The fractal mobile–immobile model (MIM) is intermediate between advection–dispersion (ADE) and fractal Fokker–Planck (FFKPE) equations. It involves two time derivatives, whose orders are 1 and γ (between 0 and 1) on the left-hand side, whereas all mentioned equations have identical right-hand sides. The fractal MIM model accounts for non-Fickian effects that occur when tracers spread in media because of through-flow, and can get trapped by immobile sites. The solid matrix of a porous material may contain such sites, so that non-Fickian spread is actually observed. Within the context of the fractal MIM model, we present a mapping that allows the computation of fluxes on the basis of the density of spreading particles. The mapping behaves as Fickian flux at early times, and tends to a fractional derivative at late times. By means of this mapping, we recast the fractal MIM model into conservative form, which is suitable to deal with sources and bounded domains. Mathematical proofs are illustrated by comparing the discretized fractal p.d.e. with Monte Carlo simulations.

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This investigation is devoted to linear stability analysis within the solutal analog of Horton-Rogers-Lapwood (HRL) problem with sorption of solid particles. The solid nanoparticles are treated as solute within the continuous approach. Therefore, the infinite horizontal porous layer saturated with mixture (carrier fluid and solute) is considered. The solute concentration difference between the layer boundaries is assumed as constant. The solute sorption is simulated in accordance with the linear mobile/immobile media model. In the first part, the instability of steady net mass flow through this layer is studied analytically. The critical values of parameters have been found. It is known that for the HRL problem the seepage makes the critical mode oscillatory, but the stability threshold remains unchanged. In contrast, if the sorption is taken into account, the stability threshold varies. In the latter case, the critical value of solutal Rayleigh-Darcy number increases versus that for the standard HRL problem. The second part is devoted to investigation of instability for modulated in time horizontal net mass flow. The ordinary differential equation has been obtained for description of the behavior near the convection instability threshold. This equation is analyzed numerically by the Floquet method; the parametric excitation of convection is observed.