Radiative shock solutions in the equilibrium diffusion limit

A semi-analytic solution is described for planar radiative shock waves in the equilibrium diffusion (1−T) limit. The solution requires finding numerically the root of a polynomial and integrating a nonlinear ordinary differential equation. This solution may be used as a test problem to verify computer codes that use the equilibrium–diffusion radiation model, or for more advanced radiation models in the optically-thick limit. The structure of the shock profiles is also discussed, including new accurate estimates on the conditions for continuous solutions. We also discuss how the Zel’dovich spike may be estimated from the equilibrium diffusion solution. Finally, results from a computer code are shown to compare well with a semi-analytic solution.      

子域精细积分及偏微分方程数值解

对于偏微分方程半解析法的方程，精细时程积分虽然能求出高度准确的解，但往往面临矩阵尺度太大的困难；另一方面差分法虽然有带宽小的优点，但有稳定性及精度方面的问题．本文提出子域精细积分法，既可利用精细积分的数值优点，又有带宽小的好处．数值例题表明了子域精细积分法的效能．     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

Nonequilibrium kinetics of the gas-phase reactions of high-velocity molecules: Description by perturbation methods and simulation by statistical methods

Data on the numerical solution of a system of kinetic Boltzmann equations for a homogeneous multicomponent mixture of reacting gases with molecules of different “colors” that change in the reactions are given. The solution is obtained using a well-known version of the direct statistical simulation (Monte-Carlo) method, namely, the majorant frequency method, under conditions when the molecules belonging to the high-velocity “tails” of the corresponding distribution functions enter into the color change reaction. The properties of the numerical solution are compared with solutions obtained within the framework of the usual perturbation methods. It is shown that to obtain correct solutions over the range of threshold molecular velocities it is necessary to modify substantially the procedure of the perturbation method, while the traditional approach can be used only on the range of thermal particle velocities. Earlier, this was definitely established for distributions of the reacting molecules over their internal degrees of freedom and for the distributions of reactant-molecules participating in a high-threshold reaction. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 176–184, May–June, 2000. The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 96-01-00573) and the Government Program for Leading Science Schools (grant No. 96-15-9603).     

Computer algebra-perturbation solution to a nonlinear wave equation

In this paper, the higher-order asymptotic solution to the Cauchy problem of a nonlinear wave equation is found by using a computer algebra-perturbation method. The secular terms in the solution from straightforward expansions are eliminated with the straining of characteristic, coordinates and the use of the renormalization technique, and the four-term uniformly valid solution is obtained with the symbolic computation by using a computer algebra system. The comparison of the derived asymptotic solution and the numerical solution shows that they coincide with each other for smaller ε and agree quite well for larger ε (e. g., ε=0.25) Project supported by the National Natural Science Foundation of China and Shanghai Municiple Natural Science Foundation     

Load transfer in fibre-reinforced composites with viscoelastic matrix: an analytical study

In A fibre-reinforced 2D composite material with elastic fibres and viscoelastic, isotropic matrix is studied. Starting from the solution of a reference-problem with elastic matrix material the elastic matrix parameters are substituted by their viscoelastic correspondents in the Laplace domain. For simplification the time-dependent solution is approximated by using limiting value theorems that give information about the time-dependent solution for t → 0 and t → ∞. Then the method of asymptotically equivalent functions is used and illustrated with examples of a steel fibre in a PMMA matrix. The analytical solutions are compared with their numerical counterparts. In summary it can be stated that this paper is a further contribution to the vast literature about the application of the correspondence principle to the solution of special problems of the linear viscoelasticity.     

A Mathematical Model and Analytical Solution for the Fixation of Bacteria in Biogrout

Biogrout is a new method for soil reinforcement, which is based on microbial-induced carbonate precipitation. Bacteria and reactants are flushed through the soil, resulting in calcium carbonate precipitation and consequent soil reinforcement. Bacteria are crucially important in the Biogrout process since they catalyse the reaction. Hence, to control the process, it is essential to know where the bacteria are located. The bacteria are possibly in suspension but can also be adsorbed or fixated on the matrix of the porous structure. In this article, a model is derived for the placement of bacteria. The model contains three phases of bacteria: bacteria in suspension, adsorbed bacteria and fixed bacteria. An analytical solution is derived for instantaneous reactions between these three phases. The analytical solution is compared to numerical simulations for finite reaction rates. For the numerical simulations the standard Galerkin Finite Element Method is used.     

