space-resolved and time-resolved diffusion-limited binary reaction-kinetics in capillaries — experimental-observation of segregation, anomalous exponents, and depletion zone, J. Stat. Phys., 65 (1991), 893-918.
doi: 10.1007/BF01049588. |
[15] |
Z. Koza, The long-time behavior of initially separated A+B$\rightarrow$0 reaction-diffusion systems with arbitrary diffusion constants, J. Stat. Phys., 85 (1996), 179-191.
doi: 10.1007/BF02175561. |
[16] |
J. Magnin, Properties of the asymptotic $nA+mB \rightarrow C$ reaction-diffusion fronts, Eur. Phys. J. B, 17 (2000), 673-678.
doi: 10.1007/s100510070107. |
[17] |
J. D. Murray, "Mathematical Biology,'' Springer-Verlag, New York, 1993.
doi: 10.1007/b98869. |
[18] |
L. Rongy, P. M. J. Trevelyan and A. De Wit, Dynamics of A+B$\rightarrow$C reaction fronts in the presence of buoyancy-driven convection, Phys. Rev. Lett., 101 (2008), 084503.
doi: 10.1103/PhysRevLett.101.084503. |
[19] |
Y. Shi and K. Eckert, Acceleration of reaction fronts by hydrodynamic instabilities in immiscible systems, Chem. Eng. Sci., 61 (2006), 5523-5533.
doi: 10.1016/j.ces.2006.02.023. |
[20] |
M. Sinder and J. Pelleg, Asymptotic properties of a reversible A+B$\leftrightarrow$C (static) reaction-diffusion process with initially separated reactants, Phys. Rev. E, 62 (2000), 3340-3348.
doi: 10.1103/PhysRevE.62.3340. |
[21] |
P. M. J. Trevelyan, D. E. Strier and A. De Wit, Analytical asymptotic solutions of nA+mB$\rightarrow$C reaction-diffusion equations in two-layer systems: A general study, Phys. Rev. E, 78 (2008), 026122.
doi: 10.1103/PhysRevE.78.026122. |
[22] |
P. M. J. Trevelyan, Higher-order large-time asymptotics for a reaction of the form nA+mB$\rightarrow$C, Phys. Rev. E, 79 (2009), 016105.
doi: 10.1103/PhysRevE.79.016105. |
[23] |
G. Van Baalen, A. Schenkel and P. Wittwer, Asymptotics of solutions in nA+nB$\rightarrow$C reaction-diffusion systems, Commun. Math. Phys., 210 (2000), 145-176.
doi: 10.1007/s002200050775. |
[24] |
G. Venzl, Pattern formation in precipitation processes. II. A postnucleation theory of Liesegang bands, J. Chem. Phys., 86 (1986), 2006-2011.
doi: 10.1063/1.451144. |
show all references
References:
[1] |
M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions,'' 10th edition, Dover, New York, 1972. |
[2] |
M. Z. Bazant and H. A. Stone, Asymptotics of reaction-diffusion fronts with one static and one diffusing reactant, Physica D, 147 (2000), 95-121.
doi: 10.1016/S0167-2789(00)00140-8. |
[3] |
E. Ben-Naim and S. Redner, Inhomogeneous 2-species annihilation in the steady-state, J. Phys. A, 25 (1992), L575-L583.
doi: 10.1088/0305-4470/25/9/012. |
[4] |
M. A. Benarde, W. B. Snow, V. P. Olivieri and B. Davidson, Kinetics and Mechanism of Bacterial Disinfection by Chlorine Dioxide, Appl. Microbiol., 15 (1967), 257-265. |
[5] |
B. Chopard, M. Droz, J. Magnin and Z. Rácz, Localization-delocalization transition of a reaction-diffusion front near a semipermeable wall, Phys. Rev. E, 56 (1997), 5343-5350.
doi: 10.1103/PhysRevE.56.5343. |
[6] |
S. Cornell, M. Droz and B. Chopard, Role of fluctuations for inhomogeneous reaction-diffusion phenomena, Phys. Rev. A, 44 (1991), 4826-4832.
