Communications on Pure and Applied Analysis (CPAA)

Macrotransport in nonlinear, reactive, shear flows

Pages: 2125 - 2146, Volume 11, Issue 5, September 2012      doi:10.3934/cpaa.2012.11.2125

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Eric S. Wright - Department of Mathematics, University of Montana-Western, 710 S. Atlantic Street Dillon, MT 59725-3598, United States (email)

Abstract: 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, macrotransport, reaction-diffusion.
Mathematics Subject Classification:  Primary: 35K55, 35K57; Secondary: 35B25, 35B40.

Received: March 2011;      Revised: October 2011;      Available Online: March 2012.