Oscillations of a vessel containing a viscous fluid

The oscillations of a rigid body having a cavity partially filled with an ideal fluid have been studied in numerous reports, for example, [1–6]. Certain analogous problems in the case of a viscous fluid for particular shapes of the cavity were considered in [6, 7]. The general equations of motion of a rigid body having a cavity partially filled with a viscous liquid were derived in [8]. These equations were obtained for a cavity of arbitrary form under the following assumptions: 1) the body and the liquid perform small oscillations (linear approximation applicable); 2) the Reynolds number is large (viscosity is small). In the case of an ideal liquid the equations of [8] become the previously known equations of [2–6]. In the present paper, on the basis of the equations of [8], we study the free and the forced oscillations of a body with a cavity (vessel) which is partially filled with a viscous liquid. For simplicity we consider translational oscillations of a body with a liquid, since even in this case the characteristic mechanical properties of the system resulting from the viscosity of the liquid and the presence of a free surface manifest themselves.The solutions are obtained for a cavity of arbitrary shape. We then consider some specific cavity shapes.     

Oscillations of liquid in a vessel in the form of a rectangular parallelepiped

We consider the forced and the free oscillations of a liquid partially filling a cavity in the form of a rectangular parallelepiped. The characteristics of these oscillations are studied for small deformations of the free surface. It is shown that for definite frequencies and amplitudes of two-dimensional translational motions of the parallelepiped the fundamental of the liquid oscillations is excited in the plane perpendicular to the plane of motion of the vessel. The effect of small linear damping of the liquid oscillations on the shape of the boundaries of the principal region of instability of the liquid oscillations is evaluated. Fairly large oscillations of a liquid in a cylinder were considered in [1]. The same problem for a cavity of arbitrary configuration was studied in [2]. We note also that the conclusions of the study presented here are in qualitative agreement with the basic results obtained by a somewhat different method in [3] for a cavity in the form of a right circular cylinder.     

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations.     

Oscillations of the fluid flow and the free surface in a cavity with a submerged bifurcated nozzle

The free surface dynamics and sub-surface flow behavior in a thin (height and width much larger than thickness), liquid filled, rectangular cavity with a submerged bifurcated nozzle were investigated using free surface visualization and particle image velocimetry (PIV). Three regimes in the free surface behavior were identified, depending on nozzle depth and inlet velocity. For small nozzle depths, an irregular free surface is observed without clear periodicities. For intermediate nozzle depths and sufficiently high inlet velocities, natural mode oscillations consistent with gravity waves are present, while at large nozzle depths long term self-sustained asymmetric oscillations occur.For the latter case, time-resolved PIV measurements of the flow below the free surface indicated a strong oscillation of the direction with which each of the two jets issue from the nozzle. The frequency of the jet oscillation is identical to the free surface oscillation frequency. The two jets oscillate in anti-phase, causing the asymmetric free surface oscillation. The jets interact through a cross-flow in the gaps between the inlet channel and the front and back walls of the cavity.     

Nonsmall liquid oscillations in moving vessels

The system of approximate nonlinear equations describing liquid oscillations in axisymmetric vessels is constructed. The equations are obtained for the case in which two coordinates belonging to the family of generalized coordinates characterizing the liquid motion are not small. This family is selected so that from the resulting nonlinear equations we can obtain as a particular case the nonlinear equations of [1–3], which are valid for the class of cylindrical vessels, and the requirements are satisfied that the resulting nonlinear equations correspond to the widely adopted linearized equations of liquid oscillations [4–6], Nonlinear equations are obtained which describe liquid oscillations in arbitrary vessels of rotation with radial baffles.     

Nonlinear Parametric Vibrations of an Elastic Structure with a Rectangular Liquid Tank

The nonlinear coupled vibrations of an elastic structure and liquidsloshing in a rectangular tank partially filled with liquid, are investigated.The structure on which the liquid tank is attached is vertically subjected to a sinusoidal excitation when the natural frequency of the structure is equal to twicethe natural frequency of one of the sloshing modes. In the theoretical analysis, the modal equations are derivedby taking nonlinear fluid force into account. Responses of the structure and the liquid surface are presented asresonance curves using the harmonic balance method. From this theoreticalanalysis the following predictions are obtained: (a) due to the nonlinearity of the fluid force, harmonic oscillations appear in the structure, while subharmonic oscillations occur on the liquid surface; (b) the shapes of the resonance curves markedly change depending on the liquid level; and (c) when the tuning condition is slightly deviated, amplitudemodulated motions and chaotic oscillations appear during a certain range of the excitation frequency. These were qualitatively in agreement with the experimental results.     

