Applied Mathematics and Mechanics (English Edition)

Effect of non-uniform temperature gradient and magnetic field on onset of Marangoni convection heated from below by a constant heat flux

S.P.ISA N.M.ARIFIN R.NAZAR M.N.SAAD

2010, 31(7): 797-804. doi:10.1007/s10483-010-1314-z

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This paper investigates the effect of non-uniform temperature gradient and magnetic field on Marangoni convection in a horizontal fluid layer heated from below and cooled from above with a constant heat flux. A linear stability analysis is performed. The influence of various parameters on the convection onset is analyzed. Six non-uniform basic temperature profiles are considered, and some general conclusions about their destabilizing effects are presented.

Minimax principle on energy dissipation of incompressible shear flow

陈波李孝伟刘高联

2010, 31(7): 805-814. doi:10.1007/s10483-010-1315-6

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The energy dissipation rate is an important concept in the theory of turbulence. Doering-Constantin’s variational principle characterizes the upper bounds (maximum) of the time-averaged rate of viscous energy dissipation. In the present study, an optimization theoretical point of view was adopted to recast Doering-Constantin’s formulation into a minimax principle for the energy dissipation of an incompressible shear flow. Then, the Kakutani minimax theorem in the game theory is applied to obtain a set of conditions, under which the maximization and the minimization in the minimax principle are commutative. The results explain the spectral constraint of Doering-Constantin, and confirm the equivalence between Doering-Constantin’s variational principle and Howard-Busse’s statistical turbulence theory.

General solutions to a class of time fractional partial differential equations

黄凤辉郭柏灵

2010, 31(7): 815-826. doi:10.1007/s10483-010-1316-9

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A class of time fractional partial differential equations is considered, which includes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier-Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine-Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.

Simple waves for two-dimensional compressible pseudo-steady Euler system

赖耕盛万成

2010, 31(7): 827-838. doi:10.1007/s10483-010-1317-7

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A simple wave is defined as a flow in a region whose image is a curve in the phase space. It is well known that “the theory of simple waves is fundamental in building up the solutions of flow problems out of elementary flow patterns” see Courant and Friedrichs’s chassical book “Supersonic Flow and Shock Waves”. This paper mainly concerned with the geometric construction of simple waves for the 2D pseudo-steady compressible Euler system. Based on the geometric interpretation, the expansion or compression simple wave flow construction around a pseudo-stream line with a bend part are constructed. It is a building block that appears in the global solution to four contact discontinuities Riemann problems.

Physical criterion study on forward stagnation point heat flux CFD computations at hypersonic speeds

李邦明鲍麟童秉纲

2010, 31(7): 839-850. doi:10.1007/s10483-010-1318-9

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In order to evaluate uncertainties in computational fluid dynamics (CFD) computations of the stagnation point heat flux, a physical criterion is developed. Based on a quasi-one-dimensional hypothesis along the stagnation line, a new stagnation flow model is applied to obtain the governing equations of the flow near the stagnation point at hypersonic speeds. From the above equations, the compatibility relations are given at the stagnation point and along the stagnation line, which consist of the physical criterion for checking the accuracy in the stagnation point heat flux computations. The verification of the criterion is made with various numerical results.

New expression for collision efficiency of spherical nanoparticles in Brownian coagulation

陈忠利游振江

2010, 31(7): 851-860. doi:10.1007/s10483-010-1319-6

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The collision efficiency of dioctyl phthalate nanoparticles in Brownian coagulation has been studied. A set of collision equations is solved numerically to find the relationship between the collision efficiency and the particle radius varying in the range of 50 nm to 500 nm in the presence of Stokes resistance, lubrication force, van der Waals force, and elastic deformation force. The calculated results are in agreement with the experimental data qualitatively. The results show that the collision efficiency decreases with the increase of the particle radii from 50 nm to 500 nm. Based on the numerical data, a new expression for collision efficiency is presented.

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

陈豫眉谢小平

2010, 31(7): 861-874. doi:10.1007/s10483-010-1320-z

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A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming (P1)2 − P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.

