The Jacobiator of Nonholonomic Systems and the Geometry of Reduced Nonholonomic Brackets

In this paper, we consider the Hamiltonian formulation of nonholonomic systems with symmetries and study several aspects of the geometry of their reduced almost Poisson brackets, including the integrability of their characteristic distributions. Our starting point is establishing global formulas for the nonholonomic Jacobiators, before and after reduction, which are used to clarify the relationship between reduced nonholonomic brackets and twisted Poisson structures. For certain types of symmetries (generalizing the Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and analyze situations in which the reduced nonholonomic brackets arise by applying a gauge transformation to these Poisson structures. We illustrate our results with mechanical examples, and in particular show how to recover several well-known facts in the special case of Chaplygin symmetries.     

Global Classical Solutions of the Relativistic Vlasov–Darwin System with Small Cauchy Data: The Generalized Variables Approach

We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov–Darwin (RVD) system globally in time. A similar result is claimed in Seehafer (Commun Math Sci 6:749–769, 2008) following the work in Pallard (Int Mat Res Not 57191:1–31, 2006). Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transversal component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and momentum variables. By doing so, we produce a simple a priori estimate on the transversal component of the electric field. We widen the functional space required for the Cauchy datum to extend the solution globally in time, and we improve decay estimates given in Seehafer (2008) on the electromagnetic field and its space derivatives. Our method extends the constructive proof presented in Rein (Handbook of differential equations: evolutionary equations, vol 3. Elsevier, Amsterdam, 2007) to solve the Cauchy problem for the Vlasov–Poisson system with a small initial datum.     

Structural Optimization of Thin Elastic Plates: The Three Dimensional Approach

The natural way to find the most compliant design of an elastic plate is to consider the three-dimensional elastic structures which minimize the work of the loading term, and pass to the limit when the thickness of the design region tends to zero. In this paper, we study the asymptotics of such a compliance problem, imposing that the volume fraction remains fixed. No additional topological constraint is assumed on the admissible configurations. We determine the limit problem in different equivalent formulations, and we provide a system of necessary and sufficient optimality conditions. These results were announced in Bouchitté et al. (C. R. Acad. Sci. Paris, Ser. I. 345:713–718, 2007). Furthermore, we investigate the vanishing volume fraction limit, which turns out to be consistent with the results in Bouchitté and Fragalà (Arch. Rat. Mech. Anal. 184:257–284, 2007; SIAM J. Control Optim. 46:1664–1682, 2007). Finally, some explicit computation of optimal plates are given.     

Experimental Observation and Measurement for Domain Switching Emission Wave and its Average Velocity Emitted from the Near Tip Zone of a Static Crack in PLZT

In this paper the strain-birefringence correlation technique (Lines and 1; 2; Scott, Ferroelectr Rev 1:1–129 3) is used to measure the optical birefringence quantities and the corresponding electric and mechanical quantities in a cracked PLZT-8/65/35 to study the electrical creep problem near a static crack tip. Experimental observations reveal that the domain switching emission wave (abbreviated as DSEW) is emitted from the crack tip as the time increases from 0 to 1,500 s. The measured DSEW spreads very slowly and the average speed of the wave is measured to be approximately1.25 μm/s. Although the wave has apparent anisotropic features emitted from the crack tip, the average wave speeds along three different directions from the tip show small differences. After 1,500 s experimental observations reveal that the DSEW occupies almost the whole region under consideration in the specimen. This phenomenon has not been reported in the literature. Some brief discussions are presented.     

A non isothermal Ginzburg-Landau model for phase transitions in shape memory alloys

A thermodynamical model for martensitic phase transitions in shape memory alloys is formulated in this paper in the framework of the Ginzburg-Landau approach to phase transitions. A single order parameter is chosen to represent the austenite parent phase and two mirror related martensite variants. A free energy previously proposed in the literature (Levitas et al. in Phys. Rev. B 66:134206, 2002; Phys. Rev. B 66:134207, 2002; Phys. Rev. B 68:134201, 2003) is employed, in its simplest form, as the main constitutive content of the model. In this paper we treat time-dependent Ginzburg-Landau equation as a balance law on the structure order and we couple it to a energy balance equation, thus allowing to account of heat transfer processes. We obtain a coupled thermo-mechanical problem whose consistency with the Second Law is verified.     