doi: 10.1103/PhysRevA.44.4826. |
[7] |
S. Cornell, Z. Koza and M. Droz, Dynamic multiscaling of the reaction-diffusion front for $mA+nB\rightarrow C$, Phys. Rev. E, 52 (1995), 3500-3505.
doi: 10.1103/PhysRevE.52.3500. |
[8] |
S. Cornell and M. Droz, Exotic reaction fronts in the steady state, Physica D, 103 (1997), 348-356.
doi: 10.1016/S0167-2789(96)00267-9. |
[9] |
P. V. Danckwerts, Unsteady-state diffusion or heat-conduction with moving boundary, Trans. Faraday Soc., 46 (1950), 701-712.
doi: 10.1039/tf9504600701. |
[10] |
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.
doi: 10.1016/S0006-3495(61)86902-6. |
[11] |
L. Gálfi and Z. Rácz, Properties of the reaction front in an A+B$\rightarrow$C type reaction-diffusion process, Phys. Rev. A, 38 (1988), 3151-3154.
doi: 10.1103/PhysRevA.38.3151. |
[12] |
M. R. Gray, "Upgrading Petroleum Residues and Heavy Oils,'' CRC Press, New York, 1994, see page 118. |
[13] |
R. A. Greinke and H. B. Mark, Jr., Conductometric analysis of binary amine mixtures reacting via apparent fractional order kinetics, Anal. Chem., 39 (1967), 1572-1576.
doi: 10.1021/ac50156a027. |
[14] |
Y-E.L. Koo and R. Kopelman, Space-resolved and time-resolved diffusion-limited binary reaction-kinetics in capillaries — experimental-observation of segregation, anomalous exponents, and depletion zone, J. Stat. Phys., 65 (1991), 893-918.
doi: 10.1007/BF01049588. |
[15] |
Z. Koza, The long-time behavior of initially separated A+B$\rightarrow$0 reaction-diffusion systems with arbitrary diffusion constants, J. Stat. Phys., 85 (1996), 179-191.
doi: 10.1007/BF02175561. |
[16] |
J. Magnin, Properties of the asymptotic $nA+mB \rightarrow C$ reaction-diffusion fronts, Eur. Phys. J. B, 17 (2000), 673-678.
doi: 10.1007/s100510070107. |
[17] |
J. D. Murray, "Mathematical Biology,'' Springer-Verlag, New York, 1993.
doi: 10.1007/b98869. |
[18] |
L. Rongy, P. M. J. Trevelyan and A. De Wit, Dynamics of A+B$\rightarrow$C reaction fronts in the presence of buoyancy-driven convection, Phys. Rev. Lett., 101 (2008), 084503.
doi: 10.1103/PhysRevLett.101.084503. |
[19] |
Y. Shi and K. Eckert, Acceleration of reaction fronts by hydrodynamic instabilities in immiscible systems, Chem. Eng. Sci., 61 (2006), 5523-5533.
doi: 10.1016/j.ces.2006.02.023. |
[20] |
M. Sinder and J. Pelleg, Asymptotic properties of a reversible A+B$\leftrightarrow$C (static) reaction-diffusion process with initially separated reactants, Phys. Rev. E, 62 (2000), 3340-3348.
doi: 10.1103/PhysRevE.62.3340. |
[21] |
P. M. J. Trevelyan, D. E. Strier and A. De Wit, Analytical asymptotic solutions of nA+mB$\rightarrow$C reaction-diffusion equations in two-layer systems: A general study, Phys. Rev. E, 78 (2008), 026122.
doi: 10.1103/PhysRevE.78.026122. |
[22] |
P. M. J. Trevelyan, Higher-order large-time asymptotics for a reaction of the form nA+mB$\rightarrow$C, Phys. Rev. E, 79 (2009), 016105.
doi: 10.1103/PhysRevE.79.016105. |
[23] |
G. Van Baalen, A. Schenkel and P. Wittwer, Asymptotics of solutions in nA+nB$\rightarrow$C reaction-diffusion systems, Commun. Math. Phys., 210 (2000), 145-176.
doi: 10.1007/s002200050775. |
[24] |
G. Venzl, Pattern formation in precipitation processes. II. A postnucleation theory of Liesegang bands, J. Chem. Phys., 86 (1986), 2006-2011.
doi: 10.1063/1.451144. |
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