应用数学和力学

2016 Vol. 37, No. 4

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Equation Systems of Generalized Hydrodynamics for Soft-Matter Quasicrystals

FAN Tian-you

2016, 37(4): 331-344. doi: 10.3879/j.issn.1000-0887.2016.04.001

Abstract(1565) PDF(996)

Abstract:
The equation systems of generalized hydrodynamics for soft-matter quasicrystals were established based on the Langevin equation, with the derivation method of Poisson bracket. The derivation was done with reference to the work on solid quasicrystals, for which there were only 2 kinds of elementary excitations: phonons and phasons. However, the generalized hydrodynamics for soft-matter quasicrystals was essentially different from that for solid ones. In the generalized hydrodynamics for solid quasicrystals the effects of solid viscosity and the interaction among viscosity, phonons and phasons were considered, without the need of state equations. Nevertheless, the fluid phonon and the interaction among fluid phonons, phonons and phasons in soft-matter quasicrystals were to be considered, where 3 kinds of elementary excitations, i.e. phonons, phasons and fluid phonons meant state equations were indispensable. This difficult problem was solved with the present method. The results show the correctness and efficiency of the theoretical analysis.

Forced Vibration Responses of Supercritical Fluid-Conveying Pipes in 3∶1 Internal Resonance

MAO Xiao-ye, DING Hu, CHEN Li-qun

2016, 37(4): 345-351. doi: 10.3879/j.issn.1000-0887.2016.04.002

Abstract(1682) PDF(679)

Abstract:
The multi-scale method was used to investigate the vibration responses of supercritical fluid-conveying pipes in the 3∶1 internal resonance condition. In view of the buckling pipe shape under the supercritical flow velocity, the partial differential-integral equation for the nonlinear vibration of continuous bodies was established and then discretized into a set of ordinary differential equations with the Galerkin truncation method. Both quadratic and cubic nonlinearities of the MDOF system were taken into consideration, and the high-order multi-scale method was applied to build the solvable conditions. 2 natural modes with 2 vibration shapes were introduced to express the approximate solution. The 3∶1 internal resonance causes the first 2 modes couple in the primary resonance, where the vibration response of the nonlinear system is soft. But in the secondary resonance the vibration response of the nonlinear system is hard. The quadratic nonlinearity makes the system response properties unpredictable. The analytical solutions are perfectly consistent with the numerical simulation results.

Flow Noise Calculation With the Viscous Acoustic Splitting Method on Curvilinear Meshes

LIU Cong-wei, WU Fang-liang, LI Huan, CHEN Can, LI Peng

2016, 37(4): 352-362. doi: 10.3879/j.issn.1000-0887.2016.04.003

Abstract(1726) PDF(753)

Abstract:
The viscous acoustic splitting method on curvilinear meshes was developed for hydrodynamic noise calculation related to the interaction between a stationary cylinder and sound waves as well as vortex waves in flow field. The 7-point dispersion-relation-preserving (DRP) difference scheme coupled with the classical 4th-order Runge-Kutta scheme, was implemented to solve the governing equations for the simulation of hydroacoustic phenomena. The propagating acoustic pulse reflected by the stationary cylinder was computed and compared with the theoretical results to demonstrate the validity of the calculation strategy. The flow noise generated by the vortex in inhomogeneous water flow and the interaction beteen the vortex and the cylinder were studied to analyze the effects of the vortex core size and the incoming flow velocity on the acoustic filed. The work lays a foundation for the precise calculation of hydrodynamic noise in flow past immersed bodies.

