On the stability of a columnar vortex to disturbances with large azimuthal wavenumber: the lower neutral points

1987 ◽  
Vol 178 ◽  
pp. 549-566 ◽  
Author(s):  
K. Stewartson ◽  
S. Leibovich
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Saddle Point Method ◽  
Marginal Stability ◽  
Columnar Vortex ◽  
The Stability ◽  
Neutral Conditions ◽  
Inviscid Instability ◽  
Limiting Values ◽  
Swirl Parameter

The inviscid instability of a columnar trailing-line vortex at large values of the azimuthal wavenumber n near neutral conditions is considered. This extends an earlier analysis (Leibovich & Stewartson 1983), which is not accurate near the limiting values of the axial wavenumber for which instabilities exist. Here an asymptotic expansion is derived for the solution in the neighbourhood of the lower neutral point and the results compared with existing computations a t moderate values of n. For these weak instabilities disturbances are centred near the axis of the vortex and the relevant equation is solved in the complex plane by a generalized saddle-point method. In addition, the marginal stability of the vortex is examined, and an estimate obtained of the value of the swirl parameter above which the vortex is stable at large values of n.

On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings

2017 ◽  
Vol 165 (2) ◽  
pp. 287-339 ◽  
Author(s):  
TOMOTADA OHTSUKI ◽  
YOSHIYUKI YOKOTA
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Expansions ◽  
Saddle Point Method ◽  
Hyperbolic Structure ◽  
3 Dimensional ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Chern Simons ◽  
Dimensional Domain

AbstractWe give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.

On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings

2017 ◽  
Vol 28 (13) ◽  
pp. 1750096 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Expansions ◽  
Technical Difficulty ◽  
Point Method ◽  
Saddle Point Method ◽  
Cambridge Philos ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Chern Simons

We give presentations of the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. As the volume conjecture states, the leading terms of the expansions present the hyperbolic volume and the Chern–Simons invariant of the complements of the knots. As coefficients of the expansions, we obtain a series of new invariants of the knots. This paper is a continuation of the previous papers [T. Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the [Formula: see text] knot, Quantum Topol. 7 (2016) 669–735; T. Ohtsuki and Y. Yokota, On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings, to appear in Math. Proc. Cambridge Philos. Soc.], where the asymptotic expansions of the Kashaev invariant are calculated for hyperbolic knots with five and six crossings. A technical difficulty of this paper is to use 4-variable saddle point method, whose concrete calculations are far more complicated than the previous papers.

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.

scholarly journals Two-Dimensional Convection Induced by Gravity and Centrifugal Forces in a Rotating Porous Layer Far Away from the Axis of Rotation

1998 ◽  
Vol 4 (2) ◽  
pp. 73-90 ◽  
Author(s):  
Peter Vadasz ◽  
Saneshan Govender
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Temperature Fields ◽  
Wave Number ◽  
Marginal Stability ◽  
Two Dimensional ◽  
Wide Range ◽  
Gravity Driven ◽  
Gravity Driven Flow ◽  
The Stability

The stability and onset of two-dimensional convection in a rotating fluid saturated porous layer subject to gravity and centrifugal body forces is investigated analytically. The problem corresponding to a layer placed far away from the centre of rotation was identified as a distinct case and therefore justifying special attention. The stability of a basic gravity driven convection is analysed. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number for different values of the gravity related Rayleigh number. For any given value of the gravity related Rayleigh number there is a transitional value of the wave number, beyond which the basic gravity driven flow is stable. The results provide the stability map for a wide range of values of the gravity related Rayleigh number, as well as the corresponding flow and temperature fields.

scholarly journals A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large

10.37236/1517 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Asymptotic Behavior ◽  
Saddle Point ◽  
Generating Functions ◽  
Single Point ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Point Method ◽  
Saddle Point Method ◽  
Numerical Verification ◽  
Height Distribution

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to $b$ items (the so called $b$-tries). We assume that such a tree is built from $n$ random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when $b$ and $n$ are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed $b$. We prove that for most $n$, the limiting distribution is concentrated at the single point $k_1=\lfloor \log_2 (n/b)\rfloor +1$ as $n,b\to \infty$. We observe that this is quite different than the height distribution for fixed $b$, in which case the limiting distribution is of an extreme value type concentrated around $(1+1/b)\log_2 n$. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

A sufficient condition for the instability of columnar vortices

1983 ◽  
Vol 126 ◽  
pp. 335-356 ◽  
Author(s):  
S. Leibovich ◽  
K. Stewartson
Keyword(s):  
Numerical Solutions ◽  
Three Dimensional ◽  
Azimuthal Velocity ◽  
Unstable Wave ◽  
Sufficient Condition ◽  
Asymptotic Results ◽  
Specific Flow ◽  
Columnar Vortex ◽  
Swirl Velocity ◽  
Inviscid Instability

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r), azimuthal velocity V(r), where r is distance from the axis, let Ω be the angular velocity V/r, and let Γ be the circulation rV. Then the flow is unstable if $ V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right] < 0.$

The stability of viscous flow between rotating cylinders in the presence of a magnetic field

1953 ◽  
Vol 216 (1126) ◽  
pp. 293-309 ◽  
Keyword(s):  
Magnetic Field ◽  
Electrical Conductivity ◽  
Viscous Flow ◽  
Thermal Instability ◽  
Horizontal Layer ◽  
Rotational Stability ◽  
Marginal Stability ◽  
Inhibiting Effect ◽  
The Stability ◽  
The Magnetic Field

In this paper the theory of the stability of viscous flow between two rotating coaxial cylinders which has been developed by Taylor, Jeffreys and Meksyn is extended to the case when the fluid considered is an electrical conductor and a magnetic field along the axis of the cylinders is present. A differential equation of order eight is derived which governs the situation in marginal stability; and a significant set of boundary conditions for the problem is formulated. The case when the two cylinders are rotating in the same direction and the difference ( d ) in their radii is small compared to their mean (R 0 ) is investigated in detail. A variational procedure for solving the underlying characteristic value problem and determining the critical Taylor numbers for the onset of instability is described. As in the case of thermal instability of a horizontal layer of fluid heated below, the effect of the magnetic field is to inhibit the onset of instability, the inhibiting effect being the greater, the greater the strength of the field and the value of the electrical conductivity. In both cases, the inhibiting effect of the magnetic field depends on the strength of the field ( H ), the density ( ρ ) and the coefficients of electrical conductivity ( σ ), kinematic viscosity ( v ) and magnetic permeability ( μ ) through the same non-dimensional combination Q =μ 2 H 2 d 2 σ/ pv ; however, the effect on rotational stability is more pronounced than on thermal instability. A table of the critical Taylor numbers for various values of Q is provided.

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