Effect of solute immobilization on the stability problem within the fractional model in the solute analog of the Horton-Rogers-Lapwood problem

The paper is devoted to the linear stability analysis within the solute analogue of the Horton-Rogers-Lapwood (HRL) problem. The solid nanoparticles are treated as solute within the continuous approach. Therefore, we consider the infinite horizontal porous layer saturated with a mixture (carrier fluid and solute). Solute transport in porous media is very often complicated by solute immobilization on a solid matrix of porous media. Solute immobilization (solute sorption) is taken into account within the fractal model of the MIM approach. According to this model a solute in porous media immobilizes within random time intervals and the distribution of such random variable does not have a finite mean value, which has a good agreement with some experiments. The solute concentration difference between the layer boundaries is assumed as constant. We consider two cases of horizontal external filtration flux: constant and time-modulated. For the constant flux the system of equations that determines the frequency of neutral oscillations and the critical value of the Rayleigh-Darcy number is derived. Neutral curves of the critical parameters on the governing parameters are plotted. Stability maps are obtained numerically in a wide range of parameters of the system. We have found that taking immobilization into account leads to an increase in the critical value of the Rayleigh-Darcy number with an increase in the intensity of the external filtration flux. The case of weak time-dependent external flux is investigated analytically. We have shown that the modulated external flux leads to an increase in the critical value of the Rayleigh-Darcy number and a decrease in the critical wave number. For moderate time-dependent filtration flux the differential equation with Caputo fractional derivatives has been obtained for the 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

Stability of uniform vertical flow through a close porous filter in the presence of solute immobilization

In the present paper we consider slow filtration of a mixture through a close porous filter. The heavy solute penetrates slowly into the porous filter due to the external vertical filtration flow and diffusion. This process is accompanied by the formation of the domain with heavy fluid near the upper boundary of the filter. The developed stratification, at which the heavy fluid is located above the light fluid, is unstable. When the mass of the heavy fluid exceeds the critical value, one can observe the onset of the Rayleigh-Taylor instability. Due to the above peculiarities we can distinguish between two regimes of vertical filtration: 1) homogeneous seepage and 2) convective filtration. When considering the filtration process it is necessary to take into account the diffusion accompanied by the immobilization effect (or sorption) of the solute. The immobilization is described by the linear MIM (mobile/immobile media) model. It has been shown that the immobilization slows down the process of forming the unstable stratification. The purpose of the paper is to find the stability conditions for homogeneous vertical seepage of he solute into the close porous filter. The linear stability problem is solved using the quasi-static approach. The critical times of instability are estimated. The stability maps are plotted in the space of system parameters. The applicability of quasi-static approach is substantiated by direct numerical simulation of the full nonlinear equations