scholarly journals Mapping the stability of stellar rotating spheres via linear response theory

2019 ◽  
Vol 487 (1) ◽  
pp. 711-728 ◽  
Author(s):  
S Rozier ◽  
J-B Fouvry ◽  
P G Breen ◽  
A L Varri ◽  
C Pichon ◽  
...  
Keyword(s):  
Parameter Space ◽  
Rotation Rate ◽  
Linear Response ◽  
Globular Clusters ◽  
Stability Boundary ◽  
Linear Response Theory ◽  
Marginal Stability ◽  
Anisotropic Spheres ◽  
Pattern Speeds ◽  
The Stability

Abstract Rotation is ubiquitous in the Universe, and recent kinematic surveys have shown that early-type galaxies and globular clusters are no exception. Yet the linear response of spheroidal rotating stellar systems has seldom been studied. This paper takes a step in this direction by considering the behaviour of spherically symmetric systems with differential rotation. Specifically, the stability of several sequences of Plummer spheres is investigated, in which the total angular momentum, as well as the degree and flavour of anisotropy in the velocity space are varied. To that end, the response matrix method is customized to spherical rotating equilibria. The shapes, pattern speeds and growth rates of the systems’ unstable modes are computed. Detailed comparisons to appropriate N-body measurements are also presented. The marginal stability boundary is charted in the parameter space of velocity anisotropy and rotation rate. When rotation is introduced, two sequences of growing modes are identified corresponding to radially and tangentially biased anisotropic spheres, respectively. For radially anisotropic spheres, growing modes occur on two intersecting surfaces (in the parameter space of anisotropy and rotation), which correspond to fast and slow modes, depending on the net rotation rate. Generalized, approximate stability criteria are finally presented.

Mathematical Analysis of Stability of a Spinning Disk Under Rotating, Arbitrarily Large Damping Forces

1996 ◽  
Vol 118 (4) ◽  
pp. 657-662 ◽  
Author(s):  
F. Y. Huang ◽  
C. D. Mote
Keyword(s):  
Periodic Solution ◽  
Parameter Space ◽  
Perturbation Technique ◽  
Stability Boundary ◽  
Rotating Disk ◽  
Stable Region ◽  
Damping Force ◽  
Analysis Of Stability ◽  
The Stability ◽  
Nontrivial Periodic Solution

Stability of a rotating disk under rotating, arbitrarily large damping forces is investigated analytically. Points possibly residing on the stability boundary are located exactly in parameter space based on the criterion that at least one nontrivial periodic solution is necessary at every boundary point. A perturbation technique and the Galerkin method are used to predict whether these points of periodic solution reside on the stability boundary, and to identify the stable region in parameter space. A nontrivial periodic solution is shown to exist only when the damping does not generate forces with respect to that solution. Instability occurs when the wave speed of a mode in the uncoupled disk, when observed on the disk, is exceeded by the rotation speed of the damping force relative to the disk. The instability is independent of the magnitude of the force and the type of positive-definite damping operator in the applied region. For a single dashpot, nontrivial periodic solutions exist at the points where the uncoupled disk has repeated eigenfrequencies on a frame rotating with the dashpot and the dashpot neither damps nor energizes these modes substantially around these points.

Three-dimensionality in the wake of a rapidly rotating cylinder in uniform flow

2013 ◽  
Vol 730 ◽  
pp. 379-391 ◽  
Author(s):  
A. Rao ◽  
J. S. Leontini ◽  
M. C. Thompson ◽  
K. Hourigan
Keyword(s):  
Steady State ◽  
Rotation Rate ◽  
Free Stream ◽  
Three Dimensional ◽  
Rotating Cylinder ◽  
Marginal Stability ◽  
Cylinder Surface ◽  
Rotation Rates ◽  
Centrifugal Instability ◽  
The Stability

AbstractThe flow around an isolated cylinder spinning at high rotation rates in free stream is investigated. The existence of two steady two-dimensional states is confirmed, as is the existence of a secondary mode of vortex shedding. The stability of the two steady states to three-dimensional perturbations is established using linear stability analysis. At lower rotation rates on the first steady state, two three-dimensional modes are confirmed, and their structure and curves of marginal stability as a function of rotation rate and Reynolds number are determined. One mode (named mode $E$) appears consistent with a hyperbolic instability in the wake, while the second (named mode $F$) appears to be a centrifugal instability of the flow very close to the cylinder surface. At higher rotation rates on the second steady state, a single three-dimensional mode due to centrifugal instability (named mode ${F}^{\prime } $) is found. This mode becomes increasingly difficult to excite as the rotation rate is increased.

