应用数学和力学

2018 Vol. 39, No. 2

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Analytical Solution for the Viscous Flow of Small Reynolds Numbers in Rough Pipes

XU Zhimin, SONG Siyuan, XIN Fengxian, YANG Xiaohu, LU Tianjian

2018, 39(2): 123-136. doi: 10.21656/1000-0887.380223

Abstract(1275) PDF(897)

Abstract:
In view of the viscous flow fields of small Reynolds numbers in rough circular tubes and petal circular tubes, the rough surface in the tube was considered as a smooth surface subjected to small disturbance. The perturbation method was used to expand the perturbation of fluid parameters under small disturbance. The boundary conditions with complex morphologies were expanded into the Taylor series, and the smooth boundary conditions were approximately obtained. Then the fluid mechanics equations were solved simultaneously to give the approximate solution of the pressure gradient under the premise of the 1storder perturbation expansion, and the static flow resistance and tortuosity of the pipeline were obtained. The results show that the fluid parameters determined with the modified perturbation method agree very well with those through numerical simulation, and the theoretical approximate solution of the flow field in the rough pipe is validated.

Fluid Infiltration Characteristics and Driving Mechanism in the Rubber-Glass Contact Interface

PANG Minghua, LIU Xiaojun, JIAO Yunlong, LIU Kun

2018, 39(2): 137-146. doi: 10.21656/1000-0887.380053

Abstract(862) PDF(906)

Abstract:
To clarify the mechanism of fluid infiltration in the rubber-glass contact interface, an experimental setup of in situ observation instrument was established. Then, the fluid infiltration process in the rubber-glass contact interface was measured. The influences of the interfacial contact characteristics and the solid-liquid interface wettability on the fluid infiltration area ratio, the path and the distribution pattern were analyzed with a MATLAB program and an image processing software. Analysis results show that, all the fluid infiltration area ratio, the path and the distribution pattern in the contact interface are influenced by the capillary force and the real contact area ratio. First, the actual contact area ratio nonlinearly increases in the form of power functions with the interfacial contact pressure. Second, the fluid infiltration path in the contact interface tends to form where the actual contact area ratio is high. The main driving mechanism of the liquid infiltration path and the velocity in the contact interface is the equilibrium relation between the capillary force and the viscous resistance. Moreover, the solid-liquid interfacial wettability is one of the evaluation indexes for the fluid infiltration area ratio in the contact interface. Reasonable regulation and control of the wettability of the solid-liquid interface can improve the infiltration effect of fluid. The research work provides an innovative approach and obtains an essential understanding of the effects of fluid infiltration on friction, lubrication and seal in the rubber contact interface.

Oscillating Characteristics of Double Diffusive Natural Convection With Soret and Dufour Effects in Square Cavities

LOU Qin, LUO Zhuqing, WANG Jun, XU Hongtao, CHEN Jian

2018, 39(2): 147-159. doi: 10.21656/1000-0887.380121

Abstract(1299) PDF(664)

Abstract:
The lattice Boltzmann method was adopted to study the oscillating characteristics of double diffusive natural convection in square cavities with Soret and Dufour effects. The inner heating cylinder with high concentration was located at the center of the cavity and the 4 surrounding walls were assumed to be at low concentrations and temperatures. The impacts of buoyancy ratio B_r(2.0≤B_r≤10.0), Soret number S_r(-0.6≤S_r≤0.0) and Dufour number D_f(-0.6≤D_f≤0.0) on the oscillating characteristics in the cavity were analyzed with the time history method and the power spectrum method. The results show that the flow state is steady at S_r=0.0 and D_f=0.0. With decreasing S_rand D_f from 0.0 to -0.6, the state of double diffusive natural convection turns into the periodic state and in turn the aperiodic oscillatory state gradually. Moreover, the positive buoyancy ratio enhances the oscillating characteristics.

