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Rapid declines in unconventional shale production arise from the poorly understood interplay between gas transport and adsorption processes in microporous organic rock. Here, we use high-fidelity molecular dynamics (MD) simulations to resolve the time-varying adsorption of methane gas in realistic organic rock samples, known as kerogen. The kerogen samples derive from various geological shale fields, with porosities ranging between 20-50%. We propose a kinetics sorption model based on a generalised solution of diffusive transport inside a nanopore, to describe the adsorption kinetics in kerogen, which gives excellent fits with all our MD results, and we demonstrate it scales with the square of the length of kerogen. The MD adsorption time constants for all samples are compared with a simplified theoretical model, which we derive from the Langmuir isotherm for adsorption capacitance and the free-volume theory for steady, highly-confined bulk transport. While the agreement with the MD results is qualitatively very good, it reveals that in the limit of low porosity, the diffusive transport term dominates the characteristic time-scale of adsorption, while the adsorption capacitance becomes important for higher pressures. This work provides the first data set for adsorption kinetics of methane in kerogen, a validated model to accurately describe this process, and a qualitative model that links adsorption capacitance and transport with the adsorption kinetics. Furthermore, this work paves the way to upscale interfacial adsorption processes to the next scale of gas transport simulations in meso-pores and macropores of shale reservoirs.

A kinetic model is proposed for the non-equilibrium flow of dense gases composed of hard-sphere molecules, which significantly simplifies the collision integral of the Enskog equation using the relaxation-time approach. The model preserves the most important physical properties of high-density gas systems, including the Maxwellian at rest as the equilibrium solution and the equation of state for hard-sphere fluids, all the correct transport coefficients, namely the shear viscosity, thermal conductivity and bulk velocity, and inhomogeneous density distribution in the presence of a solid boundary. The collision operator of the model contains a Shakhov model-like relaxation part and an excess part in low-order spatial derivatives of the macroscopic flow properties; this latter contribution is used to account for the effect arising from the finite size of gas molecules. The density inhomogeneity in the vicinity of a solid boundary in a confined flow is captured by a method based on the density functional theory. Extensive benchmark tests are performed, including the normal shock structure, the Couette, Fourier and Poiseuille flow at different reduced densities and Knudsen numbers, where the results are compared with the solutions from the Enskog equation and molecular dynamics simulations. It is shown that the proposed kinetic model provides a fairly accurate description of all these non-equilibrium dense gas flows. Finally, we apply our model to simulate forced wave propagation in a dense gas confined between two plates. The inhomogeneous density near the solid wall is found to enhance the oscillation amplitude, while the presence of bulk viscosity causes stronger attenuation of the sound wave. This shows the importance of a kinetic model to reproduce density inhomogeneity and correct transport coefficients, including bulk viscosity.

It is well known that the Poiseuille mass flow rate along microchannels shows a stationary point as the fluid density decreases, referred to as the Knudsen minimum. Surprisingly, if the flow characteristic length is comparable to the molecular size, the Knudsen minimum disappears, as reported for the first time by Wu et al. (J. Fluid Mech., vol. 794, 2016, pp. 252–266). However, there is still no fundamental understanding why the mass flow rate monotonically increases throughout the entire range of flow regimes. Although diffusion is believed to dominate the fluid transport at the nanoscale, here we show that the Fick's first law fails in capturing this behaviour, and so diffusion alone is insufficient to explain this confined flow phenomenon. Rather, we show that the Knudsen minimum disappears in tight confinements because the decay of the mass flow rate due to the decreasing density effects is overcome by the enhancing contribution to the flow provided by the fluid velocity slip at the wall.

