# American Institute of Mathematical Sciences

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

## Macrotransport in nonlinear, reactive, shear flows

 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.
Citation: Eric S. Wright. Macrotransport in nonlinear, reactive, shear flows. Communications on Pure and 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), 67-77. doi: 10.1098/rspa.1956.0065. [2] R. Aris, On the dispersion of a solute in a pulsating flow through a tube, Proc. Roy. Soc. London A, 259 (1960), 370-376. doi: 10.1098/rspa.1960.0231. [3] V. Balakotaiah and H. Chang, Dispersion of chemical solutes in cromatographs and reactors, Phil. Trans. R. Soc. Lond. A, 351 (1995), 39-75. doi: 10.1098/rsta.1995.0025. [4] B. Bloechle, "On the Taylor Dispersion of Reactive Solutes in a Parallel-Plate Fracture-Matrix System,'' PhD thesis, University of Colorado at Boulder - Department of Applied Mathematics, 2001. [5] H. Brenner and D. Edwards, "Macrotransport Processes,'' Butterworth, Boston, 1993. [6] L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. iii, J. Colloid Interface Sci., 85 (1982), 101-117. doi: 10.1016/0021-9797(82)90239-9. [7] L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. vi, J. Colloid Interface Sci., 93 (1983), 343-365. doi: 10.1016/0021-9797(83)90419-8. [8] K. D. Dorfman and H. Brenner, Generalized Taylor-Aris dispersion in spatially periodic microfluidic networks. chemical reactions, SIAM J. Appl. Math., 63 (2003), 962-986. doi: 10.1137/S0036139902401872. [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), 639-671. doi: 10.1098/rspa.1990.0077. [10] D. A. Edwards, M. Shapiro and H. Brenner, Dispersion and reaction in two-dimensional model porous media, Phys. Fluids A, 5 (1993), 837-848. doi: 10.1063/1.858631. [11] L. C. Evans, "Partial Differential Equations,'' AMS, 1998. [12] P. Fife and K. Nicholes, Dispersion in flow through small tubes, Proc. Roy. Soc. London A, 344 (1975), 131-145. doi: 10.1098/rspa.1975.0094. [13] P. C. Fife, Singular perturbation problems whose degenerate forms have many solutions, Applicable Analysis, 1 (1972), 331-358. doi: 10.1080/00036817208839022. [14] Avner Friedman, "Partial Differential Equations,'', Holt, (). [15] W. N. Gill, A note on the solution of transient dispersion problems, Proc. Roy. Soc. London A, 298 (1967), 335-339. doi: 10.1098/rspa.1967.0107. [16] W. N. Gill and R. Sankarasubramanian, Exact analysis of unsteady convective diffusion, Proc. Roy. Soc. London A, 316 (1970), 341-350. doi: 10.1098/rspa.1970.0083. [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), 1065-1067. doi: 10.1016/0009-2509(75)87008-4. [18] C. Li and E. S. Wright, Modeling chemical reactions in rivers: A three component reaction, Discrete and Continuous Dynamical Systems, 7 (2001), 377-384. doi: 10.3934/dcds.2001.7.373. [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), 77-84. [20] R. Mauri, Dispersion, convection, and reaction in porous-media, Phys. Fluids, 3 (1991), 743-756. doi: 10.1063/1.858007. [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), 1547-1565. doi: 10.1137/0150091. [22] M. Pagitsas, A. Nadim and H. Brenner, Projection operator analysis of macrotransport processes, J. Chem. Phys., 84 (1986), 2901-2807. doi: 10.1063/1.450305. [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), 1417-1433. doi: 10.1016/0009-2509(86)85228-9. [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), 551-571. doi: 10.1016/0009-2509(88)87016-7. [25] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. of London A, 219 (1953), 186-203. doi: 10.1098/rspa.1953.0139. [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), 473-477. doi: 10.1098/rspa.1954.0216. [27] S. D. Watt and A. J. Roberts, The accurate dynamic modelling of contaminant dispersion in channels, SIAM J. Appl Math, 55 (1995), 1016-1038. doi: 10.1137/S0036139993257971. [28] T. Yamanaka, Projection operator theoretical approach to unsteady convective diffusion phenomena, Journal of Chemical Engineering of Japan, 16 (1983), 29-35. doi: 10.1252/jcej.16.29. [29] T. Yamanaka and S. Inui, Taylor dispersion models involving nonlinear irreversible reactions, Journal of Chemical Engineering of Japan, 27 (1994), 434-435. doi: 10.1252/jcej.27.434.

