应用数学和力学

D Generalized Hydrodynamics of Soft-Matter Quasicrystals

FAN Tian-you, TANG Zhi-yi

2017, 38(11): 1195-1207. doi: 10.21656/1000-0887.380063

Abstract(1217) PDF(627)

Abstract:
The 3D generalized hydrodynamics of soft-matter quasicrystals was investigated, and the governing equations for observed and possibly observed soft-matter quasicrystals were derived. The solving procedure for the equations was discussed briefly. Some results obtained reveal the gigantic dissimilarities between soft-matter quasicrystals and solid ones.

Numerical Study of Unstable T-S Waves Excited by Interaction Between Free-Stream Turbulence and 3D Localized Wall Roughness in Flat-Plate Boundary Layer

SHEN Lu-yu, LU Chang-gen, ZHU Xiao-qing

2017, 38(11): 1208-1221. doi: 10.21656/1000-0887.370376

Abstract(957) PDF(460)

Abstract:
The direct numerical simulation (DNS) method was adopted to study the physical problem of unstable Tollmien-Schlichting (T-S) waves in the flat-plate boundary layer under the interaction between free-stream turbulence and 3D localized wall roughness. The numerical results show that the spatial arrays of the wave packets composed of 2D and 3D T-S waves form in the flat-plate boundary layer, with the propagation speeds of the wave packets calculated. Then it is proved that the interaction between the free-stream turbulence and the 3D localized wall roughness makes the mechanism for excitation of unstable T-S waves in the flat-plate boundary layer. Subsequently, the relations between the initial 2D & 3D T-S wave amplitudes and the free-stream turbulence intensity, the roughness length, the roughness width and the roughness height, were built. The in-depth research of this problem is conducive to understanding of the hydrodynamic stability theory.

Numerical Calculation of Liquid Crystal Cells With Free-State Upper Plates Based on the Liquid Crystalline Backflow Effects

WANG An-biao, TIAN Yong, LIU Chun-bo, CUI Fan

2017, 38(11): 1222-1229. doi: 10.21656/1000-0887.370347

Abstract(1246) PDF(586)

Abstract:
Based on the Leslie-Ericksen theory for small molecule liquid crystals, a calculation model was established for liquid crystal cells with free-state upper plates. Under the specified initial boundary conditions, the 2nd-order Runge-Kutta method and the central difference method were applied to conduct spatial-temporal discretization of the equation set. Additionally, a calculation program was compiled on MATLAB. Then, the calculation parameters were adjusted to obtain the influences of the liquid crystal cell thickness and the electric field parameters imposed at 2 ends of the cell on the liquid crystalline backflow. The results indicate that, the size of the liquid crystal director alternates with the alternation of the electric field imposed on the upper and lower plates of the liquid crystal cell. With the increment of the cell thickness, the displacement of the upper plate within a period also increases. The duty ratio of the electric field imposed at 2 ends of the cell has little impact on the upper plate speed, but has large influence on the occurring time point of the maximal upper plate speed. Compared with the experimental data, the calculated displacement values of the upper plate of the liquid crystal cell are of the same orders, and the movement loci are in good agreement.

A New-Type Counterpropagating Wave Pattern of Vertical Mirror Symmetry in Binary Fluid Convection

NING Li-zhong, QU Ya-wei, NING Bi-bo, YUAN Zhe, TIAN Wei-li, LIU Shuang

2017, 38(11): 1230-1239. doi: 10.21656/1000-0887.370367

Abstract(1081) PDF(447)

Abstract:
The 2D hydrodynamic equations for binary fluid convection were numerically simulated with the SIMPLE method. For separation ratio ψ=－0.6 of the binary fluid mixture and aspect ratio Γ=20 of the rectangular cell, a newtype counterpropagating wave pattern of vertical mirror symmetry was found for the first time and its dynamics was preliminarily studied. At the center of the counterpropagating wave pattern of vertical mirror symmetry was a standing wave, of which the wavelength extended with time. As the wavelength increased to a certain critical value, a roll split into 2 rolls, and a new roll with a 180° phase difference formed between them 2. The roll located at the center line only has phase mutation and wavelength contraction or extension, without moving toward the left or right. The convective rolls propagating toward the left or right exist on both sides of the center line. The 2 phase mutations of the standing wave form a period, and the standing wave period increases with reduced Rayleigh number Ra_r. This type of convective structure exists in the range of Ra_r∈(3.6,4.3].The convection system produces the traveling wave pattern with defect for Ra_r≤3.6. The system shifts to the traveling wave pattern for Ra_r>4.3.The work shows that the counterpropagating wave pattern of vertical mirror symmetry is a stable flow pattern between the traveling wave pattern with defect and the traveling wave pattern.

Vibration Suppression of Nonlinear Systems Under Dual-Frequency Excitations With Nonlinear Energy Sink

SUN Bin, WU Zhi-qiang

2017, 38(11): 1240-1250. doi: 10.21656/1000-0887.370379

Abstract(1442) PDF(1097)

Abstract:
In view of the dual-frequency excitation characteristics of a certain type of civil aero-engines, the dynamic model of a single DOF linear oscillator coupled with the nonlinear energy sink (NES) was established. At the low to high characteristic frequency ratio (1∶4.74) of the typical dual-rotor engine in cruise flight, the system was subjected to a dual-frequency harmonic excitation. The 4th-order Runge-Kutta algorithm was employed to study the vibration suppression effects of the system with the coupled NES. In the aspects of the influences of the external excitation frequencies on the kinetic energy of the main oscillator and the total energy of the system, etc, the numerical results were compared among the system with the NES, the one without the NES and another one coupled with a linear dynamic vibration absorber. The work shows that the NES has better vibration suppression effects on the dual-frequency excitation, and is feasible for the reduction of the vibration of aero-engines.

