Generalization of the Landau submerged jet solution

We consider a class of exact solutions of the hydrodynamic equations generalizing the Landau submerged jet solution. The obtained solutions do not vanish with the disappearance of viscosity and describe nonzero output or nonzero absorption of fluid mass from the respective sources or sinks located on one axis.     

Nonlinear instability of a viscous liquid jet

We investigate the weakly nonlinear temporal instability of an axisymmetric Newtonian liquid jet. Early nonlinear studies on the capillary instability of inviscid liquid jets were carried up to the third order contributions to the jet deformation and showed the nonlinear interaction between different modes. A recent study on the weakly nonlinear instability of planar Newtonian liquid sheets revealed the role of the liquid viscosity in the sheet stability behavior and showed a complicated influence [1]. Here, the instability of a liquid jet is examined as the axisymmetric counterpart of the sheet, in search for corresponding insight into the role of the liquid viscosity in the jet instability mechanism. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solvability of the problem on the oscillations of a fluid containing a submerged body

Conditions for unique solvability are given for the problem of steady-state oscillations of a body submerged in a heavy ideal fluid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 124–128, 1977.     

On small oscillations of a compressible stratified liquid

We study the structure of the spectrum and the completeness and basis property of a system of eigenvectors. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1614–1623, December, 2006.     

An assessment of turbulence models applied to the simulation of a two-dimensional submerged jet

This paper is concerned with the investigation of the performance of different turbulence models in the numerical prediction of transient flow caused by a confined submerged jet. Several widely used models, i.e., the standard k–ε, RNG k–ε, low Reynolds number k–ε models and the differential Reynolds stress model, as included in CFD codes, were compared with each other for a two-dimensional, incompressible, turbulent jet flow and with reported experimental data. A flapping oscillation was predicted regardless of the model used. A chosen Strouhal (St) number definition brought the fundamental frequencies from both the experiments and computations into close proximity. However, different turbulence models have exhibited quite different behaviours in terms of the frequency and regularity of the oscillation and in terms of the scale and duration of the vortices generated. All versions of the k–ε model yielded regular oscillations, which agree with experimental observations. On the other hand, the Reynolds stress (RS) model produced a complex pattern but a slower dissipation of vortices. In addition, some aspects of gridding and inflow representation are also discussed.     

Nonlinear oscillations of a variable composition liquid in a cylindrical vessel

The problem of wave motions of an ideal liquid of variable composition in a cylindrical vessel is reduced to an infinite system of nonlinear differential equations. It is shown that in the case of axisymmetric oscillations the equations obtained refine the linear analog of the problem, even if the vessel filling depth is significantly greater than the amplitude of the liquid free surface oscillations.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 104–107, 1989.     

Forced oscillations in a rotating liquid (I)

Zusammenfassung Der vorliegenden Analyse liegen die linearisierten Gleichungen für die achsensymmetrischen Störungen einer gleichförmig rotierenden Flüssigkeit zugrunde. Dabei wird aber keine Annahme hinsichtlich einer Zeitabhängigkeit der erzwungenen Bewegungen gemacht und die Entwicklung der Störungen vom Beginn der erzwungenen Bewegung an studiert. Ungeachtet der sich zeigenden analytischen Schwierigkeiten können über die allgemeinen Störbewegungen Aussagen gemacht werden.Für die relativ schnellen harmonischen Schwingungen (mit einer Störfrequenz grösser als die doppelte Rotationsfrequenz) findet man, nachdem die Einwirkung zeitlich genügend dauerte, dass die Bewegung der Flüssigkeit und jene des erzwungenen Mechanismus — wie in den früheren Arbeiten angenommen — überall phasengleich ist. Demgegenüber wird für Schwingungen mit kleinerer Relativfrequenz ein System fortschreitender Wellen vorausgesagt. Spezielle Aufmerksamkeit wird hier dem früheren Falle geschenkt; mit Hilfe einer Ähnlichkeitsbetrachtung kann die Analyse einer breiten Klasse der durch kleine harmonische Schwingungen erregten Bewegungen auf ein Problem der Potentialtheorie reduziert werden. Als Beispiele werden die axial schwingende und die radial pulsierende Kugel kurz erörtert.     

The stability of a column of liquid to torsional oscillations

Résumé On fait appel à la théorie linéaire de stabilité pour démontrer que des perturbations axisymétriques d'une colonne infinie de fluide en rotation continue et entourée d'un autre fluide noncompressible peuvent être représentées comme la somme des oscillations méridionales et des oscillations de torsion, pourvu que les nombres de Reynolds soient petits. Dans ce cas, le composant de torsion ne peut conduire à des instabilités.     

Pulsed jet of liquid under water

The Godunov numerical method is used to study the interaction of a pulsed, submerged cylindrical jet of liquid with a plane rigid barrier. The time for the parameters to establish themselves is estimated, as well as their dependence on the distance from the barrier. It is shown that the enhanced pressure zone is localized within a small region with dimensions on the order of the jet diameter near the center of the barrier. The effect of different forms of the finite difference approximation for the equations on the accuracy of the calculations near the axis of symmetry is studied. The effect of nonreflection conditions at fictional boundaries on the flow parameters along the center of the barrier is estimated. Donetsk State University. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 169–174, 1999.     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

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)     

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.     

Linear oscillations of axisymmetric viscous liquid bridges

Small amplitude free oscillations of axisymmetric capillary bridges are considered for varying values of the capillary Reynolds number C^-1 and the slenderness of the bridge L\Lambda. A semi-analytical method is presented that provides cheap and accurate results for arbitrary values of C^-1 and L\Lambda; several asymptotic limits (namely, C << 1, C >> 1, L << 1  and  |p-L| << 1C\ll 1, C\gg1, \Lambda\ll 1 \ \rm{and} \ |\pi-\Lambda|\ll 1 ) are considered in some detail, and the associated approximate results are checked. A fairly complete picture of the (fairly complex) spectrum of the linear problem is obtained for varying values of C and L\Lambda. Two kinds of normal modes, called capillary and hydrodynamic respectively, are almost always clearly identified, the former being associated with free surface deformation and the latter, only with the internal flow field; when C is small the damping rate associated with both kind of modes is comparable, and the hydrodynamic ones explain the appearance of secondary (steady or slowly-varying) streaming flows.     

The trajectory and stability of a spiralling liquid jet: Viscous theory

We examine a spiralling slender viscous jet emerging from a rapidly rotating orifice, extending Wallwork et al. [I.M. Wallwork, S.P. Decent, A.C. King, R.M.S.M. Schulkes, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory, J. Fluid Mech. 459 (2002) 43–65] by incorporating viscosity. The effects of viscosity on the trajectory of the jet and its linear instability are determined using a mixture of computational and asymptotic methods, and verified using experiments. A non-monotonic relationship between break-up length and rotation rate is demonstrated with the trend varying with viscosity. The sizes of the droplets produced by this instability are determined by considering the most unstable wave mode. It is also found that there is a non-monotonic relationship between droplet size and viscosity. Satellite droplet formation is also considered by analysing very short wavelength modes. The effects of long wavelength modes are examined, and a wave which propagates down the trajectory of the jet is identified for the highly viscous case. A comparison between theoretical and experimental results is made, with favourable agreement. In particular, a quantitative comparison is made between droplet sizes predicted from the theory with experimental observations, with encouraging agreement obtained. Four different types of break-up are identified in our experiments. The experimentally observed break-up mechanisms are discussed in light of our theory.     

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.     

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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