Reconstructing the transient,dissipative dynamics of a bistable Duffing oscillator with an enhanced averaging method and Jacobian elliptic functions

Systems characterized by the governing equation of the bistable, double-well Duffing oscillator are ever-present throughout the fields of science and engineering. While the prediction of the transient dynamics of these strongly nonlinear oscillators has been a particular research interest, the sufficiently accurate reconstruction of the dissipative behaviors continues to be an unrealized goal. In this study, an enhanced averaging method using Jacobian elliptic functions is presented to faithfully predict the transient, dissipative dynamics of a bistable Duffing oscillator. The analytical approach is uniquely applied to reconstruct the intrawell and interwell dynamic regimes. By relaxing the requirement for small variation of the transient, averaged parameters in the proposed solution formulation, the resulting analytical predictions are in excellent agreement with exact trajectories of displacement and velocity determined via numerical integration of the governing equation. A wide range of system parameters and initial conditions are utilized to assess the accuracy and computational efficiency of the analytical method, and the consistent agreement between numerical and analytical results verifies the robustness of the proposed method. Although the analytical formulations are distinct for the two dynamic regimes, it is found that directly splicing the inter- and intrawell predictions facilitates good agreement with the exact dynamics of the full reconstructed, transient trajectory.     

An Analytical Solution for 1-D Non-Darcy Flow Through Slanting Coarse Deposits

In this paper, an analytical solution to solve 1-D partial differential equation is presented for fully developed turbulent flow through highly permeable sloping deposited porous medium. The present solution will be applicable for a wide range of slopes varying from zero to relatively steep slopes. To confirm the solution, the analytical results have been validated using two sets of experimental data including rounded and crushed material. To see the compatibility of solution, a Darcy-based form of the solution is derived and compared with proposed solution and experimental data. The results showed a satisfactory agreement with experimental records from water surface profiles through rock cavities for both rounded and crushed rock materials. Finally, it may be concluded that the proposed solution could be used to analyze water surface profiles and normal depth in such slanting permeable porous media. This solution provides a reliable realization of the flow profiles in porous materials which are widely used in open-channel flow concepts.     

A Quasilinear Based Procedure for Saturated/Unsaturated Water Movement in Soils

The quasilinear form of Richards equation for one-dimensional unsaturated flow in soils can be readily solved for a wide variety of conditions. However, it cannot explain saturated/unsaturated flow and the constant diffusivity assumption, used to linearise the transient quasilinear equation, can introduce significant error. This paper presents a quasi-analytical solution to transient saturated/unsaturated flow based on the quasilinear equation, with saturated flow explained by a transformed Darcy's equation. The procedure presented is based on the modified finite analytic method. With this approach, the problem domain is divided into elements, with the element equations being solutions to a constant coefficient form of the governing partial differential equation. While the element equations are based on a constant diffusivity assumption, transient diffusivity behaviour is incorporated by time stepping. Profile heterogeneity can be incorporated into the procedure by allowing flow properties to vary from element to element. Two procedures are presented for the temporal solution; a Laplace transform procedure and a finite difference scheme. An advantage of the Laplace transform procedure is the ability to incorporate transient boundary condition behaviour directly into the analytical solutions. The scheme is shown to work well for two different flow problems, for three soil types. The technique presented can yield results of high accuracy if the spatial discretisation is sufficient, or alternatively can produce approximate solutions with low computational overheads by using large sized elements. Error was shown to be stable, linearly related to element size.     

Kramers’ problem and the Knudsen minimum: a theoretical analysis using a linearized 26-moment approach

A set of linearized 26 moment equations, along with their wall boundary conditions, are derived and used to study low-speed gas flows dominated by Knudsen layers. Analytical solutions are obtained for Kramers’ defect velocity and the velocity-slip coefficient. These results are compared to the numerical solution of the BGK kinetic equation. From the analysis, a new effective viscosity model for the Navier–Stokes equations is proposed. In addition, an analytical expression for the velocity field in planar pressure-driven Poiseuille flow is derived. The mass flow rate obtained from integrating the velocity profile shows good agreement with the results from the numerical solution of the linearized Boltzmann equation. These results are good for Knudsen numbers up to 3 and for a wide range of accommodation coefficients. The Knudsen minimum phenomenon is also well captured by the present linearized 26-moment equations.     

