Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.     

Global Solvability in W 2 2,1 -Weighted Spaces of the Two-Dimensional Navier–Stokes Problem in Domains with Strip-Like Outlets to Infinity

The time-dependent Navier–Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over cross-sections of outlets to infinity which tends in each outlet to the corresponding time-dependent Poiseuille flow. The obtained results are proved for arbitrary large norms of the data (in particular, for arbitrary fluxes) and globally in time. The authors are supported by EC FP6 MC–ToK programme SPADE2, MTKD–CT–2004–014508.     

A Note on the Existence of Solutions to the Oseen System in Lipschitz Domains

We study the boundary-value problem associated with the Oseen system in the exterior of m Lipschitz domains of an euclidean point space We show, among other things, that there are two positive constants and α depending on the Lipschitz character of Ω such that: (i) if the boundary datum a belongs to L^q(∂Ω), with q ∈ [2,+∞), then there exists a solution (u, p), with and u ∈ L^∞(Ω) if a ∈ L^∞(∂Ω), expressed by a simple layer potential plus a linear combination of regular explicit functions; as a consequence, u tends nontangentially to a almost everywhere on ∂Ω; (ii) if a ∈ W^1-1/q,q(∂Ω), with then ∇u, p ∈ L^q(Ω) and if a ∈ C^0,μ(∂Ω), with μ ∈ [0, α), then also, natural estimates holds.     

Non-linear Heat and Mass Transfer During Convective Drying of Kaolin Cylinder Under Non-steady Conditions

The self-consistent model of heat and mass transfer during convective drying of capillary porous media describing both the first and the second periods of drying is presented in Musielak (Wydawnictwo Politechniki Poznańskiej, seria Rozprawy, nr 386 (2004a); Chem. Process Eng. 25, 393–409 (2004b)). The results of simulations of processes in steady conditions are shown (Musielak Wydawnictwo Politechniki Poznańskiej, seria Rozprawy, nr 386 (2004a); Chem. Process Eng. 25, 393–409 (2004b)). The main aim of the present work is to compare experimental results with those from numerical simulations. Three convective drying processes have been performed experimentally. The first and the second periods of drying are considered, during which the humidity of air in the dryer changes due to evaporation. The first process is used to establish drying parameters, whereafter the two remaining processes are simulated. Good agreement between experimental and simulation results is found, both qualitative and quantitative.     

A constrained theory for single crystal shape memory wires with application to restrained recovery

The theory of thin wires developed in Dret and Meunier (Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 337:143–147, 2003) is adapted to phase-transforming materials with large elastic moduli in the sense discussed in James and Rizzoni (J Elast 59:399–436, 2000). The result is a one-dimensional constitutive model for shape memory wires, characterized by a small number of material constants. The model is used to analyze self-accommodated and detwinned microstructures and to study superelasticity. It also turns out that the model successfully reproduces the behavior of shape memory wires in experiments of restrained recovery (Tsoi et al. in Mater Sci Eng A 368:299–310, 2004; Tsoi in 50:3535–3544, 2002; S̆ittner et al. in Mater Sci Eng A 286:298–311, 2000; vokoun in Smart Mater Struct 12:680–685, 2003; Zheng and Cui in Intermetallics 12:1305–1309, 2004; Zheng et al. in J Mater Sci Technol 20(4):390–394, 2004). In particular, the model is able to predict the shift to higher transformation temperatures on heating. The model also captures the effect of prestraining on the evolution of the recovery stress and of the martensite volume fraction.     

