Weak and parabolic solutions of advection-diffusion equations with rough velocity field.

We study the Cauchy problem for the advection-diffusion equation ∂tu+div(ub)=Δu associated with a merely integrable divergence-free vector field b defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

Multiscale Asymptotic Analysis Reveals How Cell Growth and Subcellular Compartments Affect Tissue-Scale Hormone Transport.

Determining how cell-scale processes lead to tissue-scale patterns is key to understanding how hormones and morphogens are distributed within biological tissues and control developmental processes. In this article, we use multiscale asymptotic analysis to derive a continuum approximation for hormone transport in a long file of cells to determine how subcellular compartments and cell growth and division affect tissue-scale hormone transport. Focusing our study on plant tissues, we begin by presenting a discrete multicellular ODE model tracking the hormone concentration in each cell's cytoplasm, subcellular vacuole, and surrounding apoplast, represented by separate compartments in the cell-file geometry. We allow the cells to grow at a rate that can depend both on space and time, accounting for both cytoplasmic and vacuolar expansion. Multiscale asymptotic analysis enables us to systematically derive the corresponding continuum model, obtaining an effective reaction-advection-diffusion equation and revealing how the effective diffusivity, effective advective velocity, and the effective sink term depend on the parameters in the cell-scale model. The continuum approximation reveals how subcellular compartments, such as vacuoles, can act as storage vessels, that significantly alter the effective properties of hormone transport, such as the effective diffusivity and the induced effective velocity. Furthermore, we show how cell growth and spatial variance across cell lengths affect the effective diffusivity and the induced effective velocity, and how these affect the tissue-scale hormone distribution. In particular, we find that cell growth naturally induces an effective velocity in the direction of growth, whereas spatial variance across cell lengths induces effective velocity due to the presence of an extra compartment, such as the apoplast and the vacuole, and variations in the relative sizes between the compartments across the file of cells. It is revealed that hormone transport is faster across cells of decreasing lengths than cells with increasing lengths. We also investigate the effect of cell division on transport dynamics, assuming that each cell divides as soon as it doubles in size, and find that increasing the time between successive cell divisions decreases the growth rate, which enhances the effect of cell division in slowing hormone transport. Motivated by recent experimental discoveries, we discuss particular applications for transport of gibberellic acid (GA), an important growth hormone, within the Arabidopsis root. The model reveals precisely how membrane proteins that mediate facilitated GA transport affect the effective tissue-scale transport. However, the results are general enough to be relevant to other plant hormones, or other substances that are transported in a similar way in any type of cells.

Exploration and Gas Source Localization in Advection-Diffusion Processes with Potential-Field-Controlled Robotic Swarms.

Mobile multi-robot systems are well suited for gas leak localization in challenging environments. They offer inherent advantages such as redundancy, scalability, and resilience to hazardous environments, all while enabling autonomous operation, which is key to efficient swarm exploration. To efficiently localize gas sources using concentration measurements, robots need to seek out informative sampling locations. For this, domain knowledge needs to be incorporated into their exploration strategy. We achieve this by means of partial differential equations incorporated into a probabilistic gas dispersion model that is used to generate a spatial uncertainty map of process parameters. Previously, we presented a potential-field-control approach for navigation based on this map. We build upon this work by considering a more realistic gas dispersion model, now taking into account the mechanism of advection, and dynamics of the gas concentration field. The proposed extension is evaluated through extensive simulations. We find that introducing fluctuations in the wind direction makes source localization a fundamentally harder problem to solve. Nevertheless, the proposed approach can recover the gas source distribution and compete with a systematic sampling strategy. The estimator we present in this work is able to robustly recover source candidates within only a few seconds. Larger swarms are able to reduce total uncertainty faster. Our findings emphasize the applicability and robustness of robotic swarm exploration in dynamic and challenging environments for tasks such as gas source localization.

Rhizosphere modelling reveals spatiotemporal distribution of daidzein shaping soybean rhizosphere bacterial community.

Plant roots nurture a wide variety of microbes via exudation of metabolites, shaping the rhizosphere's microbial community. Despite the importance of plant specialized metabolites in the assemblage and function of microbial communities in the rhizosphere, little is known of how far the effects of these metabolites extend through the soil. We employed a fluid model to simulate the spatiotemporal distribution of daidzein, an isoflavone secreted from soybean roots, and validated using soybeans grown in a rhizobox. We then analysed how daidzein affects bacterial communities using soils artificially treated with daidzein. Simulation of daidzein distribution showed that it was only present within a few millimetres of root surfaces. After 14 days in a rhizobox, daidzein was only present within 2 mm of root surfaces. Soils with different concentrations of daidzein showed different community composition, with reduced α-diversity in daidzein-treated soils. Bacterial communities of daidzein-treated soils were closer to those of the soybean rhizosphere than those of bulk soils. This study highlighted the limited distribution of daidzein within a few millimetres of root surfaces and demonstrated a novel role of daidzein in assembling bacterial communities in the rhizosphere by acting as more of a repellant than an attractant.

