Taylor-Couette Instability With a Continuous Spectrum

1995 ◽  
Vol 62 (4) ◽  
pp. 915-923 ◽  
Author(s):  
L. S. Yao ◽  
S. Ghosh Moulic
Keyword(s):  
Equilibrium State ◽  
Continuous Spectrum ◽  
Bifurcation Point ◽  
Multiple Solutions ◽  
Wave Number ◽  
Taylor Vortex ◽  
Unstable Wave ◽  
Wave Numbers ◽  
Final Equilibrium State ◽  
Taylor Couette

Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wave numbers, and an integrodif-ferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly varying wave packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. In all cases, the final equilibrium state consists of a single dominant mode and its harmonics of smaller amplitudes. The predicted range of wave numbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wave numbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that a linearly stable long wave can excite a short unstable wave through nonlinear wave interaction. An important implication of the existence of nonunique equilibrium states is that the torque induced by the fluid motion cannot be determined uniquely. The numerical results show that the uncertainty, associated with nonuniqueness, of using any accurately measured Taylor-vortex torque slightly above the first bifurcation point in engineering practice can be as large as ten percent. The presence of multiple solutions at a fixed Reynolds number for a given geometry in Taylor-Couette flows has been known since Coles’ monumental contribution in 1965. A theoretical confirmation has come only 30 years later. It is worthwhile to point out that the existence of multiple solutions, found by Coles, differs from current popular bifurcation theories. The current study indicates that the state of flows on a stable bifurcation branch can involve any wave number within a finite band and can not be determined uniquely. The multiple solutions in Coles’ sense have also been found for mixed-convection flows (Yao and Ghosh Moulic, 1993, 1994) besides the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense, which complements the bifurcation theories, is a generic property for all fluid flows.

Taylor—Couette instability of travelling waves with a continuous spectrum

1996 ◽  
Vol 324 ◽  
pp. 181-198 ◽  
Author(s):  
S. Ghosh Moulic ◽  
L. S. Yao
Keyword(s):  
Continuous Spectrum ◽  
Multiple Solutions ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Fluid Flows ◽  
Taylor Vortex ◽  
Seminal Contribution ◽  
Wave Resonance ◽  
Taylor Couette ◽  
Finite Band

The nonlinear evolution of a continuous spectrum of travelling waves resulting from the growth of unstable disturbances in circular Couette flow has been investigated. Numerical solution of the governing integro-differential equations for different initial conditions shows that the equilibrium states of Taylor-vortex, wavy-vortex or spiralvortex flows are not unique, but depend on the initial disturbance. The presence of multiple solutions at a fixed Reynolds number for a given Taylor–Couette geometry has been known since Coles’ seminal contribution in 1965. The current study indicates that the equilibrium state of flows on a stable bifurcation branch is a natural consequence of nonlinear wave resonance and is dependent on the initial conditions. The resulting wavenumber can take any value within an accessible finite band. Since such multiple solutions have also been found numerically for mixed-convection flows and experimentally for several other flows, there is evidence to support the conclusion that a non-uniqueness in the sense of Coles is a generic property for all fluid flows.

Infrared spectra and substituent effects in 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones

1983 ◽  
Vol 48 (2) ◽  
pp. 586-595 ◽  
Author(s):  
Alexander Perjéssy ◽  
Pavol Hrnčiar ◽  
Ján Šraga
Keyword(s):  
Substituent Effects ◽  
Phenyl Group ◽  
Wave Number ◽  
Vibrational Coupling ◽  
Wave Numbers ◽  
Stretching Vibration ◽  
Stretching Vibrations ◽  
The Relationship ◽  
Derivatives Of ◽  
Effect Of Structure

The wave numbers of the fundamental C=O and C=C stretching vibrations, as well as that of the first overtone of C=O stretching vibration of 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones and 1,3-cycloheptanedione were measured in tetrachloromethane and chloroform. The spectral data were correlated with σ+ constants of substituents attached to phenyl group and with wave number shifts of the C=O stretching vibration of substituted acetophenones. The slope of the linear dependence ν vs ν+ of the C=C stretching vibration of the ethylenic group was found to be more than two times higher than that of the analogous correlation of the C=O stretching vibration. Positive values of anharmonicity for asymmetric C=O stretching vibration can be considered as an evidence of the vibrational coupling in a cyclic 1,3-dicarbonyl system similarly, as with derivatives of 1,3-indanedione. The relationship between the wave numbers of the symmetric and asymmetric C=O stretching vibrations indicates that the effect of structure upon both vibrations is symmetric. The vibrational coupling in 1,3-cycloheptanediones and the application of Seth-Paul-Van-Duyse equation is discussed in relation to analogous results obtained for other cyclic 1,3-dicarbonyl compounds.