Revealing the scattering behaviour of gas molecules on porous surfaces is essential to develop accurate boundary conditions for kinetic transport models that describe the gas dynamics in shale reservoirs. Here, we use high-fidelity molecular dynamics simulations to resolve the gas-surface interactions between methane molecules and realistic organic kerogen surfaces, and to assess the applicability of the widely used scattering kernels. Our results show that the tight matrix porosities have a negligible effect on the timescale and lengthscale of the scattering process, which can be considered instantaneous in time and local in space. Although reflected velocity distributions reveal that the common Maxwell, Cercignani-Lampis and Yamamoto scattering models fail to fully capture the scattering details of methane on kerogen, especially when the incident molecular speeds are high, the Maxwell model predicts best the reflected angular beam pattern and the overall reflected velocity distribution for rough kerogen surfaces. However, for low-speed impingement, more characteristic of shale applications, all scattering models give similar velocity distributions, which are driven by the high degree of gas-surface accommodation observed. We find that a Maxwell model with a calibrated tangential momentum accommodation coefficient, which approaches unity as the surface roughness increases to ∼ 2 nm, is enough to reproduce comparable velocity profiles and mass flow rates inside moderately confined kerogen mesopores. Deviations between the Maxwell model and our molecular simulations are only observed for highly rarefied transport problems, but this rarefaction lies beyond the realm of shale reservoir applications. This paper, therefore, reports the first scattering study on porous and rough kerogen surfaces, and demonstrates the applicability of the Maxwell model, which can be readily incorporated into gas kinetic solvers to predict the apparent permeability of shale with mesopore and macropore networks.

The contents of this brief Lecture Note are devoted to modeling, simulations, and applications with the aim of proposing a unified multiscale approach accounting for the physics and the psychology of people in crowds. The modeling approach is based on the mathematical theory of active particles, with the goal of contributing to safety problems of interest for the well-being of our society, for instance, by supporting crisis management in critical situations such as sudden evacuation dynamics induced through complex venues by incidents.

The heterogeneity and plasticity of macrophages have become a topic of great interest, due to their role in various diseases ranging from cancer to bacterial infections. While initial experimental studies assumed an extreme polarisation situation, with the (anti-tumour) M1 and (pro-tumour) M2 macrophages representing the two extreme cell phenotypes, more recent studies showed a continuum of macrophages polarisation phenotypes. Here, we focus on tumour-macrophage interactions and develop a mathematical model based on kinetic equations for active particles to describe (i) the dynamics of macrophages with a continuum of diverse functional states, ranging from pro-tumour to anti-tumour states; and (ii) the dynamics of tumour cells with a variety of progression (i.e. mutation) states. With the help of this model we show that the growth of solid tumours is associated with an increased clonal heterogeneity, as well as with an increased macrophages phenotypic heterogeneity (caused by a shift from an initial anti-tumour M1-like phenotype to a mixed M1–M2 phenotype). Moreover, we show that the assumption of exponential tumour/immune cell growth leads to an unbounded macrophages growth, which is biologically unrealistic. In contrast, the assumption of logistic tumour/immune cell growth can lead to tumour dormancy (under the control of immune cells), or to tumour growth towards smaller/larger sizes which depend on various model parameters. Finally, we show that tumour dormancy is associated with an increase in the clonal heterogeneity of tumour cells and in the phenotypic heterogeneity of macrophages.

Nicola Bellomo acknowledges the support of the University of Granada, Project Modeling in Nature MNat from micro to macro, https://www.modelingnature.org.This paper has been partially supported by the MINECO-Feder (Spain) research Grant Number RTI2018-098850-B-I00, the Junta de Andalucia (Spain) Project PY18-RT-2422, A-FQM-311-UGR18, and B-FQM-580-UGR20. Livio Gibelli, gratefully acknowledges the financial support of the Engineering and Physical Sciences Research Council (EPSRC) Under Grants EP/N016602/1, EP/R007438/1. Annalisa Quaini acknowledges support from the Radcliffe Institute for Advanced Study at Harvard University where she has been a 2021-2022 William and Flora Hewlett Foundation Fellow. Alessandro Reali acknowledges the partial support of the MIUR-PRIN Project XFAST-SIMS (No. 20173C478N)., The first part of our paper presents a general survey on the modeling, analytic problems, and applications of the dynamics of human crowds, where the specific features of living systems are taken into account in the modeling approach. This critical analysis leads to the second part which is devoted to research perspectives on modeling, analytic problems, multiscale topics which are followed by hints towards possible achievements. Perspectives include the modeling of social dynamics, multiscale problems and a detailed study of the link between crowds and swarms modeling., University of Granada, Project Modeling in Nature MNat from micro to macro, Spanish Government RTI2018-098850-B-I00, Junta de Andalucia, European Commission PY18-RT-2422 A-FQM-311-UGR18 B-FQM-580-UGR20, UK Research & Innovation (UKRI), Engineering & Physical Sciences Research Council (EPSRC) EP/N016602/1 EP/R007438/1, Radcliffe Institute for Advanced Study at Harvard University, Ministry of Education, Universities and Research (MIUR) 20173C478N

