Commun. Comput. Phys., 13 (2013), pp. 741-756.


Lattice Boltzmann Modeling of Advection-Diffusion-Reaction Equations: Pattern Formation Under Uniform Differential Advection

S. G. Ayodele 1*, D. Raabe 1, F. Varnik 2

1 Max-Planck Institut fur, Eisenforschung, Max-Planck Strase 1, 40237, Dusseldorf, Germany.
2 Interdisciplinary Center for Advanced Materials Simulation, Ruhr University Bochum, Stiepeler Strase 129, 44780 Bochum, Germany.

Received 31 October 2011; Accepted (in revised version) 27 January 2012
Available online 29 August 2012
doi:10.4208/cicp.441011.270112s

Abstract

A lattice Boltzmann model for the study of advection-diffusion-reaction (ADR) problems is proposed. Via multiscale expansion analysis, we derive from the LB model the resulting macroscopic equations. It is shown that a linear equilibrium distribution is sufficient to produce ADR equations within error terms of the order of the Mach number squared. Furthermore, we study spatially varying structures arising from the interaction of advective transport with a cubic autocatalytic reaction-diffusion process under an imposed uniform flow. While advecting all the present species leads to trivial translation of the Turing patterns, differential advection leads to flow induced instability characterized with traveling stripes with a velocity dependent wave vector parallel to the flow direction. Predictions from a linear stability analysis of the model equations are found to be in line with these observations.

AMS subject classifications: 76R05, 76R50, 92C15, 80A32

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Key words: Advective transport, differential advection, Turing patterns, linear stability, lattice Boltzmann.

*Corresponding author.
Email: s.ayodele@mpie.de (S. G. Ayodele), d.raabe@mpie.de (D. Raabe), fathollah.varnik@rub.de (F. Varnik)
 

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