Thermally driven acoustic oscillations,part V: Gas-liquid oscillations

Summary Thermally driven gas oscillations are considered which drive a liquid column at the cold open end. The frequency of the oscillations is lowered by the inertia of the liquid and the Stokes boundary layer is thick. Optimal conditions for thermal driving in the gas occur at correspondingly big tube radii. Experiments conducted inN ₂ confirm the theoretically calculated stability limits. With hot air driving a water column, a low critical temperature ratio for the oscillations of about 1.5 was observed.

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Zusammenfassung Thermisch getriebene akustische Schwingungen werden untersucht die eine Flüssigkeitssäule am kalten offenen Ende antreiben. Die Frequenz der Schwingungen wird durch die Trägheit der Flüssigkeit vermindert und die Stokes'sche Grenzschicht ist dick. Die optimalen Bedingungen für thermisch getriebene Schwingungen im Gas werden bei entsprechend höheren Rohr-Radien auftreten. Experimente mitN ₂ haben die theoretisch berechneten Stabilitätsgrenzen bestätigt. Bei einer Anordnung, in der heisse Luft eine Wassersäule angetrieben hat, wurde ein tiefes kritisches Temperaturverhältnis von etwa 1.5 beobachtet.

     

Non-linear oscillations of a non-viscous cylindrical liquid system in weightlessness condition

For an immiscible cylindrical liquid system in a circular cylindrical container and a liquid layer around a rigid center core consisting of incompressible and non-viscous liquid the natural non-linear frequencies are determined, if the system is in a zero-gravity environment. The frequencies are determined as a function of the surface elevation and are presented as backbone curves.It was found, that the immiscible liquid system shows hardening effect depending on the length-to-diameter ratio of the column, the density ratio and the diameter ratio. This means that with increasing interface amplitude the natural frequency increases above the linearized frequency. Decreasing density ratio (inner to outer density), increasing diameter ratio and decreasing aspect ratio yield stronger hardening effects. For a liquid layer around a rigid center core the system shows stronger effects, but exhibits softening behavior, i.e. lower natural frequencies for increasing liquid surface amplitude. Decreasing aspect ratio and decreasing layer thickness show stronger softening effects, as do increasing mode numbers. Close to the Rayleigh instability the liquid system shows extreme non-linear effects.

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Zusammenfassung Es werden die nichtlinearen Frequenzen eines nichtmischbaren Flüssigkeitssystems in einem Kreiszylinderbehälter und einer Flüssigkeitsschicht um einen Zylinder im schwerelosen Zustand bestimmt. Die Frequenzen werden als Funktion der Oberflächenauslenkung dargestellt.Es wurde festgestellt, da\ die Trennflächenfrequenzen des nichtmischbaren Systems überlinear sind, wobei die Stärke der Nichtlinearität vom Streckungs-, Dichte- und Durchmesserverhältnis abhängt. Ein kleiner werdendes Dichteverhältnis (innerer zu äu\erer Dichte), grö\er werdendes Durchmesserverhältnis und kleineres Streckungsverhältnis ergeben stärkere Überlinearität. Für eine Flüssigkeitsschicht um einen Zylinder konnte unterlineares Verhalten festgestellt werden. Kleiner werdendes Streckungs- und Dickenverhältnis sowie höhere Schwingungsformen ergeben stärkere Unterlinearität. In der Nähe der Rayleigh-Instabilität zeigt das Flüssigkeilssystem extreme Nichtlinearitäten.

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List of symbols a radius of container wall or rigid center core - b radius of interfacial or free liquid surface (see Fig. 1) - L length of liquid column - m,n angular mode number of liquid, axial mode number - I _m,K _m modified Besselfunctions of first and second kind andmth order - r,,z polar coordinates - R ₁,R ₂ radii of curvature - t time - ₁, ₂ velocity potential of liquid outside and inside - (, z, t) free liquid- or interfacial elevation - _1;2 liquid densities - liquid surface tension (or interface tension) - _mn ⁽⁰⁾ linear natural frequency of liquid - frequency of liquid (non-linear)     

Temporal analysis of a power-law liquid sheet in the presence of acoustic oscillations

The instability of a non-Newtonian liquid sheet in the presence of acoustic oscillations is investigated theoretically. The power-law model is used to describe the viscosity of the non-Newtonian liquid. The corresponding dispersion relation is obtained by linear analysis. The effects of the mean velocity of the gas, the oscillation amplitude, the oscillation frequency, and the gas density on the instability of the power-law liquid sheet are studied. The results show that the shear-thickening liquid sheet is more unstable than Newtonian and shear-thinning liquid sheets when the effects of acoustic oscillations are considered. In particular, a second unstable region appears on the shear-thickening liquid sheet at a low oscillation frequency. Especially, for the shear-thinning liquid sheet, there is a second unstable region in the dispersion curve at a high mean gas velocity. A third unstable region appears on the shear-thinning liquid sheet at a high gas density in the presence of acoustic oscillations. The unstable range of the Newtonian liquid is always the widest among these liquids.     