International audience; The modeling of living systems composed of many interacting entities is treated in this paper with the aim of describing their collective behaviors. The mathematical approach is developed within the general framework of the kinetic theory of active particles. The presentation is in three parts. First, we derive the mathematical tools, subsequently, we show how the method can be applied to a number of case studies related to well defined living systems, and finally, we look ahead to research perspectives.

The collapse of cavitation bubbles often releases high-speed liquid jets capable of surface damage, with applications in drug delivery, cancer treatment, and surface cleaning. Spherical cap-shaped surface nanobubbles have previously been found to exist on immersed substrates. Despite being known nucleation sites for cavitation, their collapsing dynamics are currently unexplored. Here, we use molecular dynamics simulations to model the shock-induced collapse of different surface nanobubble sizes and contact angles. Comparisons are made with additional collapsing spherical nanobubble simulations near a substrate, to investigate the differences in their jet formation and resulting substrate pitting damage. Our main finding is that the pitting damage in the surface nanobubble simulations is greatly reduced, when compared to the spherical nanobubbles, which is primarily caused by the weaker jets formed during their collapse. Furthermore, the pit depths for surface nanobubble collapse do not depend on bubble size, unlike in the spherical nanobubble cases, but instead depend only on their contact angle. We also find a linear scaling relationship for all bubble cases between the final substrate damage and the peak pressure impulse at the impact centre, which can now be exploited to assess the relative damage in other computational studies of collapsing bubbles. We anticipate the more controlled surface-damage features produced by surface nanobubble cavitation jets will open up new applications in advanced manufacturing, medicine, and precision cleaning.

This paper proposes a multiscale vision to human crowds which provides a consistent description at the three possible modeling scales, namely, microscopic, mesoscopic, and macroscopic. The proposed approach moves from interactions at the microscopic scale and shows how the same modeling principles lead to kinetic and hydrodynamic models. Hence, a unified framework is developed which permits to derive models at each scale using the same principles and similar parameters. This approach can be used to simulate crowd dynamics in complex environments composed of interconnected areas, where the most appropriate scale of description can be selected for each area. This offers a pathway to the development of a multiscale computational model which has the capability to optimize the granularity of the description depending on the pedestrian local flow conditions. An important feature of the modeling at each scale is that the complex interaction between emotional states of walkers and their motion is taken into account.

Surface nanobubbles have potential applications in the manipulation of nanoscale and biological materials, waste-water treatment, and surface cleaning. These spherically capped bubbles of gas can exist in stable diffusive equilibrium on chemically patterned or rough hydrophobic surfaces, under supersaturated conditions. Previous studies have investigated their long-term response to pressure variations, which is governed by the surrounding liquid's local supersaturation; however, not much is known about their short-term response to rapid pressure changes, i.e., their cavitation dynamics. Here, we present molecular dynamics simulations of a surface nanobubble subjected to an external oscillating pressure field. The surface nanobubble is found to oscillate with a pinned contact line, while still retaining a mostly spherical cap shape. The amplitude-frequency response is typical of an underdamped system, with a peak amplitude near the estimated natural frequency, despite the strong viscous effects at the nanoscale. This peak is enhanced by the surface nanobubble's high internal gas pressure, a result of the Laplace pressure. We find that accurately capturing the gas pressure, bubble volume, and pinned growth mode is important for estimating the natural frequency, and we propose a simple model for the surface nanobubble frequency response, with comparisons made to other common models for a spherical bubble, a constant contact angle surface bubble, and a bubble entrapped within a cylindrical micropore. This work reveals the initial stages of growth of cavitation nanobubbles on surfaces, common in heterogeneous nucleation, where classical models based on spherical bubble growth break down.

