A survey on mass conservation and related topics in nonlinear diffusion

We examine the validity of the principle of mass conservation for solutions of some typical equations in the theory of nonlinear diffusion, including equations in standard differential form and also their fractional counterparts. We use as main examples the heat equation, the porous medium equation and the $p$-Laplacian equation. Though these equations have the form of conservation laws, it happens that in some ranges, and posed in the whole Euclidean space, the solutions actually lose mass in time, they even disappear in finite time. This is a surprising fact. In Part 1 we pay attention to the connection between the validity of mass conservation and the existence of finite-mass self-similar solutions, as well as the role in the asymptotic behaviour of more general classes of solutions. We examine the situations when mass conservation is replaced by its extreme alternative, extinction in finite time. The conservation laws offer difficult borderline cases in the presence of critical parameters that we identify. New results are proved. We establish mass conservation in those pending cases. We also explain the disappearance of the fundamental solutions in a very graphical way. The sections of Part 2 are devoted to the discussion of mass conservation for some fractional nonlinear diffusion equations, where the situation is surveyed and a number of pending theorems are proved. We conclude with a long review of related equations and topics in Part 3. Summing up, the paper aims at surveying an important topic in nonlinear diffusion; at the time we solve a number of open problems on key issues and point out new directions.Comment: 69 page

Anomalous transport in the crowded world of biological cells

A ubiquitous observation in cell biology is that diffusion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a power-law increase of the MSD, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations, non-gaussian distributions of the displacements, heterogeneous diffusion, and immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarise some widely used theoretical models: gaussian models like FBM and Langevin equations for visco-elastic media, the CTRW model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Emphasis is put on the spatio-temporal properties of the transport in terms of 2-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even for identical MSDs. Then, we review the theory underlying common experimental techniques in the presence of anomalous transport: single-particle tracking, FCS, and FRAP. We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto- and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where model systems mimic physiological crowding conditions. Finally, computer simulations play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.Comment: review article, to appear in Rep. Prog. Phy

Non-Fickian dispersion in porous media : 2. Model validation from measurements at different scales

International audienceWe aim at testing and validating a mobile-immobile mass transfer model from a set of single-well injection withdrawal tracer tests in a heterogeneous porous aquifer. By varying the duration of the injection phase, different volumes of aquifer are investigated by the tracer. Hence, we focus the transport model validation not only on reproducing a single breakthrough curve (BTC) but also on the model's capacity to predict the amount of mixing as a function of the volume visited by the tracer. All the BTCs are strongly asymmetric, as expected when dispersion is controlled by diffusive mass transfers between the mobile water and the immobile water part of the porosity. However, the BTC cannot be modeled by a conventional mobile-immobile mass transfer model with a simple power law memory function. To account for that, we implement a continuous time random walk model in which the transition time distribution y (t), which is related to the excursion time probability of the tracer in the immobile domain, is a dual-slope power law distribution. The model best fits the BTC data set with a transitional regime controlled by y(t) t2 and an asymptotic regime characteristic of the conventional double-porosity model with y(t) t1.5 . This work emphasizes that high-resolution concentration measurement and multiple-scale tracer tests are required for assessing solute dispersion models in heterogeneous reservoirs and for subsequently obtaining reliable predictions

The Zoo of Non-Fourier Heat Conduction Models

The Fourier heat conduction model is valid for most macroscopic problems. However, it fails when the wave nature of the heat propagation or time lags become dominant and the memory or/and spatial non-local effects significant -- in ultrafast heating (pulsed laser heating and melting), rapid solidification of liquid metals, processes in glassy polymers near the glass transition temperature, in heat transfer at nanoscale, in heat transfer in a solid state laser medium at the high pump density or under the ultra-short pulse duration, in granular and porous materials including polysilicon, at extremely high values of the heat flux, in heat transfer in biological tissues. In common materials the relaxation time ranges from $10^{-8}$ to $10^{-14}$ sec, however, it could be as high as 1 sec in the degenerate cores of aged stars and its reported values in granular and biological objects varies up to 30 sec. The paper considers numerous non-Fourier heat conduction models that incorporate time non-locality for materials with memory (hereditary materials, including fractional hereditary materials) or/and spatial non-locality, i.e. materials with non-homogeneous inner structure

Solute transport in layered and heterogeneous soils

Better understanding of transport of dissolved chemicals in soils and aquifers is important to evaluate and remediate contaminated soils and aquifers. Because of the nature of heterogeneity of field porous media, studies on transport processes in non-homogeneous media are necessary. In this study, transport of solutes in layered and heterogeneous media was investigated using numerical approximations. For layered soils, transport properties were assumed homogeneous within individual layers but different between layers. For heterogeneous systems, either a time-dependent or distance-dependent dispersivity was considered to represent the effects of heterogeneity. In a series of simulations of transport in two-layered soils, we found that breakthrough curves (BTCs) were similar regardless of the layering sequence for all reversible and irreversible solute retention mechanisms. Such findings were in agreement with results from laboratory experiments using tritium as a tracer and Ca and Mg as reactive solutes. Field measured apparent dispersivity is often found to increase with time because of the heterogeneity of soils and aquifers. We proposed a fractal model to explain the time dependency of dispersivity. Our model indicates a nonlinear increase of variance of travel distance with time or mean travel distance, which implies a time-dependent dispersivity. Application of our model to three field experiments (the Cape Cod, the Borden, and the Columbus sites) indicates fractal behavior for all three cases. The term scale effects is often used in the literature to refer to the dependency of dispersivity on either mean travel distance or distance from source. We presented a critical review on the ambiguity in definition of this term. We presented comparisons between transport processes in systems with time-dependent and distance-dependent dispersivities. Our results showed that enhanced spreading in BTCs consistently occurred in systems with time-dependent dispersivities. Recently, a new governing equation, factional-order advection-dispersion equation (FADE) was proposed to describe transport processes in heterogeneous systems. We proposed a statistical method to justify the use of a FADE. A fractional order of 1.82 was confirmed to be necessary to describe the bromide plumes at the Cape Cod site

Improved modeling for fluid flow through porous media

Petroleum production is one of the most important technological challenges in the current world. Modeling and simulation of porous media flow is crucial to overcome this challenge. Recent years have seen interest in investigation of the effects of history of rock, fluid, and flow properties on flow through porous media. This study concentrates on the development of numerical models using a ‘memory’ based diffusivity equation to investigate the effects of history on porous media flow. In addition, this study focusses on developing a generalized model for fluid flow in packed beds and porous media. The first part of the thesis solves a memory-based fractional diffusion equation numerically using the Caputo, Riemann-Liouville (RL), and Grünwald-Letnikov (GL) definitions for fractional-order derivatives on uniform meshes in both space and time. To validate the numerical models, the equation is solved analytically using the Caputo, and Riemann-Liouville definitions, for Dirichlet boundary conditions and a given initial condition. Numerical and analytical solutions are compared, and it is found that the discretization method used in the numerical model is consistent, but less than first order accurate in time. The effect of the fractional order on the resulting error is significant. Numerical solutions found using the Caputo, Riemann-Liouville, and Grünwald-Letnikov definitions are compared in the second part. It is found that the largest pressure values are found from Caputo definition and the lowest from Riemann-Liouville definition. It is also found that differences among the solutions increase with increasing fractional order