Sur l'existence des petites oscillations d'une colonne liquide cylindrique en apesanteur

Résumé L'auteur considère une colonne de liquide parfait incompressible en apesanteur dans un container cylindrique infiniment long de section droite arbitraire.Quand le container est fixe, il y a une position d'équilibre de liquide pour laquelle la surface libre est un cylindre circulaire. En utilisant une méthode classique de l'analyse fonctionnelle, il prouve l'existence de petites oscillations du liquide autour de cette position.Quand le container est en rotation uniforme, il est nécessaire d'appliquer une nouvelle méthode. A partir d'une formulation variationnelle du problème, l'auteur démontre, au moyen de la théorie des semi-groupes, l'existence des petites oscillations du liquide, quelle que soit la vitesse angulaire de rotation.

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The author considers a column of inviscid incompressible liquid under zero gravity contained in a infinitely long cylindrical container, with an arbitrary cross-section.When the container is fixed, there is an equilibrium position of the liquid, the free surface being a circular cylinder. Using a classical method of the functional analysis, he proves the existence of small oscillations of the liquid about this position.When the container is in uniform rotation, it is necessary to apply a new method. From a variational formulation of the problem, the author proves, by means of semi-groups theory, the existence of small oscillations of the liquid, for an arbitrary angular velocity.

     

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

Mean oscillations and the convergence of Poisson integrals

We establish conditions for mean oscillations of a periodic summable function under which the summability of its Fourier series (conjugate series) by the Abel-Poisson method at a given point implies the convergence of Steklov means (the existence of the conjugate function) at the indicated point. Similar results are also obtained for the Poisson integral in ℝ ₊ⁿ⁺¹. Odessa University, Odessa. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 206–222, February, 1997.     

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.     

Vertical oscillations of multimass system on elastic half space with a cylindrical cavity

A problem on oscillations of a multimass system (MS) is considered on an elastic half space with a cylindrical cavity. Equations of motions of an MS are given, which are modeled by masses that are connected by springs and dampers. A motion of the half space with a cavity is characterized by a transmitting function,which is known from a solution of a contact problem with vertical oscillations of a die on the half space given. The conditions of interrelation of the MS with the base close the system of algebraic linear equations for determining amplitudes of oscillations of each element of the MS. Translated from Dinamicheskie Sistemy, No. 7, pp. 13–18, 1988.     

LINEARIZED OSCILLATIONS OF DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS

The necessary and sufficient conditions for the oscillations of every solution of the nonlinear delay equation x(t) f(x(t l)) g(x([t-k]))=0 are oblained.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Superpositions of oscillations and quasi-oscillations

If X is a topological space and (Y,d) is a metric space, then for each locally bounded function f : X→Y all possible superpositions of oscillations and quasi-oscillations give at most eight functions.     

Analysis of a model of a cocurrent packed-bed column with periodic inlet conditions

We consider a model of a process in a cocurrent packed-bed column with the simplest kinetics of the rate of liquid holdup formation. Under study is the dependence of it on the time-periodic velocities of the liquid and gas flows in a neighborhood of the stationary values of these velocities. The principal terms of the asymptotics in time are found for the liquid holdup and the flow velocities. We detect the growth of the amplitudes of the sinusoidal oscillations of the liquid holdup and the liquid and gas velocities with respect to the column length. The constructed model is used for design of the cocurrent two-phase columns.     

Mechanisms for oscillations in a biological competition model

A four-dimensional dynamical system that models perceptual bistability in the brain is analyzed. Two variables represent the activity of two competing neural populations and they evolve in fast time; other two variables are slow and they are associated with an intrinsic negative feedback to each population. The external stimulus strength I is the bifurcation parameter. We construct the normal form and prove that oscillations occur in the system through supercritical Hopf bifurcations: as I decreases from large to moderate values a limit cycle is born; then it disappears for lower values of I. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface- and interface oscillations of a rotating viscoelastic liquid column of immiscible liquids

Summary The natural damped coupled frequencies of a solidly rotating visco-elastic infinite liquid column with no axial dependency (/z=0, two-dimensional problem) have been determined. The frequency equation is numerically evaluated for a single visco-elastic liquid column, where the influence of the tension parameterTa/v ², the relaxation parameter ^*/a ² and the rotational Reynoldsnumber =Taylornumber) has been determined. It was found that the liquid column becomes unstable for a rotational speed , which is much earlier than in the case of frictionless liquid, where . In addition the stability boundary does neither depend on the magnitude of the viscosity nor the Maxwell relaxation time ^* of the liquid. The complex frequencies are presented for the modem=2, where the cross-section of the liquid column assumes during its oscillation an elliptic shape.