Electrokinetic Energy Conversion Efficiency in Rectangular Nanochannels

XING Jing-nan, JIAN Yong-jun

2016, 37(4): 363-372. doi: 10.3879/j.issn.1000-0887.2016.04.004

Abstract(1317) PDF(867)

Abstract:
The streaming potential and electrokinetic energy conversion efficiency in rigid rectangular nanochannels were studied via the variable separation approach. The analytic expressions for the streaming potential and electrokinetic energy conversion efficiency were obtained through solution of the linearized PoissonBoltzmann equation for the electric potential and the NavierStokes equation for the velocity field. By means of numerical computations, the influences of dimensionless electrokinetic width K,channel width to height ratio α and wall Zeta potential ζ on both the streaming potential and the electrokinetic energy conversion efficiency were discussed. The results show that the streaming potential exhibits monotonic decrease with K, while the electrokinetic energy conversion efficiency first increases with K for small K values, then decreases with K for large K values, when other parameters are given. In addition, the streaming potential increases with α. The electrokinetic energy conversion efficiency first increases with α for small K values, then decreases with α for large K values. Finally, both the streaming potential and the electrokinetic energy conversion efficiency increase significantly with the wall Zeta potential.

Simmulation of Mixed Electroosmotic and Pressure-Driven Flows of Power-Law Fluids in Microchannels

LUO Yan, LI Ming, YANG Da-yong

2016, 37(4): 373-381. doi: 10.3879/j.issn.1000-0887.2016.04.005

Abstract(1463) PDF(612)

Abstract:
The pressure effects on electroosmotic flows of power-law fluids in microchannels were investigated. The electric double layer (EDL) potential was described with the Poisson-Boltzmann (P-B) equation, and the flow field distribution of the power-law fluid was characterized with the Navier-Stokes (N-S) equation. Numerical simulation was carried out to discuss the influences of the dimensionless Debye-Huckel parameter, the wall Zeta potential and the flow behavior index on the flow properties and the Poiseuille number. The results reveal that, in the case of the same pressure gradient direction with the electric field direction, the velocity of a shear-thinning fluid is higher than that of a shear-thickening one, whereas the result will be opposite for a reverse pressure gradient direction. The Poiseuille number is an increasing function of the dimensionless Debye-Huckel parameter, the Zeta potential and the flow behavior index.

Solutions of Double-Layer Plates With Two-Way Spring Interlayers on Elastic Foundations

TAN Zhi-ming, GUO Jing-jing, CHEN Jing-liang

2016, 37(4): 382-390. doi: 10.3879/j.issn.1000-0887.2016.04.006

Abstract(1348) PDF(608)

Abstract:
Under axisymmetric conditions, a mechanics model for double-layer plates with two-way spring interlayers on elastic foundations was built. The Hankel transform method was used to derive the general analytical solutions of the infinite double-layer plates respectively on the Winkler foundation, the two-parameter foundation and the elastic half-space foundation under arbitrary axisymmetric load. Then the calculating formulae for the deflection, bending moment, shearing force and the interlayer reaction force and displacement were given. The analytical solutions were applied to study the effects of the interlayer conditions on the deflection and bending moment of the double-layer plates, calculate the positions of neutral axes of the upper and lower plates, and discuss the specific values of the interlayer spring coefficients. The results show that, 1) with the increase of the vertical spring coefficient, the deflection and bending stress of the upper plate decrease, while those of the lower plate increase; on the other hand, with the increase of the horizontal friction parameter, those of both the upper and lower plates decrease; 2) when the shearing coefficient and compressibility of the double-layer plates are given values of 2/3 and 3/5 respectively, the effects of shearing and compression could be well considered; 3) the neutral axes’ positions of the upper and lower plates are changeable, but respectively approach the center planes of the upper and lower plates with the increase of the distance from the load center.

Response Analysis of Thin Circular Plates in the Local Gaussian Temperature Field

LONG Lian-chun, ZHANG Chao-ya

2016, 37(4): 391-403. doi: 10.3879/j.issn.1000-0887.2016.04.007

Abstract(1241) PDF(504)

Abstract:
Based on the theory for thermal bending of thin plates, the analytical expressions for deflections and thermal stresses of circular plates were derived in the Gaussian temperature field, and the influences of boundary conditions and local temperature parameters on the deflections and thermal stresses were discussed, thus to provide a theoretical basis for the thermodynamics analysis of thin plate structures under local heating. The results show that, the maximum values of both deflections and compressive stresses occur at the circular plates’centers. Within the heat-impacted zone of the circular plate, the deflection at a point shows a trend of the Gaussian decrease with the increase of the distance from the point to the plate center. Outside the heat-impacted zone, the boundary constraints on the circular plate and the irradiation factor influence the changes of deflections. The analytical solutions of the circular plate deflections are consistent with the numerical results. The deflection of the simply supported circular plate firstly decreases linearly with the irradiation factor. Within the heat-impacted zone, thermal stresses at a point show a trend of the Gaussian decrease with the increase of the distance from the point to the plate center. The changing trends of the thermal stresses are similar between the simply supported and the clamped circular plates. Outside the heat-impacted zone, the boundary constraints and the irradiation factor influence the changing trends of thermal stresses.