Investigation of developed flow of an incompressible conductive fluid in a circular pipe by using the equation for the turbulent viscosity

The equation for the turbulent viscosity is used to investigate the developed flow of a conductive fluid in a longitudinal magnetic field. The solution of this equation is analyzed in the transition region from laminar to turbulent flow. The influence of the magnetic field on the local and integrated flow characteristics is studied. The convective heat exchange is computed in the case of liquid-metal flow with constant heat flux to the wall. It is shown that the computed results are in good agreement with the results of experiments in a broad range of variation of the governing parameters.     

Linear Stability Analysis of Infiltration,Analytical and Numerical Solution

Linear stability analysis is applied to the Richards equation by perturbing the pressure field. An analytical solution of the stability problem of flow through stratified media is presented. It is obtained under two simplifying assumptions: the quasi-steady hypothesis and the quasi-linear hypothesis. Flow is found to be unconditionally stable. A numerical experiment and the comparison with published data supports the conclusion that upscaling capillary phenomena is crucial in order to capture the essence of finger flow by continuum models.     

Generalized Plain Couette Flow and Heat Transfer in a Composite Channel

An analytical study of fluid flow and heat transfer in a composite channel is presented. The channel walls are maintained at different constant temperatures in such a way that the temperatures do not allow for free convection. The upper plate is considered to be moving and the lower plate is fixed. The flow is modeled using Darcy–Lapwood–Brinkman equation. The viscous and Darcy dissipation terms are included in the energy equation. By applying suitable matching and boundary conditions, an exact solution has been obtained for the velocity and temperature distributions in the two regions of the composite channel. The effects of various parameters such as the porous medium parameter, viscosity ratio, height ratio, conductivity ratio, Eckert number, and Prandtl number on the velocity and temperature fields are presented graphically and discussed.     

Shale-Gas Permeability and Diffusivity Inferred by Improved Formulation of Relevant Retention and Transport Mechanisms

A theoretically improved model incorporating the relevant mechanisms of gas retention and transport in gas-bearing shale formations is presented for determination of intrinsic gas permeability and diffusivity. This is accomplished by considering the various flow regimes according to a unified Hagen–Poiseuille-type equation, fully compressible treatment of gas and shale properties, and numerical solution of the non-linear pressure equation. The present model can accommodate a wide range of fundamental flow mechanisms, such as continuum, slip, transition, and free molecular flow, depending on the prevailing flow conditions characterized by the Knudsen number. The model indicates that rigorous determination of shale-gas permeability and diffusivity requires the characterization of various important parameters included in the present phenomenological modeling approach, many of which are not considered in previous studies. It is demonstrated that the improved model matches a set of experimental data better than a previous attempt. It is concluded that the improved model provides a more accurate means of analysis and interpretation of the pressure-pulse decay tests than the previous models which inherently consider a Darcian flow and neglect the variation of parameters with pressure.     

DTM-BF method and dual solutions for unsteady MHD flow over permeable shrinking sheet with velocity slip

An unsteady magnetohydrodynamic (MHD) boundary layer flow over a shrinking permeable sheet embedded in a moving viscous electrically conducting fluid is investigated both analytically and numerically. The velocity slip at the solid surface is taken into account in the boundary conditions. A novel analytical method named DTMBF is proposed and used to get the approximate analytical solutions to the nonlinear governing equation along with the boundary conditions at infinity. All analytical results are compared with those obtained by a numerical method. The comparison shows good agreement, which validates the accuracy of the DTM-BF method. Moreover, the existence ranges of the dual solutions and the unique solution for various parameters are obtained. The effects of the velocity slip parameter, the unsteadiness parameter, the magnetic parameter, the suction/injection parameter, and the velocity ratio parameter on the skin friction, the unique velocity, and the dual velocity profiles are explored, respectively.     

Computation of head–disk interface gap micro flowfields using DSMC and continuum–atomistic hybrid methods

This paper discusses computational modeling of micro flow in the head–disk interface (HDI) gap using the direct simulation Monte Carlo (DSMC) method. Modeling considerations are discussed in detail both for a stand‐alone DSMC computation and for the case of a hybrid continuum–atomistic simulation that couples the Navier–Stokes (NS) equation to a DSMC solver. The impact of the number of particles and number of cells on the accuracy of a DSMC simulation of the HDI gap is investigated both for two‐ and three‐dimensional configurations. An appropriate implicit boundary treatment method for modeling inflow and outflow boundaries is used in this work for a three‐dimensional DSMC micro flow simulation. As the flow outside the slider is in the continuum regime, a hybrid continuum–atomistic method based on the Schwarz alternating method is used to couple the DSMC model in the slider bearing region to the flow outside the slider modeled by NS equation. Schwarz coupling is done in two dimensions by taking overlap regions along two directions and the Chapman–Enskog distribution is employed for imposing the boundary condition from the continuum region to the DSMC region. Converged hybrid flow solutions are obtained in about five iterations and the hybrid DSMC–NS solutions show good agreement with the exact solutions in the entire domain considered. An investigation on the impact of the size of the overlap region on the convergence behavior of the Schwarz method indicates that the hybrid coupling by the Schwarz method is weakly dependent on the size of the overlap region. However, the use of a finite overlap region will facilitate the exchange of boundary conditions as the hybrid solution has been found to diverge in the absence of an overlap region for coupling the two models. Copyright © 2009 John Wiley & Sons, Ltd.     

Laminar boundary layers of co-current gas-liquid stratified flows—I. Theory

The flow in the developing laminar boundary layers on a flat moving gas-liquid interface is studied theoretically. Analysis of the equations shows that flows depend on two parameters, each less than unity. Using a perturbation method, an analytical solution with accuracy to the third power of the small parameters is obtained and is in close agreement with the exact solution of the problem.     

Thermal analysis of an infinite tube during quenching

 This paper deals with a numerical solution of the two-dimensional convection–diffusion equation in an infinite domain, arising out of quenching of an infinite tube. On the wetted side, upstream of the quench front, a constant heat transfer coefficient is assumed. The downstream of the quench front as well as the inside surface of the tube are assumed to be adiabatic. The solution gives the quench front temperature as a function of various model parameters such as Peclet number, Biot number and the radius ratio. The solution has been found to be in good agreement with the available analytical solutions and thus validates the numerical procedure suggested. Received on 10 July 2000     

Using the perturbation method to solve the problem of separated incompressible flow past thin airfoils

A model for separated incompressible flow past thin airfoils in the neighborhood of the “shockless entrance” condition is constructed based on the averaging of the vortex shedding flow past the airfoil edges. By approximation of the vortex shedding by two vortex curves, determination of the average hydrodynamic parameters is reduced to a twofold solution of an integral singular equation equivalent to the equation describing steady-state nonseparated airfoil flow. In this case, the calculation time is two orders of magnitude smaller than the time required for the solution of the corresponding evolution problem. The results of a test calculation using the proposed method are in fair agreement with available results of calculations and experiments. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 49–63, May–June, 2006.     

A Numerical Study of Laminar Reciprocating Flow in a Pipe of Finite Length

A numerical investigation has been carried out for a laminar incompressible reciprocating flow in a circular pipe with a finite length. An examination of the governing equations and boundary conditions indicates that a sinusoidally reciprocating flow is governed by three similarity parameters: the kinetic Reynolds number Re, the dimensionless oscillation amplitude A_o, and the length to diameter ratio L/D. The numerical solution for the velocity profiles of a developing reciprocating flow shows that at any instant of time, there exist three flow regimes in the pipe, namely, an entrance regime, a fully developed regime and an exit regime. The numerical results for the fully developed region are shown to be in excellent agreement with the analytical solution. Based on the numerical results, a correlation equation of the space-cycle averaged friction coefficient for a laminar developing reciprocating pipe flow has been obtained in terms of the three similarity parameters.     

Wave dispersion in randomly layered materials

Wave scattering in materials composed of two kinds of alternating layers with different elastic properties and randomly distributed thicknesses has been modeled. The general form of the dispersion equation is derived for the unbounded layered medium. It defines two basic macroscopic characteristics of the scattered wave: phase velocity and attenuation, which are explicit functions of wave frequency and microscopic parameters of the system: acoustic properties of the layers and stochastic characteristics of their thickness distributions. The analytical expressions are derived for three special cases: for long waves; for a periodic medium composed of layers with constant thicknesses and for random medium with uniform distribution of layer thicknesses. Special attention is paid to the analysis of the frequency dependence of the wave parameters. It was shown that the predictions of the model for long waves and for periodic medium are compatible with the results obtained in the literature.Moreover, comparison of theoretical results for frequency dependent wave parameters with numerical simulations of pulse transmission through the slab of the randomly layered medium shows good qualitative and quantitative agreement in wide frequency range.     

Analytical solution of velocity distribution for flow through submerged large deflection flexible vegetation

An analytical solution for predicting the vertical distribution of streamwise mean velocity in an open channel flow with submerged flexible vegetation is proposed when large bending occurs. The flow regime is separated into two horizontal layers: a vegetation layer and a free water layer. In the vegetation layer, a mechanical analysis for the flexible vegetation is conducted, and an approximately linear relationship between the drag force of bending vegetation and the streamwise mean flow velocity is observed in the case of large deflection, which differes significantly from the case of rigid upright vegetation. Based on the theoretical analysis, a linear streamwise drag force-mean flow velocity expression in the momentum equation is derived, and an analytical solution is obtained. For the free water layer, a new expression is presented, replacing the traditional logarithmic velocity distribution, to obtain a zero velocity gradient at the water surface. Finally, the analytical predictions are compared with published experimental data, and the good agreement demonstrates that this model is effective for the open channel flow through the large deflection flexible vegetation.     

双层石墨烯系统稳态受迫振动问题中的辛方法

针对悬臂石墨烯系统提出一种求解其稳态受迫振动问题的辛解析方法。基于Eringen非局部理论,将石墨烯层板受迫振动问题导入哈密顿体系。采用边界条件分解技术,将问题化为三种边界条件的子问题。通过辛解析方法,得到由辛本征值和辛本征解表示的双层石墨烯系统受迫振动问题的解析解表达式。数值结果表明,辛本征解级数具有很好的收敛性和精度,并与文献结果吻合;在一定的外载激励下可发生同向振动模式和反向振动模式;在一定的参数下,得到一些新的现象和结论。     

Pseudospectral calculations of stress-induced concentration changes in viscometric flows of polymer solutions

We investigate the flow of dilute polymer solutions in viscometric devices where the polymer concentration is allowed to vary due to the existence of nonzero stress gradients. A spectral collocation method is used in order to solve for the axisymmetric steady-state solution of the full set of the governing equations. Two alternative continuum formulations are considered based on a single- and two-fluid concept, respectively. For the parallel-plate flow problem and for the two-fluid model, the analytical solution obtained by Brunn (1984) is recovered. This solution describes a polymer migration toward the centerline that depends solely on the Weissenberg number. The results obtained for the single-fluid model are qualitatively similar; however, they also depend on additional molecular parameters. Variations in the geometry which initiate even a small secondary recirculation are found to result in a significant decrease in the concentration gradients, as shown in the analysis of the cone-and-plate flow for a small angular inclination and a large centerline gap thickness. When the centerline gap thickness decreases, the results are dominated by the ideal, purely azimuthal cone-and-plate flow behavior, according to which substantial variations in the radial concentration are observed mainly close to the centerline. This is in agreement with the predictions of Aubert et al. (1980) and Brunn (1984).The authors would like to acknowledge the financial support of NSF, Grant No. CTS-9114508 (to V.G.M. and A.N.B.), and the Center of Composite Materials for a Center-Industry fellowship (to A.S.). The computational support provided by the Pittsburgh Supercomputing Center is also gratefully acknowledged.