June  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 and 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), 339-391.

[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), 601-626.

[3]

S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156. 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), 405-418. 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), 237-574. 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), 328-361. doi: 10.1080/03605302.2010.505973.

[7]

M. Pinsky, "Lectures on Random Evolution,'' World Scientific, London, 1991.

[8]

G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. London A, 219 (1953), 186-203. doi: 10.1098/rspa.1953.0139.

[9]

M. Taylor, "Partial Differential Equations,'' Vols. 1-3, Springer-Verlag, New York, 1996.

show all references

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), 339-391.

[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), 601-626.

[3]

S. Goldstein, On diffusion by discontinuous movements and on the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129-156. 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), 405-418. 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), 237-574. 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), 328-361. doi: 10.1080/03605302.2010.505973.

[7]

M. Pinsky, "Lectures on Random Evolution,'' World Scientific, London, 1991.

[8]

G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. London A, 219 (1953), 186-203. doi: 10.1098/rspa.1953.0139.

[9]

M. Taylor, "Partial Differential Equations,'' Vols. 1-3, Springer-Verlag, New York, 1996.

[1]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11

[2]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266

[3]

Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure and Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161

[4]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711

[5]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056

[6]

Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200

[7]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435

[8]

Masahiro Yamamoto. Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022017

[9]

Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105

[10]

J.R. Stirling. Chaotic advection, transport and patchiness in clouds of pollution in an estuarine flow. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 263-284. doi: 10.3934/dcdsb.2003.3.263

[11]

Angelo Morro. Nonlinear diffusion equations in fluid mixtures. Evolution Equations and Control Theory, 2016, 5 (3) : 431-448. doi: 10.3934/eect.2016012

[12]

T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201

[13]

Kunimochi Sakamoto. Destabilization threshold curves for diffusion systems with equal diffusivity under non-diagonal flux boundary conditions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 641-654. doi: 10.3934/dcdsb.2016.21.641

[14]

Grégory Faye, Thomas Giletti, Matt Holzer. Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021146

[15]

Diego Berti, Andrea Corli, Luisa Malaguti. Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 6023-6046. doi: 10.3934/dcds.2021105

[16]

M. B. A. Mansour. Computation of traveling wave fronts for a nonlinear diffusion-advection model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 83-91. doi: 10.3934/mbe.2009.6.83

[17]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

[18]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701

[19]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[20]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]