Some consequences of accounting for memory and non-locality in transport systems

Due to the presence of heterogeneity transport systems can exhibit memory and non-local effects, where the instantaneous transport flux at a point in the system can be a function of system features (e.g., gradients) distributed through time and space. In diffusion transport systems, a flexible and reasonable general model for memory and non-locality can be achieved by defining the flux as an appropriate convolution integral in time and space. Here it is shown how such a model can lead to a diffusion transport equation expressed in terms of fractional time and space derivatives. In application, such a transport law can lead to some interesting physical hypothesis. Three of these are explored. (1) It is shown that in non-local sediment transport systems, the prediction of land-surface profiles is significantly influenced by the domain of the convolution integral. (2) In the modeling of geomorphic systems there can be a high degree of similarity between predictions obtained with non-local or non-linear flux treatments; a physical experiment is suggested that may be able to uncouple these two plausible transport mechanisms. (3) The Stefan melting problem is the classic moving boundary problem. Solutions can be obtained with both a strong form that explicitly tracks the melting front and a weak form that accounts for the melting in an integrated sense. In local systems with no-memory it is known that these two solutions are equivalent. Here, however, it is shown that when memory is accounted for, the equivalence between the strong and weak form is lost each exhibiting distinctly different treatments for the phase change memory; the former “lumping” it with the movement of the front the latter spatially  “distributing” it over the melt phase.

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V.R. Voller, F. Falcini, R. Garra, Fractional Stefan problems exhibiting lumped and distributed latent-heat memory effects, Phys. Rev. E 87, 042401 (2013)
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