scholarly journals Constructing the symplectic Evans matrix using maximally analytic individual vectors

2003 ◽  
Vol 133 (3) ◽  
pp. 505-526 ◽  
Author(s):  
Thomas J. Bridges ◽  
Gianne Derks
Keyword(s):  
Stability Problem ◽  
Geometric Analysis ◽  
Evans Function ◽  
Branch Points ◽  
Natural Region ◽  
Isolated Points ◽  
New Construction ◽  
The Stability ◽  
Linear Stability Problem ◽  
Symplectic Case

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

Stability of columnar convection in a porous medium

2013 ◽  
Vol 737 ◽  
pp. 205-231 ◽  
Author(s):  
Duncan R. Hewitt ◽  
Jerome A. Neufeld ◽  
John R. Lister
Keyword(s):  
Porous Medium ◽  
Linear Stability ◽  
Stability Problem ◽  
Columnar Structure ◽  
Marginal Stability ◽  
Vertically Propagating ◽  
The Stability ◽  
Rayleigh Benard ◽  
Linear Stability Problem ◽  
Dimensional Domain

AbstractConvection in a porous medium at high Rayleigh number $\mathit{Ra}$ exhibits a striking quasisteady columnar structure with a well-defined and $\mathit{Ra}$-dependent horizontal scale. The mechanism that controls this scale is not currently understood. Motivated by this problem, the stability of a density-driven ‘heat-exchanger’ flow in a porous medium is investigated. The dimensionless flow comprises interleaving columns of horizontal wavenumber $k$ and amplitude $\widehat{A}$ that are driven by a steady balance between vertical advection of a background linear density stratification and horizontal diffusion between the columns. Stability is governed by the parameter $A= \widehat{A}\mathit{Ra}/ k$. A Floquet analysis of the linear-stability problem in an unbounded two-dimensional domain shows that the flow is always unstable, and that the marginal-stability curve is independent of $A$. The growth rate of the most unstable mode scales with ${A}^{4/ 9} $ for $A\gg 1$, and the corresponding perturbation takes the form of vertically propagating pulses on the background columns. The physical mechanism behind the instability is investigated by an asymptotic analysis of the linear-stability problem. Direct numerical simulations show that nonlinear evolution of the instability ultimately results in a reduction of the horizontal wavenumber of the background flow. The results of the stability analysis are applied to the columnar flow in a porous Rayleigh–Bénard (Rayleigh–Darcy) cell at high $\mathit{Ra}$, and a balance of the time scales for growth and propagation suggests that the flow is unstable for horizontal wavenumbers $k$ greater than $k\sim {\mathit{Ra}}^{5/ 14} $ as $\mathit{Ra}\rightarrow \infty $. This stability criterion is consistent with hitherto unexplained numerical measurements of $k$ in a Rayleigh–Darcy cell.

On the stability of the asymptotic suction boundary-layer profile

1965 ◽  
Vol 23 (4) ◽  
pp. 715-735 ◽  
Author(s):  
T. H. Hughes ◽  
W. H. Reid
Keyword(s):  
Exact Solution ◽  
Stability Problem ◽  
Adjoint Problem ◽  
Analytical Continuation ◽  
Neutral Stability ◽  
The Stability ◽  
Asymptotic Suction ◽  
Linear Stability Problem ◽  
Original Treatment ◽  
General Method

This paper presents a discussion of some aspects of the linear stability problem for the asymptotic suction profile. An exact solution of the inviscid equation is first obtained in terms of the usual hypergeometric function and its analytical continuation. This exact solution provides both a corrected version of an earlier treatment by Freeman and an independent check on the more general method suggested for solving the inviscid equation numerically. Various approximations to the characteristic equation, and hence to the curve of neutral stability, are then considered. In particular, it is found that, in a consistent asymptotic treatment of the related adjoint problem, at least one viscous correction to the singular inviscid solution must be considered. Based on the present results for the adjoint problem, it is suggested that Tollmien's original treatment of the viscous corrections must be slightly modified.

Effects Of Surface Solidification on the Stability Of Multi-Layered Liquid Films

1983 ◽  
Vol 105 (1) ◽  
pp. 119-120 ◽  
Author(s):  
S. P. Lin
Keyword(s):  
Stability Problem ◽  
Film Flow ◽  
Liquid Films ◽  
Interfacial Shear ◽  
Capillary Waves ◽  
Inclined Plane ◽  
Air Interface ◽  
The Stability ◽  
Linear Stability Problem ◽  
Special Case

The linear stability problem of a n-layered liquid film with a solidified liquid-air interface is reviewed. The general formulation is applied to the special case of a two-layered film flow down an inclined plane. A stability condition is given explicitly in terms of the density, viscosity and thickness ratios. Based on this condition it is found that solidification of the free surface may have the effects of stabilizing the interfacial shear waves and destabilizing the gravity-capillary waves associated with top-heavy density stratification.

