A brief review of the homotopy analysis method

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.     

Homotopy Solutions for a Generalized Second-Grade Fluid Past a Porous Plate

The flow of a second-grade fluid past a porous plate subject to either suction or blowing at the plate has been studied. A modified model of second-grade fluid that has shear-dependent viscosity and can predict the normal stress difference is used. The differential equations governing the flow are solved using homotopy analysis method (HAM). Expressions for the velocity have been constructed and discussed with the help of graphs. Analysis of the obtained results showed that the flow is appreciably influenced by the material and normal stress coefficient. Several results of interest are deduced as the particular cases of the presented analysis.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.     

Averaging and periodic solutions in the plane and parametrically excited pendulum

We first approximate the solutions of the nonautonomous oscillating suspension point pendulum equation by the solutions of a second order autonomous differential equation. Using the strict monotonicity of the periodic solutions of the approximating equation, we prove the existence of a large number of subharmonic periodic solutions of the plane pendulum when its point of suspension is excited parametrically.     

Standing Pulse Solutions in Reaction-Diffusion Systems with Skew-Gradient Structure

The purpose of this paper is to introduce a reaction-diffusion system with skew-gradient structure and discuss the stability of standing pulse solutions. In short, the skew-gradient system is a reaction-diffusion system which resembles a gradient system but has nonlinearities with different sign. We assume the existence of a standing pulse solution and define its orientation in some geometrical manner. Then we show that the stationary solution becomes unstable if time constants satisfy some inequality. The Evans function plays a crucial role for the stability analysis.     

Buoyancy Sustained by Viscous Dissipation

The quasi-parallel regime of a Darcy–Boussinesq boundary-layer flow over a permeable vertical flat plate adjacent to a fluid saturated porous medium is considered. Quasi-parallel means here a plane flow with a constant transversal velocity v=–v ₀ directed perpendicularly towards the vertical surface, where a lateral suction with the same velocity –v ₀ is applied. The plate is held at a constant temperature T _w which coincides with the ambient temperature T of the fluid. The heat released by viscous dissipation induces a density gradient in the fluid. Thus, although T _w=T , a thermal convection occurs. The steady regime of this self-sustaining buoyant flow has been examined in detail. Wall jet-like profiles with a continuous but finite spectrum of the momentum flow have been found. These self-sustaining buoyant jets show a universal behavior, that is, there exist certain length, velocity and temperature scales such that the flow characteristics become independent of the (constant) material properties of the fluid and the porous medium as well.     

Theoretical analysis of non-uniform extensional flows in relation to processing rheology and predictions of some constitutive equations

In this paper, a characteristic equation involving the stream function, already given by one of the authors in a previous work for classifying axisymmetric incompressible flows, is re-considered. Non-uniform nearly extensional flows are derived as particular solutions from this equation. Using experimental data in the literature for polymer solutions and melts, it is proved that particular solutions of the characteristic equation lead to kinematics very close to those encountered in the fiber-spinning process. The kinematic equations satisfactorily correlating the fiber-spinning data are used in order to determine the ability of constitutive equations to predict realistic stresses in the flow domain. The rheological parameters of the fluids, obtained from experiments, are used for computation of differential and integral constitutive equations in the spinning conditions. Comparisons with the stress response of adequate constitutive equations are given and discussed.Also affiliated to: Université Joseph Fourier Grenoble I and Institut National Polytechnique de Grenoble, Associé au CNRS (URA 1510)     

Series solutions for steady unsaturated flow in irregular porous domains

For the most part, analytical solutions for steady unsaturated infiltration have been restricted to infinite and semi-infinite seepage geometries, using the quasi-linear approximation for the hydraulic conductivity. We provide analytical series methods to solve the steady quasi-linear flow equations, in finite irregular seepage geometries. Unlike the classical approach, the series method has been modified, to cater for arbitrary boundary geometry and surface recharge distributions. The matrix flux potential and the stream function both satisfy the same governing partial differential equation, and the stream function formulation is used to estimate the series coefficients. For a finite vadose zone, the stream function solution does not uniquely determine the matrix flux potential, when flux boundary conditions are used. Consequently, the stream function solution applies to a range of moisture distributions, for given infiltration and evapotranspiration rates through the surface.     

二次特征矩阵表示的特征值有界性分析

采用二次特征矩阵近似表示精确有限元和精确动态子结构分析中得出的超越或非线性动态刚度阵，并证明了在满足一定条件的前提下，二次特征阵给出的特征值是精确刚度阵特征值的上界或下界。     

Fluid Flows Through Two-Dimensional Channels of Composite Materials

The present analysis relates to the study of the full two-dimensional Brinkman equation representing the fluid flow through porous medium. The steady, incompressible fluid flow, with a negligible gravitational force, is constrained to flow in an infinitely long channel in which the height assumes a series of piecewise constant values. The control volume method is used to solve the Brinkman equation which involves the parameter, =/Da, where Da is the Darcy number and is the ratio of the fluid viscosity _f to the effective viscosity . An analytical study in the fully developed section of the composite channel is presented when the channel is of constant height and composed of several layers of porous media, each of uniform porosity. In the fully developed flow regime the analytical and numerical solutions are graphically indistinguishable. A geometrical configuration involving several discontinuities of channel height, and where the entry and exit sections are layered, is considered and the effect of different permeabilities is demonstrated. Further, numerical investigations are performed to evaluate the behaviour of fluid flow through regions which mathematically model some geological structures of various sizes, positions and permeability, for example a fault or a fracture, where the outlet channel is offset at different levels. The effect on the overall pressure gradient is also considered.