Bivariate Vector-Valued Osculatory Rational Interpolation Based on the Taylor Operator

JING Hui-qin

2016, 37(4): 404-415. doi: 10.3879/j.issn.1000-0887.2016.04.008

Abstract(1126) PDF(461)

Abstract:
A new approach based on the Taylor operator was proposed for the bivariate vectorvalued osculatory rational interpolation. First, the rational interpolation basis functions of each order were defined by means of the known nodes. Second, a new type of interpolation operator similar to the Taylor formula for bivariate functions was established with the corresponding vector values and partial derivative values of each order. At last, the combined operations were carried out, and the explicit expressions of the bivariate vectorvalued osculatory rational interpolation functions of the 1st and 2nd orders were obtained. Naturally this approach was generalized to the kth order, and the error estimates were also made. The results of an example show that, this new approach is simple and formularized in calculation, and therefore potential for application.

Error Estimates for the Complex Variable Moving Least Square Approximation in Sobolev Spaces

SUN Xin-zhi, LI Xiao-lin

2016, 37(4): 416-425. doi: 10.3879/j.issn.1000-0887.2016.04.009

Abstract(1430) PDF(797)

Abstract:
The complex variable moving least square (CVMLS) approximation is an important approach to construct shape functions in the meshless method. For the error analysis of the CVMLS-based meshless method, it is fundamental to conduct error estimates of the CVMLS approximation in Sobolev spaces. First an introduction of the CVMLS was given. Then, the error estimates of the CVMLS in Sobolev spaces with weight functions satisfying specific conditions were obtained. The error bounds of the approximation functions in H^k norm were given. Finally, a numerical example was given to verify the validity of the theoretical analysis. The results show that the errors will decrease as the nodal spacings reduce.

Soliton Travelling Wave Solutions to a Class of Nonlinear Nonlocal Disturbed LGH Equations

FENG Yi-hu, MO Jia-qi

2016, 37(4): 426-433. doi: 10.3879/j.issn.1000-0887.2016.04.010

Abstract(1340) PDF(564)

Abstract:
A class of nonlinear nonlocal Landau-Ginzburg-Higgs (LGH) differential equations were discussed with the modified functional analytic variational iteration method. Firstly, a set of travelling wave transforms were constructed and the functional was introduced, of which the variation was determined and then was made equal to 0 to obtain the conditions for and solution of the Lagrange operator. Secondly, a modified variational iteration expression was employed and the soliton solutions to the corresponding non-disturbed LGH equations were selected as the initial iteration functions. Finally, all the asymptotic solutions and the exact solutions to the nonlinear nonlocal disturbed LGH equations were obtained, successively. From an example, the proposed modified functional analytic variational iteration method is proved valid and practicable.

Bifurcations of Exact Travelling Wave Solutions to Coupled Higgs Equations and Maccari Systems

WANG Heng, WANG Han-quan, CHEN Long-wei, ZHENG Shu-hua

2016, 37(4): 434-440. doi: 10.3879/j.issn.1000-0887.2016.04.011

Abstract(1080) PDF(608)

Abstract:
With the dynamical system method, the qualitative performance of and the exact travelling wave solutions to the coupled Higgs equations and the Maccari systems were studied. Based on this method, all phase portraits of the systems in the parametric space were given. All possible bounded travelling wave solutions such as the solitary wave solutions and the periodic travelling wave solutions were obtained. Through numerical simulation, the smooth solitary wave solutions and the periodic travelling wave solutions were picturized. The results show that the present findings improve the related previous conclusions.