Identification of Thermal Diffusion Coefficients for Transient Heat Conduction Problems With Heat Sources

ZHOU Huanlin, YAN Jun, YU Bo, CHEN Haolong

2018, 39(2): 160-169. doi: 10.21656/1000-0887.380199

Abstract(1171) PDF(949)

Abstract:
An improved cuckoo search (ICS) algorithm was developed to identify the thermal diffusion coefficients for inverse transient heat conduction problems with heat sources. The heat conduction problem with heat source was transformed into one without heat source. The direct problem was solved with the boundary element method. The thermal diffusion coefficient was treated as the optimization variable, and the difference between the calculated temperature and the measured temperature was taken as the objective function. The thermal diffusion coefficient was optimized through minimization of the objective function with the ICS algorithm. Comparison between the results of the conjugate gradient method (CGM), the cuckoo search (CS) algorithm and the ICS algorithm indicates that the ICS algorithm is less sensitive to iterative initialization than the CGM, and the ICS algorithm has higher efficient convergence than the CS algorithm. The numerical examples were devoted to the influences of the measured point number, the nest number and the measured noise. The result accuracy decreases with the measured point number, and the iteration number decreases with the nest number. Moreover, the higher the measured noise goes, the lower the result accuracy will be. The results show that the ICS algorithm is accurate and efficient for the identification of thermal diffusion coefficients.

An Efficient Algorithm Based on Dynamic System Properties and Group Theory for Transient Responses of 1D Periodic Structures

LIANG Xiqiang, GAO Qiang, YAO Weian

2018, 39(2): 170-182. doi: 10.21656/1000-0887.380129

Abstract(896) PDF(999)

Abstract:
Based on the condensation technology, the dynamic periodic structure properties and the group theory, an efficient numerical method for computing the transient responses of 1D periodic structures was proposed. Efficiently solving linear equations is an issue for computing the dynamic responses. Based on the periodic properties of the structure and with the condensation technology, the scale of the linear equation corresponding to the structure was reduced. By means of the properties of linear equations for dynamic periodic systems, it was proved that the force on any chosen unit cell can only influence a finite number of adjacent unit cells within a time step. Then, the dynamic response computation of 1D periodic structures was converted into the computation of a series of small-scale substructures. Subsequently, the dynamic response computation of the substructures can be converted into the computation of the cyclic-periodic structures. Then, the cyclic-periodic structures were solved efficiently in light of the group theory. Numerical examples illustrate the high efficiency and memory saving of the proposed method.

Existence and Stability Analysis on Circular Motion of Pendulums With Uniformly Rotating Pivots

LI Shuhang, JIANG Fanghua

2018, 39(2): 183-198. doi: 10.21656/1000-0887.380028

Abstract(1405) PDF(994)

Abstract:
Authough with rich dynamic meanings, the particular circular motion and the stability of pendulums with horizontally uniformly rotating pivots have been seldom studied. Firstly, for the pendulum moving in vacuum and within a medium respectively, the general motion equations under gravity and disturbing force were established. Newton’s second law in a noninertial reference frame was used through introduction of a fictitious inertia force. Secondly, the existence of the particular motion was converted into the root finding of a quartic equation. According to Descartes’ rule of signs and analysis of the monotonicity of quartic polynomials, the relationships between the number of solutions and the physical parameters of the pendulum were given. In vacuum, the number of particular motion solution is 0, 1 or 2, and within a medium, the number is either 1 or 3. Their judging criteria were also given. Thirdly, Lyapunov’s first approximation theory was used to investigate the nonlinear stability. The motion equation was linearized around the particular solution, the stability of the particular motion was judged by the signs of the real parts of the eigenvalues related to the linear differential equation. The subsequent quartic characteristic equations were skillfully converted into quadratic equations. Thus, the linearly stability conditions in vacuum and the asymptotic stability conditions within a medium were deduced. Finally, numerical simulations were given to verify and confirm the theoretical conclusions.

