Short-Wave Instability in Extended Systems with Additional Symmetry

1997 ◽  
Vol 07 (05) ◽  
pp. 997-1006 ◽  
Author(s):  
Michael I. Tribelsky
Keyword(s):  
Seismic Waves ◽  
Short Wave ◽  
Continuous Group ◽  
Periodic Patterns ◽  
Wave Instability ◽  
Coupled Modes ◽  
Additional Symmetry ◽  
Pattern Stability ◽  
Spatially Periodic ◽  
The Stability

Stability of steady spatially periodic patterns in systems with an additional continuous group of symmetry is discussed. It is shown that different systems with the same dimensionality of the continuous group of symmetry display remarkable similarity in all qualitative features of the pattern stability problem. Attention is called to the fact that, beside an extra band of slowly varying modes, the additional symmetry may yield a mixture of different scales in the final dispersion equation for pattern's perturbations, so that the stability conditions become unusually sensitive to very fine details of the problem. A one-dimensional partial differential equation governing seismic waves in viscoelastic media is considered as a particular example. The equation exhibits short-wave instability and additional invariance under the transformation u → u + const. , where the order parameter u(x, t) is associated with the displacement velocity. The analytical study of the equation is supplemented by computer simulations. The simulations show that the system undergoes a bifurcation from the trivial state with u ≡ 0 to well-developed chaos directly and the transition to the chaos is smooth, without any discontinuity. The chaos is characterized by excitation of a big number (in a boundless system — continuum) of coupled modes localized generally in two narrow subbands, centered around the critical wavenumber for the short-wave instability and the wavenumber equals zero, respectively.

scholarly journals On the porosity influence on stability of flow over porous medium

2020 ◽  
Vol 30 (1) ◽  
pp. 134-144
Author(s):  
K.B. Tsiberkin
Keyword(s):  
Porous Medium ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Short Wave ◽  
Wave Instability ◽  
Plane Parallel ◽  
Long Wave ◽  
The Stability ◽  
Critical Wavenumber

The stability of incompressible fluid plane-parallel flow over a layer of a saturated porous medium is studied. The results of a linear stability analysis are described at different porosity values. The considered system is bounded by solid wall from the porous layer bottom. Top fluid surface is free and rigid. A linear stability analysis of plane-parallel stationary flow is presented. It is realized for parameter area where the neutral stability curves are bimodal. The porosity variation effect on flow stability is considered. It is shown that there is a transition between two main instability modes: long-wave and short-wave. The long-wave instability mechanism is determined by inflection points within the velocity profile. The short-wave instability is due to the large transverse gradient of flow velocity near the interface between liquid and porous medium. Porosity decrease stabilizes the long wave perturbations without significant shift of the critical wavenumber. Simultaneously, the short-wave perturbations destabilize, and their critical wavenumber changes in wide range. When the porosity is less than 0.7, the inertial terms in filtration equation and magnitude of the viscous stress near the interface increase to such an extent that the Kelvin-Helmholtz analogue of instability becomes the dominant mechanism for instability development. The stability band realizes in narrow porosity area. It separates the two branches of the neutral curve.

The stability of a helical vortex filament

1972 ◽  
Vol 54 (4) ◽  
pp. 641-663 ◽  
Author(s):  
Sheila E. Widnall
Keyword(s):  
Short Wave ◽  
Mutual Inductance ◽  
The Self ◽  
Vortex Filament ◽  
Wave Instability ◽  
Long Wave ◽  
Amplification Rate ◽  
Helical Vortex ◽  
Induced Motion ◽  
The Stability

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

scholarly journals Sliding Mode Control and Geometrization Conjecture in Seismic Response

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 353
Author(s):  
Ligia Munteanu ◽  
Dan Dumitriu ◽  
Cornel Brisan ◽  
Mircea Bara ◽  
Veturia Chiroiu ◽  
...  
Keyword(s):  
Sliding Mode Control ◽  
Ricci Flow ◽  
Lyapunov Functions ◽  
Seismic Waves ◽  
Sliding Mode ◽  
Flow Process ◽  
Mode Control ◽  
Finite Period ◽  
The Stability ◽  
Story Building

The purpose of this paper is to study the sliding mode control as a Ricci flow process in the context of a three-story building structure subjected to seismic waves. The stability conditions result from two Lyapunov functions, the first associated with slipping in a finite period of time and the second with convergence of trajectories to the desired state. Simulation results show that the Ricci flow control leads to minimization of the displacements of the floors.

