An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. I. Flow problems

The ADO method, an analytical version of the discrete-ordinates method, is used to solve several classical problems in the rarefied gas dynamics field. The complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. Received: July 10, 2007; revised: October 29/December 4, 2007     

An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

The ADO method, an analytical version of the discrete-ordinates method, is used here to solve a heat-transfer problem in a rarefied gas confined in a channel, as well as to solve a half-space problem in order to evaluate the temperature jump at the wall. This work is an extension of a previous work, devoted to flow problems, where the complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. In particular, numerical results for heat-flow profile, temperature and density perturbations are obtained for channels (walls), defined by different materials, on which different temperatures are imposed.     

An analytical approach to the unified solution of kinetic equations in the rarefied gas dynamics. II. Heat transfer problems

The ADO method, an analytical version of the discrete-ordinates method, is used here to solve a heat-transfer problem in a rarefied gas confined in a channel, as well as to solve a half-space problem in order to evaluate the temperature jump at the wall. This work is an extension of a previous work, devoted to flow problems, where the complete development of the solution, which is analytical in terms of the spatial variable, is presented in a way, such that, a wide class of kinetic models are considered, in an unified approach. A series of numerical results are showed and different simulations are used in order to establish a general comparative analysis based on this consistent set of results provided by the same methodology. In particular, numerical results for heat-flow profile, temperature and density perturbations are obtained for channels (walls), defined by different materials, on which different temperatures are imposed.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

由纵向振动引起的质点与圆锥形杆碰撞问题的解析解

给出细长圆锥形的截面杆受到质点纵向碰撞时的精确解析解．提出了一种新方法用于分析质点-圆锥形杆碰撞,使用了叠加法给出杆的响应．其结果可验证数值解和其他解析解．所提出方法的优点之一是响应解的解析形式简洁．结论是质量比和一些描述杆几何形状的变量,如倾斜度、杆长和半径在撞击分析中具有重要作用．     

Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid

In this paper, Adomian’s decomposition method is used to solve non-linear differential equations which arise in fluid dynamics. We study basic flow problems of a third grade non-Newtonian fluid between two parallel plates separated by a finite distance. The technique of Adomian decomposition is successfully applied to study the problem of a non-Newtonian plane Couette flow, fully developed plane Poiseuille flow and plane Couette–Poiseuille flow. The results obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving non-linear ordinary differential equations using Chebyshev spectral method. We present a comparative study between the analytical solutions and numerical solutions. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the Adomian decomposition method.     

Approximate solutions to idealized structural dynamic optimization problems

This paper presents an application of two solution methods to the least-weight optimization of a simple sandwich beam with a frequency constraint. The first method, an adaptation of a numericalshooting technique used in optimal control, is found to give good results if unknown initial conditions can be reasonably approximated. The second method, a perturbation method used widely in theoretical mechanics and aerodynamics, yields approximate analytical expressions. These expressions can be used in turn to estimate starting values for the numerical technique. Converged numerical results are presented for a least-weight cantilever beam with fixed fundamental frequency and for a beam on simple supports with fixed fundamental frequency.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.     

对流扩散方程的QC3无网格法

对流扩散方程作为偏微分运动方程的分支,在流体力学、气体动力学等领域有着重要应用.为解决对流扩散方程难以通过解析法得到解析解的难题,采用二阶一致3点积分(Quadratically Consistent 3-Point Integration,简称QC3)提高无网格法的计算效率,通过对积分点上形函数导数的修正,改善无网格...     

A low-order meshless model for multidimensional heat conduction problems

In this paper, a perturbation method is used to solve a two-dimensional unsteady heat conduction problem. Low-order transfer functions are defined. Step responses are obtained and compared to the complete numerical solutions given by a meshless method. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the used method.     

An adaptive characteristics method for advective-diffusive transport

An adaptive characteristics method is presented for the solution of advective-diffusive groundwater transport problems. The method decomposes the transport processes into advective and diffusive transport components. Advective flows are defined by using a streamtube incrementing procedure, based on the method of characteristics, to define the position of advective front. Diffusive transport orthogonal to the front is represented by an array of propagating streamtube elements, with dimension determined from analytical solution of the one-dimensional diffusion equation. Adaptive time scaling is used to moderate the dimensions and aspect ratios of the advective and diffusive streamtube elements as appropriate to the dominant transport mechanism. Finite differences are used to solve for transport ahead of the advancing front. The distribution of streamtubes are predetermined from a direct boundary element algorithm. Comparison with analytical results for a one-dimensional transport geometry indicates agreement for Peclet numbers between zero and infinity. Solution for transport in two-dimensional domains illustrates excellent agreement for Peclet numbers from zero to 25.     