On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection-diffusion equations.

The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection-diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Effect of adsorption, radioactive decay and fractal structure of matrix on solute transport in fracture.

Reservoir contamination by various contaminants including radioactive elements is an actual environmental problem for all developed countries. Analysis of mass transport in a complex environment shows that the conventional diffusion equation based on Fick's Law fails to model the anomalous character of the diffusive mass transport observed in the field and laboratory experiments. These complex processes can be modelled by non-local advection-diffusion equations with temporal and spatial fractional derivatives. In the present paper, fractional differential equations are used for modelling the transport of radioactive materials in a fracture surrounded by the porous matrix of fractal structure. A new form of fractional differential equation for modelling migration of the radioactive contaminant in the fracture is derived and justified. Solutions of particular boundary value problems for this equation were found by application of the Laplace transform. Through the use of fractional derivatives, the model accounts for contaminant exchange between fracture and surrounding porous matrix of fractal geometry. For the case of an arbitrary time-dependent source of radioactive contamination located at the inlet of the fracture, the exact solutions for solute concentration in the fracture and surrounding porous medium are obtained. Using the concept of a short memory, an approximate solution of the problem of radioactive contaminant transport along the fracture surrounded by the fractal type porous medium is also obtained and compared with the exact solution. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Mathematical modelling of cell migration: stiffness dependent jump rates result in durotaxis.

Durotaxis, the phenomena where cells migrate up a gradient in substrate stiffness, remains poorly understood. It has been proposed that durotaxis results from the reinforcement of focal adhesions on stiff substrates. In this paper we formulate a mathematical model of single cell migration on elastic substrates with spatially varying stiffness. We develop a stochastic model where the cell moves by updating the position of its adhesion sites at random times, and the rate of updates is determined by the local stiffness of the substrate. We investigate two physiologically motivated mechanisms of stiffness sensing. From the stochastic model of single cell migration we derive a population level description in the form of a partial differential equation for the time evolution of the density of cells. The equation is an advection-diffusion equation, where the advective velocity is proportional to the stiffness gradient. The model shows quantitative agreement with experimental results in which cells tend to cluster when seeded on a matrix with periodically varying stiffness.

Computational Diffusion Magnetic Resonance Imaging Based on Time-Dependent Bloch NMR Flow Equation and Bessel Functions.

Magnetic resonance imaging (MRI) uses a powerful magnetic field along with radio waves and a computer to produce highly detailed "slice-by-slice" pictures of virtually all internal structures of matter. The results enable physicians to examine parts of the body in minute detail and identify diseases in ways that are not possible with other techniques. For example, MRI is one of the few imaging tools that can see through bones, making it an excellent tool for examining the brain and other soft tissues. Pulsed-field gradient experiments provide a straightforward means of obtaining information on the translational motion of nuclear spins. However, the interpretation of the data is complicated by the effects of restricting geometries as in the case of most cancerous tissues and the mathematical concept required to account for this becomes very difficult. Most diffusion magnetic resonance techniques are based on the Stejskal-Tanner formulation usually derived from the Bloch-Torrey partial differential equation by including additional terms to accommodate the diffusion effect. Despite the early success of this technique, it has been shown that it has important limitations, the most of which occurs when there is orientation heterogeneity of the fibers in the voxel of interest (VOI). Overcoming this difficulty requires the specification of diffusion coefficients as function of spatial coordinate(s) and such a phenomenon is an indication of non-uniform compartmental conditions which can be analyzed accurately by solving the time-dependent Bloch NMR flow equation analytically. In this study, a mathematical formulation of magnetic resonance flow sequence in restricted geometry is developed based on a general second order partial differential equation derived directly from the fundamental Bloch NMR flow equations. The NMR signal is obtained completely in terms of NMR experimental parameters. The process is described based on Bessel functions and properties that can make it possible to distinguish cancerous cells from normal cells. A typical example of liver distinguished from gray matter, white matter and kidney is demonstrated. Bessel functions and properties are specifically needed to show the direct effect of the instantaneous velocity on the NMR signal originating from normal and abnormal tissues.

