Non-equilibrium thermodynamics of interfacial systems

The theory of non-equilibrium thermodynamics is applied to a multi-component system containing an interface using a method developed by Bedeaux, Albano, and Mazur. A singular mass density is allowed at the interface as well as singular densities of energy and entropy. All currents are also allowed to be singular at the interface. The conservation laws and the Gibbs relation for the entropy are used to derive the entropy production at the surface. Linear laws relating the fluxes and thermodynamic forces are presented. The theory is then applied to a two component system where one of the phases is a liquid and the other a low density gas and the boundary conditions at the free surface of the liquid derived. The boundary conditions include the conditions used by Levich in his theory of the damping of waves by surface-active substances, but include other effects as well.     

Generalized laws of reflection and refraction from real valued boundary conditions

The generalized Snell's laws are usually derived from phase-matching condition by using harmonic inhomogeneous plane waves. The inhomogeneity makes it difficult to trace curves of energy flow. Here we show that, at a lossy interface, real valued boundary conditions are valid universally. Thus a time-dependent way is developed to directly derive generalized laws of reflection and refraction from real valued boundary conditions by using harmonic homogeneous plane waves. Finally, several novel properties of transmitted wave associated with energy losses are predicted numerically, which may be applied to experimentally test our theoretical analysis.     

Boundary conditions and non-equilibrium thermodynamics

The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface. A singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account. Furthermore the momentum and the heat currents are allowed to be singular at the interface. Using the conservation laws and the Gibbs' relation for the surface, an expression for the singular entropy production density at the interface is obtained. The linear phenomenological laws between fluxes and thermodynamic forces occurring in this singular entropy production density are given. Some of these linear laws are boundary conditions for the solution of the differential equations governing the evolution of the state variables in the bulk.     

Quasi-steady-state modeling of dendritic growth

A new method of the moving-boundary problem analysis is developed. After proposed coordinate transformation, the quasi-steady-state differential equation was reduced to equivalent Laplace equation. Transformed boundary conditions of a rescaled field distribution reflect the presence of rate-dependent sources on interface. Taking into account the local rate-dependence of the Neumann's boundary conditions, non-equilibrium pattern formation was considered. The problem of dendrite fractal growth was reduced to one of interaction of conformal to tips charged particles. Conditions of the quasi-steady growth and the recurrence formula for k-order dendrite spacing were derived. Theoretically obtained scaling laws for interface shape, dendrite spacing, critical sidebranch distance are confirmed by available experimental data.     

Non-equilibrium thermodynamics of interfaces including electromagnetic effects

The application of non-equilibrium thermodynamics to a system consisting of two bulk phases and their interface is extended to include electromagnetic effects. The interface is assumed to carry singular mass, energy, and entropy densities as well as singular electric charge, electric current, polarization and magnetization. Electric and magnetic fields are allowed to be discontinuous across the interface, but not singular.Maxwell's equations are used to derive relationships among electromagnetic quantities on the surface and boundary conditions for the bulk phases. An expression for the surface entropy production including electromagnetic effects is obtained, and the resulting linear laws relating thermodynamic forces and fluxes are investigated.     

Exact boundary conditions for the initial value problem of convex conservation laws

The initial value problem of convex conservation laws, which includes the famous Burgers’ (inviscid) equation, plays an important rule not only in theoretical analysis for conservation laws, but also in numerical computations for various numerical methods. For example, the initial value problem of the Burgers’ equation is one of the most popular benchmarks in testing various numerical methods. But in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support, so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations. In this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries, which contain a finite computational domain, by using the Lax’s exact formulas for the convex conservation laws. The well-posedness of the initial-boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described. Numerical tests with the proposed artificial boundary conditions are carried out by using the Lax–Friedrichs monotone difference schemes.     