Approximate analytical solutions for Kolmogorov’s equations

This paper reports the explicit analytical solutions for Kolmogorov’s equations. Kolmogorov’s equations are commonly used to describe the structure of local isotropic turbulence, but their exact analytical solutions have not yet been found. In this paper, the closed-form solutions for two kinds of Kolmogorov’s equations are obtained. The derivations of the approximate solutions are based on the homotopy analysis method, which is a new tool for obtaining the approximate analytical solutions of both strong and weak nonlinear differential equations. To examine the validity of the approximate solutions, numerical comparisons between results from the homotopy analysis method and the fourth-order Runge-Kutta method are carried out. It is shown that the results are in good agreement.     

A small time solutions for the KdV equation using Bubnov-Galerkin finite element method

A Bubnov-Galerkin finite element method with quintic B-spline functions taken as element shape and weight functions is presented for the solution of the KdV equation. To demonstrate the accuracy, efficiency and reliability of the method three experiments are undertaken for both the evolution of a single solitary wave and the interaction of two solitary waves. The numerical results are compared with analytical solutions and the numerical results in the literature. It is shown that the method presented is accurate, efficient and can be used at small times when the analytical solution is not known.     

An approximate analytical solution for the stresses and strains in heterogeneous cubic materials

Continuous Fourier transforms (CFT) are used to derive analytical expressions for the stress and strain fields, respectively, in heterogeneous bodies consisting of cubic materials which additionally may be addressed to a uniform eigenstrain. The so‐called “equivalent inclusion method” (Mura 1987) builds the starting point of this analytical method. It allows to map the original problem onto an auxiliary problem, where as a simplification a homogeneous body is considered. Problems of this kind are effectively solved by means of the CFT. The application of this transformation results into an integral equation (IEQ) for the strains. For (cubic) anisotropic materials this equation can be further simplified by means of approximation techniques which have been demonstrated in [2,3]. For a particular geometry of the inhomogeneity it is illustrated how to derive a closed‐form solution of this approximated IEQ. This solution is compared with numerical results for different combinations of the matrix and the inhomogeneities.     

On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets

In the present work, unsteady MHD flow of a Maxwellian fluid above an impulsively stretched sheet is studied under the assumption that boundary layer approximation is applicable. The objective is to find an analytical solution which can be used to check the performance of computational codes in cases where such an analytical solution does not exist. A convenient similarity transformation has been found to reduce the equations into a single highly nonlinear PDE. Homotopy analysis method (HAM) will be used to find an explicit analytical solution for the PDE so obtained. The effects of magnetic parameter, elasticity number, and the time elapsed are studied on the flow characteristics.     

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.     

Solution of Poisson’s equation by analytical boundary element integration

The solution of Poisson’s equation is essential for many branches of science and engineering such as fluid-mechanics, geosciences, and electrostatics. Solution of two-dimensional Poisson’s equations is carried out by BEM based on Galerkin Vector Method in which the integrals that appear in the boundary element method are expressed by analytical integration. In this paper, the Galerkin vector method is developed for more general case than presented in the previous works. The integrals are computed for constant and linear elements in BEM. By employing analytical integration in BEM computation, the numerical schemes and coordinate transformations can be avoided. The presented method can also be used for the multiple domain case. The results of the analytical integration are employed in BEM code and the obtained analytical expression will be applied to several examples where the exact solution exists. The produced results are in good agreement with the exact solution.     

Analytical and numerical solutions to an electrohydrodynamic stability problem

A linear hydrodynamic stability problem corresponding to an electrohydrodynamic convection between two parallel walls is considered. The problem is an eighth order eigenvalue one supplied with hinged boundary conditions for the even derivatives up to sixth order. It is first solved by a direct analytical method. By variational arguments it is shown that its smallest eigenvalue is real and positive. The problem is cast into a second order differential system supplied only with Dirichlet boundary conditions. Then, two classes of methods are used to solve this formulation of the problem, namely, analytical methods (based on series of Chandrasekar-Galerkin type and of Budiansky-DiPrima type) and spectral methods (tau, Galerkin and collocation) based on Chebyshev and Legendre polynomials. For certain values of the physical parameters the numerically computed eigenvalues from the low part of the spectrum are displayed in a table. The Galerkin and collocation results are fairly closed and confirm the analytical results.     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.