Exact solutions of a two-fluid model of two-phase compressible flows with gravity

We aim at determining and computing a class of exact solutions of a two-fluid model of two-phase flows with/without gravity. The model is described by a non-hyperbolic system of balance laws whose characteristic fields may not be given explicitly, making it perhaps impossible to solve the Riemann problem. First, we investigate Riemann invariants in the linearly degenerate characteristic fields and obtain a surprising result on the corresponding contact waves of the model without gravity. Second, even when gravity is allowed, we show that smooth stationary solutions can be governed by a system of differential equations in divergence form, which determines jump relations for any stationary discontinuity wave. Using these relations, we establish a nonlinear equation for the pressure and propose a method to compute the pressure and then the equilibria resulted by a stationary wave.     

Asymptotic analysis and reaction-diffusion approximation forBGK kinetic models of chemical processes in multispecies gas mixtures

A BGK-type model is derived to describe the interaction between transport and chemical reactions in multispecies gas mixtures, at the kinetic level. The underlying kinetic process is modelled by a Fokker-Planck-type equation, in the Kramers-Smoluchowski limit. When the reaction terms in the kinetic equation are properly scaled, an expansion in powers of a small parameter related to the mean collison time yields a reaction-diffusion equation for the densities of the chemical species involved. For different scalings of the reaction terms, the related macroscopic equations describe the prevailing of transport processes on chemical reactions, orvice versa. The spatially homogeneous case with its own peculiarities is addressed, and the Selkov model is considered as an example.     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.     

高分辨KFVS有限体积方法及其CFD应用

1．引言文中研究三维Euler方程组的数值求解·儿1）中p,（。。,。。,。z）,p和E分别表示流体密度,流体速度矢量,压力和总能．方程组（1．1）是不封闭的,除非增加一个额外的方程一状态方程p一pk句,e表示单位质量内能．本文仅限于理想气体,此时状态方程为p一（、－1加e．队2）近H十年来,涌现了许多求解方程组（1．1）的无振荡、高分辨格式,例如TVD格式问,＊＊O格式问等,它们在一定程度上促进了航空航天和造船事业的发展．其中有一类根据双曲方程组（1．l）特征值的符号建立的迎风格式尤为突出,与中心格式相比,迎风格式的耗…     

Analysis of a kinetic cellular model for tumor-immune system interaction

Starting at a kinetic level from the equations for the evolution of dominance in populations of interacting organisms, and taking proliferative and destructive encounters into account, a simple model describing the competition between tumor cells and immune system is studied in some detail. Under reasonable assumptions, a closed set of macroscopic balance equations for macroscopic observables is derived by a moment procedure, and analyzed in the frame of the theory of dynamical systems. It is shown that a transcritical bifurcation of equilibria generates a region in the phase space in which, according to the model, the immune system defeats the tumor and leads to its depletion. Numerical results are presented and briefly commented on.     

Systematic Measures of Biological Networks I: Invariant Measures and Entropy

This paper is part I of a two‐part series devoted to the study of systematic measures in a complex biological network modeled by a system of ordinary differential equations. As the mathematical complement to our previous work with collaborators, the series aims at establishing a mathematical foundation for characterizing three important systematic measures: degeneracy, complexity, and robustness, in such a biological network and studying connections among them. To do so, we consider in part I stationary measures of a Fokker‐Planck equation generated from small white noise perturbations of a dissipative system of ordinary differential equations. Some estimations of concentration of stationary measures of the Fokker‐Planck equation in the vicinity of the global attractor are presented. The relationship between the differential entropy of stationary measures and the dimension of the global attractor is also given.© 2016 Wiley Periodicals, Inc.     

On Hashin–Shtrikman bounds for mixtures of two isotropic materials

In the context of stationary diffusion equation we calculate explicitly the optimal microstructure for the Hashin–Shtrikman energy bound in the case of two isotropic phases with prescribed ratio, in three dimensions. A similar, but more general problem arises in the study of optimal design in conductivity with multiple state equations. Here, the necessary condition of optimality leads to a finite-dimensional optimisation problem which extends the problem of Hashin–Shtrikman bounds, which can be solved explicitly, as well.These calculations have important applications to the optimality criteria method for numerical solution of optimal design problems with multiple state equations. In this iterative algorithm, the presented results enable one to calculate explicitly the update of design variables, similar to the problems with one state equation. Therefore, its implementation is simple, showing nice convergence results on a number of examples, two of them being demonstrated here.     

Kinetic Models for Chemotaxis and their Drift-Diffusion Limits

Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density.Present address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal     

Stability of rarefaction waves for the non-cutoff Vlasov–Poisson–Boltzmann system with physical boundary

In this paper, we are concerned with the Vlasov–Poisson–Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.     