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Zusammenfassung Es werden die gekoppelten gedämpften Frequenzen einer rotierenden visko-elastischen Flüssigkeitssäule ohne axiale Abhängigkeit (zweidimensionales Problem) aus nichtmischbaren Flüssigkeiten bestimmt. Die Frequenzgleichung einer einfachen viskoelastischen Säule wird numerisch ausgewertet, wobei der Einfluß des OberflächenspannungsparametersTa/v ², des Relaxationparameters ^* v/a ² und der rotierenden Reynoldszahl untersucht wird. Die Flüssigkeitssäule wird instabil bei , unabhängig von der Größe der Viskosität und der Relaxationszeit. Bei reibungsfreier Flüssigkeit tritt diese Instabilität erst für größeres auf. Es werden die komplexen Frequenzen für die elliptische Querschnitts-Schwingungsform m=2 berechnet.

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Die Veröffentlichung wurde aus Haushaltsmitteln der Universität der Bundeswehr München gefördert.     

Points of joint continuity and large oscillations

For topological spaces X and Y and a metric space Z, we introduce a new class N( X ×Y, Z ) \mathcal{N}\left( {X \times Y,\,Z} \right) of mappings f: X × Y → Z containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping f from this class and any countable-type set B in Y, the set C _B (f) of all points x from X such that f is jointly continuous at any point of the set {x} × B is residual in X: We also prove that if X is a Baire space, Y is a metrizable compact set, Z is a metric space, and f ? N( X ×Y, Z ) f \in \mathcal{N}\left( {X \times Y,\,Z} \right) , then, for any ε > 0, the projection of the set D ^ε (f) of all points p ∈ X × Y at which the oscillation ω _f (p) ≥ ε onto X is a closed set nowhere dense in X.     

Detecting and creating oscillations using multifractal methods

By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f, we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ∑_j ∑_k d_j,k ψ _j,k ∈ L ²(?) the function series f^t whose wavelet coefficients are d ^t _j,k = d_j,k 1 , for some fixed real number γ > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series f^t may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f ∈ L ²(?) such that the associated thresholded function series f^t effectively possesses oscillating singularities which were not present in the initial function f . This series f^t is a typical example of function with homogeneous non‐concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sur les petites oscillations d'une goutte de liquide caténoïdale entre deux plaques parallèles en apesanteur

Résumé L'article est consacré á l'étude mathématique du probléme des petites oscillations d'une goutte de liquide placée entre deux plaques paralleles qu'elle coupe sous un angle constant, en l'absence de pesanteur, dans le cas oú la surface libre de la goutte est un caténoïde dans la position d'équilibre.Utilisant une méthode fonctionnelle, nous retrouvons d'abord une condition de stabilité et nous obtenons ensuite une formulation variationnelle du probléme á l'aide laquelle nous démontrons l'existence des fréquences propres des petites oscillations du liquide.

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This paper deals with the mathematical study of the problem of the small oscillations of a liquid drop with a free surface held together by surface tension between two parallel plates with free contact line, the contact angle being constant, under zero gravity, in the case of a catenoïdal liquid drop in the equilibrium position.Using a functional method, we find again, at first, a condition of stability and, then, we obtain a variational formulation of the problem by means of which we prove the existence of the eigenfrequencies of the small oscillations of the liquid.

     

Sur les petites oscillations planes d'un liquide dans un container symétrique en apesanteur

Résumé L'auteur étudie le problème bidimensionnel des oscillations d'un liquide dans un container symétrique en apesanteur. II donne l'équation fonctionnelle du mouvement et prouve l'existence des fréquences propres du liquide sous des conditions convenables portant sur l'angle de contact de la ligne libre et de la paroi du container et la courbure de celle-ci.

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The author studies the two-dimensional problem of the oscillations of a liquid in a symmetrical container under zero gravity. He gives the functional equation of motion and proves the existence of the eigenfrequencies of the liquid under suitable conditions between the contact angle of the free line and of the wall of the container and the curvature of this.

     

Rectifiable oscillations in second-order linear differential equations

We study the linear differential equation , on I=(0,1), where the coefficient f(x) is strictly positive and continuous on I, and satisfies the Hartman-Wintner condition at x=0. The four main results of the paper are: (i) a criterion for rectifiable oscillations of (P), characterized by the integrability of on I; (ii) a stability result for rectifiable and unrectifiable oscillations of (P), in terms of a perturbation on f(x); (iii) the s-dimensional fractal oscillations (for which we assume also f(x)∼cx−α when x→0, α>2, and s=max{1,3/2−2/α}); and (iv) the co-existence of rectifiable and unrectifiable oscillations in the absence of the Hartman-Wintner condition on f(x). Explicit examples related to the above results are given.     

Nonlinear free oscillations of a rotating solid body in a Newtonian force field

Nonlinear free oscillations of a rotating axisymmetrical solid body are considered with respect to the center of mass and with the body moving in a Newtonian force field. To construct periodic solutions of nonlinear differential equations of the motion, some algorithms, which are based on a modification of the extension method of solution with respect to a parameter, are used. The stability of nonlinear oscillations of the rotating solid body are studied with respect to stationary motions, some amplitude-frequency characteristics and forms of oscillations of the body are formulated for different values of its inertial parameters.Translated from Dinamicheskie Sistemy, No. 8, pp. 3–8, 1989.