Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

States of classical statistical mechanical systems of infinitely many particles. I

We study a general mathematical model of a classical system of infinitely many point particles. The space X of infinite particle configurations is equipped with a natural topology as well as a measurable structure related to it. It is also connected with a family {X _A } of local spaces of finite configurations indexed by bounded open sets A in the one-particle space E. A theorem analogous to Kolmogoroff's fundamental theorem for stochastic processes is proved, according to which a consistent family { _A } of local probability measures _A defined on the X _A gives rise to a unique probability measure on X. We also study the problem of integral representation for positive linear forms defined over some linear space of real functions on X. We prove that a positive linear form F(f), defined for functions f in the class C+P, admits a uniquely determined integral representation F(f)= f () d, where is a probability measure over X.     

On a dynamical system on matrix algebra

LetM(n) be the algebra of alln×n complex matrices. We consider a dynamical system onM(n) defined by the vector fieldV(X)=[[X ^*,X],X], (X M(n)). It arises as the gradient flow for two kinds of variational problems onM(n). Given anyX ₀ M(n), letX(t) be the trajectory starting atX ₀. We study the global behavior ofX(t) ast . We show that, ifX ₀ is semisimple, thenX(t) converges exponentially to a normal matrix. IfX ₀ is not semisimple, then the behavior ofX(t) is completely different and difficult to analyze. We give some results also in this case. Furthermore, we discuss about a center manifold approach to our dynamical system.     

Interactions in general continua with microstructure

This paper studies the behavior of the one dimensional Broadwell model of a discrete three velocity gas on a bounded domain 0 x 1 with specularly reflective boundary condition at x = 0, 1. For smooth initial data we show that the initial boundary value problem possesses a unique smooth solution which tends as t to a free state consisting of traveling waves f _1e (x – ct), f _2e (x + ct), f _3e (x) where each f _ie is 2-periodic. The convergence is in the weak* topology of an appropriate Orlicz-Banach state space. No smallness assumptions are made on the data.In memory of Ronald J. DiPerna     

An experimental investigation of the sources of propeller noise due to the ingestion of turbulence at low speeds

Noise radiation from a four bladed, 10 in. diameter propeller operating in air at a rotational speed of 3000 RPM and a freestream velocity of 33 ft/s was experimentally analyzed using hot-wire and microphone measurements in an anechoic wind tunnel. Turbulence levels from 0.2 to 5.5% at the propeller location were generated by square-mesh grids upstream of the propeller. Autobicoherence measurements behind the blade trailing edges near the hub and tip showed regions of high phase-coherence between the blade-passage harmonics and the broadband frequencies. Inflow turbulence reduced this coherence. By relating the fluctuation velocities in the propeller wake to the unsteady blade forces, the primary regions of tonal noise generation have been identified as the hub and tip regions, while the midspan has been identified as a region responsible for broadband noise generation. These measurements were complimented by cross-spectra between the propeller wake-flow and the measured sound. The effect of turbulence on the radiated noise level showed an overall increase of 2 dB in the broadband levels for every 1% increase in turbulence. This effect varied for different frequency bands in the acoustic spectrum.List of Symbols b ² (f _k, f _l) Bicoherence - B (k, l) Bispectrum - B Number of Blades - c Speed of Sound - C _T Thrust coefficient=T/n ²D⁴ - D Propeller diameter - E [ ] Expected value - f Frequency, Hz - G _xx (f) One sided autospectral density function. - G _xy (f) One sided cross-spectral density function - J Advance ratio, J=U/nD - j, k Fourier component indices - m Grid mesh length - M _o Rotational mach number at a radial location M _o =2nr/a _o - M _c Axial convection mach number - n Rotational speed, revolutions per second - r Propeller radial location - R Propeller radius - R _e Reynolds number - T Propeller thrust - U Freestream velocity - U _i Induced axial velocity from propeller - u, w RMS of fluctuating velocity, u=(U–u)² - X(f) Fourier transform of x(t) Symbols _xy ^/2 (f) Coherence function, - Observer angle, measured from propeller thrust direction - _f Longitudinal Eulerian dissipation length scale - _f Longitudinal Eulerian integral length scale - Air density - Blade azimuthal location This research was performed at the Hessert Center for Aerospace Research, Department of Aerospace and Mechanical Engineering, University of Notre Dame, and was sponsored by the U.S. Navy, Office of Naval Research, Arlington, Virginia under Contract No. N00014-89-J-1783. The authors would like to thank the program manager, and technical manager, Dr. E. P. Rood. The authors would also like to thank Dr. William Blake of the David Taylor Research Center and Dr. Flint O. Thomas and Dr. Huang-Chang Chu of Notre Dame for their help and comments at various stages of this research     

Some closure results for inertial manifolds

Suppose that the family of evolution equationsdu/dt+Au+f _N (u)=0 possesses inertial manifolds of the same dimension for a sequence of nonlinear termsf _N withf _N f in the C⁰ norm. Conditions are found to ensure that the limiting equationdu/dt+Au+f(u)=0 also possesses an inertial manifold. There are two cases. The first, where the manifolds for the family have a bounded Lipschitz constant, is straightforward and leads to an interesting result on inertial manifolds for Bubnov-Galerkin approximations. When the Lipschitz constant is unbounded, it is still possible to prove the existence of an exponential attractor of finite Hausdorff dimension for the limiting equation. This more general result is applied to a problem in approximate inertial manifold theory discussed by Sell (1993).For Paul Glendinning, with thanks.     