The Remarkable Nature of Cylindrically Anisotropic Elastic Materials Exemplified by an Anti-Plane Deformation

A material is cylindrically anisotropic when its elastic moduli referred to a cylindrical coordinate system are constants. Examples of cylindrically anisotropic materials are tree trunks, carbon fibers [1], certain steel bars, and manufactured composites [2]. Lekhnitskii [3] was the first one to observe that the stress at the axis of a circular rod of cylindrically monoclinic material can be infinite when the rod is subject to a uniform radial pressure (see also [4]). Ting [5] has shown that the stress at the axis of the circular rod can also be infinite under a torsion or a uniform extension. In this paper we first modify the Lekhnitskii formalism for a cylindrical coordinate system. We then consider a wedge of cylindrically monoclinic elastic material under anti-plane deformations. The stress singularity at the wedge apex depends on one material parameter γ. For a given wedge angle α, one can choose a γ so that the stress at the wedge apex is infinite. The wedge angle 2α can be any angle. It need not be larger than π, as is the case when the material is homogeneously isotropic or anisotropic. In the special case of a crack (2α=2π) there can be more than one stress singularity, some of them are stronger than the square root singularity. On the other hand, if γ < there is no stress singularity at the wedge apex for any wedge angle, including the special case of a crack. The classical paradox of Levy [6] and Carothers [7] for an isotropic elastic wedge also appears for a cylindrically anisotropic elastic wedge. There can be more than one critical wedge angle and, again, the critical wedge angle can be any angle. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new approach for the ellipsoidal statistical model

In this paper we aim to introduce a systematic way to derive relaxation terms for the Boltzmann equation based on the minimization problem for the entropy under moments constraints (Levermore in J. Stat. Phys. 83:1021–1065, 1996; Schneider in M2AN 38:541–561, 2004). In particular the moment constraints and corresponding coefficients are linked with the eigenfunctions and eigenvalues of the linearized collision operator through the Chapman–Enskog expansion. Then we deduce from this expansion a single relaxation term of BGK type. Here we stop the moments constraints at order two in the velocity v and recover the ellipsoidal statistical model (Holway in Rarefied Gas Dynamics, vol I, pp 193–215, 1966).      

Comment on the Paper: The CRONE Suspension: Management of the Dilemma Comfort-Road Holding (by Moreau et al.)

The linearisation procedure used in the modelling of the hydro-pneumatic suspension system by Moreau et al. contains an inaccuracy. Hence, the conclusion that the weight of the sprung mass is normally absent of the dynamic behaviour of the sprung mass is demonstrated to be false.In Nonlinear Dynamics 38: 461–484, 2004, Kluwer Academics Publishers.     

On variational features of vortex flows

Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones, the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997; Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion is discussed.      

Motion types and chaos of multi-body systems vibrating with impacts

In this paper, the limit sets theory for an autonomous dynamical system is generalized to a multi-body system vibrating with impacts. We discover that if every motion of the system is bounded, it has only four different types: periodic motion γ1, non·periodic recurrent motion γ2, and non-Poisson stable motions γ3 and γ4 approaching γ1 and γ2, respectively. γ2 is the source of chaos. It is very interesting that chaotic motions seem stochastic but possess the character of recurrence. By way of example, we discuss chaotic motions of a small ball bouncing vertically on a massive vibrating table. The result obtained by us is different from that obtained by Holmes. The project supported by National Natural Science Foundation of China     

Dynamics of a Non-Autonomous ODE System Occurring in Coagulation Theory

We consider a constant coefficient coagulation equation with Becker–D?ring type interactions and power law input of monomers J ₁(t) = α t ^ω, with α > 0 and . For this infinite dimensional system we prove solutions converge to similarity profiles as t and j converge to infinity in a similarity way, namely with either or constants, where is a function of t only. This work generalizes to the non-autonomous case a recent result of da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398. and provides a rigorous derivation of formal results obtained by Wattis J. Phys. A: Math. Gen. 37, 7823–7841. The main part of the approach is the analysis of a bidimensional non-autonomous system obtained through an appropriate change of variables; this is achieved by the use of differential inequalities and qualitative theory methods. The results about rate of convergence of solutions of the bidimensional system thus obtained are fed into an integral formula representation for the solutions of the infinite dimensional system which is then estimated by an adaptation of methods used by da Costa et al. (2004). Markov Processes Relat. Fields 12, 367–398.      

