American Institute of Mathematical Sciences

Advanced
September  2012, 11(5): 2125-2146. doi: 10.3934/cpaa.2012.11.2125

Macrotransport in nonlinear, reactive, shear flows

Eric S. Wright 1,

1. 

Department of Mathematics, University of Montana-Western, 710 S. Atlantic Street Dillon, MT 59725-3598, United States

Received  March 2011 Revised  October 2011 Published  March 2012

In 1953, G.I. Taylor published his paper concerning the transport of a contaminant in a fluid flowing through a narrow tube. He demonstrated that the transverse variations in the fluid's velocity field and the transverse diffusion of the solute interact to yield an effective longitudinal mixing mechanism for the transverse average of the solute. This mechanism has been dubbed ``Taylor Dispersion.'' Since then, many related studies have surfaced. However, few of these addressed the effects of nonlinear chemical reactions upon a system of solutes undergoing Taylor Dispersion. In this paper, I present a mathematical model for the evolution of the transverse averages of reacting solutes in a fluid flowing down a pipe of arbitrary cross-section. The technique for deriving the model is a generalization of an approach by introduced by P.C. Fife. The key outcome is that while one still finds an effective mechanism for longitudinal mixing, there is also a effective mechanism for nonlinear advection.
Keywords: Partial differential equations, asymptotic analysis, Taylor dispersion, reaction-diffusion., macrotransport.
Mathematics Subject Classification: Primary: 35K55, 35K57; Secondary: 35B25, 35B4.
Citation: Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2125-2146. doi: 10.3934/cpaa.2012.11.2125
References:
[1]

R. Aris, On the dispersion of a solute in a fluid flowing through a tube,, Proc. Roy. Soc. London A, 235 (1956), 67.  doi: 10.1098/rspa.1956.0065.  Google Scholar

[2]

R. Aris, On the dispersion of a solute in a pulsating flow through a tube,, Proc. Roy. Soc. London A, 259 (1960), 370.  doi: 10.1098/rspa.1960.0231.  Google Scholar

[3]

V. Balakotaiah and H. Chang, Dispersion of chemical solutes in cromatographs and reactors,, Phil. Trans. R. Soc. Lond. A, 351 (1995), 39.  doi: 10.1098/rsta.1995.0025.  Google Scholar

[4]

B. Bloechle, "On the Taylor Dispersion of Reactive Solutes in a Parallel-Plate Fracture-Matrix System,'', PhD thesis, (2001).   Google Scholar

[5]

H. Brenner and D. Edwards, "Macrotransport Processes,'', Butterworth, (1993).   Google Scholar

[6]

L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. iii,, J. Colloid Interface Sci., 85 (1982), 101.  doi: 10.1016/0021-9797(82)90239-9.  Google Scholar

[7]

L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. vi,, J. Colloid Interface Sci., 93 (1983), 343.  doi: 10.1016/0021-9797(83)90419-8.  Google Scholar

[8]

K. D. Dorfman and H. Brenner, Generalized Taylor-Aris dispersion in spatially periodic microfluidic networks. chemical reactions,, SIAM J. Appl. Math., 63 (2003), 962.  doi: 10.1137/S0036139902401872.  Google Scholar

[9]

S. R. Dungan, M. Shapiro and H. Brenner, Convective-diffusive-reactive Taylor dispersion processes in particulate multiphase systems,, Proc. R. Soc. Lond. A, 429 (1990), 639.  doi: 10.1098/rspa.1990.0077.  Google Scholar

[10]

D. A. Edwards, M. Shapiro and H. Brenner, Dispersion and reaction in two-dimensional model porous media,, Phys. Fluids A, 5 (1993), 837.  doi: 10.1063/1.858631.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,'', AMS, (1998).   Google Scholar

[12]

P. Fife and K. Nicholes, Dispersion in flow through small tubes,, Proc. Roy. Soc. London A, 344 (1975), 131.  doi: 10.1098/rspa.1975.0094.  Google Scholar

[13]

P. C. Fife, Singular perturbation problems whose degenerate forms have many solutions,, Applicable Analysis, 1 (1972), 331.  doi: 10.1080/00036817208839022.  Google Scholar

[14]

Avner Friedman, "Partial Differential Equations,'', Holt, ().   Google Scholar

[15]

W. N. Gill, A note on the solution of transient dispersion problems,, Proc. Roy. Soc. London A, 298 (1967), 335.  doi: 10.1098/rspa.1967.0107.  Google Scholar

[16]

W. N. Gill and R. Sankarasubramanian, Exact analysis of unsteady convective diffusion,, Proc. Roy. Soc. London A, 316 (1970), 341.  doi: 10.1098/rspa.1970.0083.  Google Scholar

[17]