Numerical solutions of the Enskog-Vlasov (EV) equation are used to determine the velocity distribution function of atoms spontaneously evaporating into near-vacuum conditions. It is found that an accurate approximation is provided by a half-Maxwellian including a drift velocity combined with different characteristic temperatures for the velocity components normal and parallel to the liquid-vapor interface. The drift velocity and the temperature anisotropy reduce as the liquid bulk temperature decreases but persist for relatively low temperatures corresponding to a vapor behaviour which is only slightly non-ideal. Deviations from the undrifted isotropic half-Maxwellian are shown to be consequences of collisions in the liquid-vapor interface which preferentially backscatter atoms with lower normal-velocity component., Comment: 28 pages, 9 figures

Modelling dense gas flows inside channels with sections comparable to the diameter of gas molecules is essential in porous media applications, such as in non-conventional shale reservoir management and nanofluidic separation membranes. In this paper, we perform the first verification study of the Enskogequation by using particle simulation methods based on the same hard-sphere collisions dynamics. Our in-house Event-Driven Molecular Dynamics (EDMD) code and a pseudo-hard-sphere Molecular Dynamics (PHS-MD) solver are used to study force-driven Poiseuille flows, in the limit of high gas densitiesand high confinements. Our results showed (a) very good agreement between EDMD, PHS-MD, and Enskog solutions across density, velocity, and temperature profiles for all the simulation conditions, and (b) numerical evidence that deviations exist in the normalized mass flow rate versus Knudsen number curve compared to the standard curve without confinement. While we observe slight deviations in the Enskog density and velocity profiles from the MD when the reduced density is greater than 0.2, this limit is well above practical engineering applications, such as in shale gas. The key advantages of promoting the Enskog equation for upscaling flows in porous media lie in its ability to capture the non-equilibrium physics of tightly confined fluids, while being computationally more efficient than fundamental simulation approaches, such as molecular dynamics and derivative solvers.

We consider the two-dimensional steady channel flow of a rarefied gas over a backward facing step in the limit of large Knudsen numbers. The free-molecular problem is solved analytically for both diffuse and specular-reflecting channel boundaries, and the solutions are validated through comparison with direct simulation Monte Carlo calculations. Prescribing the density and temperature differences between the inlet and outlet external equilibrium conditions, the results for the density- and temperature-drop-driven flows are analysed and contrasted, revealing higher flow velocities and mass flow rates in the former. While the flow rate is unaffected by the step geometry in the specular case, it increases with the step size in the diffuse-reflecting set-up. At conditions where small flow velocities occur, flow detachment is observed in the form of streamlines connecting the step edge stagnation points. Considering the problem at finite Knudsen numbers, the collisionless-flow regime breaks down at higher Knudsen numbers for lower gas speed flows, followed by the occurrence of step flow separation and recirculation.

Developing unconventional shale gas resource has increased rapidly in recent years. However, while methane adsorbed inside organic kerogen matter is a source of shale production, it is still not a fully understood process. Here, we use molecular simulations to investigate methane adsorption in local micropores that are less than 1 nm inside realistic kerogen samples. We find an exponential scaling law for the local pore adsorption capacity and rationalise the pore density with the effective pore diameter, reservoir pressure, and sample porosity. This scaling law is determined from four kerogen samples at different porosities, each taken from a different shale reservoir, which have been experimentally validated in previous work. We find that pores closer to methane’s diameter are responsible for ~ 20% of the adsorption inside the sample and it is these small pores and lower pressures that dictate the largest adsorption capacity inside kerogen. Predictions of adsorption isotherms from properties of the kerogen structures are now possible using a proposed numerical percolation model by means of this scaling law. Adsorption predictions using our model show remarkably good agreement with molecular dynamics results in this work and isotherms in the literature, at a fraction of the computational cost. This work opens up a new route for determining adsorption isotherms of dense porous media from knowledge about their local pore structure, and can be scaled efficiently to support experimental campaigns, where molecular simulations would be intractable.