The Influence of Topography on the Stability of Jets

2005 ◽  
Vol 35 (5) ◽  
pp. 811-825 ◽  
Author(s):  
F. J. Poulin ◽  
G. R. Flierl
Keyword(s):  
Shallow Water ◽  
Stability Problem ◽  
Small Amplitude ◽  
Fluid Transport ◽  
Nonlinear Evolution ◽  
Flat Bottom ◽  
The Stability ◽  
The Right ◽  
Linear Stability Problem ◽  
Topographic Parameters

Abstract In this article, the effect shelflike topography has on the stability of a jet that flows along the smooth shelf is addressed. The linear stability problem is solved to determine for which nondimensional parameters a shelf can either destabilize or stabilize a jet. These calculations reveal an intricate dependence of growth rate on topography. In particular, the authors determine that retrograde topography (with the shallow water on the left) always stabilizes the jet (in relation to the flat-bottom equivalent), whereas prograde topography (with the shallow water on the right) can either stabilize or destabilize the jet depending on the particular values of the Rossby number and topographic parameters. For Rossby numbers of order 1 and larger, prograde topography is strictly stabilizing. For small Rossby numbers, small-amplitude topography destabilizes whereas large topography stabilizes. The nonlinear evolution of these instabilities is explored to confirm the predictions from the linear theory and, also, to illustrate how stabilization is directly related to fluid transport across the shelf.

Experiments on the stability of viscous flow between rotating cylinders - VI. Finite-amplitude experiments

1965 ◽  
Vol 283 (1395) ◽  
pp. 531-556 ◽  
Keyword(s):  
Stability Problem ◽  
High Speed ◽  
Amplification Factor ◽  
Finite Amplitude ◽  
Taylor Number ◽  
The Difference ◽  
The Stability ◽  
Speed Computer ◽  
Linear Stability Problem

The ion technique is applied to a variety of experiments in Couette flow: determination of the critical Taylor number; determination of wavenumber and waveform of the vortices; establishment of the laws that the square of the amplitude and the amplification factor both vary as the difference between the Taylor number and the critical Taylor number; and finally certain studies of the growth and decay of vortices as the Taylor number is changed. The experimental data are compared with the results of integration of the exact equations for the linear stability problem by means of a high-speed computer. These theo­retical values were obtained by P. H. Roberts and are presented in an appendix to this paper.

Effect of Modulation on Thermal Convection Instability

2000 ◽  
Vol 55 (11-12) ◽  
pp. 957-966 ◽  
Author(s):  
P. K. Bhatia ◽  
B. S. Bhadauria
Keyword(s):  
Fourier Series ◽  
Rayleigh Number ◽  
Temperature Gradient ◽  
Stability Problem ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Time Dependent ◽  
The Stability ◽  
Linear Stability Problem ◽  
Oscillatory Temperature

Abstract The linear stability problem for a fluid in a classic Benard configuration is considered. The applied temperature gradient is the sum of a steady component and a time-dependent periodic component. Only infinitesimal disturbances are considered. The time-dependent perturbation is expressed in Fourier series. The shift in critical Rayleigh number is calculated and the modulating effect of the oscillatory temperature gradient on the stability of the fluid layer is examined. Some comparison is made with known results.

scholarly journals The two-dimensional hydrodynamical theory of moving aerofoils. IV

1947 ◽  
Vol 188 (1015) ◽  
pp. 439-463
Keyword(s):  
Stability Problem ◽  
Two Dimensional ◽  
Centre Of Gravity ◽  
First Approximation ◽  
Von Karman ◽  
Moving Cylinder ◽  
Period Equation ◽  
The Stability ◽  
Von Kármán ◽  
Hydrodynamical Theory

In the first part of this paper opportunity has been taken to make some adjustments in certain general formulae of previous papers, the necessity for which appeared in discussions with other workers on this subject. The general results thus amended are then applied to a general discussion of the stability problem including the effect of the trailing wake which was deliberately excluded in the previous paper. The general conclusion is that to a first approximation the wake, as usually assumed, has little or no effect on the reality of the roots of the period equation, but that it may introduce instability of the oscillations, if the centre of gravity of the element is not sufficiently far forward. During the discussion contact is made with certain partial results recently obtained by von Karman and Sears, which are shown to be particular cases of the general formulae. An Appendix is also added containing certain results on the motion of a vortex behind a moving cylinder, which were obtained to justify certain of the assumptions underlying the trail theory.

Some Structural Properties of Systolic Tree Automata

1989 ◽  
Vol 12 (4) ◽  
pp. 571-585
Author(s):  
E. Fachini ◽  
A. Maggiolo Schettini ◽  
G. Resta ◽  
D. Sangiorgi
Keyword(s):  
Structural Properties ◽  
Stability Problem ◽  
Sufficient Condition ◽  
Tree Automata ◽  
Necessary And Sufficient Condition ◽  
The Stability ◽  
Necessary And Sufficient

We prove that the classes of languages accepted by systolic automata over t-ary trees (t-STA) are always either equal or incomparable if one varies t. We introduce systolic tree automata with base (T(b)-STA), a subclass of STA with interesting properties of modularity, and we give a necessary and sufficient condition for the equivalence between a T(b)-STA and a t-STA, for a given base b. Finally, we show that the stability problem for T(b)-ST A is decidible.

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