Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

A solution of the E-ε model for nearly homogeneous turbulence with a mean shear

An analytical solution of the E- model for the downstream evolution of a stationary and nearly homogeneous turbulent shear flow is presented. In case that the turbulent time scale has adjusted itself to the time scale imposed by the shear, an asymptotic solution can be derived from the full solution, which shows that both E and increase downstream exponentially. By comparing this asymptotic solution with experimental data a value for the unknown constant c _l , in the -equation, is derived. Moreover, we find an expression for the downstream development of the variance of a scalar, which is also compared with experimental data. The analytical solution shows that a homogeneous shear flow with a uniform velocity gradient can only be obtained if the shear is sufficiently small. In the experiments this condition is not always satisfied. A discussion is given on how a nearly homogeneous shear flow can be obtained over a limited downstream interval by changing the initial conditions in E and , and a comparison is made with experimental data. Finally it is shown that better transverse homogeneity can be obtained by taking an exponential velocity profile instead of a linear profile.     

Scheme of Investigation of the Spectrum of a Family of Perturbed Operators and Its Application to Spectral Problems in Thick Junctions

We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {A: > 0} that act in different spaces and lose their compactness in the limit case 0. We prove the Hausdorff convergence of the spectrum of the operator A to the spectrum of the limit operator A₀, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A₀, and prove asymptotic estimates for eigenvectors of A. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an -periodic system of thin rods of fixed length.     

Homogenization of the Process of Phase Transitions in Multidimensional Heterogeneous Periodic Media

The homogenization of the Stefan multidimensional problem is carried out in the case where the medium is a composite consisting of two different substances with an –periodic structure. The averaged problem is deduced by asymptotic methods. It is shown that its solution is the limit of solutions of –problems.     

Resonance Capture in a Three Degree-of-Freedom Mechanical System

We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane. In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor. These equations, valid in the neighborhood of the resonance, possess a small parameter which is related to the imbalance. In the limit 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance. Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for 0. It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur. We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape.     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

Near-wall turbulence models for 3D boundary layers

Calculations of the three-dimensional boundary layer in an S shaped duct are performed with various – models. Three different near-wall models are used for the – model, of which one is using a new set of near-wall damping functions deduced from direct numerical simulations of turbulent channel flow available in the literature. The results show that it is possible to obtain damping functions giving better agreement, especially for and , with direct simulation data and experiments than with damping functions deduced from trial and error.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

On the indentation of an inhomogeneous plastic solid

Summary The problem considered here is that of the indentation of a semi infinite, inhomogeneous rigid-plastic solid by a smooth, flat ended punch under conditions of plane strain. It is assumed that the yield stress of the solid k(x, y) has the form k ⁰+k(x, y) where k ⁰ is a constant and is small. A perturbation method of solution developed by Spencer [1] is used, and general results are obtained for arbitrary values of k(x, y). Some particular cases are then considered.     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

Chaos and bifurcation of phase-locking loops under periodic perturbation

This paper discusses the chaos and bifurcation for equation x+εcosxx+αsinx =εbsint. By use of the Melnikov method the conditions to have the chaotic behavior and to have subharmonic oscillations are given.     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

The eddy viscosity of turbulent equilibrium layers

Summary Similarity laws for the mean flow and scaling laws for the turbulent motion are used in an attempt to obtain a general expression for the eddy viscosity of equilibrium layers. It is found that =0.09 w ² /w*, in which w ² is a Reynolds stress representative for the region of overlap between the law of the wall and the velocity-defect law, while w* is the logarithmic slope of the mean velocity profile in that region. The distinction between w and w* is related to the strong inhomogeneity of the mean rate of strain in the inner layer. The results of the theory agree with experimental evidence obtained from transpired equilibrium layers.     

Square and Pulse Waves with Two Delays

This work is concerned with the effects on the dynamics of a differential difference equation with two delays as the delays become unbounded in a fixed direction. This leads to a singularly perturbed delay differential equation with singular parameter and delays (1, d). We study in detail d=2 for the case when =0 yields the Hénon map. In a neighborhood of a generic period doubling point for the Hénon map, we show that there can be either a stable square wave or an unstable pulse wave even though the period two point for the map is always stable.     