This paper proposes a conceptual revisiting of population dynamics to include heterogeneous behaviors of individuals, mutations, and selection. The first part of the paper focuses on the derivation of a general mathematical structure which permits to describe systems composed of individuals whose interactions are stochastic. Hybrid models where some of the populations follow a deterministic dynamics are also discussed. The second part deals with two specific applications, namely the effect of the cellular aging in the virus infection process and the dynamics of virus mutation and competition with the immune system. Sample simulations are presented and classical models of population dynamics are critically analyzed in light of the proposed approach.

This paper presents a computational modeling approach to the dynamics of human crowds, where social interactions can have an important influence on the behavioral dynamics of pedestrians. The modeling of the contagion and propagation of emotional states is carried out by looking at real physical situations where safety problems might arise in some specific circumstances. The approach is based on the methods of the kinetic theory of active particles. The evacuation of a metro station is simulated to enlighten the role of the emotional state in the overall dynamics.

This paper aims to settle disputes in the literature about the spatial variation of the molecular mean free path (MFP) in confined geometries. The MFP of a gas is determined by using both molecular dynamics (MD) and the direct simulation Monte Carlo (DSMC) technique. In spatially-homogeneous cases, the numerical results exactly recover the kinetic theory predictions of a constant MFP. However, in microchannels, the MFP is found to vary near to the bounding walls and reduce at the surfaces to half of its bulk value as long as collisions between gas molecules and wall atoms are taken into account in the calculation of the MFP.

The paper aims at presenting Direct Simulation Monte Carlo (DSMC) extensions and applications to dense fluids. A succinct review of past and current research topics is presented, followed by a more detailed description of DSMC simulations for the numerical solution of the Enskog-Vlasov equation, applied to the study of liquid-vapor flows. Results about simulations of evaporation of a simple liquid in contact with a dense vapor are presented as an example.

Thin-film evaporation from nanoporous membranes is a promising cooling technology employed for the thermal management of modern electronic devices. We propose an effective one-dimensional analytical approach that can accurately predict the temperature and density jump relations, and evaporation rates, for arbitrary nanoporous membrane configurations. This is accomplished through the specification of an effective evaporation coefficient that encompasses the influence of different system parameters, such as porosity, meniscus shape, evaporation coefficient, and receding height. Our proposed approach can accurately predict all the typical output evaporation parameters of interest like mass flux, and temperature and density jumps, without the need to carry out computationally demanding numerical simulations. Several exemplar cases comprising of nanoporous configurations with a wide range of parameters have been considered to demonstrate the feasibility and accuracy of this analytic approach. This work thus enables a quick, efficient, and accurate means of aiding the design and engineering analysis of nanoporous membrane-based cooling devices.\ud

A mesoscopic model of behavioral crowds is developed within the framework of the kinetic theory for active particles. An analytic long-time equilibrium solution is obtained which gives a fundamental density-velocity diagram consistent with the empirical evidence. Numerical simulations based on a Monte Carlo particle method show that the proposed model has the capability to qualitatively depict emerging behaviors and to provide a realistic description of the crowd dynamics in complex evacuation scenarios.

This paper proposes an essay concerning the understanding of human behaviours and crisis management of crowds in extreme situations, such as evacuation through complex venues. The first part focuses on the understanding of the main features of the crowd viewed as a living, hence complex system. The main concepts are subsequently addressed, in the second part, to a critical analysis of mathematical models suitable to capture them, as far as it is possible. Then, the third part focuses on the use, toward safety problems, of a model derived by the methods of the mathematical kinetic theory and theoretical tools of evolutionary game theory. It is shown how this model can depict critical situations and how these can be managed with the aim of minimizing the risk of catastrophic events.