Research on the Flutter of Micro-Scale Cantilever Pipes——A Finite-Dimensional Analysis

GUO Yong, XIE Jianhua

2018, 39(2): 199-214. doi: 10.21656/1000-0887.370400

Abstract(934) PDF(733)

Abstract:
Based on the modified couple stress theory, the integro-differential equations of motion for micro-scale cantilever pipes were derived by means of Hamilton’s principle. The geometric nonlinearity, arising from the Lagrangian strain tensor, was taken into account. The integro-differential equations were transformed into ordinary differential equations with the Galerkin method. With different numbers of modes in the Galerkin discretization, the diagrams of critical flow velocity vs. mass ratio were given. The difference between the Galerkin approximation results and the exact solutions to the 2-point boundary problem was investigated and the effect of the internal material length scale parameter on the graphs of critical flow velocity vs. mass ratio was studied. For different numbers of modes, the first Lyapunov’s coefficient was calculated and the critical eigenvalue with respect to the flow velocity was derived with the projection method based on the center manifold theory and the normal form method, therefrom, the bifurcation model was analyzed and the effect of the number of modes on the dynamical behaviors was examined. The dynamics of hysteresis and intersection points of the curves of critical flow velocity vs. mass ratio was also investigated and then bifurcation diagrams in different directions were found. Finally, the 6-mode ordinary differential equations of the Galerkin discretization were employed to construct the bifurcation diagrams and verify the relevant results obtained, and the natural frequencies of flutter were calculated through the theoretical analysis and with the numerical method, respectively.

A Class of Exponential Outer Synchronization Between Uncertain Spatiotemporal Networks With Different Numbers of Nodes

RONG Tingting, GAO Yan, YAN Zhe

2018, 39(2): 215-225. doi: 10.21656/1000-0887.380230

Abstract(690) PDF(520)

Abstract:
The problem of exponential outer synchronization between uncertain spatiotemporal networks with different numbers of nodes was studied. Firstly, based on the Lyapunov stability theorem, an appropriate controller was designed to realize exponential outer synchronization between uncertain spatiotemporal networks with different numbers of nodes. The adaptive law of the coupling matrix elements representing the topological structure of the network and the feedback strength was further identified. Finally, with the spatiotemporal network composed of the 1D Burgers system and the Logistic system as an example for numerical simulation. The results show that there exist stable external synchronization phenomena in the whole network. Furthermore, the synchronization speed depends on the adjustable parameters, and the number of network nodes does not affect the stability of the whole network synchronization. The proposed synchronization scheme has certain universality.

Spatial Dynamics of Periodic ReactionDiffusion Epidemic Models With Delay and Logistic Growth

WANG Shuangming, ZHANG Mingjun, FAN Xinman

2018, 39(2): 226-238. doi: 10.21656/1000-0887.370301

Abstract(1302) PDF(760)

Abstract:
The dynamics of periodic reactiondiffusion epidemic models with delay and logistic growth was investigated based on the theory of dynamic systems. Firstly, the existence of the global attractor of the ω operator associated with the periodic semiflow was proved. Next, the basic reproduction number of the model was introduced via the next generation operator. Finally, by means of the persistence theory and the comparison principle, the sufficient conditions for the disease persistence and extinction were obtained. If the basic reproduction number is less than 1, the diseasefree periodic solution will be globally asymptotically stable and the disease will go extinct. If the basic reproduction number is greater than 1, the system will be uniformly persistent and the disease will become endemic.

Modified-Projective-Synchronization of Memristor-Based Fractional-Order Delayed Neural Networks

ZHANG Weiwei, CHEN Dingyuan, WU Ranchao, CAO Jinde

2018, 39(2): 239-248. doi: 10.21656/1000-0887.370359

Abstract(1053) PDF(653)

Abstract:
The discussion of fractional-order memristor-based neural networks with time delay is a hot topic. The modified projective synchronization of fractional-order memristor-based neural networks with time delay was investigated. By means of the fractional-order inequality, sufficient conditions for the modified projective synchronization of drive-response systems were achieved. The results obtained here are more general. The corresponding numerical simulations show the feasibility of the theoretical results.