The thrombotic potential of circulating tumor microemboli: computational modeling of circulating tumor cell-induced coagulation.

Thrombotic events can herald the diagnosis of cancer, preceding any cancer-related clinical symptoms. Patients with cancer are at a 4- to 7-fold increased risk of suffering from venous thromboembolism (VTE), with ∼7,000 patients with lung cancer presenting from VTEs. However, the physical biology underlying cancer-associated VTE remains poorly understood. Several lines of evidence suggest that the shedding of tissue factor (TF)-positive circulating tumor cells (CTCs) and microparticles from primary tumors may serve as a trigger for cancer-associated thrombosis. To investigate the potential direct and indirect roles of CTCs in VTE, we characterized thrombin generation by CTCs in an interactive numerical model coupling blood flow with advection-diffusion kinetics. Geometric measurements of CTCs isolated from the peripheral blood of a lung cancer patient prior to undergoing lobectomy formed the basis of the simulations. Singlet, doublet, and aggregate circulating tumor microemboli (CTM) were investigated in the model. Our numerical model demonstrated that CTM could potentiate occlusive events that drastically reduce blood flow and serve as a platform for the promotion of thrombin generation in flowing blood. These results provide a characterization of CTM dynamics in the vasculature and demonstrate an integrative framework combining clinical, biophysical, and mathematical approaches to enhance our understanding of CTCs and their potential direct and indirect roles in VTE formation.

Quantitative analysis of molecular transport in the extracellular space using physics-informed neural network.

The brain extracellular space (ECS), an irregular, extremely tortuous nanoscale space located between cells or between cells and blood vessels, is crucial for nerve cell survival. It plays a pivotal role in high-level brain functions such as memory, emotion, and sensation. However, the specific form of molecular transport within the ECS remain elusive. To address this challenge, this paper proposes a novel approach to quantitatively analyze the molecular transport within the ECS by solving an inverse problem derived from the advection-diffusion equation (ADE) using a physics-informed neural network (PINN). PINN provides a streamlined solution to the ADE without the need for intricate mathematical formulations or grid settings. Additionally, the optimization of PINN facilitates the automatic computation of the diffusion coefficient governing long-term molecule transport and the velocity of molecules driven by advection. Consequently, the proposed method allows for the quantitative analysis and identification of the specific pattern of molecular transport within the ECS through the calculation of the Péclet number. Experimental validation on two datasets of magnetic resonance images (MRIs) captured at different time points showcases the effectiveness of the proposed method. Notably, our simulations reveal identical molecular transport patterns between datasets representing rats with tracer injected into the same brain region. These findings highlight the potential of PINN as a promising tool for comprehensively exploring molecular transport within the ECS.

An efficient technique based on cubic B-spline functions for solving time-fractional advection diffusion equation involving Atangana-Baleanu derivative.

The present paper deals with cubic B-spline approximation together with θ -weighted scheme to obtain numerical solution of the time fractional advection diffusion equation using Atangana-Baleanu derivative. To discretize the Atangana-Baleanu time derivative containing a non-singular kernel, finite difference scheme is utilized. The cubic basis functions are associated with spatial discretization. The current discretization scheme used in the present study is unconditionally stable and the convergence is of order O ( h 2 + Δ t 2 ) . The proposed scheme is validated through some numerical examples which reveal the current scheme is feasible and quite accurate.

A three-dimensional fractional solution for air contaminants dispersal in the planetary boundary layer.

In this study, we investigated some closed-form solutions for solving atmospheric dispersion issues under variable atmospherical hypothesis, in a vertically positioned non-homogeneous planetary boundary-layer. In our context, a nonidentical expansion for the solution of the fractional advection-diffusion equation in a non-integer dimensional-space was examined. In a nutshell, a Sturm-Liouville eigenvalue problem with more reliable information concerning the initial value problem is discussed. The performance of the model was estimated by presenting numerical simulations against experimental data. Under these meteorological conditions, fractional-order models performed far better than those of the classical integer-order ones.

Distinctive distribution and migration of global fallout plutonium isotopes in an alpine lake and its implications for sediment dating.