An extension of CVDV model to Zeldovich exchange reactions for hypersonic non-equilibrium air flows

A new preferential vibration-dissociation-exchange reactions coupling model – labelled CVDEV – resulting from an extension of the well-known Treanor and Marrone CVDV model, has been derived to take into account the coupling between the vibrational excitation of the and molecules and the two Zeldovich exchange reactions. Analytical expressions for the exchange reactions coupling factor and for the average vibrational energy lost – or gained – by a molecule through an exchange reaction have been developed. The influence of such a coupling has been shown by means of numerical simulations of hypersonic air flows through normal and bow shock waves. Code-to-code comparisons between our model and other recent approaches have been conducted. The infrared radiation of nitric oxide behind a normal shock wave resulting from computations with the CVDEV model has been compared with other coupling model results and to recent shock tube experimental data. These comparisons have shown a good agreement of our model results with the experimental data. In this context, the results show the prominent influence of vibration coupling on the first Zeldovich reaction, and the absence of vibration coupling effects on the second Zeldovich reaction. Received 30 June 1997 / Accepted 3 December 1997     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

There has been a dramatic increase in the number of research publications using the population balance equation (PBE). The PBE allows the prediction of the spatial distribution of the dispersed phase size for an accurate estimation of the flow fields in multiphase flows. A few recent studies have proposed new efficient numerical methods to solve non-homogeneous multivariate PBE and implemented the same in computational fluid dynamics (CFD) codes. However, these codes are generally benchmarked against other numerical methods and applied without verification. To address this gap, an analytical solution for a three-dimensional non-homogeneous bivariate PBE is presented here for the first time. The method of manufactured solutions (MMS) has been used to construct a solution of the non-homogeneous PBE containing breakage and coalescence terms, and an additional source term appearing as a result of this method. The analytical solution presented in this work can be used for the rigorous verification of computer codes written to solve the non-homogeneous bivariate PBE. Quantification of the errors due to different numerical schemes will also become possible with the availability of this analytical solution for the PBE.     

Material forces and phase transitions in elasticity

A computational scheme for the determination of the interface in a strain-induced phase-transition problem for an elastic bar is proposed. The algorithm is based on the material force notion and more specifically on the simultaneous solution of equilibrium equations for the physical and material forces. The weak form of both equations is derived with the aid of a variational principle that accounts for the variations of the dependent and the independent variables. The whole scheme concludes in a nonlinear algebraic system which is numerically solved by the Newton method. The numerical results thus derived seem to be quite encouraging for further application of the concept of material forces in computations related to phase transition problems. The austenite–martensite transformation could be a possible application of the proposed model.     

Hyperbolicity of Velocity-Stress Equations for Waves in Anisotropic Elastic Solids

This paper reports mathematical properties of the three-dimensional, first-order, velocity-stress equations for propagating waves in anisotropic, linear elastic solids. The velocity-stress equations are useful for numerical solution. The original equations include the equation of motion and the elasticity relation differentiated by time. The result is a set of nine, first-order partial differential equations (PDEs) of which the velocity and stress components are the unknowns. Cast into a vector-matrix form, the equations can be characterized by three Jacobian matrices. Hyperbolicity of the equations is formally proved by analyzing (i) the spectrum of a linear combination of the three Jacobian matrices, and (ii) the eigenvector matrix for diagonalizing the linearly combined Jacobian matrices. In the three-dimensional space, linearly combined Jacobian matrices are shown to be connected to the classic Christoffel matrix, leading to a simpler derivation for the eigenvalues and eigenvectors. The results in the present paper provide critical information for applying modern numerical methods, originally developed for solving conservation laws, to elastodynamics.     

Stability and implementable ℋ<Subscript>∞</Subscript> filters for singular systems with nonlinear perturbations

In this paper, we investigate the problem of designing ℋ_∞ filter for a class of continuous-time uncertain singular systems with nonlinear perturbations, which can be realized in practice. The perturbation is a time-varying function of the system state and satisfies a Lipschitz constraint. The design objective is to guarantee that a prescribed upper bound on an ℋ_∞ performance of the robust filter is attained for all possible energy-bounded input disturbances and all admissible uncertainties and which can be implemented on-line to get a good replica of the state. We first establish sufficient condition for the existence and uniqueness of solution to the singular system connected with the normal filter. Using a linear matrix inequality (LMI) format, we then provide a sufficient condition for the asymptotic stability of the realizable ℋ_∞ filter. Then by means of a convex analysis procedure the filter gain matrices are derived and an important special case is readily deduced. Finally, a numerical example is presented to illustrate the theoretical developments.     