On an analogue of the Euler-Cauchy polygon method for the numerical solution of ux y = f(x,y, u,ux, uy)

This paper develops, with an eye on the numerical applications, an analogue of the classical Euler-Cauchy polygon method (which is used in the solution of the ordinary differential equation dy/dx=f(x, y), y(x ₀)=y ₀) for the solution of the following characteristic boundary value problem for a hyperbolic partial differential equation u _xy =f(x, y, u, u _x , y _y ), u(x, y ₀)=(x), u(x ₀, y)=(y), where (x ₀)=(y ₀). The method presented here, which may be roughly described as a process of bilinear interpolation, has the advantage over previously proposed methods that only the tabulated values of the given functions (x) and (y) are required for its numerical application. Particular attention is devoted to the proof that a certain sequence of approximating functions, constructed in a specified way, actually converges to a solution of the boundary value problem under consideration. Known existence theorems are thus proved by a process which can actually be employed in numerical computation.

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Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Let be a domain. Suppose that f ∈ W^1,1_loc(Ω,R²) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J_f. We show that f^-1 ∈ W^1,1_loc(f(Ω),R²) and that Df⁻¹(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f⁻¹ ∈ W^1,q(f(Ω),R²) for some 1<q≤2 are also given.     

Die Stabilität der Strömung in einem gekrümmten Kanal

Zusammenfassung Die Wirbelinstabilität einer laminaren Strömung durch einen gekrümmten, unendlich hohen Kanal, dessen Breite klein ist gegenüber den Krümmungsradien der Kanalwände, wird durch die beiden Differentialgleichungen (2) beschrieben. Das System (2) stellt ein Eigenwertproblem dar, bei dem vor allem die kleinsten Eigenwerte S ₁ in Abhängigkeit von dem Parameter mit den zugehörigen Eigenfunktionen interessieren; dabei beziehen wir uns bereits auf (3), wo neutrale Störungen (=) betrachtet werden, und S ₁() ist bis auf einen Zahlenfaktor die kritische Kurve des Parameters 2 Re² d/R ₁ in Abhängigkeit von der Dicke /2 der angesetzten Wirbel. Mit Hilfe Greenscher Funktionen werden die Differentialgleichungen (3) in die Integralgleichungen (4) verwandelt, die es zunächst erlauben, untere Schranken für S ₁() anzugeben (Fig. 2). Das Iterationsverfahren nach [9] liefert S ₁() mit hinreichender Genauigkeit nach wenigen Iterationsschritten (Fig. 3). Ebenso ergeben sich damit die Eigenfunktionen (Fig. 4), also die Wirbelkomponenten, so daß in Fig. 5 das Aussehen der neutralen Wirbel angegeben werden kann. Der kritische Wert — aus dem kleinsten unter den Eigenwerten. S ₁() — ist 2Re² d/R ₁=5740, während Dean [2] 1928 den ebenfalls richtigen Wert 5832±2,5% gefunden hatte. Die zugehörige kritische Wirbeldicke ist /2=0,79 d. Die Ergebnisse von Yih und Sangster [3] erweisen sich als falsch. Die Arbeit verfolgte zwei Ziele: Eine Neubehandlung der Differentialgleichungen (5) mit exakt begründeten Methoden, um eine Beurteilung der Ergebnisse von Dean und von Yih und Sangster möglich zu machen. Dann die Angabe der Eigenfunktionen, die es gestatten, das Aussehen der entstehenden Wirbel zu beschreiben.Diese Arbeit wurde durch das Wirtschaftsministerium des Landes Baden-Württemberg gefördert. Vorgelegt von H. Görtler     

