On the Jacobian conjecture for global asymptotic stability

An old conjecture says that, for the two-dimensional system of ordinary differential equationsx=f(x), wheref: ^{2 2},f C ¹, andf(0)=0 the originx=0 should beglobally asymptotically stable (i.e., a stable equilibrium and all trajectoriesx(t) converge to it ast +) whenever the following conditions on the Jacobian matrixJ(x) off hold: trJ(x) < 0, detJ(x) > 0, x ² It is known that if such anf is globallyone-to-one as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the systemx =f(x). In this paper we outline a new strategy to tackle the injectivity off, based on anauxiliary boundary value problem. The strategy is shown to be successful if the norm of the matrixJ(x) ^T J(x)t/det J(x) is bounded or, at least, grows slowly (for instance, linearly) as ¦x¦ t.     

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     

Homogenization of the Robin problem in a thick multilevel junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction that is the union of a domain ₀ and a large number 2N of thin rods with variable thickness of order = (N ^–1). The thin rods are divided into two levels, depending on their length. In addition, the thin rods from each level are -periodically alternated. We investigate the asymptotic behavior of the solution as 0 under the Robin conditions on the boundaries of the thin rods. By using some special extension operators, a convergence theorem is proved.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 336–355, July–September, 2004.     

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.     

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.     

Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes II: Non-critical sizes of the holes for a volume distribution and a surface distribution of holes

This paper is devoted to the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. For obstacles of critical size it was established in Part I that the limit problem is described by a law of Brinkman type. Here we prove that for smaller obstacles, the limit problem reduces to the Stokes or Navier-Stokes equations, and for larger obstacles, to Darcy's law. We also apply the abstract framework of Part I to the case of a domain containing tiny obstacles, periodically distributed on a surface. (For example, in three dimensions, consider obstacles of size ², located at the nodes of a regular plane mesh of period .) This provides a mathematical model for fluid flows through mixing grids, based on a special form of the Brinkman law in which the additional term is concentrated on the plane of the grid.     

Discretized ordinary differential equations and the Conley index

LetN be a compact isolating neighborhood of an isolated invariant setK with respect to an ODEx=f(x) (C) and(h) x=x + h(x, h) be a consistent one-step-discretization of (C). It is proved in this paper that for someh ₀ > 0 and allh ]0, h₀[, the setN isolates an invariant setK(h) of(h) and the discrete Conley index ofK(h) coincides with the continuous Conley index ofK.     

Stress-intensity factors for a semi-infinite plane crack with a wavy front

The stress-intensity factors for a semi-infinite plane crack with a wavy front are determined when the crack faces are subjected to normal and shearing tractions. The results are derived using asymptotic methods and are valid to O(²) where =A/1; A is the amplitude and is the wavelength of the wavy front. The normal and shearing tractions are in the form of line loads parallel to the crack front.The results are then used to evaluate, in a qualitative manner, the growth characteristics of a semi-infinite plance crack with a wavy front under combined mode loading. This provides a possible explanation of crack front segmentation observed experimentally.     

Deformationsbeständige,elastische und Relaxations-Eigenschaften der flüssigplastischen kolloiden Systeme

Zusammenfassung Es wurde gezeigt, daß es zweckmäßig ist, die strukturierten kolloiden Systeme in fest-plastische und flüssig-plastische Systeme einzuteilen, weil beide Systeme einen übereinstimmenden Eigenschaftskomplex aufweisen, sich jedoch quantitativ durch die RelaxationszeitenP >P _k im Schubspannungsbereich unterhalb der unteren FließgrenzeP < P _k unterscheiden. Es wurde darauf hingewiesen, daß für beide Systeme die Spannungsdeformationskurven sehr charakteristisch sind.Die Kurven werden unter der Bedingung der konstanten Deformationsgeschwindigkeit erhalten, wobei ihre Form von der gegenseitigen Beziehung von und abhängt und mit den Strukturelementetypen, die durch und _i charakterisiert sind, im Zusammenhang stehen.Die Methoden,die zur Messung der elastischen Deformation im breiten Bereich längs der KurveP () bei sowohl kleiner als auch größer _r entsprechend der kritischen SchubspannungP _r angewandt werden können, wurden entwickelt. Dabei wurde gezeigt, daß die Kurveine() durch das Maximum bei _m hindurchgeht.Der Einfluß von auf die kritische Deformation _r der Strukturzerstörung und auf die maximale Rückfederung _e max, die ihrerseits wiederum von der Gelkonzentration abhängen, wurde eingehend untersucht.Es wurden die Zahlenwerte der Grenzviskosität der Nachwirkung bestimmt und die Abhängigkeit der Geschwindigkeit (der Zeit) der Relaxation der elastischen Deformation von der gesamten und der elastischen Deformation ermittelt.Weiter wurde gezeigt, daß die größte elastische Deformation _e max des Systems größer als die kritische Deformation _r der Strukturzerstörung, die dem Maximum der kritischen Schubspannung der Struktur entspricht, sein kann.     

Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation

In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + [ + (cos ₁ t + cos ₂ t)] x + x³ = 0.We assume that 1 and that the coefficient of the nonlinearterm, , is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the usual trigonometric functions – to study subharmonic resonances associated with orbits in 2m:1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (, , ₁, ₂) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov's overlap criterion to obtain an overview of the O() global behavior of equation (1) as a function of and ₂ with ₁, , and fixed.     