Instability of a liquid jet in a high-frequency alternating electric field

Stability of a liquid (electrolyte) jet in a tangential electric field harmonically oscillating with a high frequency is considered under an assumption of an ideal liquid. It is demonstrated that it is possible to solve the electrodynamic and hydrodynamic parts of the problem inside the jet separately if the Peclet number based on the Debye layer thickness is small. Linear stability of the trivial solution of the problem is studied. A dispersion relation is derived, which is used to study the effect of the amplitude and frequency of electric field oscillations on jet stability. An increase in the amplitude of oscillations is demonstrated to exert a stabilizing effect, whereas an increase in frequency leads to insignificant destabilization of the jet.     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

Unsteady phenomena of an oscillating turbulent jet flow inside a cavity: Effect of aspect ratio

Self-sustained oscillatory phenomena in confined flow may occur when a turbulent plane jet is discharging into a rectangular cavity. An experimental set-up was developed and the flow analysis has been made using mainly hot-wire measurements, which were complemented by visualisation data. Previous studies confirmed that periodic oscillations may occur, depending on the location of the jet exit nozzle inside the cavity, and also the distance between the side-walls. The present study deals with the symmetrical interaction between a turbulent plane jet and a rectangular cavity and the influence of the geometrical characteristics of the cavity on the oscillatory motion. The size and aspect ratio of the cavity were varied together with the jet width compared to that of the cavity. The study is carried out both numerically and experimentally. The numerical method solves the unsteady Reynolds averaged Navier–Stokes equations (URANS) together with the continuity equation for an incompressible fluid. The closure of the flow equations system is achieved using a two-scale energy-flux model at high Reynolds number in the core flow coupled with a wall function treatment in the vicinity of the wall boundaries. The fundamental frequency of the oscillatory flow was found to be practically independent of the cavity length. Moreover, the oscillations are attenuated as the cavity width increases, until they disappear for a critical value of the cavity width. Contour maps of the instantaneous flow field are drawn to show the flow pattern evolution at the main phases of oscillation. They are given for several aspect ratios of the cavity, keeping constant values for the cavity width and the jet thickness. The proposed approach may help to investigate further the oscillation mechanisms and the entrainment process occurring in pressure driven jet–cavity interactions.     

Finite amplitude oscillations of a hyperelastic spherical cavity

We present numerical results for the finite oscillations of a hyperelastic spherical cavity by employing the governing equations for finite amplitude oscillations of hyperelastic spherical shells and simplifying it for a spherical cavity in an infinite medium and then applying a fourth-order Runge-Kutta numerical technique to the resulting non-linear first-order differential equation.The results are plotted for Mooney-Rivlin type materials for free and forced oscillations under Heaviside type step loading. The results for Neo-Hookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the problem is also discussed in length.     

Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity

The nonlinear dynamical behavior of a micromechanical resonator acting as one of the mirrors in an optical resonance cavity is investigated. The mechanical motion is coupled to the optical power circulating inside the cavity both directly through the radiation pressure and indirectly through heating that gives rise to a frequency shift in the mechanical resonance and to thermal deformation. The energy stored in the optical cavity is assumed to follow the mirror displacement without any lag. In contrast, a finite thermal relaxation rate introduces retardation effects into the mechanical equation of motion through temperature dependent terms. Using a combined harmonic balance and averaging technique, slow envelope evolution equations are derived. In the limit of small mechanical vibrations, the micromechanical system can be described as a nonlinear Duffing-like oscillator. Coupling to the optical cavity is shown to introduce corrections to the linear dissipation, the nonlinear dissipation and the nonlinear elastic constants of the micromechanical mirror. The magnitude and the sign of these corrections depend on the exact position of the mirror and on the optical power incident on the cavity. In particular, the effective linear dissipation can become negative, causing self-excited mechanical oscillations to occur as a result of either a subcritical or supercritical Hopf bifurcation. The full slow envelope evolution equations are used to derive the amplitudes and the corresponding oscillation frequencies of different limit cycles, and the bifurcation behavior is analyzed in detail. Finally, the theoretical results are compared to numerical simulations using realistic values of various physical parameters, showing a very good correspondence.     

Attenuation of cavity internal pressure oscillations by shear layer forcing with pulsed micro-jets

Experimental results concerning the pressure oscillations induced by a grazing flow over a deep cavity like a Helmholtz resonator are presented. The study deals with the forcing of the neck shear layer instability in an opened-loop control scheme by means of pulsed micro-jets. The effects of the frequency and amplitude are investigated. It is found that efficient attenuation of the pressure oscillations can be reached when the forcing frequency is larger than the cavity resonance frequency. Then the shear layer is locked with the forcing and resonance with the cavity is lost, inducing a significant decrease of the acoustic pressure level in the cavity. Effects of the jet amplitude are weak, a very small amplitude being capable of forcing the shear layer. By contrast, when the forcing frequency is lower than the cavity resonance frequency (the forcing wave length is greater than twice the neck length) the forcing is ineffective.     