L. E. Johns and A. E. Degance, Dispersion approximations to the multicomponent convective diffusion equation for chemically active systems,, Chemical Engineering Science, 30 (1975), 1065.  doi: 10.1016/0009-2509(75)87008-4.  Google Scholar

[18]

C. Li and E. S. Wright, Modeling chemical reactions in rivers: A three component reaction,, Discrete and Continuous Dynamical Systems, 7 (2001), 377.  doi: 10.3934/dcds.2001.7.373.  Google Scholar

[19]

C. Li and E. S. Wright, Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics,, Communications on Pure and Applied Analysis, 1 (2002), 77.   Google Scholar

[20]

R. Mauri, Dispersion, convection, and reaction in porous-media,, Phys. Fluids, 3 (1991), 743.  doi: 10.1063/1.858007.  Google Scholar

[21]

G. N. Mercer and A. J. Roberts, A center manifold description of contaminant dispersion in channels with varying flow properties,, SIAM J. appl. Math, 50 (1990), 1547.  doi: 10.1137/0150091.  Google Scholar

[22]

M. Pagitsas, A. Nadim and H. Brenner, Projection operator analysis of macrotransport processes,, J. Chem. Phys., 84 (1986), 2901.  doi: 10.1063/1.450305.  Google Scholar

[23]

M. Shapiro and H. Brenner, Taylor dispersion of chemically reactive species: irreversible first order reactions in bulk and on boundaries,, Chem. Engng Sci., 41 (1986), 1417.  doi: 10.1016/0009-2509(86)85228-9.  Google Scholar

[24]

M. Shapiro and H. Brenner, Taylor dispersion of chemically reactive solute in a spatially periodic model of a porous medium,, Chem. Engng Sci., 43 (1988), 551.  doi: 10.1016/0009-2509(88)87016-7.  Google Scholar

[25]

G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube,, Proc. Roy. Soc. of London A, 219 (1953), 186.  doi: 10.1098/rspa.1953.0139.  Google Scholar

[26]

G. I. Taylor, Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular dispersion,, Proc. Roy. Soc. London A, 225 (1954), 473.  doi: 10.1098/rspa.1954.0216.  Google Scholar

[27]

S. D. Watt and A. J. Roberts, The accurate dynamic modelling of contaminant dispersion in channels,, SIAM J. Appl Math, 55 (1995), 1016.  doi: 10.1137/S0036139993257971.  Google Scholar

[28]

T. Yamanaka, Projection operator theoretical approach to unsteady convective diffusion phenomena,, Journal of Chemical Engineering of Japan, 16 (1983), 29.  doi: 10.1252/jcej.16.29.  Google Scholar

[29]

T. Yamanaka and S. Inui, Taylor dispersion models involving nonlinear irreversible reactions,, Journal of Chemical Engineering of Japan, 27 (1994), 434.  doi: 10.1252/jcej.27.434.  Google Scholar

show all references

References:
[1]

R. Aris, On the dispersion of a solute in a fluid flowing through a tube,, Proc. Roy. Soc. London A, 235 (1956), 67.  doi: 10.1098/rspa.1956.0065.  Google Scholar

[2]

R. Aris, On the dispersion of a solute in a pulsating flow through a tube,, Proc. Roy. Soc. London A, 259 (1960), 370.  doi: 10.1098/rspa.1960.0231.  Google Scholar

[3]

V. Balakotaiah and H. Chang, Dispersion of chemical solutes in cromatographs and reactors,, Phil. Trans. R. Soc. Lond. A, 351 (1995), 39.  doi: 10.1098/rsta.1995.0025.  Google Scholar

[4]

B. Bloechle, "On the Taylor Dispersion of Reactive Solutes in a Parallel-Plate Fracture-Matrix System,'', PhD thesis, (2001).   Google Scholar

[5]

H. Brenner and D. Edwards, "Macrotransport Processes,'', Butterworth, (1993).   Google Scholar

[6]

L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. iii,, J. Colloid Interface Sci., 85 (1982), 101.  doi: 10.1016/0021-9797(82)90239-9.  Google Scholar

[7]

L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. vi,, J. Colloid Interface Sci., 93 (1983), 343.  doi: 10.1016/0021-9797(83)90419-8.  Google Scholar

[8]

K. D. Dorfman and H. Brenner, Generalized Taylor-Aris dispersion in spatially periodic microfluidic networks. chemical reactions,, SIAM J. Appl. Math., 63 (2003), 962.  doi: 10.1137/S0036139902401872.  Google Scholar

[9]

S. R. Dungan, M. Shapiro and H. Brenner, Convective-diffusive-reactive Taylor dispersion processes in particulate multiphase systems,, Proc. R. Soc. Lond. A, 429 (1990), 639.  doi: 10.1098/rspa.1990.0077.  Google Scholar

[10]