Dielectric properties of heterogeneous mixtures with a polar constituent

Summary After defining the boundaries for the dielectric constant of a heterogeneous mixture, the behaviour of such a mixture is studied as a function of the frequency, when one of its components is polar. Deviations from a semicircle are to be expected for the function _m =f( _m ) even when the dielectric properties of the polar constituent can be described with a semicircular Cole-Cole-arc. The relaxation time of the mixture is shorter than that of the polar constituent.     

A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter , we study thedynamics of this system at the limit 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.     

Analysis of a method and a system to determine heat transfer and properties of fluids in the absence of temperature measurements and a modified Fourier Equation

The technique to determine by capacitance measurements heat transfer, thermal transport and dielectric properties of fluids introduced recently is now analyzed for a simple system of spherical geometry. The temperature distribution under programmed heat input to a fluid annulus between solid walls is computed by finite difference method for the determination of the capacitance time function of the arrangement. A system of heavy wall structure and heated long enough will produce a capacitance-time curve which is a function of thermal conductivity only. Thermal diffusivity is of influence in thin wall systems. The capacitance change of a heavy wall arrangement is related to the thermal conductivity of the test fluid by a modified Fourier equation. This equation describes the heat flow through the fluid layer but includes the thermal expansion of the solid walls. The change of geometry with T is therefore accounted for. For other multicomposite structures the Fourier equation must be further modified by including the thermal expansion of all materials of the structure and possibly also their compressibilities.

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Zusammenfassung Die kürzlich eingeführte Methode der Bestimmung von Wärmeübergang, thermischen Transport und dielektrischen Größen mittels Kapazitäts-Zeit-Messung wird analysiert für ein einfaches kugeliges System. Die Temperaturverteilung in der Flüssigkeit im Kugelspalt zwischen zwei festen Körpern wird für konstante Wärmezufuhr von außen mittels der Differenzmethode bestimmt und daraus die Kapazitäts-Zeit-Funktion ermittelt. Es wird gezeigt, daß die Kapazitäts-Zeit-Kurve nur eine Funktion der Wärmeleitzahl ist für den Fall dickwandiger Anordnungen. Für dünnwandige Systeme wird sie auch abhängig von der Temperaturleitzahl. Es wird eine modifizierte Fourier-Gleichung eingeführt, die den Wärmetransport durch die Flüssigkeit beschreibt, dabei aber die Änderung der Geometrie der Schicht berücksichtigt, die sich wegen der thermischen Ausdehnung der festen Wände bei der Einstellung der Temperaturdifferenz ergibt. Für andere mehrschichtige Körper muß die Fourier-Gleichung weiterhin modifiziert werden durch Berücksichtigung der thermischen Ausdehnungskoeffizienten aller beteiligten Materialien und möglicherweise auch ihrer Kompressibilitäten.

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Nomenclature A average cross-sectional area of fluid layer - A coefficient matrix - B matrix defined by Eq. (20) - B₀ geometric constant of fluid layer (A/L) at reference temperature - C capacitance of arrangement - C_i, C_r capacitance of layer of fluid i and reference fluid at temperature T - capacitances at reference temperature - C_H, c_l specific heats of outer and inner wall - FA...FE constants defined in Eqs. (13 ... 17) - L thickness of fluid layer - M_H, M_L mass of outer and inner wall - P power input to the system - R constant defined by Eq. (24) - T temperature - T_ref reference temperature - T (O, t), T (L, t) temperatures of outer and inner wall at time t - T _i ⁿ , T _i+0 ^n+m temperatures at location i and time n (m=number of t's; 0=number of x's) - T temperature difference across fluid layer - T apparent temperature difference - t_h, T_l temperature increases of outer and inner wall - T_max temperature change of system from one to another thermal equilibrium condition a thermal diffusivity - k, k_i, k_r thermal qonductivity of fluids and of fluid i and reference fluid - q heat flow through fluid layer - ^rh,^rl inner radius of outer wall and outer radius of inner wall - ^rOH,^rOL radii at reference temperature - t time - t time interval - x coordinate - ¯x vector of unknown T_i ⁿ⁺¹ - x length interval Greek symbols linear thermal expansion coefficient - _H, _L linear thermal expansion coefficient of materials of outer and inner wall - dielectric constant - _i, _ref dielectric constant of fluid i and reference fluid - ₀ permittivity of free space - multiplyer of conduction Eq. (7) in finite difference form - time needed to establish quasi-steady state conditions in the system heated by a constant power input In honor of Prof. Dr. E. Schmidt to his 80th Birthday.     