A Saint-Venant principle for nonlinear elasticity

This paper proves the global boundedness in time of higher moments to a weak solution of the non-linear space-homogeneous Boltzmann equation for inverse k^th-power forces with k ≧ 5. In the course of the proof a new collisional estimate of is obtained, where v, v_* are the velocities before and v′, v′_* are the velocities after a binary collision.     

Stability and Freezing of Nonlinear Waves in First Order Hyperbolic PDEs

It is a well-known problem to derive nonlinear stability of a traveling wave from the spectral stability of a linearization. In this paper we prove such a result for a large class of hyperbolic systems. To cope with the unknown asymptotic phase, the problem is reformulated as a partial differential algebraic equation for which asymptotic stability becomes usual Lyapunov stability. The stability proof is then based on linear estimates from (Rottmann-Matthes, J Dyn Diff Equat 23:365–393, 2011) and a careful analysis of the nonlinear terms. Moreover, we show that the freezing method (Beyn and Thümmler, SIAM J Appl Dyn Syst 3:85–116, 2004; Rowley et al. Nonlinearity 16:1257–1275, 2003) is well-suited for the long time simulation and numerical approximation of the asymptotic behavior. The theory is illustrated by numerical examples, including a hyperbolic version of the Hodgkin–Huxley equations.     

On the Mechanisms Behind Chaos

Chaotic systems are observed everywhere. Electronic circuit analogues based on the differential equations of the models for the chaotic systems are often used to study the nature of chaotic systems. This tutorial is an attempt to classify electronic chaotic oscillators according to the mechanism behind the chaotic behavior, e.g. one group is based on the sudden interrupt of inductive currents, another group is based on the sudden parallel coupling of capacitors with different voltages, and a third group may be based on multiplication of signals. An example of chaos based on disturbance of integration is discussed in details. Paper presented at ‘NDES 2004‘, Evora, Portugal, May 2004. IEEE life member.     

Use of particle image velocimetry to study heterogeneous drag reduction

Large polymer filaments can form when drag reducing polymers are injected through wall slots. The presence of these structures enhances the performance of the drag reducing function by mechanisms which are not understood. This paper shows how particle image velocimetry (PIV) techniques can be used to study changes in the configuration of the injected polymer and in the structure of the velocity field with increasing drag reduction. The filaments are found to behave as solid bodies which break up in high shear regions close to a boundary. The breakup process provides an explanation of why the filaments are not observed close to a wall and offers the possibility of providing a heterogeneous distribution of small aggregates of polymers which could be more effective than uniformly distributed molecules as suggested by Hoyer and Gyr (J Non-Newton Fluid Mech 65:221–240, 1996; J Fluids Eng 120:818–823, 1998), Dunlop and Cox (Phys Fluids 20:203–213, 1977) and Vlachogiannis et al. (Phys Fluid 15:3786–3794, 2004). PIV measurements show dramatic qualitative changes in the velocity patterns at maximum drag reduction.     

Systematic multiscale models for deep convection on mesoscales

This paper builds on recent developments of a unified asymptotic approach to meteorological modeling [ZAMM, 80: 765–777, 2000, SIAM Proc. App. Math. 116, 227–289, 2004], which was used successfully in the development of Systematic multiscale models for the tropics in Majda and Klein [J. Atmosph. Sci. 60: 393–408, 2003] and Majda and Biello [PNAS, 101: 4736–4741, 2004]. Biello and Majda [J. Atmosph. Sci. 62: 1694–1720, 2005]. Here we account for typical bulk microphysics parameterizations of moist processes within this framework. The key steps are careful nondimensionalization of the bulk microphysics equations and the choice of appropriate distinguished limits for the various nondimensional small parameters that appear. We are then in a position to study scale interactions in the atmosphere involving moist physics. We demonstrate this by developing two systematic multiscale models that are motivated by our interest in mesoscale organized convection. The emphasis here is on multiple length scales but common time scales. The first of these models describes the short-time evolution of slender, deep convective hot towers with horizontal scale ~ 1 km interacting with the linearized momentum balance on length and time scales of (10 km/3 min). We expect this model to describe how convective inhibition may be overcome near the surface, how the onset of deep convection triggers convective-scale gravity waves, and that it will also yield new insight into how such local convective events may conspire to create larger-scale strong storms. The second model addresses the next larger range of length and time scales (10 km, 100 km, and 20 min) and exhibits mathematical features that are strongly reminiscent of mesoscale organized convection. In both cases, the asymptotic analysis reveals how the stiffness of condensation/evaporation processes induces highly nonlinear dynamics. Besides providing new theoretical insights, the derived models may also serve as a theoretical devices for analyzing and interpreting the results of complex moist process model simulations, and they may stimulate the development of new, theoretically grounded sub-grid-scale parameterizations.     