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     

Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W ^s() and W ^u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W ^s()W ^u(); W ^s()W ^u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W ^s() and W ^u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O( ²(log)²) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W ^s()W ^u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.     

A class of quasilinear differential inequalities whose solutions are ultimately constant

Let be a domain in R ⁿ with compact complement and let T be a quasilinear elliptic or degenerate elliptic operator associated with functions u C ²(). This paper is a study of solutions of (sgn u) Tuf(¦u¦, ¦ grad u¦) where f belongs to a class of functions here termed bifurcation functions. The main condition on f is that uniqueness fails for the ordinary differential equation y'=f(y, y) with the initial condition y(0)=y(0)=0. The conclusion is that u is constant for large ¦x¦ and hence, under mild supplementary hypotheses, u has compact support. Examples show that the results fail if the assumptions on f are only slightly weakened, so that the class of f is, essentially, the largest class for which the results can be stated truly.To James Serrin, in celebration of his 60^th birthday     

Flow model for velocity distribution in fixed porous beds under isothermal conditions

In a brief survey of the previous work the limitations of the modified Darcy equation and of the vectorial form of the Ergun equation are discussed. To include the effect of wall friction on the flows the viscous resistance term is added to the vectorial form of the Ergun equation. Using the generalized Ergun equation a one-dimensional formulation is presented for flow of fluids through packed beds taking into account the variation of porosity along the radial direction. It is found that there is a reasonable agreement between the numerical and the experimental results and it is observed that the variation of porosity with radial position has greater influence on channeling of velocity near the walls. For the assumption of constant porosity the velocity profiles exhibit similar nature as the plug flow profiles with a thin boundary layer near the wall.

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Modell der Geschwindigkeitsverteilung in einem isotherm durchströmten Festbett

Zusammenfassung In der vorliegenden Arbeit werden eingangs die Anwendbarkeitsgrenzen der modifizierten Darcy-Gleichung und der in vektorieller Form geschriebenen Ergun-Gleichung diskutiert. Um Einflüsse der Wandreibung auf eine Strömung mit in der Ergun-Gleichung berücksichtigen zu können, wird ein Reibungsterm hinzugefügt. Die so generalisierte Gleichung kann benutzt werden, um die eindimensional gerichtete Strömung durch eine Kugel schüttung zu berechnen. Eine radiale Veränderung der Schüttungsporosität ist dabei mit in die Betrachtung eingeschlossen. Das nichtlineare Grenzwertproblem wird numerisch gelöst und mit experimentellen Daten aus der Literatur verglichen. Die mit Meßwerten zufriedenstellend übereinstimmenden Rechenergebnisse zeigen, daß die radiale Porositätsverteilung in einem Festbett einen erheblichen Einfluß auf die Durchströmungsgeschwindigkeit in Wandnähe ausübt; die Berechnungen geben die Strömungsrandgängigkeit wieder. Wird die Bettporosität als unveränderlich angenommen, erhält man pfropfenströmungsähnliche Geschwindigkeitsprofile mit einer dünnen Wandgrenzschicht, in welcher die Geschwindigkeit auf den Wert null abfällt.

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Nomenclature A Tridiagonal matrix defined in Eq. (20) - a Bed radius - d_p Particle diameter - f₁ 150 (1–)²/(³d _p ² ) Darcy resistance term - f₂ 1,75(1–)/(³d_p) Parameter of resistance due to inertial effects - ¯f₁ 150(1–)²/³ - ¯f₂ 1,75(1–)/₃ - G Column vector defined in Eq. (20) - k Permeability, /f₁ - L Length of the bed - P Pressure - r Radial co-ordinate - R_p Reynolds number based on particle diameter, v₀d_p/ - , v_z Superficial velocity vector, axial component - v_1z Average superficial velocity defined in Eq. (20) - V Absolute magnitude of velocity - ¯v The average velocity - v₀ The velocity at the centre of the tube - X Column vector defined in Eq. (20) - r^* Dimensionless radial co-ordinate, r/a - p^* Dimensionless pressure, p/v ₀ ² - v _z ^* Dimensionless axial component of velocity, v_z/v₀ - ¯v^* Dimensionless average velocity defined in Eq. (20) - z^* Dimensionless axial co-ordinate, z/L Greek letters Ratio of tube radius to particle diameter, a/d_p - Porosity or void fraction - ₀ Porosity at the axis of the container - Dynamical viscosity - Kinematic viscosity - p Density - Distance from the wall of the container, defined in Eq. (16)     

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.     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

A new representation of energy spectra in fully developed turbulence

It is well known that the Kolmogorov 1941 theory is based on global invariance, in the limit of Reynolds number tending to infinity. Experimentally, it is well verified only for very high Reynolds numbers, namelyR 2000 (Monin and Yaglom 1975).We propose a new experimental representation for energy spectra. Using the Kolmogorov scales, a compilation of dimensionless spectra (E= (k)/(v ⁵)^1/4 andK=k(v ³/)^1/4) shows that log(0.154E)/log(R /R*) is a universal function of log(5.42K)/log(R /R*) withR*=75. This new representation is not compatible with neither local nor global scaling invariance. The constant 5.42 takes into account the small scale intermittency. Similar results have been obtained for velocity structure functions of order 2, 3 and 6. In particular the wavenumber constant 5.42 is independent on the order of the moments.