The flow in a narrow duct with an indentation or hump on one wall

This paper deals with the flow in a narrow duct with an indentation or hump on one wall, on the assumption that , the ratio of the duct width to the length of the indentation or hump, is small. This enables the governing equations to be simplified and an analytic solution is derived on the assumption thatRe (the Reynolds number based on duct width) is of 0 (1). This simple solution breaks down whenRe is of 0 (^–1) and numerical solutions are obtained for the case whenR=Re is of 0 (1). These show, forR sufficiently large, that there are regions of reversed flow both in the indentation and on the plane wall opposite to it, and for humps, regions of reversed flow downstream.

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Strömung in einem schmalen Kanal mit einer Einbuchtung oder Wölbung an einer Wand

Zusammenfassung Diese Untersuchung befaßt sich mit der Strömung in einem schmalen Kanal mit einer Einbuchtung oder Wölbung an einer Wand, mit der Annahme, daß das Verhältnis der Kanallänge zu der Länge der Einbuchtung oder Wölbung klein ist. Dies ermöglicht, daß die beschreibenden Gleichungen vereinfacht werden können und eine analytische Lösung mit der AnnahmeRe (Re basierend auf der Kanallänge) gleich 0 (1) erhalten werden kann. Die einfache Lösung ist nicht mehr anwendbar, wennRe gleich 0 (^–1) ist. Numerische Lösungen werden für den Fall erhalten, daßR=Re gleich 0 (1) ist. Dies zeigt, daß es für hinreichend großesR Gebiete mit Gegenstrom sowohl in der Einbuchtung als auch an der flachen Wand gegenüber gibt und für den Fall einer Wölbung, daß diese stromabwärts existieren.

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Nomenclature h width of duct - h ₀ maximum height of hump/indentation - H the shape of the hump/indentation - l length of the hump/indentation - p pressure of the fluid - Q applied flow rate - Re Reynolds number based on duct width,Re=Q/ - R modified Reynolds number,R= Re - u velocity in thex-direction - v velocity in they-direction - x coordinate along the duct - y coordinate across the duct - duct width/length ratio,=h/l - modified transverse coordinate,=y/H - kinematic viscosity - ₀ skin friction ony=0 - stream function     

Asymptotic study of viscous compressible gas flow past a blunt body

Supersonic viscous gas flow past a blunt body is examined. A method is proposed which permits constructing the asymptotic expansion of any order in the small parameter , which characterizes the viscosity and thermal conductivity coefficients. The asymptotic solution is constructed, including terras of zero, first, and second orders of . Acomparison is made with results of other authors who have studied various particular aspects of the subject problem using the method of inner and outer expansions [1–3].     

Integral technique solutions to a class of simple ablation problems

Summary The integral technique is applied to a class of ablation problems of a semi-infinite solid subjected to a heat flux of the form q = At ^m . The governing equations are highly simplified by normalizing the variables with respect to the values obtained at the onset of ablation and by introducing the asymptotic values obtained for large times. The results are discussed in terms of a parameter expressing the ratio of heat capacities between the heat storable in the solid and the latent heat of ablation.

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Sommario La tecinca integrale viene applicata nella ricerca di soluzioni al problema dell'ablazione di un solido semiinfinito soggetto ad un flusso di calore del tipo q = At ^m . Le equazioni risolventi sono notevolmente semplificate introducendo un a normalizzanzione rispetto ai valori relativi all'inizio del processo ed alla condizione asintotica. I risultati sono espressi in termini di un parametro rapporto delle capacità termiche.