• Previous Article
    Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys
  • DCDS-S Home
  • This Issue
  • Next Article
    Experimental data for solid tumor cells: Proliferation curves and time-changes of heat shock proteins
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 & 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, (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

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

[1]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[2]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[3]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[4]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

[5]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

[6]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255

[7]

A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure & Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65

[8]

Congming Li, Eric S. Wright. Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics. Communications on Pure & Applied Analysis, 2002, 1 (1) : 77-84. doi: 10.3934/cpaa.2002.1.77

[9]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[10]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[11]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[12]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

[13]

Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415

[14]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-13. doi: 10.3934/dcdsb.2019114

[15]

Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173

[16]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526

[17]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69

[18]

Antonio Carlos Fernandes, Marcela Carvalho Gonçcalves, Jacson Simsen. Non-autonomous reaction-diffusion equations with variable exponents and large diffusion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1485-1510. doi: 10.3934/dcdsb.2018217

[19]

Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815

[20]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]