Direct measurements of cold and hot plasma composition and EMIC waves in the outer magnetosphere: Implications for inner magnetosphere wave-particle interactions

2021 ◽  
Author(s):  
Justin Lee ◽  
Drew Turner ◽  
Sarah Vines ◽  
Robert Allen ◽  
Sergio Toledo-Redondo
Keyword(s):  
Unstable Wave ◽  
Linear Wave ◽  
Plasma Composition ◽  
Inner Magnetosphere ◽  
Hot Plasma ◽  
Emic Wave ◽  
Direct Measurements ◽  
Wave Numbers ◽  
Emic Waves

<p>Although thorough characterization of magnetospheric ion composition is rare for EMIC wave studies, convective processes that occur more frequently in Earth’s outer magnetosphere have allowed the Magnetospheric Multiscale (MMS) satellites to make direct measurements of the cold and hot plasma composition during EMIC wave activity. We will present an observation and linear wave modeling case study conducted on EMIC waves observed during a perturbed activity period in the outer dusk-side magnetosphere. During the two intervals investigated for the case study, the MMS satellites made direct measurements of cold plasmaspheric plasma in addition to multiple hot ion components at the same time as EMIC wave emissions were observed. Applying the in-situ plasma composition data to wave modeling, we find that wave growth rate is impacted by the complex interactions between the cold as well as the hot ion components and ambient plasma conditions. In addition, we observe that linear wave properties (unstable wave numbers and band structure) can significantly evolve with changes in cold and hot ion composition. Although the modeling showed the presence of dense cold ions can broaden the range of unstable wave numbers, consistent with previous work, the hot heavy ions that were more abundant nearer storm main phase could limit the growth of EMIC waves to smaller wave numbers. In the inner magnetosphere, where higher cold ion density is expected, the ring current heavy ions could also be more intense near storm-time, possibly resulting in conditions that limit the interactions of EMIC waves with trapped radiation belt electrons to multi-MeV energies. Additional investigation when direct measurements of cold and hot plasma composition are available could improve understanding of EMIC waves and their interactions with trapped energetic particles in the inner magnetosphere.</p>

Ostwald´s Rule and the Principle of the Shortest Way

2010 ◽  
Vol 224 (06) ◽  
pp. 929-934 ◽  
Author(s):  
Herbert W. Zimmermann
Keyword(s):  
Melting Point ◽  
Equilibrium State ◽  
Entropy Production ◽  
Thermodynamic Stability ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Geodesic Line ◽  
Final Equilibrium State ◽  
Relevant Variables ◽  
Non Equilibrium

AbstractWe consider a substance X with two monotropic modifications 1 and 2 of different thermodynamic stability ΔH1 < ΔH2. Ostwald´s rule states that first of all the instable modification 1 crystallizes on cooling down liquid X, which subsequently turns into the stable modification 2. Numerous examples verify this rule, however what is its reason? Ostwald´s rule can be traced back to the principle of the shortest way. We start with Hamilton´s principle and the Euler-Lagrange equation of classical mechanics and adapt it to thermodynamics. Now the relevant variables are the entropy S, the entropy production P = dS/dt, and the time t. Application of the Lagrangian F(S, P, t) leads us to the geodesic line S(t). The system moves along the geodesic line on the shortest way I from its initial non-equilibrium state i of entropy Si to the final equilibrium state f of entropy Sf. The two modifications 1 and 2 take different ways I1 and I2. According to the principle of the shortest way, I1 < I2 is realized in the first step of crystallization only. Now we consider a supercooled sample of liquid X at a temperature T just below the melting point of 1 and 2. Then the change of entropy ΔS1 = Sf 1 - Si 1 on crystallizing 1 can be related to the corresponding chang of enthalpy by ΔS1 = ΔH1/T. Now it can be shown that the shortest way of crystallization I1 corresponds under special, well-defined conditions to the smallest change of entropy ΔS1 < ΔS2 and thus enthalpy ΔH1 < ΔH2. In other words, the shortest way of crystallization I1 really leads us to the instable modification 1. This is Ostwald´s rule.

