One-dimensional Perturbations, Asymptotic Expansions, and Spectral Gaps

2008 ◽  
pp. 149-173
Author(s):  
Seppo Hassi ◽  
Adrian Sandovici ◽  
Henk de Snoo ◽  
Henrik Winkler
Keyword(s):  
Asymptotic Expansions ◽  
Spectral Gaps ◽  
One Dimensional

Spectral Gaps in One-Dimensional Random Binary Alloys

1973 ◽  
Vol 8 (8) ◽  
pp. 3773-3777
Author(s):  
M. Fowler ◽  
C. T. Papatriantafillou
Keyword(s):  
Binary Alloys ◽  
Spectral Gaps ◽  
One Dimensional

A quasi-steady approach to the instability of time-dependent flows in pipes

2002 ◽  
Vol 465 ◽  
pp. 301-330 ◽  
Author(s):  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Experimental Data ◽  
Laminar Flow ◽  
Asymptotic Expansions ◽  
Unstable Mode ◽  
Landau Equation ◽  
Base Flow ◽  
Neutral Stability ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Weakly Nonlinear Analysis

Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

scholarly journals Mathematical Models of Ice Shelves

1976 ◽  
Vol 17 (77) ◽  
pp. 419-432 ◽  
Author(s):  
P. A. Shumskiy ◽  
M. S. Krass
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansions ◽  
Internal Heat ◽  
Heat Accumulation ◽  
Rheological Equation ◽  
Ice Shelves ◽  
One Dimensional ◽  
Integro Differential Equation ◽  
The One ◽  
Reduced Temperature

For flat external ice shelves, expanding freely in all directions, the problem of thermodynamics is one-dimensional. In the affine dimensionless system of coordinates, equations of the dynamics together with the rheological equation lead to the non-linear integro-differential equation involving the reduced temperature. In the quasi-steady case the boundary problem for this equation is solved by means of the method of combining asymptotic expansions. It is shown that if ice is coming from the upper and lower surfaces in the opposite directions the regime is unsteady because of the internal heat accumulation.The integro-differential equation for the temperature in the case of thinning internal ice shelves is more complicated, but it can be solved by a method analogous to the one mentioned above.

scholarly journals A theory of spectral partitions of metric graphs

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
James B. Kennedy ◽  
Pavel Kurasov ◽  
Corentin Léna ◽  
Delio Mugnolo
Keyword(s):  
Spectral Gaps ◽  
Qualitative Properties ◽  
Metric Graphs ◽  
One Dimensional ◽  
Graph Partitions ◽  
Planar Domains ◽  
Open Questions ◽  
The One ◽  
Abstract Framework ◽  
Special Case

AbstractWe introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band et al. (Commun Math Phys 311:815–838, 2012) as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic—rather than numerical—results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in Conti et al. (Calc Var 22:45–72, 2005), Helffer et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:101–138, 2009), but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

scholarly journals Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

2011 ◽  
Vol 468 (2138) ◽  
pp. 545-562 ◽  
Author(s):  
Kaori Nagatou ◽  
Michael Plum ◽  
Mitsuhiro T. Nakao
Keyword(s):  
Periodic Function ◽  
Essential Spectrum ◽  
Schrodinger Equation ◽  
Spectral Gaps ◽  
Numerical Approximations ◽  
One Dimensional ◽  
Spectral Bands ◽  
Computer Assistance ◽  
Computational Errors ◽  
Perturbed Periodic

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at . It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

Near-frozen quasi-one-dimensional flow I. The reservoir problem

1967 ◽  
Vol 262 (1125) ◽  
pp. 203-224 ◽  
Keyword(s):  
Relaxation Time ◽  
Entropy Production ◽  
Asymptotic Expansions ◽  
Flow Time ◽  
Stagnation Zone ◽  
Rate Parameter ◽  
One Dimensional ◽  
Frozen Solution ◽  
Nozzle Flows ◽  
Relaxing Gas

Non-equilibrium quasi-one-dimensional nozzle flows are considered in the limit when the relaxation time is large compared with some characteristic flow time. Non-uniformities which arise in the reservoir region, for convergent-divergent nozzles, are treated by the method of matched asymptotic expansions (see, for example, Van Dyke 1964). It is shown that even away from this stagnation zone the solution does not proceed simply in integral powers of the rate parameter. The correct solution is deduced for a vibrationally relaxing gas. It is noted, however, that this near-frozen solution does not necessarily remain valid at downstream infinity where the overall entropy production may become important. Solutions valid in this region are presented in part II of this paper.

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