Compressible Navier–Stokes Model with Inflow-Outflow Boundary Conditions

In the paper [7], author gives a definition of weak solution to the nonsteady Navier–Stokes system of equations which describes compressible and isentropic flows in some bounded region Ω with influx of fluid through a part of the boundary ∂Ω. Here, we present a way for proving existence of such solutions in the same situation as in [7] under the sole hypothesis γ > 3/2 for the adiabatic constant.     

L 1 and BV-type stability of the inelastic Boltzmann equation near vacuum

The L ¹ and BV-type stability to mild solutions of the inelastic Boltzmann equation is given in this paper. The result is an extension of the stability of the classical solution of the elastic Boltzmann equation proved in Ha (Arch. Ration. Mech. Anal. 173:25–42, 2004 [16]). The observation relies on the energy loss of the inelastic Boltzmann equation. This is a continuity work of Alonso (Indiana Univ. Math. J. [1]), where the author obtained the global existence of a mild solution for the inelastic Boltzmann equation. The proof is based on the mollification method and constructing some functionals as the one in Chae and Ha (Contin. Mech. Thermodyn. 17(7):511–524, 2006 [9]).     

High-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation

The dynamics and stability of the high-speed fiber spinning process with spinline flow-induced crystallization and neck-like deformation have been studied using a simulation model equipped with governing equations of continuity, motion, energy, and crystallinity, along with the Phan-Thien–Tanner constitutive equation. Despite the fact that a simple one-phase model was incorporated into the governing equations to describe the spinline crystallinity, as opposed to the best-known two-phase model [Doufas et al. J Non-Newton Fluid Mech, 92:27–66, 2000a]; [Kohler et al. J Macromol Sci Phys, 44:185–202, 2005] that treats amorphous and crystalline phases separately in computing the spinline stress, the simulation has successfully portrayed the typical nonlinear characteristic of the high-speed spinning process called neck-like spinline deformation. It has been found that the criterion for the neck-like deformation to occur on the spinline is for the extensional viscosity to decrease on the spinline, so that the spinning is stabilized by the formation of the spinline neck-like deformation. The accompanying linear stability analysis explains this stabilizing effect of the spinline neck-like deformation, corroborating a recent experimental finding [Takarada et al. Int Polym Process, 19:380–387, 2004].This paper was presented at the 2nd Annual European Rheology Conference 2005 on April 21–23, 2005, in Grenoble, France.     

Full 3D correlation tensor computed from double field stereoscopic PIV in a high Reynolds number turbulent boundary layer

The turbulence structure near a wall is a very active subject of research and a key to the understanding and modeling of this flow. Many researchers have worked on this subject since the fifties Hama et al. (J Appl Phys 28:388–394, 1957). One way to study this organization consists of computing the spatial two-point correlations. Stanislas et al. (C R Acad Sci Paris 327(2b):55–61, 1999) and Kahler (Exp Fluids 36:114–130, 2004) showed that double spatial correlations can be computed from stereoscopic particle image velocimetry (SPIV) fields and can lead to a better understanding of the turbulent flow organization. The limitation is that the correlation is only computed in the PIV plane. The idea of the present paper is to propose a new method based on a specific stereoscopic PIV experiment that allows the computation of the full 3D spatial correlation tensor. The results obtained are validated by comparison with 2D computation from SPIV. They are in very good agreement with the results of Ganapthisubramani et al. (J Fluid Mech 524:57–80, 2005a).