scholarly journals The band spectrum of hydrogen

1924 ◽  
Vol 106 (735) ◽  
pp. 69-82 ◽  
Keyword(s):  
Molecular Hydrogen ◽  
Hydrogen Molecule ◽  
Spectral Lines ◽  
Wave Number ◽  
Band Spectrum ◽  
Wave Numbers ◽  
Experimental Side ◽  
The Subject ◽  
The Moment ◽  
Theoretical Side

Many attempts have been made to detect regularities amongst the numerous lines which constitute the secondary or many-lined spectrum of hydrogen. The extreme complexity of the spectrum may be realised from the fact that in the Bakerian Lecture of 1922 Merton and Barratt record some 750 lines in the interval between Hα (wave-number v = 5233.216) and Hβ ( v = 20564.793). Three methods of investigation may be employed in the search for regularities. (1) The lines may be classified according to their physical characteristics, such as intensity or mode of excitation, as in the tables of Merton and Barrat ( loc. cit .). (2) Lines may be grouped together by the discovery of relations between their wave-lengths or wave-numbers, as in the important groups of lines which have been arranged in bands by Fulcher. (3) Lastly, the question may be attacked from the theoretical side, and a model of the hydrogen molecule may be imagined, which will give rise to the emission of certain characteristic spectral lines. Thus Sutherland, working on the foundation of the classical mechanical laws, more than twenty years ago, came to the conclusion that spectral series must arise from kinematical considera­tions, and explained them by considering the nodal sub-divisions of a circle. At the present time we may expect more successful results to follow from the application of the quantum theory, and in this paper an endeavour will be made to examine the secondary spectrum of hydrogen, and more particularly the Fulcher bands, from this standpoint. I may add that my interest in the subject was aroused when attempting to construct a model of the hydrogen molecule, for it seemed that the most likely method of obtaining reliable information from the experimental side as to the moment of inertia of the molecule would be from a study of the spectrum of molecular hydrogen.

scholarly journals Small-Scale Solar Magnetic Fields

1976 ◽  
Vol 71 ◽  
pp. 69-99 ◽  
Author(s):  
J. O. Stenflo
Keyword(s):  
Magnetic Fields ◽  
Magnetic Flux ◽  
Solar Surface ◽  
Life Time ◽  
Wave Number ◽  
Small Scale ◽  
Solar Magnetic Fields ◽  
Field Amplification ◽  
Wave Numbers ◽  
Diffuse Form

The observed properties of small-scale solar magnetic fields are reviewed. Most of the magnetic flux in the photosphere is in the form of strong fields of about 100–200 mT (1–2 kG), which have remarkably similar properties regardless of whether they occur in active or quiet regions. These fields are associated with strong atmospheric heating. Flux concentrations decay at a rate of about 107 Wb s-1, independent of the amount of flux in the decaying structure. The decay occurs by smaller flux fragments breaking loose from the larger ones, i.e. a transfer of magnetic flux from smaller to larger Fourier wave numbers, into the wave-number regime where ohmic diffusion becomes significant. This takes place in a time-scale much shorter than the length of the solar cycle.The field amplification occurs mainly below the solar surface, since very little magnetic flux appears in diffuse form in the photosphere, and the life-time of the smallest flux elements is very short. The observations further suggest that most of the magnetic flux in quiet regions is supplied directly from below the solar surface rather than being the result of turbulent diffusion of active-region magnetic fields.

Inviscid instability of an unbounded heterogeneous shear layer

1971 ◽  
Vol 48 (2) ◽  
pp. 405-415 ◽  
Author(s):  
S. A. Maslowe ◽  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Froude Number ◽  
Mixing Layer ◽  
Density Variation ◽  
Wave Number ◽  
Unstable Wave ◽  
Number Range ◽  
Spatial Growth ◽  
Wave Number Range ◽  
Low Froude Number

Stability curves are computed for both spatially and temporally growing disturbances in a stratified mixing layer between two uniform streams. The low Froude number limit, in which the effects of buoyancy predominate, and the high Froude number limit, in which the effects of density variation are manifested by the inertial terms of the vorticity equation, are considered as limiting cases. For the buoyant case, although the spatial growth rates can be predicted reasonably well by suitable use of the results for temporal growth, spatially growing disturbances appear to have high group velocities near the lower cutoff wave-number. For the inertial case, it is demonstrated that density variations can be destabilizing. More precisely, when the stream with the higher velocity has the lower density, both the wave-number range of unstable disturbances and the maximum spatial growth rate are increased relative to the case of homogeneous flow. Finally, it is shown how the growth rate of the most unstable wave in the inertial case diminishes as buoyancy becomes important.

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