On the energy instability of liquid crystals

1998 ◽  
Vol 9 (1) ◽  
pp. 23-36
Author(s):  
V. A. VLADIMIROV ◽  
K. I. ILIN
Keyword(s):  
Growth Rate ◽  
Initial Data ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Upper And Lower Bounds ◽  
Direct Lyapunov Method ◽  
Distortion Energy ◽  
The Stability ◽  
Energy Instability ◽  
General Equilibria

The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

Resistive Ballooning Modes in Three-Dimensional Configurations

1982 ◽  
Vol 37 (8) ◽  
pp. 848-858 ◽  
Author(s):  
D. Correa-Restrepo
Keyword(s):  
Growth Rates ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Coupled System ◽  
Small Constant ◽  
Ballooning Mode ◽  
The Stability ◽  
The Magnetic Field ◽  
Symmetric Systems ◽  
The Ideal

Resistive ballooning modes in general three-dimensional configurations are studied on the basis of the equations of motion of resistive MHD. Assuming small, constant resistivity and perturbations localized transversally to the magnetic field, a stability criterion is derived in the form of a coupled system of two second-order differential equations. This criterion contains several limiting cases, in particular the ideal ballooning mode criterion and criteria for the stability of symmetric systems. Assuming small growth rates, analytical results are derived by multiple-length-scale expansion techniques. Instabilities are found, their growth rates scaling as fractional powers of the resistivity

The Influence of Internal Friction on the Stability of High Speed Rotors With Anisotropic Supports

1969 ◽  
Vol 91 (4) ◽  
pp. 1105-1113 ◽  
Author(s):  
E. J. Gunter ◽  
P. R. Trumpler
Keyword(s):  
Internal Friction ◽  
High Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Analog Computer ◽  
Three Dimensional Model ◽  
Stiffness Properties ◽  
Stiffness Characteristics ◽  
The Stability ◽  
Internal Rotor

This paper evaluates the stability of the single mass rotor with internal friction on damped, anisotropic supports. The paper shows under what conditions the rotor stability may be improved by an undamped support with anisotropic stiffness properties. A three dimensional model is presented to show the influence of rotor and support stiffness characteristics on stability. Curves are also presented on how support damping may also improve or even reduce rotor stability. An analog computer solution of the governing equations of motion is presented showing the shaft transient motion for various speed ranges, and also plots of the rotor steady state motion are given for various speeds up to and including the stability threshold. The analysis is used to explain many of the experimental observations of B. L. Newkirk concerning stability due to internal rotor friction.

scholarly journals Latitudinal libration driven flows in triaxial ellipsoids

2015 ◽  
Vol 771 ◽  
pp. 193-228 ◽  
Author(s):  
S. Vantieghem ◽  
D. Cébron ◽  
J. Noir
Keyword(s):  
Numerical Simulations ◽  
Equations Of Motion ◽  
Liquid Core ◽  
Three Dimensional ◽  
Base Flow ◽  
Upper And Lower Bounds ◽  
Earth Planet ◽  
Ekman Layers ◽  
Core Dynamics ◽  
Inertial Instability

Motivated by understanding the liquid core dynamics of tidally deformed planets and moons, we present a study of incompressible flow driven by latitudinal libration within rigid triaxial ellipsoids. We first derive a laminar solution for the inviscid equations of motion under the assumption of uniform vorticity flow. This solution exhibits a resonance if the libration frequency matches the frequency of the spin-over inertial mode. Furthermore, we extend our model by introducing a reduced model of the effect of viscous Ekman layers in the limit of low Ekman number (Noir & Cébron, J. Fluid Mech., vol. 737, 2013, pp. 412–439). This theoretical approach is consistent with the results of Chan et al. (Phys. Earth Planet. Inter., vol. 187, 2011, pp. 404–415) and Zhang et al. (J. Fluid Mech., vol. 692, 2012, pp. 420–445) for spheroidal geometries. Our results are validated against systematic three-dimensional numerical simulations. In the second part of the paper, we present the first linear stability analysis of this uniform vorticity flow. To this end, we adopt different methods (Lifschitz & Hameiri, Phys. Fluids A, vol. 3, 1991, p. 2644; Gledzer & Ponomarev, Acad. Sci., USSR, Izv., Atmos. Ocean. Phys., vol. 13, 1977, pp. 565–569) that allow us to deduce upper and lower bounds for the growth rate of an instability. Our analysis shows that the uniform vorticity base flow is prone to inertial instabilities caused by a parametric resonance mechanism. This is confirmed by a set of direct numerical simulations. Applying our results to planetary settings, we find that neither a spin-over resonance nor an inertial instability can exist within the liquid core of the Moon, Io and Mercury.

Three-dimensional stability of solitary shear kinetic Alfvén waves in a low-beta plasma

1989 ◽  
Vol 41 (1) ◽  
pp. 171-184 ◽  
Author(s):  
K. P. Das ◽  
L. P. J. Kamp ◽  
F. W. Sluijter
Keyword(s):  
Growth Rate ◽  
Dimensional Stability ◽  
Three Dimensional ◽  
Alfven Waves ◽  
Magnetized Plasma ◽  
Alfvén Waves ◽  
Ion Acoustic Solitons ◽  
The Stability ◽  
External Uniform Magnetic Field ◽  
Kinetic Alfvén Waves

The three-dimensional stability of solitary shear kinetic Alfvén waves in a low-β plasma is investigated by the method of Zakharov & Rubenchik (1974). It is found that there is no instability if the direction of perturbation falls within a certain region of space. The growth rate of the instability for the unstable region is determined. This growth rate is found to decrease with increasing angle between the direction of propagation of the solitary wave and the direction of the external uniform magnetic field. A particular case of the present analysis gives results on the stability of ion-acoustic solitons in a magnetized plasma.

scholarly journals On the stability of solutions of fractional non conformable differential equations

2020 ◽  
Vol 65 (4) ◽  
pp. 495-502
Author(s):  
Paulo M. Guzman ◽  
Luciano M. Lugo Motta Bittencurt ◽  
Juan E. Napoles Valdes
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Method ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Direct Lyapunov Method ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
The Stability

In this note we obtain sufficient conditions under which we can guarantee the stability of solutions of a fractional differential equations of non conformable type and we obtain some fractional analogous theorems of the direct Lyapunov method for a given class of equations of motion.

