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February  2012, 5(1): 219-234. doi: 10.3934/dcdss.2012.5.219

Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$

Philip M. J. Trevelyan 1,

1. 

Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, 1050 Brussels, Belgium

Received  March 2009 Revised  November 2009 Published  February 2011

Consider the reaction front formed when two initially separated reactants A and B are brought into contact and react at a rate proportional to $A^n B^m$ when the concentrations $A$ and $B$ are positive. Further, suppose that both $n$ and $m$ are less than unity. Then the leading order large time asymptotic reaction rate has compact support, i.e. the reaction zone where the reaction takes place has a finite width and the reaction rate is identically zero outside of this region. In the large time asymptotic limit an analytical approximate solution to the reactant concentrations is constructed in the vicinity of the reaction zone. The approximate solution is found to be in good agreement with numerically obtained solutions. For $n \ne m$ the location of the maximum reaction rate does not coincide with the centre of mass of the reaction, and further for $n>m$ this local maximum is shifted slightly closer to the zone that initially contained species A, with the reverse holding when $m>n$. The three limits $m\rightarrow 0$, $n\rightarrow 1$ and $m,n\rightarrow 1$ are given special attention.
Keywords: approximate solutions., large time asymptotic, fractional order kinetics, Reaction-diffusion.
Mathematics Subject Classification: Primary: 35K57, 41A60; Secondary: 33C0.
Citation: Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219
References:
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M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions,'', 10th edition, (1972).   Google Scholar

[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.  doi: 10.1016/S0167-2789(00)00140-8.  Google Scholar

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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.  doi: 10.1103/PhysRevE.52.3500.  Google Scholar

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S. Cornell and M. Droz, Exotic reaction fronts in the steady state,, Physica D, 103 (1997), 348.  doi: 10.1016/S0167-2789(96)00267-9.  Google Scholar

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R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

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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.  doi: 10.1103/PhysRevA.38.3151.  Google Scholar

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M. R. Gray, "Upgrading Petroleum Residues and Heavy Oils,'', CRC Press, (1994).   Google Scholar

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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.  doi: 10.1021/ac50156a027.  Google Scholar

[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.  doi: 10.1007/BF01049588.  Google Scholar

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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.  doi: 10.1007/BF02175561.  Google Scholar

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J. Magnin, Properties of the asymptotic $nA+mB \rightarrow C$ reaction-diffusion fronts,, Eur. Phys. J. B, 17 (2000), 673.  doi: 10.1007/s100510070107.  Google Scholar

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J. D. Murray, "Mathematical Biology,'', Springer-Verlag, (1993).  doi: 10.1007/b98869.  Google Scholar

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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).  doi: 10.1103/PhysRevLett.101.084503.  Google Scholar

[19]

Y. Shi and K. Eckert, Acceleration of reaction fronts by hydrodynamic instabilities in immiscible systems,, Chem. Eng. Sci., 61 (2006), 5523.  doi: 10.1016/j.ces.2006.02.023.  Google Scholar

[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.  doi: 10.1103/PhysRevE.62.3340.  Google Scholar

[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).  doi: 10.1103/PhysRevE.78.026122.  Google Scholar

[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).  doi: 10.1103/PhysRevE.79.016105.  Google Scholar

[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.  doi: 10.1007/s002200050775.  Google Scholar

[24]

G. Venzl, Pattern formation in precipitation processes. II. A postnucleation theory of Liesegang bands,, J. Chem. Phys., 86 (1986), 2006.  doi: 10.1063/1.451144.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions,'', 10th edition, (1972).   Google Scholar

[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.  doi: 10.1016/S0167-2789(00)00140-8.  Google Scholar

[3]

E. Ben-Naim and S. Redner, Inhomogeneous 2-species annihilation in the steady-state,, J. Phys. A, 25 (1992).  doi: 10.1088/0305-4470/25/9/012.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1103/PhysRevE.56.5343.  Google Scholar

[6]

S. Cornell, M. Droz and B. Chopard, Role of fluctuations for inhomogeneous reaction-diffusion phenomena,, Phys. Rev. A, 44 (1991), 4826.  doi: 10.1103/PhysRevA.44.4826.  Google Scholar

[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.  doi: 10.1103/PhysRevE.52.3500.  Google Scholar

[8]

S. Cornell and M. Droz, Exotic reaction fronts in the steady state,, Physica D, 103 (1997), 348.  doi: 10.1016/S0167-2789(96)00267-9.  Google Scholar

[9]

P. V. Danckwerts, Unsteady-state diffusion or heat-conduction with moving boundary,, Trans. Faraday Soc., 46 (1950), 701.  doi: 10.1039/tf9504600701.  Google Scholar

[10]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane,, Biophys. J., 1 (1961), 445.  doi: 10.1016/S0006-3495(61)86902-6.  Google Scholar

[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.  doi: 10.1103/PhysRevA.38.3151.  Google Scholar

[12]

M. R. Gray, "Upgrading Petroleum Residues and Heavy Oils,'', CRC Press, (1994).   Google Scholar

[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.  doi: 10.1021/ac50156a027.  Google Scholar

[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.  doi: 10.1007/BF01049588.  Google Scholar

[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.  doi: 10.1007/BF02175561.  Google Scholar

[16]

J. Magnin, Properties of the asymptotic $nA+mB \rightarrow C$ reaction-diffusion fronts,, Eur. Phys. J. B, 17 (2000), 673.  doi: 10.1007/s100510070107.  Google Scholar

[17]

J. D. Murray, "Mathematical Biology,'', Springer-Verlag, (1993).  doi: 10.1007/b98869.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.101.084503.  Google Scholar

[19]

Y. Shi and K. Eckert, Acceleration of reaction fronts by hydrodynamic instabilities in immiscible systems,, Chem. Eng. Sci., 61 (2006), 5523.  doi: 10.1016/j.ces.2006.02.023.  Google Scholar

[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.  doi: 10.1103/PhysRevE.62.3340.  Google Scholar

[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).  doi: 10.1103/PhysRevE.78.026122.  Google Scholar

[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).  doi: 10.1103/PhysRevE.79.016105.  Google Scholar

[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.  doi: 10.1007/s002200050775.  Google Scholar

[24]

G. Venzl, Pattern formation in precipitation processes. II. A postnucleation theory of Liesegang bands,, J. Chem. Phys., 86 (1986), 2006.  doi: 10.1063/1.451144.  Google Scholar

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