Vortex Interaction in a Ducted Flow

The objective of this study is to describe the structure of pipe flow by considering it as a superposition of many axisymmetric vortex rings. In knowing the unsteady gross feature of pipe flow, the investigation on vortex interactions is important. As a first step to the goal, we investigate the nonlinear interaction among vortex rings whose number is three at most. The interaction among vortex rings of equal circulation is here investigated. Momentum and energy conservation of the present vortex ring system are also discussed to know a better understanding of the perturbed pipe flow. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large‐κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of the renormalized energy (the free energy less the vortex self‐induction energy). A rigorous study of the full time‐dependent Ginzburg‐Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.     

Space‐Time Finite Elements for Fluid‐Structure Interaction

A numerical method for the analysis of fluid‐structure interaction in the areas of building aeroelasticity and building‐water interaction is presented. In order to achieve optimal convergence of the solution a consistent space‐time discretization and a strong coupling algorithm is applied to the highly nonlinear problem.     

Bilinear Form and N‐Shock‐Wave Solutions for a (2+1)‐Dimensional Breaking Soliton Equation in Certain Fluids with the Bell Polynomials and Auxiliary Function

In this paper, we will investigate a (2+1)‐dimensional breaking soliton (BS) equation for the (2+1)‐dimensional collision of a Riemann wave with a long wave in certain fluids. Using the Bell polynomials and an auxiliary function, we derive a new bilinear form for the (2+1)‐dimensional BS equation, which is different from those in the previous literatures. One‐, two‐ and N‐shock‐wave solutions are obtained with the Hirota method and symbolic computation. One shock wave is found to be able to stably propagate. Two shock waves are observed to have the parallel collision, oblique collision, and stable propagation of the V‐type structure. In addition, we present the collision between one shock wave and V‐type structure, and the collision between two V‐type structures.     

Vortex energy and 360° Néel walls in thin‐film micromagnetics

We study the vortex pattern in ultrathin ferromagnetic films of circular crosssection. The model is based on the following energy functional: for in‐plane magnetizations m: B² → S¹ in the unit disc . The avoidance of volume charges ? · m ≠ 0 in B² and surface charges m · ν ≠ 0 on δB² leads to the formation of a vortex in the limit ε → 0. At the level ε > 0 the vortex is regularized by the formation of a 360° Néel wall (a one‐dimensional transition layer with core of scale ε) concentrated along a radius of B². We derive the limiting energy of the vortex by matching upper and lower bounds. Our analysis on the lower bound is based on a dynamical system argument and an interpolation inequality with sharp leading‐order constant, while the upper bound uses the leading‐order energy for 360° Néel walls. © 2010 Wiley Periodicals, Inc.     

Universal and Integrable Aspects of an Elliptic Vortex Representation in 2+1‐Dimensional Magneto‐Gasdynamics

Integrable substructure of a 2+1‐dimensional magneto‐gasdynamic system is investigated via a general elliptic vortex ansatz. Certain universal and Hamiltonian aspects of the admitted representation are uncovered. A class of confined magneto‐gasdynamic flows with an elliptic cylindrical boundary is isolated with time‐dependent semiaxes determined by an integrable Ermakov–Ray–Reid system.     

Nonlinear Stability of the Current‐Vortex Sheet to the Incompressible MHD Equations

In this paper, we solve a long‐standing open problem: nonlinear stability of the current‐vortex sheet in the ideal incompressible magnetohydrodynamics under the Syrovatskij stability condition. This result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin‐Helmholtz instability.© 2017 Wiley Periodicals, Inc.     

Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

We analyse the evolution of a system of finite faults by considering the non‐linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction. We consider the anti‐plane shearing on a system of finite faults under a slip‐dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non‐linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non‐linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non‐linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two‐faults system, and a comparison between the non‐linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.     

On Multi‐component Ermakov Systems in a Two‐Layer Fluid: Integrable Hamiltonian Structures and Exact Vortex Solutions

By introducing an elliptic vortex ansatz, the 2+1‐dimensional two‐layer fluid system is reduced to a finite‐dimensional nonlinear dynamical system. Time‐modulated variables are then introduced and multicomponent Ermakov systems are isolated. The latter is shown to be also Hamiltonian, thereby admitting general solutions in terms of an elliptic integral representation. In particular, a subclass of vortex solutions is obtained and their behaviors are simulated. Such solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is proved that the Hamiltonian system is equivalent to the stationary nonlinear cubic Schrödinger equations coupled with a Steen‐Ermakov‐Pinney equation.     

Fully Coupled Fluid‐Structure Interaction using Weak Coupling

We consider the procedure of computing the response of a coupled fluid‐structure interaction problem. Often we have existing software to solve each subproblem separately, and want to couple both. This setting seems to allow only the so‐called weak coupling, which is not sufficient for some problems. We show how to solve the so‐called strong coupling – a totally implicit formulation – by using only the solvers for the subproblems.     