Cavitation structures formed during the rebound of a sphere from a wetted surface

We use high-speed imaging to observe the dynamics of cavitation, caused by the impact and subsequent rebound of a sphere from a solid surface covered with a thin layer of highly viscous liquid. We note marked qualitative differences between the cavitation structures with increase in viscosity, as well as between Newtonian and non-Newtonian liquids. The patterns observed are quite unexpected and intricate, appearing in concentric ring formations around the site of impact. In all cases, we identify a distinct radius from which the primary bubbles emanate. This radius is modelled with a modified form of Hertz contact theory. Within this radius, we show that some fine cavitation structure may exist or that it may be one large cavitation bubble. For the non-Newtonian fluids, we observe foam-like structures extending radially with diminishing bubble sizes with increase in radial position. Whereas for the Newtonian fluids, the opposite trend is observed with increasing bubble size for increasing radial position. Finally, we compare our experimental observations of cavitation to the maximum tension criterion proposed by Joseph (J Fluid Mech 366:367–378, 1998) showing that this provides the lower limit for the onset of cavitation in our experiments.     

Elastic Fields for the Ellipsoidal Cavity Problem

Lur’e (Three-dimensional Problem of the Theory of Elasticity. Interscience, New York, 1964, §6.9) presented an approach to solve the problem of an ellipsoidal cavity in a linear, elastic and isotropic medium loaded by uniform principal stresses at infinity. In this paper we show that the approach by Lur’e may have no solution. Derivation mistakes are first pointed out in his (6.9.22), (6.9.23), (6.9.30) and (6.9.31). With the correct expressions, we then prove the coefficient matrix in his (6.9.32) to be singular. Therefore constants A,A ₄,A ₅ may have no solution. The problem lies in the harmonic functions chosen by Lur’e for the Papkovich-Neuber solution. From the solutions obtained by the Eshelby equivalent inclusion method, the present paper derives new Papkovich-Neuber harmonic functions for the ellipsoidal cavity problem.     

The weak shear kinetic phase diagram for nematic polymers

We study the shear problem for nematic polymers as modeled by the molecular kinetic theory of Doi (1981), focusing on the anomalous slow flow regime. We provide the kinetic phase diagram of monodomain (MD) attractors and phase transitions vs normalized nematic concentration (N) and weak normalized shear rate (Peclet number, Pe). We then overlay all rheological features typically reported in experiments: alignment properties, normal stress differences and shear stress. These features play a critical role in the synthesis between theory and experiment for nematic polymers (Larson 1999; Doi and Edwards 1986). MD type is routinely used for rheological shear characterization: cf., flow-aligning 5CB (Mather et al. 1996a), tumbling PBT (Srinivasarao and Berry 1991), and 8CB (Mather et al. 1996b), evidence for a wagging regime (Mewis et al. 1997), out-of-plane kayaking modes (Larson and Ottinger 1991), and evidence for chaotic major director dynamics (Bandyopadhyay et al. 2000). MD transitions correlate with sign changes in normal stresses (Larson and Ottinger 1991; Magda et al. 1991; Kiss and Porter 1978, 1980). Furthermore, structure formation in shear devices appears to be correlated with monodomain precursor dynamics (Tan and Berry 2003; Forest et al. 2002a). In this paper we combine seminal kinetic theory results (Kuzuu and Doi 1983, 1984; Larson 1990; Larson and Ottinger 1991; Faraoni et al. 1999; Grosso et al. 2001), symmetry observations (Forest et al. 2002b), and mesoscopic results on the fate of orientational degeneracy in weak shear (Forest and Wang 2003; Forest et al. 2003a), together with our resolved numerical simulations, to provide the kinetic flow-phase diagram of Doi theory in the weak shear regime, 0<Pe<1, for infinitely thin rods. We report the "birth" of key rheological features at the onset of flow: sign changes and local maxima and minima in normal stress differences (N₁ and N₂) associated with MD transitions. These results serve as the basis for continuation of the kinetic phase diagram to Pe>1 ; as the definitive benchmark for any mesoscopic or continuum model; and experimental data can be compared in order to determine accuracy and limitations of the Doi theory in weak shear.     