D. A. Edwards, M. Shapiro and H. Brenner, Dispersion and reaction in two-dimensional model porous media,, Phys. Fluids A, 5 (1993), 837.  doi: 10.1063/1.858631.  Google Scholar

[11]

L. C. Evans, "Partial Differential Equations,'', AMS, (1998).   Google Scholar

[12]

P. Fife and K. Nicholes, Dispersion in flow through small tubes,, Proc. Roy. Soc. London A, 344 (1975), 131.  doi: 10.1098/rspa.1975.0094.  Google Scholar

[13]

P. C. Fife, Singular perturbation problems whose degenerate forms have many solutions,, Applicable Analysis, 1 (1972), 331.  doi: 10.1080/00036817208839022.  Google Scholar

[14]

Avner Friedman, "Partial Differential Equations,'', Holt, ().   Google Scholar

[15]

W. N. Gill, A note on the solution of transient dispersion problems,, Proc. Roy. Soc. London A, 298 (1967), 335.  doi: 10.1098/rspa.1967.0107.  Google Scholar

[16]

W. N. Gill and R. Sankarasubramanian, Exact analysis of unsteady convective diffusion,, Proc. Roy. Soc. London A, 316 (1970), 341.  doi: 10.1098/rspa.1970.0083.  Google Scholar

[17]

L. E. Johns and A. E. Degance, Dispersion approximations to the multicomponent convective diffusion equation for chemically active systems,, Chemical Engineering Science, 30 (1975), 1065.  doi: 10.1016/0009-2509(75)87008-4.  Google Scholar

[18]

C. Li and E. S. Wright, Modeling chemical reactions in rivers: A three component reaction,, Discrete and Continuous Dynamical Systems, 7 (2001), 377.  doi: 10.3934/dcds.2001.7.373.  Google Scholar

[19]

C. Li and E. S. Wright, Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics,, Communications on Pure and Applied Analysis, 1 (2002), 77.   Google Scholar

[20]

R. Mauri, Dispersion, convection, and reaction in porous-media,, Phys. Fluids, 3 (1991), 743.  doi: 10.1063/1.858007.  Google Scholar

[21]

G. N. Mercer and A. J. Roberts, A center manifold description of contaminant dispersion in channels with varying flow properties,, SIAM J. appl. Math, 50 (1990), 1547.  doi: 10.1137/0150091.  Google Scholar

[22]

M. Pagitsas, A. Nadim and H. Brenner, Projection operator analysis of macrotransport processes,, J. Chem. Phys., 84 (1986), 2901.  doi: 10.1063/1.450305.  Google Scholar

[23]

M. Shapiro and H. Brenner, Taylor dispersion of chemically reactive species: irreversible first order reactions in bulk and on boundaries,, Chem. Engng Sci., 41 (1986), 1417.  doi: 10.1016/0009-2509(86)85228-9.  Google Scholar

[24]

M. Shapiro and H. Brenner, Taylor dispersion of chemically reactive solute in a spatially periodic model of a porous medium,, Chem. Engng Sci., 43 (1988), 551.  doi: 10.1016/0009-2509(88)87016-7.  Google Scholar

[25]

G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube,, Proc. Roy. Soc. of London A, 219 (1953), 186.  doi: 10.1098/rspa.1953.0139.  Google Scholar

[26]

G. I. Taylor, Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular dispersion,, Proc. Roy. Soc. London A, 225 (1954), 473.  doi: 10.1098/rspa.1954.0216.  Google Scholar

[27]

S. D. Watt and A. J. Roberts, The accurate dynamic modelling of contaminant dispersion in channels,, SIAM J. Appl Math, 55 (1995), 1016.  doi: 10.1137/S0036139993257971.  Google Scholar

[28]

T. Yamanaka, Projection operator theoretical approach to unsteady convective diffusion phenomena,, Journal of Chemical Engineering of Japan, 16 (1983), 29.  doi: 10.1252/jcej.16.29.  Google Scholar

[29]

T. Yamanaka and S. Inui, Taylor dispersion models involving nonlinear irreversible reactions,, Journal of Chemical Engineering of Japan, 27 (1994), 434.  doi: 10.1252/jcej.27.434.  Google Scholar

[1]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515
[2]

Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090
[3]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
[4]

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
[5]

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
[6]

Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118
[7]

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
[8]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
[9]

Alan E. Lindsay. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 189-215. doi: 10.3934/dcdsb.2014.19.189
[10]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211
[11]

Linghai Zhang. Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1025-1042. doi: 10.3934/dcds.2002.8.1025
[12]

Jimmy Garnier, FranÇois Hamel, Lionel Roques. Transition fronts and stretching phenomena for a general class of reaction-dispersion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 743-756. doi: 10.3934/dcds.2017031
[13]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[14]

Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147
[15]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[16]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[17]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020264
[18]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481
[19]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703
[20]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences