Continuum Kinetic Theory Research Articles

Моделирование движения обвальной массы горной породы с использованием двухжидкостной математической модели

Introduction. Rock falls are considered to be very dangerous slope processes that can lead to serious consequences. In addition to real observations, numerical (mathematical) or experimental (physical) modeling methods are used to study slope processes. Mathematical modeling is used in the cases where experimental data is insufficient. There are quite a lot of models to describe slope processes. The choice of model to describe a particular slope process lies in the approach on which the model is based. In this work, the debris flow taking into account the fluidization of the collapse mass during its movement along the hillside was described. We used a two-fluid model based on the continuum approach and the kinetic theory of granular gas. The model takes into account the fluidization of the collapse mass, and its implementation does not require the use of powerful computing resources. Purpose of the study. Investigation of the collapse mass motion along a hillside using a two-fluid model based on continuum approach and kinetic theory of granular gas. Estimation of the maximum height of the collapse mass flow layer and the average flow velocity. Materials and methods. The two-fluid model based on the continuum approach and the kinetic theory of granular gas was used for theoretical study of the collapse mass flow. The motion of the collapse mass flow consisting of rock granules (average diameter 7.5 cm) along the hillside (slope angle of 30 degrees) was considered. The 3D simulation results obtained using the two-fluid model were compared with experimental data and calculated data obtained using the discrete element method. The results and discussion. Three-dimensional modeling was applied to study the motion of the collapse mass flow along the hillside using the two-fluid model. The obtained calculations of the maximum height of the flow layer and the average flow velocity are compared with experimental data and calculated data using the discrete element method. The comparison showed that the averaged value (obtained as the arithmetic mean) of the experimental data of the average flow velocity coincides with the result of calculations using the two-fluid model. And the average value of the calculated data using the discrete element method differs slightly from the calculated value using the two-fluid model. The paper proposes a method for calculating the maximum height of the flow layer using graphs of the distribution of the particles volume fraction along the normal to the inclined surface. Using this technique, the maximum value of the layer height was obtained. This value coincides with the maximum value obtained using the discrete element method and differs only slightly from the maximum value obtained in the experiments. Using the two-fluid model, additional calculations were carried out at different values of the slope angle in the range from 20 to 60 degrees. The calculation results of the maximum height of the flow layer and average flow velocity were compared with the calculated data obtained using a model based on the discrete element method. It was found that the calculation results obtained using two different models agree better with each other when the dependence of the average flow velocity on the slope angle is studied. Conclusion. In general, it can be concluded that the results of calculations of the maximum layer height and average flow velocity obtained using the two-fluid model are in fairly good agreement with experimental data and calculated data obtained using the discrete element method. It can also be concluded that the two-fluid model can be used to describe the motion of the flows of the collapse mass with a relatively low liquid content (up to 18 %) and fine inclusions (up to 27 %). Resume. The research results can be useful in estimating the maximum height of the flow layer of the collapse mass and the average flow velocity. The direction of future research is to improve the two-fluid model based on the continuum approach and the kinetic theory of granular gas for describing of real slope processes.

Hydrodynamic Anisotropy of Depletion in Nonequilibrium.

Active colloids in a bath of inert particles of smaller size cause anisotropic depletion. The active hydrodynamics of this nonequilibrium phenomenon, which is fundamentally different from its equilibrium counterpart and passive particles in an active bath, remains scarcely understood. Here we combine mesoscale hydrodynamic simulation as well as theoretical analysis to examine the physical origin for the active depletion around a self-propelled noninteractive colloid. Our results elucidate that the variable hydrodynamic effect critically governs the microstructure of the depletion zone. Three characteristic states of anisotropic depletion are identified, depending on the strength and stress of activity. This yields a state diagram of depletion in the two-parameter space, captured by developing a theoretical model with the continuum kinetic theory and leading to a mechanistic interpretation of the hydrodynamic anisotropy of depletion. Furthermore, we demonstrate that such depletion in nonequilibrium results in various clusters with ordered organization of squirmers, which follows a distinct principle contrary to that of the entropy scenario of depletion in equilibrium. The findings might be of immediate interest to tune the hydrodynamics-mediated anisotropic interactions and active nonequilibrium organizations in the self-propulsion systems.

Thermodynamically consistent coarse-graining of polar active fluids

Starting from continuum kinetic theory, we derive a mean-field theory and entropy-based closure model for a polar active fluid. Analysis and simulation show the model closely approximates both the transient and nonequilibrium dynamics of the kinetic theory. Taking advantage of the model's reduced degrees of freedom, we perform large-scale simulations of turbulent active suspensions in two and three dimensions.

Lattice Boltzmann simulations of plasma wakefield acceleration

We explore a novel simulation route for Plasma Wakefield Acceleration (PWFA) by using the computational method known as the Lattice Boltzmann Method (LBM). LBM is based on a discretization of the continuum kinetic theory while assuring the convergence toward hydrodynamics for coarse-grained fields (i.e., density, velocity, etc.). LBM is an established numerical analysis tool in computational fluid dynamics, able to efficiently bridge between kinetic theory and hydrodynamics, but its application in the context of PWFA has never been investigated so far. This paper takes a step forward to fill this gap. Results of LBM simulations for PWFA are discussed and compared with those of a code (Architect) implementing a Cold Fluid (CF) model for the plasma. In the hydrodynamic framework, we discuss the importance of regularization effects related to diffusion properties intrinsic of the LBM, allowing to go beyond the CF approximations. Issues on computational efficiency are also addressed.

Spontaneous directional flow of active magnetic particles

We predict the emergence of large-scale polar order and spontaneous directional flows in a class of self-propelled autonomous particles that interact via passive repulsion between off-center sites. The coupling of active motion with the passive torque acting about the particle centers results in hybrid active-passive interactions responsible for a macroscopic phase transition from an isotropic state to a polar-aligned state in systems of particles with front interaction sites. We employ a continuum kinetic theory to explain that the emergence of long-ranged orientational order, which occurs in unbounded domains at finite densities, can be externally activated independently of the self-propulsion mechanism and drives a macroscopic particle flow in a direction selected by symmetry breaking.

Microfluidic flow actuation using magnetoactive suspensions

The rheological behavior of magnetotactic bacterial suspensions is analyzed using a continuum kinetic theory. In both unbounded and confined geometries, the response of these suspensions under simple external flows can be controlled by applying a magnetic field and hinges in a subtle way on the interplay of magnetic alignment, rotation under shear, and wall-induced accumulation under confinement. By tuning magnetic field strength and direction, the apparent viscosity can either be enhanced or reduced, and the mechanisms for these trends are elucidated. In the absence of any applied flow, we further demonstrate the ability of magnetoactive suspensions to internally drive steady unidirectional flows upon application of a magnetic field, thus suggesting novel avenues for the design of microfluidic pumps and flow actuation devices.

A Micro-Macro Framework for Analyzing Steric and Hydrodynamic Interactions in Gliding Assays

Macroscopic flows of filament-motor mixtures, driven by the hydrolysis of ATP, are important to many cellular processes such as cytoplasmic streaming in Drosophila oocytes and cortical flow in the first cell division of C. elegans. Gliding assays, reduced in vitro model systems where motor proteins adsorbed onto a planar substrate bind to and move filaments, recreate large-scale dynamic patterns like coherent swarming motion and density waves. These systems are sensitive to the microscopic behavior such as motor protein binding and unbinding dynamics, which take place on a faster timescale than the direct and fluid-mediated filament interactions. In this work, we present a multiscale modeling and simulation framework for gliding assays that allows detailed microscopic motor modeling as well as both steric and hydrodynamic interactions between filaments. Our model is based on continuum kinetic theory, and our implementation utilizes CPU and GPU parallelism to track the sparse but high-dimensional state spa...

Lattice Boltzmann based simulation of gas flow regimes in low permeability porous media: Klinkenberg’s region and beyond

Tight gas reservoirs are of prominent importance among unconventional petroleum reservoirs. Due to the complicated flow dynamics of gas in tight porous media and compressibility effects, corrections have to be applied to the Darcy equation so that it can properly predict gas flow dynamics. As the mean pore diameter of tight rocks (e.g. shales) decreases, the ratio of the mean free path to the pore diameter, known as the non-dimensional Knudsen number, rises, the gas rarefaction effects becomes important, and the condition of no-slip on the pore surface is violated. Proper treatments of these complexities have been attempted by using classical continuum fluid mechanics and kinetic theory of gases approaches. The most well-known slip condition correction was presented by Klinkenberg in 1941. Many researchers have tried to correct and extend Klinkenberg's model beyond the slip flow regime. In this research the slip condition of the rarefied gas in low permeability simple porous media is studied by the lattice Boltzmann method and new corrections to Klinkenberg's coefficients are investigated. These corrections allow the extension of Klinkenberg model to higher ranges of Knudsen numbers. All the simulations in this work were performed by codes developed by the authors.