Hamiltonian structures of dynamics of a gyrostat in a gravitational field

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.     

The equations of motion for a nonholonomic system under the action of impulsive forces

The equations of impact for a nonholonomic system described with generalized coordinated have been discussed in detail in the general references of classical dynamics. But these equations contain undetermined multipliers which made the problem complicated. Through the appropriate treatment of mathematics, using the -function and expression of matrix in this paper, the author derived equations of impact for a nonholonomic system without undetermined multipliers. Therefore, the problem can be solved more simply.     

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E _∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.     

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.     

The Cauchy Problem for the 3-D Vlasov–Poisson System with Point Charges

In this paper we establish global existence and uniqueness of the solution to the three-dimensional Vlasov–Poisson system in the presence of point charges with repulsive interaction. The present analysis extends an analogous two-dimensional result (Caprino and Marchioro in Kinet. Relat. Models 3(2):241–254, 2010).     

Reynolds Number Effects on the Coherent Dynamics of the Turbulent Horseshoe Vortex System

The adverse pressure gradient induced by a surface-mounted obstacle in a turbulent boundary layer causes the approaching flow to separate and form a dynamically rich horseshoe vortex system (HSV) in the junction of the obstacle with the wall. The Reynolds number of the flow (Re) is one of the important parameters that control the rich coherent dynamics of the vortex, which are known to give rise to low-frequency, bimodal fluctuations of the velocity field (Devenport and Simpson, J Fluid Mech 210:23–55, 1990; Paik et al., Phys Fluids 19:045107, 2007). We carry out detached eddy simulations (DES) of the flow past a circular cylinder mounted on a rectangular channel for Re = 2.0 × 10⁴ and 3.9 × 10⁴ (Dargahi, Exp Fluids 8:1–12, 1989) in order to systematically investigate the effect of the Reynolds number on the HSV dynamics. The computed results are compared with each other and with previous experimental and computational results for a related junction flow at a much higher Reynolds number (Re = 1.15 × 10⁵) (Devenport and Simpson, J Fluid Mech 210:23–55, 1990; Paik et al., Phys Fluids 19:045107, 2007). The computed results reveal significant variations with Re in terms of the mean-flow quantities, turbulence statistics, and the coherent dynamics of the turbulent HSV. For Re = 2.0 × 10⁴ the HSV system consists of a large number of necklace-type vortices that are shed periodically at higher frequencies than those observed in the Re = 3.9 × 10⁴ case. For this latter case the number of large-scale vortical structures that comprise the instantaneous HSV system is reduced significantly and the flow dynamics becomes quasi-periodic. For both cases, we show that the instantaneous flowfields are dominated by eruptions of wall-generated vorticity associated with the growth of hairpin vortices that wrap around and disorganize the primary HSV system. The intensity and frequency of these eruptions, however, appears to diminish rapidly with decreasing Re. In the high Re case the HSV system consists of a single, highly energetic, large-scale necklace vortex that is aperiodically disorganized by the growth of the hairpin mode. Regardless of the Re, we find pockets in the junction region within which the histograms of velocity fluctuations are bimodal as has also been observed in several previous experimental studies.     

Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model

Linear viscoelastic (LVE) measurements as well as non-linear elongation measurements have been performed on stoichiometrically imbalanced polymeric networks to gain insight into the structural influence on the rheological response (Jensen et al., Rheol Acta 49(1):1–13, 2010). In particular, we seek knowledge about the effect of dangling ends and soluble structures. To interpret our recent experimental results, we exploit a molecular model that can predict LVE data and non-linear stress–strain data. The slip-link model has proven to be a robust tool for both LVE and non-linear stress–strain predictions for linear chains (Khaliullin and Schieber, Phys Rev Lett 100(18):188302–188304, 2008, Macromolecules 42(19):7504–7517, 2009; Schieber, J Chem Phys 118(11):5162–5166, 2003), and it is thus used to analyze the experimental results. Initially, we consider a stoichiometrically balanced network, i.e., all strands in the ensemble are attached to the network in both ends. Next we add dangling strands to the network representing the stoichiometric imbalance, or imperfections during curing. By considering monodisperse network strands without dangling ends, we find that the relative low-frequency plateau, G₀/G_N⁰G_0/G_N^0, decreases linearly with the average number of entanglements. The decrease from G_N⁰G_N^0 to G ₀ is a result of monomer fluctuations between entanglements, which is similar to “longitudinal modes” in tube theory. It is found that the slope of G′ is dependent on the fraction of network strands and the structural distribution of the network. The power-law behavior of G ^″ is not yet captured quantitatively by the model, but our results suggest that it is a result of polydisperse dangling and soluble structures.     

Blast Wave Mitigation by a Particulate Foam Barrier

Blast waves mitigate in foam due to various mechanisms, whose contribution to the final result is not fully understood yet. Actually, blast waves can destroy the foam barrier that is usually prone to decay and thus subsides with time. Fortunately, different time scales allow separating between these processes. The foam shattering, for example, could be completed within several milliseconds, while the foam decay lasts minutes and even hours. Recently, an increasing interest in this area has emerged, because particle-laden foams are much more stable and thus, could be applied for blast wave protection. To explore the full advantage of these new foams, the relationship between the micro-properties of the foam structure and the blast wave mitigation has to be clarified. In order to specify this relationship, little has been done. Information available in the literature on this subject clearly shows that during the test, the foam structure could be changed in a wide range, which is not usually controlled. This complicates the analysis of the occurring processes and ensures that the new factor involved in the studied problem has to be tested one by one, after the result of the previous step is well understood. To follow this strategy, this study continues our previous investigation (Britan et al in Colloid Surf A Physicochem Eng Aspects 309:137–150, 2007; Colloid Surf A Physicochem Eng Aspects 344:15–23, 2009; 2011), while mainly focusing on a single new factor, namely blast-shaped profile. To separate out the effect of the foam decay, which was discussed elsewhere (Britan et al in 2011), a special effort has been spent to ensure that the tested foam is homogeneous over its height. To exclude the bubble shattering, preference was given to weak impact conditions (Mach number of the shock generated inside the shock tube was about M _S  = 1.05). Under these circumstances, the blast wave mitigation inside the tested foam barrier solely depended on the concentration of the solid additives.     

Marangoni instabilities in small circular containers under microgravity

Circular containers of various aspect ratios a with flat free upper liquid surfaces were heated from below under microgravity to generate the Marangoni instability (MI). We realized “liquid lateral sidewalls” for the containers to come near to the “slippery sidewalls” introduced by Rosenblat et al. (J Fluid Mech 120:91–122, 1982a) and Echebarría et al. (Physica D 99:487–502, 1997), henceforth referred to as RHD and EKP, respectively. The flow structure was visualized by aluminium flakes and recorded on videotape. The MI was clearly observed in all containers above a critical Marangoni number Ma ^c which depends on a. In the first microgravity experiment in a container with a=7.5, we found significant convective heat transport and reported a Nusselt number Nu=1.8 for Ma=4×Ma ^c . In a second microgravity experiment with containers with a=0.5, 0.75, 1.0, 1.5, 2.0, 4.0 and 5.0, various flow structures (azimuthal and radial wave numbers) were observed, depending on a and Ma. The observed scenario compares qualitatively well with the stability curves calculated by RHD and EKP. Frequent switching between modes (2,1) and (1,1) was observed in the container with a=2 at supercritical Ma that is exactly the case for which this behaviour was predicted by EKP for reduced gravity.     

论轮式移动结构的非完整约束及其运动控制

当质点沿光滑曲线运动时,必须满足横向速度为零的条件.同样地,不同轮式移动结构在平面上做光滑曲线运动时都需要满足该非完整约束条件.本文结合轮子转速和它们运动速度的完整约束关系,理清各轮式移动结构的完整和非完整约束,然后利用 Euler-Lagrange 方程方便地推导出相应的动力学方程.另外,通过该非完整约束,将目标轨迹曲线转化为速度目标的形式,然后引入目标轨迹曲线的相对曲率设计合适的动态跟踪目标.进一步,通过采用该动态跟踪目标可以将轮式移动结构的运动规律和动力学方程有机结合起来,并将原运动任务简化为一般的 轨迹 控制问题.基于该动态跟踪目标可以为轮式移动结构设计合适的鲁棒跟踪控制器,通过跟踪目标轨迹曲线的相对曲率来实现对目标曲线的精确跟踪.最后,理论分析和仿真结果显示,采用动态目标跟踪方法能够从根本上解决初始速度误差过大和位置误差不断被累积的问题,即使前向速度误差系统不稳定的,实际运动轨迹也几乎能和目标轨迹曲线重合.      

Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints

This paper is concerned with the dynamics of a mechanical system subject to nonintegrable constraints. In the first part, we prove the equivalence between the classical nonholonomic equations and those derived from the nonholonomic variational formulation, proposed by Kozlov in [10–12], for a class of constrained systems with constraints transverse to a foliation. This result extends the equivalence between the two formulations, proved for holonomic constraints, to a class of linear nonintegrable ones. In the second part, we derive the nonholonomic variational reduced equations for a constrained system with symmetry and constraint transverse to a principal bundle fibration, using a reduction procedure similar to the one developed in [5]. The resulting equations are compared with the nonholonomic reduced ones through mechanical examples.     

Equivalent Fracture Network Permeability of Multilayer-Complex Naturally Fractured Reservoirs

Modeling naturally fractured reservoirs (NFRs) requires an accurate representation of fracture network permeability (FNP). Conventionally, logs, cores, seismic, and pressure transient tests are used as a data base for this. Our previous attempts showed that a strong correlation exists between the fractal parameters of 2-D fracture networks and their permeability (Jafari and Babadagli, SPE 113618, Western Regional and Pacific Section AAPG joint meeting, 2008; Jafari and Babadagli, SPE Reserv Eval Eng 12(3):455–469, 2009a). We also showed that 1-D well (cores-logs) and 3-D reservoir data (well test) may not be sufficient in FNP mapping and that 2-D (outcrop) characteristics are needed (Jafari and Babadagli, SPE 124077, SPE/EAGE reservoir characterization and simulation conference, 2009b). This paper is an extension of those studies, where only 2-D (single-layer, uniform fracture characteristics in z direction) representations were used. In this paper, we considered a more complex and realistic 3-D network system. Two-dimensional random fractures with known fractal and statistical characteristics were distributed in the x- and y directions. A variation of fracture network characteristics in the z direction was presented by a multilayer system representing three different facieses with different fracture properties. Wells were placed in different locations of the model to collect 1-D fracture density and pressure transient data. In addition, five different fractal and statistical properties of the network of each layer were measured. The equivalent FNP was calculated using a commercial software package as the base case. Using available 1-D, 2-D, and 3-D data, multivariable regression analyses were performed to obtain equivalent FNP correlations for many different fracture network realizations. The derived equations were validated against a new set of synthetic fracture networks, and the conditions at which 1-D, 2-D and 3-D data are sufficient to map FNP were determined. The importance of the inclusion of each data type, i.e., 1-D, 2-D and 3-D, in the correlations was discussed. It was shown that using only 3-D data are insufficient to predict the FNP due to wide spatial heterogeneity of the fracture properties in the reservoir, which cannot be captured from single-well tests. Incorporating all types of data (1-D, 2-D, and 3-D) would result in better prediction. Also, it is recommended that the 2-D data of the most conductive layer in reservoir, which has longer fractures with a higher density, should be incorporated in the correlations.