Transition to time-dependent convection

1974 ◽  
Vol 65 (4) ◽  
pp. 625-645 ◽  
Author(s):  
R. M. Clever ◽  
F. H. Busse
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Horizontal Layer ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Prandtl Numbers ◽  
Steady Solutions ◽  
The Stability ◽  
Convection Rolls ◽  
Good Agreement

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

scholarly journals Stability of three-dimensional Gaussian vortices in an unbounded, rotating, vertically stratified, Boussinesq flow: linear analysis

2017 ◽  
Vol 824 ◽  
pp. 97-134 ◽  
Author(s):  
Mani Mahdinia ◽  
Pedram Hassanzadeh ◽  
Philip S. Marcus ◽  
Chung-Hsiang Jiang
Keyword(s):  
Growth Rate ◽  
Parameter Space ◽  
Accretion Disks ◽  
Growth Rates ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Small Region ◽  
Giant Planets ◽  
Ocean Eddies ◽  
The Stability

The linear stability of three-dimensional vortices in rotating, stratified flows has been studied by analysing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby numbers ($-0.5<Ro<0.5$) and Burger numbers ($0.02<Bu<2.3$) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of $Ro{-}Bu$. We have found neutrally stable vortices only over a small region of the $Ro{-}Bu$ parameter space: cyclones with $Ro\sim 0.02{-}0.05$ and $Bu\sim 0.85{-}0.95$. However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g. $Ro<0$ and $0.5\lesssim Bu\lesssim 1.3$) is slower than 50 turnaround times of the vortex (which often corresponds to several years for ocean eddies). For cyclones, the region with such slow growth rates is confined to $0<Ro<0.1$ and $0.5\lesssim Bu\lesssim 1.3$. While most calculations have been done for $f/\bar{N}=0.1$ (where $f$ and $\bar{N}$ are the Coriolis and background Brunt–Väisälä frequencies), we have numerically verified and explained analytically, using non-dimensionalized equations, the insensitivity of the results to reducing $f/\bar{N}$ to the more ocean-relevant value of 0.01. The results of our stability analysis of Gaussian vortices both support and contradict the findings of earlier studies with QG or multilayer models or with other families of vortices. The results of this paper provide a stepping stone to study the more complicated problems of the stability of geophysical (e.g. those in the atmospheres of giant planets) and astrophysical vortices (in accretion disks).

Investment Boolean problem with Savage risk criteria under uncertainty

2020 ◽  
Vol 30 (3) ◽  
pp. 159-168
Author(s):  
Sergei E. Bukhtoyarov ◽  
Vladimir A. Emelichev
Keyword(s):  
Initial Data ◽  
Minimization Problem ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Portfolio Theory ◽  
Stability Radius ◽  
Risk Criteria ◽  
The Stability ◽  
Three Dimensional Space

AbstractThe portfolio theory is used to formulate a multicriteria investment Boolean escaped gain minimization problem for searching all extreme portfolios. Stability aspects of this set against perturbed parameters of minimax Savage criteria are studied. We give lower and upper estimates for the stability radius for arbitrary Hölder norms on the three-dimensional space of initial data.

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

2009 ◽  
Vol 37 (2) ◽  
pp. 62-102 ◽  
Author(s):  
C. Lecomte ◽  
W. R. Graham ◽  
D. J. O’Boy
Keyword(s):  
Internal Pressure ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Prediction Method ◽  
Analytical Models ◽  
Beam Model ◽  
Impedance Boundary ◽  
Bending Waves ◽  
Tire Rotation ◽  
Three Dimensional Models

Abstract An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

A Liquid Metal Flow Between Two Coaxial Cylinders System

2019 ◽  
Vol 11 (1) ◽  
pp. 79-86
Author(s):  
Abdelkrim Merah ◽  
Ridha Kelaiaia ◽  
Faiza Mokhtari
Keyword(s):  
Couette Flow ◽  
Metal Flow ◽  
Three Dimensional ◽  
Liquid Sodium ◽  
Taylor Vortex ◽  
Turbulent Regime ◽  
Coaxial Cylinders ◽  
Concentric Cylinders ◽  
Ideal Tool ◽  
The Stability

Abstract The Taylor-Couette flow between two rotating coaxial cylinders remains an ideal tool for understanding the mechanism of the transition from laminar to turbulent regime in rotating flow for the scientific community. We present for different Taylor numbers a set of three-dimensional numerical investigations of the stability and transition from Couette flow to Taylor vortex regime of a viscous incompressible fluid (liquid sodium) between two concentric cylinders with the inner one rotating and the outer one at rest. We seek the onset of the first instability and we compare the obtained results for different velocity rates. We calculate the corresponding Taylor number in order to show its effect on flow patterns and pressure field.

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