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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 $A^n B^m$

 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.
Citation: Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219
##### 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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