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Effect of spatial concentration fluctuations on effective kinetics in diffusion‐reaction systems
Author(s) -
Tartakovsky A. M.,
Anna P.,
Le Borgne T.,
Balter A.,
Bolster D.
Publication year - 2012
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2011wr010720
Subject(s) - scaling , diffusion , thermodynamics , statistical physics , moment (physics) , curse of dimensionality , kinetics , reaction–diffusion system , function (biology) , mathematics , physics , statistics , classical mechanics , geometry , biology , evolutionary biology
The effect of spatial concentration fluctuations on the reaction of two solutes, A + B ⇀ C, is considered. In the absence of fluctuations, the concentration of solutes decays as A det = B det ∼ t −1 . Contrary to this, experimental and numerical studies suggest that concentrations decay significantly slower. Existing theory suggests a t − d /4 scaling in the asymptotic regime ( d is the dimensionality of the problem). Here we study the effect of fluctuations using the classical diffusion‐reaction equation with random initial conditions. Initial concentrations of the reactants are treated as correlated random fields. We use the method of moment equations to solve the resulting stochastic diffusion‐reaction equation and obtain a solution for the average concentrations that deviates from ∼ t −1 to ∼ t − d /4 behavior at characteristic transition time t *. We also derive analytical expressions for t * as a function of Damköhler number and the coefficient of variation of the initial concentration.