Macrotransport in nonlinear, reactive, shear flows

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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

Abstract
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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

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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. 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

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