Laminar thermally developing flow inside right-angularly triangular ducts

An analytical approach based on the generalized integral transform technique is presented, for the solution of laminar forced convection within the thermal entry region of ducts with arbitrarily shaped cross-sections. The analysis is illustrated through consideration of a right triangular duct subjected to constant wall temperature boundary condition. Critical comparisons are made with results available in the literature, from direct numerical approaches. Numerical results for dimensionless average temperature and Nusselt numbers are presented for different apex angles.Nomenclature a,b sides of right triangular duct - A _c cross-sectional area of duct - c _p specific heat of fluid - D _h =4A _c /p hydraulic diameter, with P the wet perimeter - h(z) heat transfer coefficient at duct wall - k thermal conductivity - Pe=c _p D _h /k Peclet number - T(x, y, z) temperature distribution - T _o inlet temperature - T _w prescribed wall temperature - u(x, y); U(X, Y) dimensional and dimensionless velocity profile - average flow velocity - x; X dimensional and dimensionless normal coordinate (Fig. 1) - x ₁(y); X ₁(Y) dimensional and dimensionless position at irregular boundary (Fig. 1) - y; Y dimensional and dimensionless normal coordinate (Fig. 1) - z; Z dimensional and dimensionless axial coordinate Greek letters side of right triangular duct in X direction (dimensionless) - side of right triangular duct in Y direction (dimensionless) - density of fluid - (X, Y, Z) dimensionless temperature distribution - * apex angle of triangular duct (Fig. 1) - ** apex angle of triangular duct (Fig. 1)     

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as 0+ of the chemical potentials associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ²¦¦²+ W(), where is the density. The main result is that is asymptotically equal to E/d+o(), with E the interfacial energy, per unit surface area, of the interface between phases, the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.     

Feasibility study of three-dimensional PIV by correlating images of particles within parallel light sheet planes

One approach to obtain information about the out-of-plane velocity component from PIV recordings is to analyze the height of the peak in the correlation plane. This value depends on the portion of paired particle images, which itself depends on the out-of-plane velocity component and on other parameters. To circumvent problems with other influences (e.g. background light, amount and size of images), images from another light sheet plane parallel to the first one were also captured for peak height normalization. Our experimental results show the feasibility of an out-of-plane velocity estimation by analyzing images of particles within parallel light sheets by spatial cross-correlation.List of Symbols C particle density in the flow - d particle image diameter - f ₀, f ₁ frames containing images of particles within the first light sheet at t=t ₀ (frame f ₀) and at t=t ₀ + t (frame f ₁) - f ₂ frame containing images of particles within the second light sheet parallel to the first one at t=t ₀ + 2t - F ₁ estimator of the loss of image pairs due to in-plane motion - F ₀ estimator of the loss of image pairs due to out-plane motion - F convolution of the particle image intensity distributions - K factor containing constant parameters in the correlation plane - M imaging magnification (image size/object size) - n ₀ number of particles in the measurement volume at t=t ₀ - n _0,1 number of particle image pairs in interrogation windows of f ₀ andf ₁ - n _1,2 number of particle image pairs in interrogation windows off ₁ and f ₂ - O _z overlap of the light sheets - R _C (s) convolution of the mean intensity distributions - R _D (s) correlation which gives the image displacement - R _F (s) fluctuating noise component of the cross correlation estimator - R _0,1(s _D ) cross-correlation peak height of interrogation windows off ₀ and f ₁ - R _1,2(s _iuD) cross-correlation peak height of interrogation windows of f ₁ and f ₂ - s two-dimensional separation vector in the correlation plane - s _D mean particle image displacement in the interrogation cell - t _e light pulse duration - t _f frame-transfer time of the video camera - u three-dimensional local flow velocity vector (u,v,w) - X _i position of the center of an interrogation window in the image plane (2d) - x _i position of the center of an interrogation volume in the flow (3d) - (z ₂ — Z ₁) displacement of the light sheets in z-direction - t separation time of the light pulses - x ₀ x-extension of an interrogation volume - y ₀ y-extension of an interrogation volume - z ₀ light sheet thickness The authors would like to thank DLR for supporting Markus Raffel's and Olaf Ronneberger's visit to Caltech (Center for Quantitative Visualisation), and the Office of Naval Research through the URI grant ONR-URI-N00014-92-J-1610. Dr. Alexander Weigand's generous offer of his experimental set-up and stimulating discussions with Dr. Jerry Westerweel and Dr. Thomas Roesgen are greatly appreciated. Special thanks also to Dr. Christian Willert for his advice regarding the modifications to the DPIV software.     

Heat transfer for laminar flow in internally finned pipes with different fin heights and uniform wall temperature