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##### References:
 [1] R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. London A, 235 (1956), 67-77. doi: 10.1098/rspa.1956.0065. [2] R. Aris, On the dispersion of a solute in a pulsating flow through a tube, Proc. Roy. Soc. London A, 259 (1960), 370-376. doi: 10.1098/rspa.1960.0231. [3] V. Balakotaiah and H. Chang, Dispersion of chemical solutes in cromatographs and reactors, Phil. Trans. R. Soc. Lond. A, 351 (1995), 39-75. doi: 10.1098/rsta.1995.0025. [4] B. Bloechle, "On the Taylor Dispersion of Reactive Solutes in a Parallel-Plate Fracture-Matrix System,'' PhD thesis, University of Colorado at Boulder - Department of Applied Mathematics, 2001. [5] H. Brenner and D. Edwards, "Macrotransport Processes,'' Butterworth, Boston, 1993. [6] L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. iii, J. Colloid Interface Sci., 85 (1982), 101-117. doi: 10.1016/0021-9797(82)90239-9. [7] L. H. Dill and H. Brenner, A general theory of Taylor dispersion phenomena. vi, J. Colloid Interface Sci., 93 (1983), 343-365. doi: 10.1016/0021-9797(83)90419-8. [8] K. D. Dorfman and H. Brenner, Generalized Taylor-Aris dispersion in spatially periodic microfluidic networks. chemical reactions, SIAM J. Appl. Math., 63 (2003), 962-986. doi: 10.1137/S0036139902401872. [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), 639-671. doi: 10.1098/rspa.1990.0077. [10] D. A. Edwards, M. Shapiro and H. Brenner, Dispersion and reaction in two-dimensional model porous media, Phys. Fluids A, 5 (1993), 837-848. doi: 10.1063/1.858631. [11] L. C. Evans, "Partial Differential Equations,'' AMS, 1998. [12] P. Fife and K. Nicholes, Dispersion in flow through small tubes, Proc. Roy. Soc. London A, 344 (1975), 131-145. doi: 10.1098/rspa.1975.0094. [13] P. C. Fife, Singular perturbation problems whose degenerate forms have many solutions, Applicable Analysis, 1 (1972), 331-358. doi: 10.1080/00036817208839022. [14] Avner Friedman, "Partial Differential Equations,'', Holt, (). [15] W. N. Gill, A note on the solution of transient dispersion problems, Proc. Roy. Soc. London A, 298 (1967), 335-339. doi: 10.1098/rspa.1967.0107. [16] W. N. Gill and R. Sankarasubramanian, Exact analysis of unsteady convective diffusion, Proc. Roy. Soc. London A, 316 (1970), 341-350. doi: 10.1098/rspa.1970.0083. [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), 1065-1067. doi: 10.1016/0009-2509(75)87008-4. [18] C. Li and E. S. Wright, Modeling chemical reactions in rivers: A three component reaction, Discrete and Continuous Dynamical Systems, 7 (2001), 377-384. doi: 10.3934/dcds.2001.7.373. [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), 77-84. [20] R. Mauri, Dispersion, convection, and reaction in porous-media, Phys. Fluids, 3 (1991), 743-756. doi: 10.1063/1.858007. [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), 1547-1565. doi: 10.1137/0150091. [22] M. Pagitsas, A. Nadim and H. Brenner, Projection operator analysis of macrotransport processes, J. Chem. Phys., 84 (1986), 2901-2807. doi: 10.1063/1.450305. [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), 1417-1433. doi: 10.1016/0009-2509(86)85228-9. [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), 551-571. doi: 10.1016/0009-2509(88)87016-7. [25] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. of London A, 219 (1953), 186-203. doi: 10.1098/rspa.1953.0139. [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), 473-477. doi: 10.1098/rspa.1954.0216. [27] S. D. Watt and A. J. Roberts, The accurate dynamic modelling of contaminant dispersion in channels, SIAM J. Appl Math, 55 (1995), 1016-1038. doi: 10.1137/S0036139993257971. [28] T. Yamanaka, Projection operator theoretical approach to unsteady convective diffusion phenomena, Journal of Chemical Engineering of Japan, 16 (1983), 29-35. doi: 10.1252/jcej.16.29. [29] T. Yamanaka and S. Inui, Taylor dispersion models involving nonlinear irreversible reactions, Journal of Chemical Engineering of Japan, 27 (1994), 434-435. doi: 10.1252/jcej.27.434.
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