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2013年 第34卷 第9期    刊出日期：2013-09-02

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论文

Series solutions of annular axisymmetric stagnation flow and heat transfer on moving cylinder

A. MASTROBERARDINO

2013, 34(9):  1043-1054.  doi:10.1007/s10483-013-1726-7

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The steady, laminar, incompressible flow and heat transfer of a viscous fluid between two circular cylinders for two different types of thermal boundary conditions are investigated. The governing Navier-Stokes and thermal equations of the flow are reduced to a nonlinear system of ordinary differential equations. The equations are solved analytically using the homotopy analysis method (HAM). Convergence of the HAM solutions is discussed in detail. These solutions are then compared with recently obtained numerical and perturbative solutions. Plots of the velocity and temperature profiles are provided for various values of the relevant parameters.

3D numerical simulation of avascular tumour growth: effect of hypoxic micro-environment in host tissue

蔡彦 吴洁 龙泉 许世雄 李志勇

2013, 34(9):  1055-1068.  doi:10.1007/s10483-013-1727-x

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A three-dimensional (3D) mathematical model of tumour growth at the avascular phase and vessel remodelling in host tissues is proposed with emphasis on the study of the interactions of tumour growth and hypoxic micro-environment in host tissues. The hybrid based model includes the continuum part, such as the distributions of oxygen and vascular endothelial growth factors (VEGFs), and the discrete part of tumour cells (TCs) and blood vessel networks. The simulation shows the dynamic process of avascular tumour growth from a few initial cells to an equilibrium state with varied vessel networks. After a phase of rapidly increasing numbers of the TCs, more and more host vessels collapse due to the stress caused by the growing tumour. In addition, the consumption of oxygen expands with the enlarged tumour region. The study also discusses the effects of certain factors on tumour growth, including the density and configuration of preexisting vessel networks and the blood oxygen content. The model enables us to examine the relationship between early tumour growth and hypoxic micro-environment in host tissues, which can be useful for further applications, such as tumour metastasis and the initialization of tumour angiogenesis.

Numerical simulation on evolution of subharmonic low-speed streaks in minimal channel turbulent flow

李健 董刚

2013, 34(9):  1069-1082.  doi:10.1007/s10483-013-1728-9

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The evolution of low-speed streaks in the turbulent boundary layer of the minimum channel flow unit at a low Reynolds number is simulated by the direct numerical simulation (DNS) based on the standard Fourier-Chebyshev spectral method. The subharmonic sinuous (SS) mode for two spanwise-aligned low-speed streaks is excited by imposing the initial perturbations. The possibilities and the physical realities of the turbulent sustaining in the minimal channel unit are examined. Based on such a flow field environment, the evolution of the low-speed streaks during a cycle of turbulent sustaining, including lift-up, oscillation, and breakdown, is investigated. The development of streamwise vortices and the dynamics of vortex structures are examined. The results show that the vortices generated from the same streak are staggered along the streamwise direction, while the vortices induced by different streaks tilt toward the normal direction due to the mutual induction effect. It is the spatial variations of the streamwise vortices that cause the lift-up of the streaks. By resolving the transport dynamics of enstrophy, the strength of the vortices is found to continuously grow in the logarithmic layer through the vortex stretching mechanism during the evolution of streaks. The enhancement of the vortices contributes to the spanwise oscillation and the following breakdown of the low-speed streaks.

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

陈刚 冯民富 何银年

2013, 34(9):  1083-1096.  doi:10.1007/s10483-013-1729-x

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This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart (CR) element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.

Oscillatory flow of second grade fluid in cylindrical tube

A. ALI S. ASGHAR H. H. ALSULAMI

2013, 34(9):  1097-1106.  doi:10.1007/s10483-013-1730-8

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The unsteady oscillatory flow of an incompressible second grade fluid in a cylindrical tube with large wall suction is studied analytically. Flow in the tube is due to uniform suction at the permeable walls, and the oscillations in the velocity field are due to small amplitude time harmonic pressure waves. The physical quantities of interest are the velocity field, the amplitude of oscillation, and the penetration depth of the oscillatory wave. The analytical solution of the governing boundary value problem is obtained, and the effects of second grade fluid parameters are analyzed and discussed.

