Limitations and generalizations of the classical phenomenological model for diffusion in fluids

Authors: Aranovich, G. L.; Donohue, M. D.

Source: Molecular Physics, Volume 105, Number 8, April 2007 , pp. 1085-1093(9)

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Abstract:

It is shown that the classical derivation of the diffusion equation uses two incompatible assumptions: (1) the diffusion coefficient, [image omitted], is assumed to be finite ( [image omitted] is a characteristic velocity and λ is the mean-free path); but (2) the flux is approximated in the limit of λ → 0. The second assumption results in unphysical instantaneous propagation which disappears as this assumption is relaxed and the flux term is represented exactly (in the framework of the classical model). With this correction, the diffusion equation becomes (∂n/∂t) = (∂/∂x) [image omitted] where n is the density as a function of coordinate, x, and time, t. Keeping D finite and letting λ → 0 gives the classical equation (∂n/∂t) = (∂/∂x)[D(∂n/∂x)] and classical solution [image omitted]. However, this solution is valid only in the limits where |∂n/∂x| >> (λ2/3!)|∂3n/∂x3|; otherwise the classical approximation is not correct. Here, we have shown that this inequality translates into 24D2t2/|6Dt - x2| >> λ2 and gives limits for x2: at any fixed t, x2 cannot exceed [image omitted]. Relaxing the requirement of λ → 0 gives a new (finite-difference) type of diffusion equation for fluids [image omitted]. This equation and its more general (2D/3D) analogues reveal a qualitatively new, more complex diffusion paradigm than the classical 'propagation of chaos' concept.

Document Type: Research article

DOI: http://dx.doi.org/10.1080/00268970701348758

Affiliations: 1: The Johns Hopkins University, Baltimore, MD, USA

Publication date: 2007-04-01

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