On the equilibrium and stability of a row of point vortices

The equilibrium and stability of a single row of equidistantly spaced identical point vortices is a classical problem in vortex dynamics, which has been addressed by several investigators in different ways for at least a century. Aspects of the history and the essence of these treatments are traced, stating some in more accessible form, and pointing out interesting and apparently new connections between them. For example, it is shown that the stability problem for vortices in an infinite row and the stability problem for vortices arranged in a regular polygon are solved by the same eigenvalue problem for a certain symmetric matrix. This result also provides a more systematic enumeration of the basic instability modes. The less familiar theory of equilibria of a finite number of vortices situated on a line is also recalled.

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On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon

Mathematics ◽

10.3390/math8061033 ◽

2020 ◽

Vol 8 (6) ◽

pp. 1033

Author(s):

Leonid Kurakin ◽

Irina Ostrovskaya

Keyword(s):

Circular Cylinder ◽

Hamiltonian Systems ◽

Nonlinear Stability ◽

Vortex Dynamics ◽

Stability Problem ◽

Orbital Stability ◽

Stability And Instability ◽

The Stability ◽

Parameter Values ◽

Linear Setting

The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation Γ around the circle is arbitrary. The problem has three parameters N, q, Γ , where q = R 2 / R 0 2 . This old problem of vortex dynamics is posed by Havelock (1931) and is a generalization of the Kelvin problem (1878) on the stability of a regular vortex polygon (Thomson N-gon) on the plane. In the case of Γ = 0 , the problem has already been solved: in the linear setting by Havelock, and in the nonlinear setting in the series of our papers. The contribution of this work to the solution of the problem consists in the analysis of the case of non-zero circulation Γ ≠ 0 . The linearization matrix and the quadratic part of the Hamiltonian are studied for all possible parameter values. Conditions for orbital stability and instability in the nonlinear setting are found. The parameter areas are specified where linear stability occurs and nonlinear analysis is required. The nonlinear stability theory of equilibria of Hamiltonian systems in resonant cases is applied. Two resonances that lead to instability in the nonlinear setting are found and investigated, although stability occurs in the linear approximation. All the results obtained are consistent with those known for Γ = 0 . This research is a necessary step in solving similar problems for the case of a moving circular cylinder, a model of vortices inside an annulus, and others.

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A Method for Approximate Stability Analysis and Its Application to Circular Cylindrical Shells Under Circumferentially Varying Edge Loads

Journal of Applied Mechanics ◽

10.1115/1.3423196 ◽

1973 ◽

Vol 40 (4) ◽

pp. 971-976 ◽

Cited By ~ 12

Author(s):

A. Libai ◽

D. Durban

Keyword(s):

Stress Distribution ◽

Boundary Conditions ◽

Stability Problem ◽

Circular Cylindrical Shell ◽

Symmetric Matrix ◽

Circumferential Stress ◽

Characteristic Roots ◽

Circular Cylindrical Shells ◽

The Stability ◽

Simple Stress

A procedure is presented which relates the solution of the stability problem for an elastic structure subjected to a complex prebuckling stress distribution, to that of a solved stability problem for the same structure but with a “simple” stress distribution. The procedure reduces to obtaining the characteristic roots of a symmetric matrix. It is then applied to the buckling problem of a circular cylindrical shell subjected to a circumferentially varying compressive stress field and boundary conditions of the SS1-2 types. The main result is that for the boundary conditions and number of harmonic terms in the edge load taken, the buckling parameter approaches, as h/R → 0, that of a cylinder uniformly subjected to the maximum circumferential stress.

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On bifurcation of the four Liouville tori in one generalized integrable model of the vortex dynamics

Доклады Академии наук ◽

10.31857/s0869-56524874376-380 ◽

2019 ◽

Vol 487 (4) ◽

pp. 376-380

Author(s):

P. E. Ryabov

Keyword(s):

Bifurcation Diagram ◽

Vortex Dynamics ◽

Integrable Model ◽

Vortex Pair ◽

Einstein Condensate ◽

Point Vortices ◽

Physical Parameters ◽

Bose Einstein Condensate ◽

Liouville Tori ◽

The Stability

The article deals with a generalized mathematical model of the dynamics of two point vortices in the Bose-Einstein condensate enclosed in a harmonic trap, and of the dynamics of two point vortices in an ideal fluid bounded by a circular region. In the case of a positive vortex pair, which is of interest for physical experimental applications, a new bifurcation diagram is obtained, for which the bifurcation of four tori into one is indicated. The presence of bifurcations of three and four tori in the integrable model of vortex dynamics with positive intensities indicates a complex transition and the connection of bifurcation diagrams of both limit cases. Analytical results of this publication (the bifurcation diagram, the reduction to a system with one degree of freedom, the stability analysis) form the basis of computer simulation of absolute dynamics of vortices in a fixed coordinate system in the case of arbitrary values of the physical parameters of the model (the intensities, the vortex interaction and etc.).

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Global stability result for parabolic Cauchy problems

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0026 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mourad Choulli ◽

Masahiro Yamamoto

Keyword(s):

New York ◽

Partial Differential Equations ◽

Global Stability ◽

Classical Problem ◽

Stability Result ◽

Cauchy Problems ◽

Smooth Solutions ◽

Linear Partial Differential Equations ◽

Ill Posed ◽

The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

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The two-dimensional hydrodynamical theory of moving aerofoils. IV

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1947.0019 ◽

1947 ◽

Vol 188 (1015) ◽

pp. 439-463

Keyword(s):

Stability Problem ◽

Two Dimensional ◽

Centre Of Gravity ◽

First Approximation ◽

Von Karman ◽

Moving Cylinder ◽

Period Equation ◽

The Stability ◽

Von Kármán ◽

Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

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Some Structural Properties of Systolic Tree Automata

Fundamenta Informaticae ◽

10.3233/fi-1989-12409 ◽

1989 ◽

Vol 12 (4) ◽

pp. 571-585

Author(s):

E. Fachini ◽

A. Maggiolo Schettini ◽

G. Resta ◽

D. Sangiorgi

Keyword(s):

Structural Properties ◽

Stability Problem ◽

Sufficient Condition ◽

Tree Automata ◽

Necessary And Sufficient Condition ◽

The Stability ◽

Necessary And Sufficient

We prove that the classes of languages accepted by systolic automata over t-ary trees (t-STA) are always either equal or incomparable if one varies t. We introduce systolic tree automata with base (T(b)-STA), a subclass of STA with interesting properties of modularity, and we give a necessary and sufficient condition for the equivalence between a T(b)-STA and a t-STA, for a given base b. Finally, we show that the stability problem for T(b)-ST A is decidible.

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Two-sided estimates in the stability problem for compressed elastic blocks

Mechanics of Solids ◽

10.3103/s0025654410010073 ◽

2010 ◽

Vol 45 (1) ◽

pp. 41-50

Author(s):

S. A. Panteleev

Keyword(s):

Stability Problem ◽

The Stability

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Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

Journal of Applied Mechanics ◽

10.1115/1.3169122 ◽

1985 ◽

Vol 52 (3) ◽

pp. 686-692 ◽

Cited By ~ 4

Author(s):

L. A. Month ◽

R. H. Rand

Keyword(s):

Angular Velocity ◽

Rigid Body ◽

Classical Problem ◽

Moment Of Inertia ◽

Model Parameters ◽

Stability Chart ◽

The Stability ◽

Transition Curves ◽

Minimum Moment ◽

Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

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