On the stability of viscous flow between eccentric rotating cylinders

1968 ◽  
Vol 32 (1) ◽  
pp. 131-144 ◽  
Author(s):  
G. S. Ritchie
Keyword(s):  
Viscous Flow ◽  
Stability Boundary ◽  
Outer Cylinder ◽  
Marginal Stability ◽  
Important Conclusion ◽  
Periodic Disturbances ◽  
Eccentric Cylinders ◽  
Relative Eccentricity ◽  
The Difference ◽  
The Stability

The stability of viscous flow between eccentric cylinders is analysed for the case in which the inner cylinder rotates while the outer cylinder remains stationary, and where the difference in radii of the cylinders is small in comparison with their mean radius. The linearised equations governing the marginal stability of axially periodic disturbances are derived in general for the case where the cylinders are infinitely long, and are solved approximately to give estimates of the critical Taylor number at which vortex flow occurs for a range of relative eccentricity of the cylinders.The results give an upper bound to the stability boundary, and certain results of DiPrima are used to establish a lower bound, and consequently the stability boundary is well established for eccentricity ratios less than about 0·6. One important conclusion is that for a considerable range of eccentricity ratio the flow is less stable than when the cylinders are concentric.

scholarly journals The role of three-body stability in tidally interacting globular clusters

2014 ◽  
Vol 10 (S312) ◽  
pp. 231-234
Author(s):  
Gareth F. Kennedy
Keyword(s):  
Globular Clusters ◽  
Milky Way ◽  
Velocity Dispersion ◽  
Stability Boundary ◽  
Three Body Problem ◽  
Stability Method ◽  
The Stability ◽  
Boundary Radius ◽  
Three Body

AbstractThe role of stability in the general three-body problem is investigated with regard to the tidal radius of a globular cluster (GC) in a galactic potential. This proceedings is a summary of two papers which outline the stability method (Kennedy 2014a) and compare the predicted stability boundary radius to observations of velocity dispersion profiles in Milky Way GCs (Kennedy 2014b).

Theoretical Study on the Flow Instability of Supercritical Water in the Parallel Channels

2012 ◽  
Author(s):  
Yali Su ◽  
Jian Feng ◽  
Wenxi Tian ◽  
Suizheng Qiu ◽  
Guanghui Su
Keyword(s):  
Supercritical Water ◽  
Flow Instability ◽  
Stability Boundary ◽  
Physical Models ◽  
Marginal Stability ◽  
Inlet Temperature ◽  
Flow And Heat Transfer ◽  
System Pressure ◽  
Parallel Channels ◽  
The Stability

For the flow of the supercritical water (SCW), the fierce variation of density and specific volume possibly cause flow instability. Based on the structure of parallel channels, mathematical and physical models were established to simulate the flow and heat transfer characteristics of the supercritical water in the parallel channels with semi-implicit scheme and staggered mesh scheme. Flow instability of super-critical water was obtained by using the little perturbation method. Pseudo-subcooling number (NSUB) and pseudo-phase change number (NPCH) are defined based on the property of SCW. The marginal stability boundary (MSB) is obtained with using the NSUB and NPCH. The effects of mass flow rate, inlet temperature and system pressure on the flow instability boundary were also investigated. When increasing the mass flows and system pressure, decreasing the heat flux, the stability in the parallel channels increases. The effect of inlet temperature in the low pseudo-subcooling number region is different from that in high pseudo-subcooling number region.