Short-Lived Lattice Quasiparticles for Strongly Interacting Fluids

It is shown that lattice kinetic theory based on short-lived quasiparticles proves very effective in simulating the complex dynamics of strongly interacting fluids (SIF). In particular, it is pointed out that the shear viscosity of lattice fluids is the sum of two contributions, one due to the usual interactions between particles (collision viscosity) and the other due to the interaction with the discrete lattice (propagation viscosity). Since the latter is negative, the sum may turn out to be orders of magnitude smaller than each of the two contributions separately, thus providing a mechanism to access SIF regimes at ordinary values of the collisional viscosity. This concept, as applied to quantum superfluids in one-dimensional optical lattices, is shown to reproduce shear viscosities consistent with the AdS-CFT holographic bound on the viscosity/entropy ratio. This shows that lattice kinetic theory continues to hold for strongly coupled hydrodynamic regimes where continuum kinetic theory may no longer be applicable.

Charging dynamics of supercapacitors with narrow cylindrical nanopores

We present a coarse-grained, continuum kinetic theory for charging supercapacitors with narrow cylindrical nanopores. The theory reveals that the occupancy of a nonpolarized pore and the energy barrier for ion–ion interdiffusion are the key issues controlling the different regimes of dynamic response. For ‘ionophobic’ pores, where the pore is empty at no applied voltage, charge density advances into the pore via diffusion-like dynamics. The mechanism of charging an ‘ionophilic’ pore is starkly different: for moderate ionophilicities, co-ions are expelled from the pore in a front-like manner, with significant ‘congestion’ at the pore entrance predicted for strong ionophilicity. We thus show that pore ionophilicity is detrimental to the speed of charging/discharging cycles, whereas making pores more ionophobic can substantially accelerate charging and cyclic recharging.

Flow of granular materials with slip boundary condition: A continuum–kinetic theory approach

We study the steady fully developed flow of granular materials between two horizontal flat plates, subject to slip at the walls. The constitutive model for the stress tensor is based on ideas in continuum mechanics and kinetic theory. The constitutive equation used in our study is a model proposed by Rajagopal et al. (1994) [24], and the material properties such as viscosity and the normal stress coefficients are derived using the kinetic theory approximation proposed by Boyle and Massoudi (1990) [2] which includes the effect of the gradient of volume fraction. The slip boundary condition is based on the particle dynamics simulation results of Rosato and Kim (1994) [30]. The governing equations are non-dimensionalized, and the resulting system of non-linear differential equations is solved numerically. The results for the velocity profiles and the volume fraction profiles are presented.

Sublimation, chemical decomposition, and melting inside an elastoplastic material: General continuum thermodynamic and kinetic theory

General thermodynamic and kinetic approaches for sublimation inside an elastoplastic material are developed for large strains. Various conceptual problems related to the effect of irreversible plastic deformation and dissipation, path-dependence of the appearance of a critical nucleus, and the presence of large strains are considered. Two transformation paths are studied: nucleation via homogeneous transformation in the nucleus of fixed mass and nucleation via continuous interface propagation. For both paths, the expressions for the thermodynamic driving forces and activation energies are derived. The activation energy is equal to the negative driving force for the appearance of a nucleus maximized with respect to nucleus mass and minimized over the nucleus shape, transformation path, and position. This definition corresponds to the principle of the minimum of transformation time and reduces to the traditional one in the limit of elastic materials. An Arrhenius-type kinetic equation for nucleation time and kinetic nucleation criterion are formulated. Algorithms for the determination of the critical nucleus are suggested. After appearance of the nucleus via homogeneous transformation, the possibility of its growth should be checked. Growth may occur by further sublimation or by mechanical expansion without phase transformation due to mechanical instability. Because the driving force for forward and reverse transformations maybe different, several scenarios are possible. The nucleus can grow, disappear, or be arrested; in the last case, it represents a stable rather than a critical nucleus. It is demonstrated that with small modifications, our approach to sublimation can be applied to chemical decomposition and melting inside an elastoplastic material. In the accompanying paper (Levitas and Altukhova, 2012) we will apply the developed theory to nucleation of a spherical gas bubble inside an elastoplastic material.