This work investigated plutonium (Pu) isotopes in sediment cores collected from an alpine lake (Lake Heinongpo with 3779 m above sea level) in Southwestern China. 240Pu/239Pu atom ratios in all sediment samples showed the typical global fallout values of ∼0.18 without any influences from other Pu contaminant sources. 239+240Pu activities with surface and subsurface maximums followed by exponential decline with sediment depth were respectively observed in the two sediment cores. The distinctive depth distributions of 239,240Pu in the lake sediments was attributed to the very slow sediment deposition rate due to the lack of terrestrial sediment input, while the alpine snowmelt input was the primary source of Pu in the lake sediments in addition to the direct atmospheric deposition. The total Pu inventory was estimated to be 56.3 ± 1.4 and 63.9 ± 0.8 Bq/m2 respectively in the two sediment cores. The generally higher Pu inventory in the Lake Heinongpo compared with other reported lakes in similar latitude should be mainly attributed to their different Pu input passages. The advection-diffusion equation was further applied to evaluate the downward migration of Pu isotopes in the sediment cores and predict the future evolution of Pu distribution in the sediment cores. The fitted results indicated that the diffusion effect controlled the downward migration of Pu in the sediments, but this diffusive migration will not prevent the peak of global fallout Pu in undisturbed sediment cores from being a valuable time marker for sediment dating.

Comparison of Finite Difference and Finite Volume Simulations for a Sc-Drying Mass Transport Model.

Different numerical solutions of a previously developed mass transport model for supercritical drying of aerogel particles in a packed bed [Part 1: Selmer et al. 2018, Part 2: Selmer et al. 2019] are compared. Two finite difference discretizations and a finite volume method were used. The finite volume method showed a higher overall accuracy, in the form of lower overall Euclidean norm (l2 ) and maximum norm (l∞ ) errors, as well as lower mole balance errors compared to the finite difference methods. Additionally, the finite volume method was more efficient when the condition numbers of the linear systems to be solved were considered. In case of fine grids, the computation time of the finite difference methods was slightly faster but for 16 or fewer nodes the finite volume method was superior. Overall, the finite volume method is preferable for the numerical solution of the described drying model for aerogel particles in a packed bed.

Dispersion of traffic derived air pollutants into urban parks.

Scarce land resources in dense cities means that small urban parks are important as a leisure and amenity resource for the urban population. However, streets with heavily traffic often surround these fragmented parks and increase the potential user exposure to air pollutants from vehicles. The dispersion profiles of PM2.5 and black carbon from roadside into urban parks at pedestrian level, in Hong Kong, were measured using mobile high time resolution instruments. In the downwind direction, pollutant concentrations decreased rapidly from roadside and by some tens of metres reached relatively constant values. An even sharper gradient is found in the upwind direction, with a rapid increase detected within 2m of the road edge. The distinct decay profiles were explained with an analytical dispersion model formulated based on the gradient transport theory using an Eulerian approach. The simulations using the dispersion model suggest 17m as a typical halving distance under normal urban conditions, which is introduced to simplify the description of dispersion profiles. Using Hong Kong as an example, ~90% of urban parks, to different extent, overlap with the 17m halving distance from roads, which means few urban parks in Hong Kong avoid the impact from nearby traffic emissions. Thus, from the perspective of human exposure to air pollutants in urban parks, this study provides observations of relevance for future park design in dense cities.

Semi-analytical solution for the in-vitro sedimentation, diffusion and dosimetry model: surveying the impact of the Peclet number.

Reducing size of the particles to the nanoscale range gives them new physicochemical properties. Several experiments have shown cytotoxic effects for different kinds of engineered nanoparticles (ENP). In-vitro cell culture assays are widely utilized by researchers to evaluate cytotoxic effects of the ENPs. The present paper deals with the "In vitro Sedimentation, Diffusion and Dosimetry (ISDD)" model. This mathematical model uses an advection-diffusion equation with specific assumptions and coefficients to estimate the dose of the particles delivered to the cells monolayer in the culture medium. In the present work, utilizing the generalized integral transform technique (GITT), a semi-analytical solution is developed for the ISDD model. The parameters affecting the ISDD predictions are integrated into two dimensionless numbers, Pe and τ. The Pe number shows the ratio of the convective to the diffusive mass transport rates and τ is a dimensionless time parameter. The quality of the results for an extensive range of Pe and τ numbers is surveyed through application of the developed formula to two series of test cases. A comparison of the results with those obtained from numerical methods shows deviations in the numerical results at high Pe numbers. Applying the developed formula, ISDD predictions for a wide practical range of Pe and τ numbers are calculated and plotted in two- and three-dimensional plots. The curves and formula obtained in this study facilitate the achievement of ISDD predictions with higher accuracies and capabilities for verification of the results.