On the modulation of water waves on shear flows

1976 ◽  
Vol 347 (1651) ◽  
pp. 537-546 ◽  
Keyword(s):  
Water Waves ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Vries Equation ◽  
Harmonic Wave ◽  
Short Wave ◽  
Long Wave ◽  
Stokes Waves ◽  
The Stability ◽  
Wave Limit

The method of multiple scales is used to examine the slow modulation of a harmonic wave moving over the surface of a two dimensional channel. The flow is assumed inviscid and incompressible, but the basic flow takes the form of an arbitrary shear. The appropriate nonlinear Schrödinger equation is derived with coefficients that depend, in a complicated way, on the shear. It is shown that this equation agrees with previous work for the case of no shear; it also agrees in the long wave limit with the appropriate short wave limit of the Korteweg-de Vries equation, the shear being arbitrary. Finally, it is remarked that the stability of Stokes waves over any shear can be examined by using the results derived here.

scholarly journals Sliding Mode Control and Geometrization Conjecture in Seismology

2021 ◽  
Author(s):  
Ligia Munteanu ◽  
Dan Dumitriu ◽  
Cornel Brisan ◽  
Mircea Bara ◽  
Veturia Chiroiu ◽  
...  
Keyword(s):  
Sliding Mode Control ◽  
Ricci Flow ◽  
Lyapunov Functions ◽  
Seismic Waves ◽  
Sliding Mode ◽  
Flow Process ◽  
Mode Control ◽  
Finite Period ◽  
The Stability ◽  
Story Building

The purpose of this paper is to study the sliding mode control as a Ricci flow process in the context of a three-story building structure subjected to seismic waves. The stability conditions result from two Lyapunov- functions, the first associated with slipping in a finite period of time, and the second with convergence of trajectories to the desired state. Simulation results show that the Ricci flow control leads to the minimization of the displacements of the floors. 3D Ricci solitons projection via a semi-conformal mapping to a surface is also studied.

Effects of the Coriolis force on the stability of Stuart vortices

1998 ◽  
Vol 356 ◽  
pp. 353-379 ◽  
Author(s):  
STÉPHANE LEBLANC ◽  
CLAUDE CAMBON
Keyword(s):  
Coriolis Force ◽  
Three Dimensional ◽  
Basic Mechanism ◽  
Short Wave ◽  
Stagnation Points ◽  
Floquet Modes ◽  
The Stability ◽  
Simple Flows ◽  
Stuart Vortices ◽  
Wave Breakdown

A detailed investigation of the effects of the Coriolis force on the three-dimensional linear instabilities of Stuart vortices is proposed. This exact inviscid solution describes an array of co-rotating vortices embedded in a shear flow. When the axis of rotation is perpendicular to the plane of the basic flow, the stability analysis consists of an eigenvalue problem for non-parallel versions of the coupled Orr–Sommerfeld and Squire equations, which is solved numerically by a spectral method. The Coriolis force acts on instabilities as a ‘tuner’, when compared to the non-rotating case. A weak anticyclonic rotation is destabilizing: three-dimensional Floquet modes are promoted, and at large spanwise wavenumber their behaviour is predicted by a ‘pressureless’ analysis. This latter analysis, which has been extensively discussed for simple flows in a recent paper (Leblanc & Cambon 1997) is shown to be relevant to the present study. The basic mechanism of short-wave breakdown is a competition between instabilities generated by the elliptical cores of the vortices and by the hyperbolic stagnation points in the braids, in accordance with predictions from the ‘geometrical optics’ stability theory. On the other hand, cyclonic or stronger anticyclonic rotation kills three-dimensional instabilities by a cut-off in the spanwise wavenumber. Under rapid rotation, the Stuart vortices are stabilized, whereas inertial waves propagate.

Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

1995 ◽  
Vol 352 (1698) ◽  
pp. 125-168 ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Pattern ◽  
Invariant Equilibrium ◽  
Periodic Patterns ◽  
Planar Lattice ◽  
Diffusive Convection ◽  
Spatially Periodic ◽  
Spatio Temporal ◽  
Boussinesq Fluid ◽  
Time Periodic

In Rayleigh-Bénard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms . We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular dif­ferential equation.

scholarly journals Peculiarities in the Director Reorientation and the NMR Spectra Evolution in a Nematic Liquid Crystals under the Effect of Crossed Electric and Magnetic Fields

Crystals ◽  
2019 ◽  
Vol 9 (5) ◽  
pp. 262 ◽  
Author(s):  
Alex V. Zakharov ◽  
Izabela Sliwa
Keyword(s):  
Nematic Phase ◽  
Viscosity Coefficient ◽  
Threshold Value ◽  
Periodic Patterns ◽  
Director Field ◽  
Electric And Magnetic Fields ◽  
Spatially Periodic ◽  
Rotational Viscosity ◽  
Director Reorientation ◽  
Reorientational Dynamics

The illustrative description of the field-induced peculiarities of the director reorientation in the microsized nematic volumes under the effect of crossed magnetic B and electric E fields have been proposed. The most interesting feature of such configuration is that the nematic phase becomes unstable after applying the strong E . The theoretical analysis of the reorientational dynamics of the director field provides an evidence for the appearance of the spatially periodic patterns in response to applied large E directed at an angle α to B . The feature of this approach is that the periodic distortions arise spontaneously from a homogeneously aligned nematic sample that ultimately induces a faster response than in the uniform mode. The nonuniform rotational modes involve additional internal elastic distortions of the conservative nematic system and, as a result, these deformations decrease of the viscous contribution U vis to the total energy U of the nematic phase. In turn, that decreasing of U vis leads to decrease of the effective rotational viscosity coefficient γ eff ( α ) . That is, a lower value of γ eff ( α ) , which is less than one in the bulk nematic phase, gives the less relaxation time τ on ( α ) ∼ γ eff ( α ) , when α is bigger than the threshold value α th . The results obtained by Deuterium NMR spectroscopy confirm theoretically obtained dependencies of τ on ( α ) on α .

The stability of spatially periodic flows

1981 ◽  
Vol 108 ◽  
pp. 461-474 ◽  
Author(s):  
D. N. Beaumont
Keyword(s):  
Velocity Profile ◽  
Phase Speed ◽  
Neutral Curve ◽  
Long Waves ◽  
Inviscid Fluid ◽  
Periodic Flows ◽  
A Value ◽  
Parallel Flows ◽  
Spatially Periodic ◽  
The Stability

The stability characteristics for spatially periodic parallel flows of an incompressible fluid (both inviscid and viscous) are studied. A general formula for the determination of the stability characteristics of periodic flows to long waves is obtained, and applied to approximate numerically the stability curves for the sinusoidal velocity profile. The neutral curve for the sinusoidal velocity profile is obtained analytically. The stability of two broken-line velocity profiles in an inviscid fluid is studied and the results are used to describe the overall pattern for the sinusoidal velocity profile in the case of long waves. In an inviscid fluid it is found that all periodic flows (other than the trivial flow in which the basic velocity is constant) are unstable to long waves with a value of the phase speed determined by simple integrals of the basic flow. In a viscous fluid it is found that the sinusoidal velocity profile is very unstable with the inviscid solution being a good approximation to the solution of the viscous problem when the value of the Reynolds number is greater than about 20.

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