Phragmén-Lindelöf and continuous dependence type results in a Stokes flow

J.C.SONG

2010, 31(7): 875-882. doi:10.1007/s10483-010-1321-z

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This paper investigates the asymptotic behavior of end effects for a Stokes flow defined on a three-dimensional semi-infinite cylinder. With homogeneous Dirichlet conditions of the velocity on the lateral surface of the cylinder, solutions either grow or decay exponentially in the distance from the finite end of the cylinder. In the case of decay, the effect of perturbing the equation parameters is also investigated.

General analytical solution to bending of composite laminated beams with delaminations

韩海涛张铮卢子兴

2010, 31(7): 883-894. doi:10.1007/s10483-010-1322-7

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Based on the first-order shear deformable beam theory, a refined model for composite beams containing a through-the-width delamination is presented, and the deformation at the delamination front is considered. Different from the ordinary delaminated beam theory, each of the perfectly bonded portions of the new model is constructed as two separated beams along the interface without assuming a plane section at the delamination front. The governing equations of the delaminated portions and bonded ones are established, combined with continuity conditions of displacements and internal forces. Solutions of delaminated composite beams with different boundary conditions, delamination locations and sizes are shown in excellent agreement with the finite element results, showing efficiency and applicability of the present model.

Elastic-plastic analysis of antiplane crack near crack surface region

易志坚谷建义何小兵马银华杨庆国彭凯黄锋黄宗明

2010, 31(7): 895-902. doi:10.1007/s10483-010-1323-x

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The elastic-plastic stress distribution and the elastic-plastic boundary configuration near a crack surface region are significant but hard to obtain by means of the conventional analysis. A crack line analysis method is developed in this paper by considering the crack surface as an extension of the crack line. The stresses in the plastic zone, the length, and the unit normal vector of the elastic-plastic boundary near a crack surface region are obtained for an antiplane crack in an elastic-perfectly plastic solid. The usual small scale yielding assumptions are not needed in the analysis.

Some qualitative properties of incompressible hyperelastic spherical membranes under dynamic loads

袁学刚张洪武任九生朱正佑

2010, 31(7): 903-910. doi:10.1007/s10483-010-1324-6

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Some nonlinear dynamic properties of axisymmetric deformation are examined for a spherical membrane composed of a transversely isotropic incompressible Rivlin-Saunders material. The membrane is subjected to periodic step loads at its inner and outer surfaces. A second-order nonlinear ordinary differential equation approximately describing radially symmetric motion of the membrane is obtained by setting the thickness of the spherical structure close to one. The qualitative properties of the solutions are discussed in detail. In particular, the conditions that control the nonlinear periodic oscillation of the spherical membrane are proposed. In certain cases, it is proved that the oscillating form of the spherical membrane would present a homoclinic orbit of type “∞”, and the amplitude growth of the periodic oscillation is discontinuous. Numerical results are provided.

Group classification for path equation describing minimum drag work and symmetry reductions

M.PAKDEMIRLI Y.AKSOY

2010, 31(7): 911-916. doi:10.1007/s10483-010-1325-x

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The path equation describing the minimum drag work first proposed by Pakdemirli is reconsidered (Pakdemirli, M. The drag work minimization path for a flying object with altitude-dependent drag parameters. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science 223(5), 1113–1116 (2009)). The Lie group theory is applied to the general equation. The group classification with respect to an altitude-dependent arbitrary function is presented. Using the symmetries, the group-invariant solutions are determined, and the reduction of order is performed by the canonical coordinates.

Quadratic minimization for equilibrium problem variational inclusion and fixed point problem

张石生李向荣陈志坚

2010, 31(7): 917-928. doi:10.1007/s10483-010-1326-6

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The purpose of this paper is to find the solutions to the quadratic minimization problem by using the resolvent approach. Under suitable conditions, some new strong convergence theorems are proved for approximating a solution of the above minimization problem. The results presented in the paper extend and improve some recent results.

A new auxiliary equation method for finding travelling wave solutions to KdV equation

庞晶边春泉朝鲁

2010, 31(7): 929-936. doi:10.1007/s10483-010-1327-z

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In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solutions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.

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