Inertial instability of intense stratified anticyclones. Part 2. Laboratory experiments

2013 ◽  
Vol 732 ◽  
pp. 485-509 ◽  
Author(s):  
Ayah Lazar ◽  
A. Stegner ◽  
R. Caldeira ◽  
C. Dong ◽  
H. Didelle ◽  
...  
Keyword(s):  
Parameter Space ◽  
Large Scale ◽  
Laboratory Experiments ◽  
Stability Limit ◽  
Stability Diagram ◽  
Relative Vorticity ◽  
Marginal Stability ◽  
Rotating Platform ◽  
Coriolis Parameter ◽  
The Stability

AbstractLarge-scale laboratory experiments were performed on the Coriolis rotating platform to study the stability of intense vortices in a thin stratified layer. A linear salt stratification was set in the upper layer on top of a thick barotropic layer, and a cylinder was towed in the upper layer to produce shallow cyclones and anticyclones of similar size and intensity. We focus our investigations on submesoscale eddies, where the radius is smaller than the baroclinic deformation radius. Towing speed, cylinder size and stratification were changed in order to cover a large range of the parameter space, staying in a relatively high horizontal Reynolds number ($Re= 2000{{\unicode{x2013}}}7000$). The Rayleigh criterion states that inertial instabilities should strongly destabilize intense anticyclonic eddies if the vorticity in the vortex core is negative enough ${\zeta }_{0} / f\lt - 1$, where ${\zeta }_{0} $ is the relative vorticity in the core of the vortex, and $f$ is the Coriolis parameter. However, we found that some anticyclones remain stable even for very intense negative vorticity values, up to ${\zeta }_{0} / f= - 3. 5$, when the Burger number is large enough. This is in agreement with the linear stability analysis performed in part 1 (J. Fluid Mech., vol. 732, 2013, pp. 457–484), which shows that the combined effect of a strong stratification and a moderate vertical dissipation may stabilize even very intense anticyclones, and the unstable eddies we found were located close to the marginal stability limit. Hence, these experimental results agree well with the simple stability diagram proposed in the Rossby, Burger and Ekman parameter space for inertial destabilization of viscous anticyclones within a shallow and stratified layer.

Instability Mechanisms of a Spinning Disk Under Rotating Damping Forces

1995 ◽  
Vol 48 (11S) ◽  
pp. S127-S131
Author(s):  
F. Y. Huang ◽  
C. D. Mote
Keyword(s):  
Periodic Solution ◽  
Parameter Space ◽  
Stability Boundary ◽  
Rotating Disk ◽  
Positive Definite ◽  
Stable Region ◽  
Damping Force ◽  
Galerkin Solution ◽  
The Stability ◽  
Definition Of

Stability of a rotating disk under rotating positive-definite damping forces is investigated analytically. The stability boundary is located exactly in parameter space by the criterion that at least one non-trivial periodic solution is necessary at every boundary point. The stable region of parameter space is identified through perturbation of a Galerkin solution. A non-trivial periodic solution is shown to exist only when damping forces are not generated with respect to that solution. Instability in the disk-damping system occurs when the wave speed of any mode in the undamped disk, when observed on the disk, is less that the speed of the damping force relative the disk. The instability occurs independent of the magnitude of the force and the definition of the positive-definite damping operator.

Stability of bounded rapid shear flows of a granular material

1996 ◽  
Vol 308 ◽  
pp. 31-62 ◽  
Author(s):  
Chi-Hwa Wang ◽  
R. Jackson ◽  
S. Sundaresan
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Granular Material ◽  
Bulk Density ◽  
Particulate Material ◽  
Dominant Mode ◽  
Marginal Stability ◽  
Sheared Layer ◽  
The Stability ◽  
Unusual Type

This paper presents a linear stability analysis of a rapidly sheared layer of granular material confined between two parallel solid plates. The form of the steady base-state solution depends on the nature of the interaction between the material and the bounding plates and three cases are considered, in which the boundaries act as sources or sinks of pseudo-thermal energy, or merely confine the material while leaving the velocity profile linear, as in unbounded shear. The stability analysis is conventional, though complicated, and the results are similar in all cases. For given physical properties of the particles and the bounding plates it is found that the condition of marginal stability depends only on the separation between the plates and the mean bulk density of the particulate material contained between them. The system is stable when the thickness of the layer is sufficiently small, but if the thickness is increased it becomes unstable, and initially the fastest growing mode is analogous to modes of the corresponding unbounded problem. However, with a further increase in thickness a new mode becomes dominant and this is of an unusual type, with no analogue in the case of unbounded shear. The growth rate of this mode passes through a maximum at a certain value of the thickness of the sheared layer, at which point it grows much faster than any mode that could be shared with the unbounded problem. The growth rate of the dominant mode also depends on the bulk density of the material, and is greatest when this is neither very large nor very small.

Sign in / Sign up

Export Citation Format

Share Document