Oscillatory Buckling Mode in Thin-Film Nucleation

We are interested in switching of elongated thin-film elements as described by the micromagnetic model. Nucleation occurs when the saturation branch becomes unstable at a critical field. It is characterized by a degeneracy of the Hessian of the micromagnetic energy. The degenerate subspace describes the unstable mode. In a prior work , we showed that there are four regimes for nucleation. We proved this by identifying the scaling of the critical field in the non-dimensional parameters. This contradicts a claim by Aharoni that there are at most three regimes. Two of these regimes are buckling regimes, where the magnetization is pinned at the lateral facets of the sample and buckles in-plane. As shown in our prior work, these regimes differ in the scaling of the critical field. Here we show that the unstable modes are also qualitatively different: Only in one of the regimes do they oscillate in the long direction. The unstable modes are asymptotically identified by Γ-convergence, which is applied to the Rayleigh quotient of the Hessian.     

The Concertina Pattern: A Bifurcation in Ferromagnetic Thin Films

The concertina pattern is a metastable stage in the switching process of elongated thin-film elements. It is an approximately periodic structure of domains, separated by walls perpendicular to the long axis of the element. In this paper, we give arguments in favor of our claim that the period is frozen-in at nucleation, i.e., at the critical external field. In prior work, R. Cantero-Alvarez and F. Otto, Journal of Nonlinear Science (2006), we argued that there are four qualitatively different regimes for nucleation. In one of these asymptotic regimes, the unstable mode displays an oscillatory behavior in the direction of the long axis. In this work, we derive a scaling limit of the micromagnetic energy near the bifurcation point in the above regime. We also prove that the scaling limit is coercive for all values of the reduced external field. Because of this coercivity, there exists a branch of nontrivial local minimizers. Numerical minimization of the scaling limit reveals that this branch is indeed a continuous branch of concertina pattern. The scaling limit is derived by Γ-convergence of the suitably rescaled micromagnetic energy. This robust procedure combines the limit of an asymptotic parameter regime with a zoom-in in configuration space. The coercivity of the scaling limit is derived by suitable nonlinear interpolation estimates.     

Travelling Waves with a Singularity in Magnetic Nanowires

We model the evolution of the magnetization in an infinite cylinder by harmonic map heat flow with an additional external field. Using variational methods, we prove the existence of corotationally symmetric travelling wave solutions with a moving vortex. We moreover show that for weak and strong fields the travelling waves connect the original state anti-parallel to the external magnetic field with the totally reversed state in direction of the external field. Our results match numeric simulations. For thicker wires several groups have found a reversal mode where a domain wall with a corotational symmetry and a vortex is propagating through the wire.     

柱状纤维上超薄液膜的线性稳定性分析

运用线性理论分析了粘性超薄液膜沿柱状纤维垂直下落的稳定性特征,研究了厚度低于100 nm的薄膜在外力驱动下的流动以及van der Waals力的影响．结果表明随着薄膜相对厚度的下降,纤维表面的曲率将使得线性扰动的发展得到抑制,而van der Waals力促进扰动的增长,这一竞争机制导致了增长率随薄膜相对厚度非单调的变化．还得到了流动的绝对和对流不稳定分区．结果表明van der Waals力扩大绝对不稳定流动区域,表面张力也会有利于绝对不稳定的发展,而外驱动力正好起到相反的作用．     

Bistability of rotational modes in a system of coupled pendulums

The main goal of this research is to examine any peculiarities and special modes observed in the dynamics of a system of two nonlinearly coupled pendulums. In addition to steady states, an in-phase rotation limit cycle is proved to exist in the system with both damping and constant external force. This rotation mode is numerically shown to become unstable for certain values of the coupling strength. We also present an asymptotic theory developed for an infinitely small dissipation, which explains why the in-phase rotation limit cycle loses its stability. Boundaries of the instability domain mentioned above are found analytically. As a result of numerical studies, a whole range of the coupling parameter values is found for the case where the system has more than one rotation limit cycle. There exist not only a stable in-phase cycle, but also two out-of phase ones: a stable rotation limit cycle and an unstable one. Bistability of the limit periodic mode is, therefore, established for the system of two nonlinearly coupled pendulums. Bifurcations that lead to the appearance and disappearance of the out-ofphase limit regimes are discussed as well.     