Derivation and thermodynamics of a lattice Boltzmann model with soluble amphiphilic surfactant

We derive a lattice Boltzmann model of a ternary fluid mixture, one component of which consists of amphiphilic molecules with a rotational degree of freedom. The model was first introduced elsewhere where it was constructed on the basis of the earlier Shan-Chen type lattice Boltzmann model for binary fluids. We provide a rigorous derivation of the model from an underlying continuum kinetic theory. In particular, we show how the model can be interpreted as a discretization of a Vlasov-Boltzmann type kinetic theory for a fluid composed of two species of oppositely charged monopoles and a composite, dipolar molecule. We also derive a free-energy functional for the model, including the contribution from the amphiphiles, and confirm that the free energy is lowered during phase separation and during the formation of a stable emulsion.

Multicomponent lattice Boltzmann model from continuum kinetic theory

We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermolecular interaction. The resulting model is found to be consistent with the model previously derived from a lattice-gas cellular automaton [X. Shan and H. Chen, Phys. Rev. E 47, 1815 (1993)] but applies in a much broader domain. A number of important insights are gained from the kinetic theory perspective. First, it is shown that even in the isothermal case, the energy equipartition principle dictates the form of the equilibrium distribution function. Second, thermal diffusion is shown to exist and the corresponding diffusivities are given in terms of macroscopic parameters. Third, the ordinary diffusion is shown to satisfy the Maxwell-Stefan equation at the ideal-gas limit.

Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria

Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non-ideal equation of state.

Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations.

Lattice Boltzmann equations for the isothermal Navier-Stokes equations have been constructed systematically using a truncated moment expansion of the equilibrium distribution function from continuum kinetic theory. Applied to the shallow water equations, with its different equation of state, the same approach yields discrete equilibria that are subject to a grid scale computational instability. Different and stable equilibria were previously constructed by Salmon [J. Marine Res. 57, 503 (1999)]. The two sets of equilibria differ through a nonhydrodynamic or "ghost" mode that has no direct effect on the hydrodynamic behavior derived in the slowly varying limit. However, Salmon's equilibria eliminate a coupling between hydrodynamic and ghost modes, one that leads to instability with a growth rate increasing with wave number. Previous work has usually assumed that truncated moment expansions lead to stable schemes. Such instabilities have implications for lattice Boltzmann equations that simulate other nonideal equations of state, or that simulate fully compressible, nonisothermal fluids using additional particles.

A continuum–kinetic theory approach to the rapid flow of granular materials: the effects of volume fraction gradient

We study the fully developed flow of granular materials down an inclined plane. The model uses ideas from both continuum and kinetic theories, where the form of the constitutive relation for the stress tensor is derived using standard techniques in continuum mechanics (cf. Rajagopal and Massoudi, DOE Report, DOE/PETC/TR-90/3, 1990); the material properties are derived using the kinetic theory approximation proposed by Boyle and Massoudi (Int. J. Eng. Sci. 28 (1990) 1261–1275), which includes the effects of the gradient of the volume fraction. The equations are non-dimensionalized and the system of non-linear ordinary differential equations is solved numerically.

Correlations and renormalization in lattice gases.

An exact kinetic theory is formulated for lattice gases. This kinetic theory makes possible the calculation of corrections to the usual combined Boltzmann and Chapman-Enskog analysis of lattice gases due to the buildup of correlations. Cluster expansion methods are used to show that renormalized transport coefficients for lattice gases satisfying semidetailed balance can be calculated perturbatively by summing terms in an infinite series. A diagrammatic notation for the terms in this series is given, in analogy with the diagrammatic expansions of continuum kinetic theory and quantum field theory. A closed-form expression for the coefficients associated with the vertices of these diagrams is given. This method is applied to several standard lattice gases, and the results are shown to correctly predict observed deviations from the Boltzmann analysis.