Interface Shear Stress Analysis of Bond FRP Tendon Anchorage under Different Boundary Conditions

This article describes a study of an analytic interfacial stresses solution of FRP bond tendon anchorage, under different boundary conditions. The analytic solution was obtained with the cohesive zone model (CZM): the concept of the minimum relative interface displacement s _mis introduced and used as the fundamental variable to express all other parameters. The presented analytic solution agrees well with the result of experiment and that of finite element analysis (FEA). Furthermore, the interface shear stress distributions under two kinds of boundary conditions are discussed. It is indicated that the boundary conditions affected distribution of interfacial stress greatly. Under different boundary conditions, at the same load level, the peak interface shear stress corresponding to the first boundary condition is smaller than it is corresponding to the second boundary condition. The FRP tendon anchor under the first boundary condition can alleviate the peak bond stress, resulting in better uniformity in bond stress distribution.     

Interface model for non-equilibrium evaporation

A microscopic interface condition for condensing/evaporating interfaces is developed by combining a velocity dependent condensation probability [T. Tsuruta, H. Tanaka, T. Masuoka, Int. J. Heat Mass Transfer 42 (1999) 4107] and Maxwell type interface conditions with accommodation. Using methods from kinetic theory, macroscopic interface conditions for mass and energy transport across the phase boundary are derived. This model only applies to simple substances, where diffusive effects in the bulk phases are not present. The results are compared to classical non-equilibrium thermodynamics. The interface conditions are considered for the limit of small deviation from equilibrium, and the corresponding Onsager coefficients are computed. These results are useful as boundary conditions for non-equilibrium evaporation and condensation problems, as done previously by our group [M. Bond, H. Struchtrup, Phys. Rev. E 70 (2004) 061605].     

探地雷达数值模拟中广义完全匹配层吸收边界条件的模拟分析

探讨探地雷达正演模拟中,应用广义完全匹配层(GPML)吸收边界条件时的模拟效果.利用时域有限差分(FDTD)法将麦克斯韦方程离散化,再采用Mur超吸收边界条件和GPML吸收边界条件吸收向外的电磁波,模拟充水岩溶的雷达图像;同时,采用GPML吸收边界条件模拟多个圆柱体异常模型和倾斜界面异常模型的探地雷达剖面图像.对模拟结果的分析可知,GPML吸收边界条件可有效地吸收向外的电磁波,其吸收效果优于Mur超吸收边界条件,在探地雷达的应用中可推广应用.     

Initial Boundary Value Problem for Conservation Laws

This paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions. The main assumption is that the system under study admits a convex entropy extension. It is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interior solutions are generated by Godunov schemes, any Lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values. This in particular gives a trace formula for the flux functions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws. It is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general. The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. For example, we give example which show that Godunov schemes produce numerical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleinik entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of × quadratic system with a uniformly characteristic boundary condition. Received: 23 July 1996 / Accepted: 28 October 1996     

Effects of surface and interface energies on the bending behavior of nanoscale multilayered beams

A modified continuum model of the nanoscale multilayered beams is established by incorporating surface and interface energies. Through the principle of minimum potential energy, the governing equations and boundary conditions are obtained. The closed-form solutions are presented and the overall Young's modulus of the beam is studied. The surface and interface energies are found to have a major influence on the bending behavior and the overall Young's modulus of the beam. The effect of surface and interface energies on the overall Young's modulus depends on the boundary condition of the beam, the values of the surface/interface elasticity constants and the initial surface/interface energy of the system. The results can be used to guide the determinations of the surface/interface elasticity properties and the initial surface/interface energies of the nanoscale multilayered materials through nanoscale beam bending experiments.     

Renormalization of boundary conditions for distribution functions of quasiparticles obeying quantum statistics at interfaces between crystalline grains

Boundary conditions for distribution functions of quasiparticles scattered by an interface between two crystalline grains are presented. In contradistinction to former formulations where the Maxwell-Boltzmann statistics was considered, the present boundary conditions take into account the quantum (Fermi-Dirac or Bose-Einstein) statistics of the quasiparticles. Provided that small deviations from the thermodynamical equilibrium only are present, the boundary conditions are linearized, and then their “renormalization” is investigated in case of the elastic scattering. The final results of the renormalization, which are obtained for a simplified model of an interface, suggest that the portion of the Fermi (Bose)-quasiparticles reflected or transmitted specularly is decreased (increased) in comparison with the case of quasiparticles obeying the Maxwell-Boltzmann statistics.     