Variational Principles for Dual and Triple Mixed Variables of Linear Elasticity With Finite Displacements and the Application

FU Bao-lian

2017, 38(11): 1251-1268. doi: 10.21656/1000-0887.380004

Abstract(1186) PDF(620)

Abstract:
Variational principles for dual and triple mixed variables of linear elasticity with finite displacements were proposed. Considering the variation of prescribed boundary conditions and using the reciprocal theorem of finite displacements played the key and bridging roles in derivation of the above variational principles. First, in view of the variation of the prescribed geometrical boundary conditions and based on the reciprocal theorem, the principle of minimum potential energy with dual mixed variables was derived. In a similar way, the principle of stationary complementary energy with dual mixed variables was also given. Then the relation between the strain energy density and the complementary energy density was applied to the above 2 principles, and the variational principle with triple mixed variables was deduced. In turn, the principles of virtual work and virtual complementary work with dual and triple mixed variables were directly given. Meantime, the generalized variational principles were derived with the Lagrangian multiplier method. Through an example the Lagrangian multiplier method in certain cases was proved to be ineffective. The semiinverse method for construction of the functionals for generalized variational principles was also introduced. Finally, a cantilever beam with large deflection was calculated by means of the principle of minimum potential energy for dual mixed variables.

Analysis on Transmission Potential and Control Strategies of Zika Virus

FAN Lin-xuan, TANG San-yi

2017, 38(11): 1269-1278. doi: 10.21656/1000-0887.380031

Abstract(1309) PDF(706)

Abstract:
Currently, Zika virus has spread in more than 65 countries and regions. To estimate the transmission potential of Zika virus and evaluate the effectiveness of the control strategies in Singapore, the classical infectious disease model was employed, and both the least square method and the MCMC method were used to estimate the unknown parameters which can fit the cumulative number of reported cases very well. With the nextgeneration matrix method the basic reproduction number was calculated and its value and confidence interval were evaluated according to the estimated parameter values, which can be verified through comparison between the results obtained from 2 different estimation methods. Furthermore, the effectiveness of different control measures was discussed in more details through sensitivity analyses, which can help verify the key parameters related to the cumulative number of cases and the Zika outbreak. The results show that, for the control of Zika virus in Singapore, the number of screening and the screening rate shall be increased, the quarantine and isolation of infected patients and the mosquito control shall be effectively implemented, and the number of tourists shall be reduced.

Eigenvalues of the Deformation Tensor and Regularity Estimates for the Boussinesq Equations

WANG Zhen, DENG Da-wen

2017, 38(11): 1279-1288. doi: 10.21656/1000-0887.370355

Abstract(1090) PDF(474)

Abstract:
The blow-up possibility of local regular solutions to the initial-boundary-value problems with periodic boundary conditions for 2D and 3D Boussinesq systems was discussed. In the 2D case, an L² estimate of the temperature gradient was given in terms of the eigenvalues of the deformation tensor. From this estimate it is found that if the deformation rate of a fluid element is large, the regular solution is more likely to blow up. In the 3D case, an L² estimate of the vorticity was given in terms of the eigenvalues of the deformation tensor and the derivatives of temperature. From this estimate it is shown that if for most of the time, most of the fluid elements are stretched in plane and the temperature gradient is small, the regular solution is more likely to blow up. On the contrary, if linear stretching dominates and the temperature gradient is bounded, the solution is less likely to blow up.

Error Analysis of the Scaled Moving Least Squares Approximation

WANG Qing-qing, LI Xiao-lin

2017, 38(11): 1289-1299. doi: 10.21656/1000-0887.370260

Abstract(1233) PDF(658)

Abstract:
Compared with the moving least squares (MLS) approximation, the scaled moving least squares (SMLS) approximation can avoid the issue of ill-conditioned matrices involved in the MLS approximation. Error estimates of the SMLS approximation were conducted for the approximation function and its arbitrary-order derivatives. Finally, some numerical examples were given. The numerical results indicate that the SMLS approximation provides monotonic convergence and higher accuracy with higher computational stability in comparison with the MLS approximation.

A Class of 2-DOF Coupled Systems With Fractional-Order Derivatives

GE Zhi-xin, CHEN Xian-jiang, CHEN Song-lin

2017, 38(11): 1300-1308. doi: 10.21656/1000-0887.370333

Abstract(1280) PDF(542)

Abstract:
The vibration problems of a class of 2-DOF coupled systems with fractional-order derivatives and small perturbations were studied. First, the asymptotic solutions of the vibration equations with Riemann-Liouville fractional-order derivatives were constructed. With the multi-scale method, the solvability conditions for the asymptotic solutions to the vibration problems were obtained. Then, under the solvability conditions for the solutions, the influences of the fractional-order derivatives, their coefficients and the small parameters on the vibration were discussed, and the asymptotic solutions were also given. Finally, the stability properties of the 1st-order approximate solutions were studied. It is found that all the steady-state solutions are stable.

2017 Vol. 38, No. 11