Dynamics of Vortex Rossby Waves in Tropical Cyclones,Part 1: Linear Time‐Dependent Evolution on an f‐Plane

Analyses of observational data on hurricanes in the tropical atmosphere indicate the existence of spiral rainbands which propagate outward from the eye and affect the structure and intensity of the hurricane. These disturbances may be described as vortex Rossby waves. This paper describes the evolution of barotropic vortex Rossby waves in a cyclonic vortex in a two‐dimensional configuration where the variation of the Coriolis force with latitude is ignored. The waves are forced by a constant‐amplitude boundary condition at a fixed radius from the center of the vortex and propagate outward. The mean flow angular velocity profile is taken to be a quadratic function of the radial distance from the center of the vortex and there is a critical radius at which it is equal to the phase speed of the waves. For the case of waves with steady amplitude, an exact solution is derived for the steady linearized equations in terms of hypergeometric functions; this solution is valid in the outer region away from the critical radius. For the case of waves with time‐dependent amplitude, asymptotic solutions of the linearized equations, valid for late time, are obtained in the outer and inner regions. It is found that there are strong qualitative similarities between the conclusions on the evolution of the vortex waves in this configuration and those obtained in the case of Rossby waves in a rectangular configuration where the latitudinal gradient of the Coriolis parameter is taken into account. In particular, the amplitude of the steady‐state outer solution is greatly attenuated and there is a phase change of across the critical radius, and in the linear time‐dependent configuration, the outer solution approaches a steady state in the limit of infinite time, while the amplitude of the inner solution grows on a logarithmic time scale and the width of the critical layer approaches zero.     

Unsteady Cavitation in Injection Nozzles ‐ Dynamic Interaction of Internal Flow and Jet Formation ‐

This actual extension of our previous work demonstrates the strong interaction of cavitation inside injection nozzles with the external jet flow. Due to the high sensitivity of cavitation on the imposed initial and boundary conditions, simulations with restriction to the internal nozzle flow are qualitatively and quantitatively incorrect. Results of this very first simulations indicate the potential of cavitation for enhancement of atomization and spray formation.     

Dynamics of Vortex Rossby Waves in Tropical Cyclones,Part 2: Nonlinear Time‐Dependent Asymptotic Analysis on a β‐Plane

Vortex Rossby waves in cyclones in the tropical atmosphere are believed to play a role in the observed eyewall replacement cycle, a phenomenon in which concentric rings of intense rainbands develop outside the wall of the cyclone eye, strengthen and then contract inward to replace the original eyewall. In this paper, we present a two‐dimensional configuration that represents the propagation of forced Rossby waves in a cyclonic vortex and use it to explore mechanisms by which critical layer interactions could contribute to the evolution of the secondary eyewall location. The equations studied include the nonlinear terms that describe wave‐mean‐flow interactions, as well as the terms arising from the latitudinal gradient of the Coriolis parameter. Asymptotic methods based on perturbation theory and weakly nonlinear analysis are used to obtain the solution as an expansion in powers of two small parameters that represent nonlinearity and the Coriolis effects. The asymptotic solutions obtained give us insight into the temporal evolution of the forced waves and their effects on the mean vortex. In particular, there is an inward displacement of the location of the critical radius with time which can be interpreted as part of the secondary eyewall cycle.     

Energy Estimates and Cavity Interaction for a Critical‐Exponent Cavitation Model

We consider the minimization of in a perforated domain of among maps that are incompressible (det ) and invertible, and satisfy a Dirichlet boundary condition u = g on ?Ω. If the volume enclosed by g (?Ω) is greater than |Ω|, any such deformation u is forced to map the small holes B_ε( a _i) onto macroscopically visible cavities (which do not disappear as ε → 0). We restrict our attention to the critical exponent p = n, where the energy required for cavitation is of the order of and the model is suited, therefore, for an asymptotic analysis (v₁,…, v_M denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg‐Landau theory, we obtain estimates for the “renormalized” energy showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points a ₁,…, a _M, and on the distance from these points to the outer boundary ?Ω. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence. © 2012 Wiley Periodicals, Inc.     

Vortex analysis of the periodic Ginzburg–Landau model

We study the vortices of energy minimizers in the London limit for the Ginzburg–Landau model with periodic boundary conditions. For applied fields well below the second critical field we are able to describe the location and number of vortices. Many of the results presented appeared in [H. Aydi, Doctoral Dissertation, Université Paris-XII, 2004], others are new.     

Oblique projections and frames

We characterize those frames on a Hilbert space which can be represented as the image of an orthonormal basis by an oblique projection defined on an extension of . We show that all frames with infinite excess and frame bounds are of this type. This gives a generalization of a result of Han and Larson which only holds for normalized tight frames.

     

Oblique Dual Fusion Frames

We introduce and develop the concept of oblique duality for fusion frames. This concept provides a mathematical framework to deal with problems in distributed signal processing where the signals considered as elements in a Hilbert space are, under certain requirements, analyzed in one subspace and reconstructed in another subspace. The requirements are, on one side, the uniqueness of the reconstructed signal, and on the other what we call consistency of the sampling for fusion frames. Both conditions are naturally related to oblique projections. We study the main properties of oblique dual fusion frames and oblique dual fusion frame systems introduced in this work and present several results that provide alternative methods for their construction.