The Dual-Reciprocity Boundary Element Analysis for Hydraulically Fractured Shale Gas Reservoirs Considering Diffusion and Sorption Kinetics

This work presents a new application of boundary element method (BEM) to model fluid transport in unconventional shale gas reservoirs with discrete hydraulic fractures considering diffusion, sorption kinetics and sorbed-phase surface diffusion. The fluid transport model consists of two governing partial differential equations (PDEs) written in terms of effective diffusivities for free and sorbed gases, respectively. Boundary integral formulations are analytically derived using the fundamental solution of the Laplace equation for the governing PDEs and Green’s second identity. The domain integrals arising due to the time-dependent function and nonlinear terms are transformed into boundary integrals employing the dual-reciprocity method. This transformation retains the domain-integral-free, boundary-integral-only character of standard BEM approaches. In the proposed solution, the free- and sorbed-gas flow in the shale matrix is solved simultaneously after coupling the fracture flow equation of free gas. Well production performance under the effect of relaxation phenomenon due to delayed responses of sorbed gas under nonequilibrium sorption condition is rigorously captured by imposing the zero-flux condition at fracture–matrix interface for the sorbed-gas transport equation. The validity of proposed solution is verified using several case studies through comparison against a commercial finite-element numerical simulator.

     

Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach

A set of linearized 26 moment equations, along with their wall boundary conditions, are derived and used to study low-speed gas flows dominated by Knudsen layers. Analytical solutions are obtained for Kramers’ defect velocity and the velocity-slip coefficient. These results are compared to the numerical solution of the BGK kinetic equation. From the analysis, a new effective viscosity model for the Navier–Stokes equations is proposed. In addition, an analytical expression for the velocity field in planar pressure-driven Poiseuille flow is derived. The mass flow rate obtained from integrating the velocity profile shows good agreement with the results from the numerical solution of the linearized Boltzmann equation. These results are good for Knudsen numbers up to 3 and for a wide range of accommodation coefficients. The Knudsen minimum phenomenon is also well captured by the present linearized 26-moment equations.     

Frictional Hertzian contact problems under cyclic loading using static reduction

In this study we investigate an axisymmetric Hertzian contact problem of a rigid sphere pressing into an elastic half-space under cyclic loading. A numerical solution is sought to obtain a steady state, which demands a large amount of computer memory and computing speed. To achieve a tractable problem, the current numerical model uses a “static reduction” technique, which employs only a contact stiffness matrix rather than the entire stiffness of the problem and is more accurate than the approach used by most finite element codes. Investigation of the tendency of contact behavior in the transient and steady states confirms that a steady state exists, showing converged energy dissipation. The dependence of dissipation on load amplitude shows a power law of load amplitude less than 3, which may explain some deviations in the experimental findings.     

Singularities in a Solution to a Rotating Orthotropic Disk with Temperature Gradient

A semi-analytical solution is obtained for a rotating stress-free edge disk of constant thickness and density. In the plastic range, the Hill’s quadratic orthotropic yield criterion is adopted. In the elastic range, the Hooke’s law holds with thermal effects included. The analysis of singularities performed may be used for correct implementation of numerical codes and preliminary engineering design.     

Natural vibration of a sandwich beam on an elastic foundation

The natural vibration of an elastic sandwich beam on an elastic foundation is studied. Bernoulli’s hypotheses are used to describe the kinematics of the face layers. The core layer is assumed to be stiff and compressible. The foundation reaction is described by Winkler’s model. The system of equilibrium equations is derived, and its exact solution for displacements is found. Numerical results are presented for a sandwich beam on an elastic foundation of low, medium, or high stiffness __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 57–63, May 2006.     

Finite element solution of multi-dimensional two-phase flow through porous media with arbitrary heating conditions

Mechanistic models for flow regime transitions and drag forces proposed in an earlier work are employed to predict two-phase flow characteristics in multi-dimensional porous layers. The numerical scheme calls for elimination of velocities in favor of pressure and void fraction. The momentum equations for vapor and liquid then can be reduced to a system of two partial differential equations (PDEs) which must be solved simultaneously for pressure and void fraction.

Solutions are obtained both in two-dimensional cartesian and in axi-symmetric coordinate systems. The porous layers in both cases are composed of regions with different permeabilities. The finite element method is employed by casting the PDEs in their equivalent variational forms. Two classes of boundary conditions (specified pressure and specified fluid fluxes) can be incorporated in the solution. Volumetric heating can be included as a source term. The numerical procedure is thus suitable for a wide variety of geometry and heating conditions. Numerical solutions are also compared with available experimental data.