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We investigate the linear stability of the Bickley jet in the framework of the beta-plane approximation. Because singular inviscid neutral modes exist in the retrograde case , it is necessary to add viscosity to interpret them. One of these modes was found in closed form by Howard and Drazin [1] . However, its critical point is at the center of the jet and it was therefore not possible for these authors to ascertain the relationship of this mode to the stability problem or to discuss how to continue the eigenfunction across the singularity.
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber α2 ≤ 9/2 ), that is, it corresponds to both an odd and an even mode. This conclusion is confirmed by direct numerical solution of the Orr–Sommerfeld equation which shows, in addition, that viscosity is destabilizing along portions of the stability boundary. For a retrograde jet, instability is found to occur beyond the inviscid critical value of β, that is, in the region where the flow would be stable according to the Rayleigh–Kuo condition. 相似文献
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber α
5.
Let
d−1{(x1,…,xd)
d:x21+···+x2d=1} be the unit sphere of the d-dimensional Euclidean space
d. For r>0, we denote by Brp (1p∞) the class of functions f on
d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on
d−1,
and Pλk(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of Brp in Lq(
d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces XN of Lq(
d−1).The main result in this paper is that for r2(d−1)2,where ANBN means that there exists a positive constant C, independent of N, such that C−1ANBNCAN.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions. 相似文献
6.
Rebecca L Brown 《Journal of sound and vibration》2003,262(3):591-611
Most structural dynamic systems are of high order; however, they often exhibit phenomena that can be dealt with effectively using low order models. This paper presents a method for describing certain kinds of damage evolution in mechanical systems. The method relies on a simple principle that as damage evolves in a structural dynamic system, the damage indicator (i.e., diagnostic feature) behaves like a stable quasi-stationary equilibrium point in a subsidiary non-linear bifurcating system within the so-called damage center manifold. It is shown that just as linear normal modes govern the behavior of linear structures with idealized damping, so too do non-linear normal forms govern the evolution of damage within structures in many instances. The method is justified with citations from the literature on certain types of mechanical failure and then applied in an experimental case involving reversible damage in a bolted fastener. Off-line experiments on a rotorcraft fuselage show that the evolution of damage is sensitive to both temporal and spatial bifurcation parameters. A diagnostic sensing strategy whereby output-only transmissibility features are used to decrease the order of high order structural dynamic measurements is also described. 相似文献
7.
Richard Brown 《Geometriae Dedicata》2003,97(1):129-150
In this note, we show that given a closed, orientable genus-g surface S
g
, any hyperbolic toral automorphism has a positive power which induces a quadratic, orientable pseudo-Anosov homeomorphism on S
g
. To show this, we lift Anosov toral automorphisms through a ramified topological covering and present the lifted homeomorphism via a standard set of Lickorish twists. This construction provides a general method of producing pseudo-Anosov maps of closed surfaces with predetermined orientable foliations and quadratic dilatation. Since these lifted automorphisms have orientable foliations, this construction is a sort of converse to that of Franks and Rykken [Trans. Amer. Math. Soc. 1999], who established that one can associate to a quadratic pseudo-Anosov homeomorphism with oriented unstable foliation a hyperbolic toral automorphism. 相似文献
8.
Synthetic routes to BEDT-TTF derivatives bearing side chain carboxylic ester and amide groups are reported. Methyl ET-ethanoate was prepared in five steps from vinylacetic acid; amide groups were installed early in the synthesis by mixed anhydride methods before the final coupling reaction. 相似文献
9.
Anna Arkhireeva John N. Hay Jackie M. Lane Miguel Manzano Helen Masters Wilberforce Oware Steve J. Shaw 《Journal of Sol-Gel Science and Technology》2004,31(1-3):31-36
The synthesis of ORganically MOdified SILica (ORMOSIL) particles has been carried out using both the hydrolytic and non-hydrolytic sol-gel routes. The hybrid (nano)composites are organically modified with an alkyl or aryl group covalently bonded to silicon. Hybrids have been synthesised in an aqueous sol-gel process by a modified Stöber route, producing spherical nanoparticles with diameters in the range 50–300 nm. The size of the particles can be controlled by control of certain reaction parameters. Smaller ormosil nanoparticles can be synthesised by a base-catalysed emulsion polymerisation route, by varying the type and concentration of surfactant and precursor feed rate. In this case, particles in the size range 3.5–10 nm can be obtained. Hybrids have been synthesised from hyperbranched polyesters by encapsulation in a silica matrix using the hydrolytic sol-gel route. Optimisation of the reaction conditions allows the hybrids to be produced as isolated sub-micron spherical particles. Ormosil particles have also been synthesised using the non-hydrolytic sol-gel route, which may lead to products of different morphologies because of the different polarity of the reaction medium. Different reaction conditions were studied in order to optimise the size and shape of the particles, including choice of solvent, use of surfactants and addition of polystyrene. Dimethylsulfoxide acts as a novel oxygen donor for the catalyst-free formation of colourless silsesquioxanes. 相似文献
10.