On Trapped Modes of Rotating Fluids in Spherical Shells

The question of existence of axisymmetric, equatorially trapped modes in rotating spherical shells (Stern, 1963; Stewartson and Rickard, 1969) is approached by means of a numerical simulation. The existence of one trapped mode is confirmed, and the dependence of its frequency on the thickness of the shell is investigated. The ray theoretical approach of Bretherton (1964) is also reconsidered, and it is found that in a shell of given thickness there are only a limited number of closed ray patterns which are confined to the vicinity of the equator. A continuous band of frequencies is associated with each one of these rays. It is found that the frequencies derived by the numerical simulation for thin shells agree with the maximum frequency in the first of these bands. It is conjectured that this fact may be associated with the viscous boundary conditions driving the forced oscillations inside the shell.     

On the Existence of Trapped Modes in Channels of Arbitrary Cross-section

This article establishes the existence of trapped-mode solutions of a linearized water-wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls. A trapped mode corresponds to an eigenvalue of a non-local Neumann–Dirichlet operator for an elliptic boundary-value problem in the fluid domain. The existence of such an eigenvalue is established by generalizing previous results concerning spectral theory for differential operators to this non-local operator. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

The S-Matrix and Acoustic Signal Structure in Simple and Compounds Waveguides

AMS(MOS) 35P25 Secondary: 76Q05     

Thermal Effects in a Two-Dimensional Hybrid Elastic Structure

In this paper we consider a two-dimensional hybrid thermo-elastic structure consisting of a thermo-elastic plate which has a beam attached to its free end. We show that the initial-boundary-value problem for the interactive system of partial differential equations which take account of the mechanical strains/stresses and the thermal stresses in the plate and the beam, can be associated with a uniformly bounded evolution operator. It turns out that the interplay of parabolic dynamics due to the thermal effects in the hybrid structure and the hyperbolic dynamics associated with the elasticity of the structure yields analyticity for the entire system. This result yields solvability for the problem under optimal initial freedom of the displacement, velocity, and temperature in the plate and the beam, while uniform stability is readily available.     

Branch Structure and Implementation of Lambert W

We begin with a discussion of general design decisions made in implementing the Lambert W function in Maple . Many of these decisions are not system-specific and apply to any implementation of W; also they touch some of the fundamental issues in computer-algebra systems. A specific topic is the choice of a branch structure for W, and a new approach is presented that allows us to extend the definition of a function from the real line into the complex plane.     

Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating-Disk

In this article, a theoretical study is pursued to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating-disk. Special attention is focused on to the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [ 1 ] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [ 1 ] for the incompressible stationary mode and in [ 2 ] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. The asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.     

Two-Dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions

This paper examines the bifurcation and structure of the bifurcated solutions of the two-dimensional infinite Prandtl number convection problem. The existence of a bifurcation from the trivial solution to an attractor Σ _R was proved by Park (Disc. Cont. Dynam. Syst. B [2005]). We prove in this paper that the bifurcated attractor Σ _R consists of only one cycle of steady-state solutions and that it is homeomorphic to S¹. By thoroughly investigating the structure and transitions of the solutions of the infinite randtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In turn, this will corroborate and justify the suggested results with the physical findings about the presence of the roll structure. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using a new geometric theory of incompressible flows. Both theories were developed by Ma and Wang; see Bifurcation Theory and Applications (World Scientific, 2005) and Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics (American Mathematical Society, 2005).     

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.     

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [ 1 ], it was demonstrated that the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [ 2 ] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameter     , which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [ 1 ] for zero-suction compressible modes.
The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.     

Riccati方程周期解的分支

证明带参数λ的Riccati方程x′=x~2+(λ+Q(t))存在周期解的分支点λ_0,当λλ_0时有且仅有两个周期解,当λ=λ_0时有且仅有一个周期解,当λλ_0时所有解无界.     

一类二维系统的极限环的个数

本文利用Dulac函数方法讨论一类二维系统在环城上的包围多个奇点的极限环的唯一性及在n连城上极限环的唯n-1性，并给出了两个多项式的例子，讨论了极限环的唯一性和唯二性．     

The Exchange Value Embedded in a Transport System

This paper shows that a well designed transport system has an embedded exchange value by serving as a market for potential exchange between consumers. Under suitable conditions, one can improve the welfare of consumers in the system simply by allowing some exchange of goods between consumers during transportation without incurring additional transportation cost. We propose an explicit valuation formula to measure this exchange value for a given compatible transport system. This value is always nonnegative and bounded from above. Criteria based on transport structures, preferences and prices are provided to determine the existence of a positive exchange value. Finally, we study a new optimal transport problem with an objective taking into account of both transportation cost and exchange value.     

The Spectrum of the Two-Dimensional Periodic Schrödinger Operator

The absence of eigenvalues (of infinite multiplicity) for the two-dimensional periodic Schrödinger operator with a variable metric is proved. The method of proof does not use the change of variables reducing the metric to a scalar form.     

The Solution of Two-Dimensional Oseen Flow Problems Using Integral Conditions

A general method of solving Oseen's linearized equations fortwo-dimensional steady flow of a viscous incompressible fluidpast a cylinder in an unbounded field is developed. The analysisis developed in terms of the scalar vorticity and stream functionand it is shown that the vorticity for Oseen flow problems canbe obtained separately from the stream function. The determinationof the vorticity can be effected using conditions of an integralcharacter deduced from the no-slip condition at the cylindersurface together with the conditions at large distances. Theindependent determination of the vorticity seems to be a newstep in Oseen theory. The method enables one to obtain manyproperties of the flow in terms ofthe Reynolds number by usingonly the vorticity without the necessity of finding the streamfunction. The use of integral conditions makes the detailedcalculations straightforward, systematic, and elementary. Themethod is tested by applying it to the case of uniform flowpast an elliptic cylinder at an arbitrary angle of incidenceand also to cases of symmetrical and asymmetrical flows pastcircular cylinders. The leading approximation for small Reynoldsnumber is obtained where possible. In the case of flow pasta rotating cylinder, the only possible solution is the Oseensolution for the nonrotating case with the addition of a potentialvortex.     

The Eigenvalues of Mathieu's Equation and their Branch Points

A comprehensive account is given of the behavior of the eigenvalues of Mathieu's equation as functions of the complex variable q. The convergence of their small-q expansions is limited by an infinite sequence of rings of branch points of square-root type at which adjacent eigenvalues of the same type become equal. New asymptotic formulae are derived that account for how and where the eigenvalues become equal. Known asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.