An analysis is presented for fully developed laminar convective heat transfer in a pipe provided with internal longitudinal fins, and with uniform outside wall temperature. The fins are arranged in two groups of different heights. The governing equations have been solved numerically to obtain the velocity and temperature distributions. The results obtained for different pipe-fins geometries show that the fin heights affect greatly flow and heat transfer characteristics. Reducing the height of one fin group decreases the friction coefficient significantly. At the same time Nusselt number decreases inappreciably so that such reduction is justified. Thus, the use of different fin heights in internally finned pipes enables the enhancement of heat transfer at reasonably low friction coefficient.Nomenclature A_f dimensionless flow area of the finned pipe, Eq. (8) - a_f flow area of the finned pipe - C_p specific heat at constant pressure - f coefficient of friction, Eq. (12) - H₁, H₂ dimensionless fin height h₁/r_o h₂/r_o - h₁, h₂ fin heights - average heat transfer coefficient at solid-fluid interface - KR fin conductance parameter, k_s/k_f - k_f thermal conductivity of fluid - k_s thermal conductivity of fin - l pipe length - mass flow rate - N number of fins - Nu Nusselt number, Eqs. (15) and (16) - P pressure - Q total heat transfer rate at solid fluid interface - Q_f1, Q_f2 heat transfer rate at fin surface - q_w average heat flux at pipe-wall, Q/(2 r_ol) - R dimensionless radial coordinate r/r_o - Re Reynolds Number, Eq. (13) - r radial coordinate - r_o radius of pipe - r₁, r₂ radii of fin tips - T temperature - T_b bulk temperature - U dimensionless velocity, Eq. (2) - U_b dimensionless bulk velocity - u_z axial velocity - z axial coordinate - angle between the flanks of two adjacent fins - half the angle subtended by a fin - angle between the center-lines of two adjacent fins - angular coordinate - dynamic viscosity - density - dimensionless temperature, Eq. (6) - _b dimensionless bulk temperature     

Bulk modulus of particulate composites: A comparison between a statistical and micro-mechanical approach

Two models are compared. One is based on the theory of elastic continua, and describes the interaction between filler and matrix in terms of an interfacial layer of varying volume fraction and elastic properties. The other derives from an equation of state for the constituents and the composite, based on molecular considerations. The filler-matrix interaction is then expressed in terms of segmental attractions and repulsions. We examine the dependence of the bulk modulusK _c ( _f ) on the volume fraction _f of filler and then show the correspondence between the two theories in terms of the infinite dilution limit of the ratio [K _c ( _f ) –K _m ]/(K _{m f} ) where the indexm refers to the matrix.Dedicated to Prof. Dr. F. R. Schwarzl on the occasion of his 60th birthdayOn leave from Rajdhani College, University of Delhi, India     

Regularity of invariant manifolds

Under mild conditions it is proved that an invariant submanifold ofX 0<1 for the equationdx/dt+Ax=f(x), A sectorial,fC'(X ,X),0<1, is a submanifold ofX ¹ as well. In addition, conditions are given for the semiflow of the equation to extend fromX toX and a new inertial manifold theorem is proved for the scalar reaction diffusion equation.     

An optical analysis of an induced flow ejector using light polarization properties

This paper presents an optical analysis of an induced flow ejector by means of plane laser sheets. The visualization method, which is developed here, takes advantage of the polarization properties of the light scattered by the fine droplets produced by condensation within the flow. This optical analysis shows that the droplets scatter near the Rayleigh scattering regime, thereby proving that their mean radius does not exceed 0.05 m. Furthermore, the injection of depolarizing tracers into the induced stream makes it is possible to distinguish visually between the supersonic primary jet and the subsonic induced stream, and to obtain information about the mixing of the two streams.List of symbols A polarization angle - D inner diameter of the mixing tube - d exit diameter of the primary nozzle - d _c throat diameter of the primary nozzle - E electric field - F transformation matrix - I intensity - L _m length of the mixing tube - m mass flow rate - M Mach number - P static pressure - r radius - S Stokes vector - U entrainment ratio - X penetration of the primary nozzle - (I, Q, U, V) Stokes parameters - OU _o propagating direction of the incident wave - (r _o, l _o) vibrating plane - (U _o, l _o) scattering plane - ejector throat area ratio - wavelength - scattering (or observation) angle Indices a atmospheric value - i stagnation value - 0 incident value - 1 primary (or central) jet - 2 secondary (or induced) jet     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Asymptotic behavior of one-dimensional discrete-velocity models in a slab

We prove results on the asymptotic behavior of solutions to discrete-velocity models of the Boltzmann equation in the one-dimensional slab 0x<1 with=" general=" stochastic=" boundary=" conditions=" at=" x="0" and=" x="1." assuming=" that=" there=" is=" a=" constant=">wall Maxwellian M=(M _i) compatible with the boundary conditions, and under a technical assumption meaning strong thermalization at the boundaries, we prove three types of results:

A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

The paper is concerned with the asymptotic behavior as t of solutions u(x,t) of the equation in the case f(0)=f(1)=0, with f(u) non-positive for u(>0) sufficiently close to zero and f(u) non-negative for u(<1) sufficiently close to 1. This guarantees the uniqueness (but not the existence) of a travelling front solution u;U(x–ct), U(–);0, U();, and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front, if it exists, and to a stacked combination of travelling fronts if it does not. The approach is to use the monotonicity to take u and t as independent variables and p = u _x as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.