Effects of aspect ratio on unsteady solutions through curved duct flow

R. N. MONDAL S.ISLAM K. UDDIN A. HOSSAIN

2013, 34(9):  1107-1122.  doi:10.1007/s10483-013-1731-8

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The effects of the aspect ratio on unsteady solutions through the curved duct flow are studied numerically by a spectral based computational procedure with a temperature gradient between the vertical sidewalls for the Grashof number 100  Gr  2 000. The outer wall of the duct is heated while the inner wall is cooled and the top and bottom walls are adiabatic. In this paper, unsteady solutions are calculated by the time history analysis of the Nusselt number for the Dean numbers Dn = 100 and Dn = 500 and the aspect ratios 1  γ  3. Water is taken as a working fluid (Pr = 7.0). It is found that at Dn = 100, there appears a steady-state solution for small or large Gr. For moderate Gr, however, the steady-state solution turns into the periodic solution if γ is increased. For Dn = 500, on the other hand, it is analyzed that the steady-state solution turns into the chaotic solution for small and large Gr for any γ lying in the range. For moderate Gr at Dn = 500, however, the steady-state flow turns into the chaotic flow through the periodic oscillating flow if the aspect ratio is increased.

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

史志杰 汪越胜 张传增

2013, 34(9):  1123-1144.  doi:10.1007/s10483-013-1732-6

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A multiple monopole (or multipole) method based on the generalized multipole technique (GMT) is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure is obtained. Some numerical examples are presented to validate the proposed method.

Nonlinear progressive damage model for composite laminates used for low-velocity impact

郭卫 薛璞 杨军

2013, 34(9):  1145-1154.  doi:10.1007/s10483-013-1733-7

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In order to effectively describe the progressively intralaminar and interlaminar damage for composite laminates, a three dimensional progressive damage model for composite laminates to be used for low-velocity impact is presented. Being applied to three-dimensional (3D) solid elements and cohesive elements, the nonlinear damage model can be used to analyze the dynamic performance of composite structure and its failure behavior. For the intralaminar damage, as a function of the energy release rate, the damage model in an exponential function can describe progressive development of the damage. For the interlaminar damage, the damage evolution is described by the framework of the continuum mechanics through cohesive elements. Coding the user subroutine VUMAT of the finite element software ABAQUS/Explicit, the model is applied to an example, i.e., carbon fiber reinforced epoxy composite laminates under low-velocity impact. It is shown that the prediction of damage and deformation agrees well with the experimental results.

Global structure stability of impact-induced tensile waves in phase-transforming materials

黄守军 王静静

2013, 34(9):  1155-1166.  doi:10.1007/s10483-013-1734-6

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The global structure stability of the impact-induced tensile waves mentioned by Huang (Huang, S. J. Impact-induced tensile waves in a kind of phase-transforming materials. IMA Journal of Applied Mathematics, 76, 847–858 (2011)) is considered. By introducing Riemann invariants, the governing equations of motion are reduced into a 2×2 diagonally strictly hyperbolic system. Then, with the aid of the theory on the typical free boundary problem and maximally dissipative kinetics, the global structure stability of the impact-induced tensile waves propagating in a phase-transforming material is proved.

Chaotification for switched linear systems with controllers

解玲丽

2013, 34(9):  1167-1174.  doi:10.1007/s10483-013-1735-9

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This paper shows that two or more switched linear systems can generate chaotic dynamical behaviors by an appropriate switching rule as they at least consist of a controllable system and an unstable system with the expanding property. According to the results in the reference (Xie, L. L., Zhou, Y., and Zhao, Y. Criterion of chaos for switched linear systems with antrollers. International Journal of Bifurcation and Chaos, 20(12), 4105–4109 (2010)), a nonlinear feedback gain is needed to generate chaotic dynamics. A linear feedback control is usually used to approximate the nonlinear one for simulation. In order to obtain the exact control, as a main result of this paper, the controller is constructed by Russell’s result, and a block diagram is included to interpret the realization of the controller. Numerical simulations are given to illustrate the generated chaotic dynamical behavior of the switched linear systems with some parameters and show the effects of the constructed controller.

Comments on “Non-existence of Shilnikov chaos in continuous-time systems”

A. ALGABA F.FERN′ANDEZ-S′ANCHEZ M. MERINO A.J.RODR′IGUEZ-LUIS

2013, 34(9):  1175-1176.  doi:10.1007/s10483-013-1736-7

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A paper, “Non-existence of Shilnikov chaos in continuous-time systems” was published in the journal Applied Mathematics and Mechanics (English Edition). The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.