KINETIC FLUX VECTOR SPLITTING FOR THE EULER EQUATIONS WITH GENERAL PRESSURE LAWS

This paper attempts to develop kinetic flux vector splitting(KFVS)for the Euler equa-tions with general pressure laws.It is well known that the gas distribution function forthe local equilibrium state plays an important role in the construction of the gas-kineticschemes.To recover the Euler equations with a general equation of state(EOS),a newlocal equilibrium distribution is introduced with two parameters of temperature approx-imation decided uniquely by macroscopic variables.Utilizing the well-known connectionthat the Euler equations of motion are the moments of the Boltzmann equation wheneverthe velocity distribution function is a local equilibrium state,a class of high resolutionMUSCL-type KFVS schemes are presented to approximate the Euler equations of gas dy-namics with a general EOS.The schemes are finally applied to several test problems for ageneral EOS.In comparison with the exact solutions,our schemes give correct location andmore accurate resolution of discontinuities.The extension of our idea to multidimensionalcase is natural.     

A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows

In this paper, an alternative approach to the traditional continuum analysis of flow problems is presented. The traditional methods, that have been popular with the CFD community in recent times, include potential flow, Euler and Navier–Stokes solvers. The method presented here involves solving the governing equation of the molecular gas dynamics that underlies the macroscopic behaviour described by the macroscopic governing equations. The equation solved is the Boltzmann kinetic equation in its simplified collisionless and BGK forms. The algorithm used is a discontinuous Taylor–Galerkin type and it is applied to the 2D problems of a highly rarefied gas expanding into a vacuum, flow over a vertical plate, rarefied hypersonic flow over a double ellipse, and subsonic and transonic flow over an aerofoil. The benefit of this type of solver is that it is not restricted to continuum regime (low Knudsen number) problems. However, it is a computationally expensive technique.     

Numerical simulation of biofilm growth in porous media

Biofilms are very important in controlling pollution in aquifers. The bacteria may either consume the contaminant or form biobarriers to limit its spread. In this paper we review the mathematical modeling of biofilm growth at the microscopic and macroscopic scales, together with a scale-up technique. At the pore-scale, we solve the Navier-Stokes equations for the flow, the advection-diffusion equation for the transport, together with equations for the biofilm growth. These results are scaled up using network model techniques, in order to have relations between the amount and distribution of the biomass, and macroscopic properties such as permeability and porosity. A macroscopic model is also presented. We give some results.     

A FLUID-PARTICLE MODEL WITH ELECTRIC FIELDS NEAR A LOCAL MAXWELLIAN WITH RAREFACTION WAVE

The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions, Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces. Precisely, based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system, which was first developed in [16], we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.     

A boundary value problem for a kinetic model describing electron flow in a semiconductor

A kinetic model describing the evolution of an electron gas in a semiconductor device is analysed. It arises from the Boltzmann equation by using a spherical harmonic expansion, and it involves difference‐partial differential equations. A boundary value problem is proposed and an existence and uniqueness theorem is proved for the stationary one‐dimensional case. A simple asymptotic model is derived and for this a maximum principle is shown. Copyright © 2000 John Wiley & Sons, Ltd.     

Stochastic integral equations applied to telecommunications traffic without delay

Modern telecommunication techniques case the problem of traffic handling in the framework of fairly general networks, as applied to traffic without delay but with virtually arbitrary service-time distributions. In this paper we use stochastic integral equations to deal with the case involving the most general input process and lost calls. For this purpose, Fortet's equation, unsolved so far in the general case, is solved to analyze the single trunk group model. The stationary case is then treated as a special case. Finally we study networks which satisfy a certain assumption of symmetry. The same general stochastic assumptions are maintained throughout the paper.     

Lattice-Boltzmann Models for heat transfer

A multispeed heat transfer lattice Boltzmann model is presented. The model possesses the perfect gas state equation with arbitrary special heat ratio. The macroscopic conservation equations are derived by the Chapman-Enskog method. The one dimensional simulation for the sinusoidal energy distributions are compared with the theoretical results, showing good agreement. The theoretical conductivity in the energy equation is in accordance with the simulations.     

Homogenization of long-range auxin transport in plant tissues

A model for the cell-to-cell transport of the plant hormone auxin is presented. Auxin is a weak acid which dissociates into ions in the aqueous cell compartments. A microscopic model is defined by diffusion-reaction equations and a Poisson equation for a given charge distribution. The microscopic properties of the plant cell were taken into account through oscillating coefficients in the model. Via formal asymptotic expansion a macroscopic model was obtained. The effective diffusion coefficients and transport velocities are expressed by the solution of unit cell problems. Published experimental values of diffusivity and permeability were used to determine numerically the effective transport coefficients and the calculated transport velocity was shown to be of the same order as measured values.     

Homogenization of Linear Transport Equations in a Stationary Ergodic Setting

We study the homogenization of a linear kinetic equation which models the evolution of the density of charged particles submitted to a highly oscillating electric field. The electric field and the initial density are assumed to be random and stationary. We identify the asymptotic microscopic and macroscopic profiles of the density, and we derive formulas for these profiles when the space dimension is equal to one.