# American Institute of Mathematical Sciences

2012, 17(4): 1261-1287. doi: 10.3934/dcdsb.2012.17.1261

## Random walks, random flows, and enhanced diffusivity in advection-diffusion equations

 1 Mathematics Department, University of North Carolina, Chapel Hill, NC 27599, United States

Received  June 2010 Revised  August 2011 Published  February 2012

We study the phenomenon of enhanced diffusivity, introduced by G. I.Taylor, for a class of advection-diffusion equations, modeling, for example, the spread of an ink drop in a fluid engaged in Poiseuille flow. We consider such flow in a pipe of general cross section, and compute variances and covariances of certain random flows associated with the advection-diffusion. We examine both long time behavior, including a central limit theorem, and short time asymptotics.
Citation: Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261
##### References:
 [1] M. Avellaneda and A. Majda, An integral representation and bounds on the effective diffusivity in passive advective diffusion by laminar and turbulent flows,, Commun. Math. Phys., 138 (1991), 339. [2] R. Camassa, Z. Lin and R. McLaughlin, The exact evolution of the scalar variance in pipe and channel flow,, Commun. Math. Sci., 8 (2010), 601. [3] S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation,, Quart. J. Mech. Appl. Math., 4 (1951), 129. doi: 10.1093/qjmam/4.2.129. [4] R. Griego and R. Hersh, Theory of random evolutions with applications to partial differential equations,, Trans. AMS, 156 (1971), 405. doi: 10.1090/S0002-9947-1971-0275507-7. [5] A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,, Physics Reports, 314 (1999), 237. doi: 10.1016/S0370-1573(98)00083-0. [6] A. Mazzucato and M. Taylor, Vanishing viscosity limits for a class of circular pipe flows,, Commun. PDE, 36 (2012), 328. doi: 10.1080/03605302.2010.505973. [7] M. Pinsky, "Lectures on Random Evolution,'', World Scientific, (1991). [8] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube,, Proc. Roy. Soc. London A, 219 (1953), 186. doi: 10.1098/rspa.1953.0139. [9] M. Taylor, "Partial Differential Equations,'' Vols. 1-3,, Springer-Verlag, (1996).

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##### References:
 [1] M. Avellaneda and A. Majda, An integral representation and bounds on the effective diffusivity in passive advective diffusion by laminar and turbulent flows,, Commun. Math. Phys., 138 (1991), 339. [2] R. Camassa, Z. Lin and R. McLaughlin, The exact evolution of the scalar variance in pipe and channel flow,, Commun. Math. Sci., 8 (2010), 601. [3] S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation,, Quart. J. Mech. Appl. Math., 4 (1951), 129. doi: 10.1093/qjmam/4.2.129. [4] R. Griego and R. Hersh, Theory of random evolutions with applications to partial differential equations,, Trans. AMS, 156 (1971), 405. doi: 10.1090/S0002-9947-1971-0275507-7. [5] A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena,, Physics Reports, 314 (1999), 237. doi: 10.1016/S0370-1573(98)00083-0. [6] A. Mazzucato and M. Taylor, Vanishing viscosity limits for a class of circular pipe flows,, Commun. PDE, 36 (2012), 328. doi: 10.1080/03605302.2010.505973. [7] M. Pinsky, "Lectures on Random Evolution,'', World Scientific, (1991). [8] G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube,, Proc. Roy. Soc. London A, 219 (1953), 186. doi: 10.1098/rspa.1953.0139. [9] M. Taylor, "Partial Differential Equations,'' Vols. 1-3,, Springer-Verlag, (1996).
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