This paper proposes a systems approach to social sciences based on a mathematical framework derived from a generalization of the mathematical kinetic theory and of theoretical tools of game theory. Social systems are modeled as a living evolutionary ensemble composed of many individuals, who express specific strategies, cooperate, compete and might aggregate into groups which pursue a common interest. A critical analysis on the complexity features of social system is developed and a differential structure is derived to provide a general framework toward modeling. Then, a case study shows how the systems approach is applied. Moreover, it is shown how the theory leads to the interpretation and use of the so-called big data. Finally some research perspectives are brought to the attention of readers.

The evaporation of polyatomic liquids into near-vacuum conditions is investigated by using the Enskog–Vlasov model. Molecules are approximated as classical rigid rotators, and the collisional energy exchanges between the translational and rotational degrees of freedom are dealt with by the Borgnakke–Larsen method. The distribution function of evaporated molecules and the evaporation coefficient are evaluated in a wide range of liquid bulk temperatures and inelastic collision fractions. It is found that the translational velocity distribution function is well approximated by a drifted bi-Maxwellian, while the rotational energy follows the Boltzmann distribution at a temperature that varies between the separation and the parallel temperatures as the inelastic collision fraction increases. The evaporation coefficient based on the separation temperature turns out to be independent of the inelastic collision fraction and only mildly dependent on the liquid bulk temperature.

Evaporation of a binary liquid into near vacuum conditions has been studied using numerical solutions of a system of two coupled Enskog-Vlasov equations. Liquidvapor coexistence curves have been mapped out for different liquid compositions. The evaporation process has been investigated at a range of liquid temperatures sufficiently lower than the critical one for the vapor not to significantly deviate from the ideal behaviour. It is found that the shape of the distribution functions of evaporating atoms is well approximated by an anisotropic Maxwellian distribution with different characteristic temperatures for velocity components normal and parallel to the liquid-vapor interface. The anisotropy reduces as the evaporation temperature decreases. Evaporation coefficients are computed based on the separation temperature and the maximum concentration of the less volatile component close to the liquid-vapor interface. This choice leads to values which are almost constant in the simulation conditions.

This chapter provides an introduction to the contents of Bellomo and Gibelli (Crowd dynamics, volume 1 – theory, models, and safety problems. Modeling and simulation in science, engineering, and technology. Birkhauser, New York, 2018) and a general critical analysis on crowd modeling. The presentation is organized in three parts: firstly, a general framework and rationale toward the modeling and simulations of human crowds are proposed; subsequently the contents of Chaps. 2, 3, 4 , 5 , 6 , 7 , 8 and 9 are summarized by referring to the existing literature; finally, by taking advantage of the contents of the whole book, some speculations are proposed on possible research perspectives. Five key problems are presented, and hints are given to tackle them within a multiscale vision which appears to be the most looking forward idea to be pursued in research projects.

Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional square driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.

A kinetic model for the study of capillary flows in devices with microscale geometry is presented. The model is based on the Enskog-Vlasov kinetic equation and provides a reasonable description of both fluid-fluid and fluid-wall interactions. Numerical solutions are obtained by an extension of the classical Direct Simulation Monte Carlo (DSMC) to dense fluids. The equilibrium properties of liquid menisci between two hydrophilic walls are investigated and the validity of the Laplace-Kelvin equation at the microscale is assessed. The dynamical process which leads to the meniscus breakage is clarified.

This chapter shows how the mathematical tools presented in Chapter 3 can be developed to derive mathematical models in social sciences and to understand how interactions at the microscopic scale can lead to the collective social behaviors.

The contents of this chapter are motivated by the second key question posed in Section 1.3, namely by the search of mathematical tools suitable to model living systems (specifically, we deal with systems composed of many interacting entities).