Investigations of chaotic dynamics of multi-layer beams taking into account rotational inertial effects

We propose a novel mathematical model of a vibrating multi-layer Timoshenko-type beam. We show that the introduced model essentially changes the type of partial differential equations allowing inclusion of rotational inertial effects. We illustrate and discuss the influence of boundary conditions, the beam layers and parameters of the external load on the non-linear dynamics of this composite beam including a study of its regular, bifurcation and chaotic behavior.The originally derived infinite problem is reduced to the finite one using either Finite Difference Method (FDM) or Finite Element Method (FEM) which guarantees validity and reliability of the obtained numerical results. In addition, a comparative study is carried out aiming at a proper choice of the efficient wavelet transform. In particular, scenarios of transition into chaos are studied putting emphasis on novel phenomena. Charts of the system dynamical regimes are also constructed with respect to the control parameters regarding thickness and composition of the beam layers.     

Breakup of finite fluid films

We study the dewetting process of thin fluid films that partially wet a solid surface. Using long wave (lubrication) approximation, we formulate a nonlinear partial differential equation governing the evolution of the film thickness, h. This equation includes the effects of capillarity, gravity, and additional conjoining/disjoining pressure term to account for intermolecular forces. We perform standard linear stability analysis of an infinite flat film, and identify the corresponding stable, unstable and metastable regions. Within this framework, we analyze the evolution of a semi-infinite film of length L in one direction. The numerical simulations show that for long and thin films, the dewetting fronts of the film generate a pearling process involving successive formation of ridges at the film ends and consecutive pinch-off behind these ridges. On the other hand, for shorter and thicker films, the evolution ends up by forming a single drop. The time evolution as well as the final drops pattern shows a competition between the dewetting mechanisms caused by nucleation and by free surface instability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a cylindrical plasma in the presence of a monotonic field—II

The stability of an incompressible inviscid, perfectly conducting cylindrical plasma against azimuthal disturbances in the presence of a monotonic decreasing magnetic field having a constant pitch is discussed by using energy principle. The results obtained by this principle are compared form = 1 mode (which is a dangerous mode in which there is a lateral shift of the entire column) with that obtained by normal mode analysis. It is found thatm = 1 mode is always unstable. Further, an axial line current, external axial field and the surface tension tend to stabilisem ≠ modes.     

Commutator subgroup of Vershik–Kerov group

We describe a commutator subgroup of Vershik–Kerov group over an infinite field and find the bound for its commutator width. This gives a partial solution of the problem posed by Sushchanskii in 2010. We also describe the lower central series of the group of infinite upper triangular matrices over an infinite field and find the bound for its commutator width.     

Films courbés minces ferromagnétiques

We consider a thin curved ferromagnetic film not submitted to an external magnetic field. The behavior of the film is described by an energy depending on the magnetization of the film verifying the saturation constraint. The energy is composed of an induced magnetostatic energy and an energy term with density including the exchange energy and the anisotropic energy. We study the behavior of this energy when the thickness of the curved film goes to zero. We show with Γ-convergence arguments that the minimizers of the free energy converge to the minimizers of a local energy depending on a two-dimensional magnetization. To cite this article: H. Zorgati, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Research in the flexural buckling modes of a cylindrical sandwich shell under the action of a temperature field inhomogeneous across its thickness

A solution to the problem on the stability according to the flexural buckling mode is given for a cylindrical sandwich shell with a transversely soft core of arbitrary thickness. The shell is under the action of a temperature field inhomogeneous across the thickness, and its end faces are fastened in such a way (in the axial direction, the face sections of the external layer are fixed, but of the internal one are free) that an inhomogeneous subcritical stress-strain state arises in the shell across the thickness of its layers. It is shown that, under such conditions, the buckling mode of the shell is mixed flexural. To reveal and investigate this mode, equations of subcritical equilibrium and stability of a corresponding degree of accuracy are needed.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 715–730, November–December, 2004.     

Spectral numerical schemes for time-dependent convection with viscosity dependent on temperature

This article proposes spectral numerical methods to solve the time evolution of convection problems with viscosity strongly dependent on temperature at infinite Prandtl number. Although we verify the proposed techniques solely for viscosities that depend exponentially on temperature, the methods are extensible to other dependence laws. The set-up is a 2D domain with periodic boundary conditions along the horizontal coordinate which introduces a symmetry in the problem. This is the O(2) symmetry, which is particularly well described by spectral methods and motivates the use of these methods in this context. We examine the scope of our techniques by exploring transitions from stationary regimes towards time dependent regimes. At a given aspect ratio, stable stationary solutions become unstable through a Hopf bifurcation, after which the time-dependent regime is solved by the spectral techniques proposed in this article.     

How to excite the internal modes of sine-Gordon solitons

We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton behaves as a deformable (non-rigid) object. A soliton moving in an array of inhomogeneities can also present sustained oscillations of its width. There are very important phenomena (like the soliton–antisoliton collisions) where the existence of internal modes plays a crucial role. We show that, under some conditions, the dynamics of the soliton shape modes can be chaotic. A short report of some of our results has been published in [Phys. Rev. E 65 (2002) 065601(R)].     

Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

     

Resonance in viscous film flow over topography

We study viscous gravity-driven films flowing over periodically undulated substrates. Linear analysis describes resonance in steady flow along small bottom corrugations for films of arbitrary thickness. Depending on the dimensionless film thickness we find different regimes for the resonance, which is associated with the interaction of the undulated film with capillarygravity waves traveling against the flow. Nonlinear resonance produces higher harmonics and bistable resonance. The higher harmonics are due to higher harmonics resonance and due to the nonlinear coupling to lower harmonics. For the bistable resonance we derive a minimum model showing that the bistability corresponds essentially to that of the driven Duffing oscillator. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal investment strategies with a reallocation constraint

We study a Merton type optimization problem under a reallocation constraint. Under this restriction, the stock holdings can not be liquidated faster than a certain rate. This is a common restriction in certain type of investment firms. Our main objective is to study the large time optimal growth rate of the expected value of the utility from wealth. We also consider a discounted infinite horizon problem as a step towards understanding the first problem. A numerical study is done by solving the dynamic programming equations. Under the assumption of a power utility function, an appropriate dimension reduction argument is used to reduce the original problem to a two dimensional one in a bounded domain with convenient boundary conditions. Computation of the optimal growth rate introduces additional numerical difficulties as the straightforward approach is unstable. In this direction, new analytical results characterizing the growth rate as the limit of a sequence of finite horizon problems with continuously derived utility are proved.     

Predictions of the gas–liquid flow in wet electrostatic precipitators

Based on Computational Fluid Dynamics (CFD), the present paper aims to simulate several important phenomena in a wet type ESP from the liquid spray generation to gas-droplet flow in electric field. A single passage between the adjacent plates is considered for the simulation domain. Firstly, the electric field intensity and ion charge density are solved locally around a corona emitter of a barbed wire electrode, which are applied to the entire ESP using periodic conditions. Next, the Euler–Lagrange method is used to simulate the gas-droplet flow. Water droplets are tracked statistically along their trajectories, together with evaporation and particle charging. Finally, the deposition density on the plate is taken as the input for the liquid film model. The liquid film is simulated separately using the homogenous Eulerian approach in ANSYS-CFX. In the current case, since the free surface of the thin water film is difficult to resolve, a special method is devised to determine the film thickness.As parametric study, the variables considered include the nozzle pressure, initial spray spreading patterns (solid versus hollow spray) and plate wettability. The droplet emission rate and film thickness distribution are the results of interest. Main findings: electric field has strong effect on the droplet trajectories. Hollow spray is preferred to solid spray for its lower droplet emission. The liquid film uniformity is sensitive to the plate wettability.     

Bound state solutions of the elliptic sine-Gordon equation

We present a detailed investigation of finite-energy solutions with point-like singularities of the elliptic sine-Gordon equation in a plane. Such solutions are of the bound-state type in the sense of scalar field theory. If the solution has a unique singularity, then it behaves as a soliton-like annular wave packet at a large distance from the singularity. The effective radius of this wave packet is evaluated both analytically and numerically for axially symmetric solutions. The analytical investigation is based on the method of isomonodromy deformations for the third Painlevé equation, which singles out these solutions as separatrices of the manifold of general solutions (with infinite energy). Exact analytical estimates provide a tool for investigating bound-state solutions of the nonintegrable sine-Gordon equation with a nonzero right-hand side. More precisely, for large-intensity fields at the singularity, we derive the critical forcing that allows the existence and stability of a bound state. As an illustration, we consider two applications: a large-area Josephson junction and a nematic liquid crystal in a rotating magnetic field. For each of the examples, we evaluate the critical values of the field that allow finite-energy regimes. These are in good agreement with numerical and experimental data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 1, pp. 15–31, April, 1997.     

Non-Archimedean Probability

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by a different type of infinite additivity.     

Closure planes

We introduce a simple algebraic method for constructing infinite affine (and projective) planes from an infinite set of finite planes of prime power order stemming from a “root” plane. The construction uses finite fields and infinite extensions of finite fields in a critical way. We obtain a classical-looking result which states that if the construction succeeds over the algebraic closure of a finite field, then both the infinite plane and the original root plane must be Desarguesian. The Lenz–Barlotti types for these planes are then linked to the Lenz–Barlotti type of the root plane. Examples are then given. These show that under suitable conditions, the method can yield infinitely many non-isomorphic infinite planes. These examples are of Lenz–Barlotti types II.1 and V.1.