The Linear KdV Equation with an Interface

The interface problem for the linear Korteweg–de Vries (KdV) equation in one-dimensional piecewise homogeneous domains is examined by constructing an explicit solution in each domain. The location of the interface is known and a number of compatibility conditions at the boundary are imposed. We provide an explicit characterization of sufficient interface conditions for the construction of a solution using Fokas’s Unified Transform Method. The problem and the method considered here extend that of earlier papers to problems with more than two spatial derivatives.     

Thermodynamic basis for a variational model for crystal growth

Variational models provide an alternative approach to standard sharp interface models for calculating the motion of phase boundaries during solidification. We present a correspondence between objective functions used in variational simulations and specific thermodynamic functions. We demonstrate that variational models with the proposed identification of variables are consistent with nonequilibrium thermodynamics. Variational models are derived for solidification of a pure material and then generalized to obtain a model for solidification of a binary alloy. Conservation laws for internal energy and chemical species and the law of local entropy production are expressed in integral form and used to develop variational principles in which a "free energy," which includes an interfacial contribution, is shown to be a decreasing function of time. This free energy takes on its minimum value over any short time interval, subject to the laws of conservation of internal energy and chemical species. A variational simulation based on this model is described, and shown for small time intervals to provide the Gibbs-Thomson boundary condition at the solid-liquid interface.     

Elastic Wave Reflection at the Interface at Preset Stresses

Russian Physics Journal - The problem of elastic wave reflection at the interface at preset constant stresses is discussed. Under the conditions where the reflection laws are fulfilled, analytical...     

Plane-parallel nonisothermal gas filtration: The role of thermodynamics

The influence of the mathematical model parameters and the type of boundary condition on the dynamics of temperature and pressure fields in non-isothermal gas filtration is studied in a computational experiment. The process is described by a nonlinear system of partial differential equations, which is derived from the laws of mass and energy conservation, and the Darcy law, the closure relationships are physical and caloric equations of state. The boundary conditions correspond to gas injection for the given mass flow rate with varying intensity.     

The wave flow effect on a plane boundary between vacuum and an optical medium with a quasi-zero refractive index

A boundary problem in which a plane electromagnetic wave is reflected and refracted at a plane boundary of a semi-infinite optical medium with a quasi-zero refractive index has been solved. Such a medium has a random refractive index taking values in an interval from zero to some finite value less than unity. It means that the concept of a sharp interface between two media loses its meaning, the boundary of the medium becomes inhomogeneous, and laws of reflection and refraction of light become non-Fresnelian. Formulas for non-Fresnelian amplitudes of reflection and refraction have been derived. It is shown that a surface wave arises near the boundary of a medium with a quasi-zero refractive index. The wave propagates both on the inside and outside of the boundary.     

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

     

Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming

We present a 2D- and 3D-lattice Boltzmann model for the treatment of free surface flows including gas diffusion. Interface advection and related boundary conditions are based on the idea of the lattice Boltzmann equation. The fluid dynamic boundary conditions are approximated by using the mass and momentum fluxes across the interface, which do not require explicit calculation of gradients. A similar procedure is applied to fulfill the diffusion boundary condition. Simple verification tests demonstrate the correctness of the algorithms. 2D- and 3D-foam evolution examples demonstrate the potential of the method.     

基于求解Riemann问题的界面处理方法

通过在界面处构造Riemann问题,根据流体的法向速度和压力在界面(接触间断)处连续的特性,利用Riemann问题的解不仅定义了ghost流体的值,而且对真实流体中邻近界面的点值进行了更新,使得在界面处的流体的状态满足接触间断的性质,给出了更加精确的界面边界条件,守恒误差分析表明该方法在界面计算过程中引入较小的误差.数值试验表明该方法能准确地捕捉界面和激波的位置.