Effect of viscoelastic properties of a liquid on the dynamics of small oscillations of a gas bubble

A series of papers has been devoted to questions of gas bubble dynamics in viscoeiastic liquids. Of these papers we mention [1–4]. The radial oscillations of a gas bubble in an incompressible viscoeiastic liquid have been studied numerically in [1, 2] using Oldroyd's model [5]. Anexact solution was found in [3], and independently in [4], for the equation of small density oscillations of a cavity in an Oldroyd medium when there is a periodic pressure change at infinity. The analysis of bubble oscillations in a viscoeiastic liquid is complicated by properties of limiting transitions in the rheological equation of the medium. These properties are of particular interest for the problem under investigation. These properties are discussed below, and characteristics of the small oscillations of a bubble in an Oldroyd medium are investigated on the basis of a numerical analysis of the exact solution obtained in [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 82–87, May–June, 1976.The authors are grateful to V. N. Nikolaevskii for useful advice and for discussing the results.     

Approximate method of solving the problem of characteristic oscillations of a liquid in cavities of revolution

The problem of the characteristic oscillations of a liquid in axisymmetric cavities of rotation has been fairly fully studied [1–5], its solution in the general case being found by the variational method. Analysis of numerical results using the variational method shows that to achieve acceptable accuracy it is necessary to retain an appreciable number of coordinate functions, which entails the solution of a matrix eigenvalue problem of high order, this applying especially to the case when it is necessary to determine several eigenfrequencies and the shapes of the oscillations. In the present paper, a method proposed earlier by Shmakov [6] is developed, the velocity potential being sought in the form of a sun of two potentials. The first (base) potential is a solution to the problem of the characteristic oscillations of a liquid in a cavity whose free surface coincides with the free surface of the original cavity, and the second (correcting) potential is chosen in the form of a system of harmonic functions, this system being complete and orthogonal on the wetted surface of the cavity. Cavities of revolution are analyzed as examples, and a detailed investigation of numerical results is made for a spherical cavity. The numerical analysis shows that a sufficiently accurate result in the determination of a frequency is obtained if one term of the base problem is retained and only the correcting potential is used to make this more accurate. As a result, it is only necessary to solve an algebraic equation of first degree in the square of the frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 3–8, September–October, 1983.     

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

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.     

Potential flow around a deforming bubble in a Venturi

The differential pressure between the entrance and throat of a Venturi will fluctuate if the liquid flowing through the Venturi contains bubbles. This paper reports computations of the pressure fluctuation due to the passage of a single bubble. The liquid is assumed inviscid and its velocity, assumed irrotational, is computed by means of a boundary integral technique. The liquid velocity at the entrance to the Venturi is assumed constant and uniform across the pipe, as is the pressure at the outlet. The bubble is initially far upstream of the Venturi and moves with velocity equal to that of the liquid. Buoyancy is neglected. If the bubble is sufficiently small that interactions with the Venturi walls may be neglected, a simple one-dimensional model for the bubble velocity is in good agreement with the full boundary integral computations. The differential pressure (taken to be positive) decreases when the bubble enters the converging section of the Venturi, and then increases to a value higher than for liquid alone as the bubble passes the pressure measurement position within the throat. The changes can be estimated using the one-dimensional model, if the bubble is small. The bubble is initially spherical (due to surface tension) but is perturbed by the low pressure within the Venturi throat. In the absence of viscosity, the bubble oscillates after leaving the Venturi. The quadrupole oscillations of the bubble are similar in frequency to those of a bubble in unbounded fluid; the frequency of the monopole oscillations is modified by the presence of the pipe walls. Numerical results for the frequency of monopole oscillations of a bubble in a uniform tube of finite length are in good agreement with analytic predictions, as is the computed drift of the oscillating bubble.     

Stability of a spherical cavity in a sound field

The linear theory of the stability of the spherical shape of a cavity and the stability of its radial oscillations in a sound field are discussed. An equation is derived for the amplitudes of the spherical harmonics with allowance for surface tension, viscosity, and compressibility of the surrounding liquid in the Herring-Flynn approximation. The radial pulsation stability is analyzed in the same approximation. The equations derived in the article are subjected to numerical analysis.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 109–114, November–December, 1973.     

Lift Force Acting on a Heavy Solid in a Rotating Liquid-Filled Cavity with a Time-Varying Rotation Rate

The dynamics of a heavy cylindrical body in a liquid-filled horizontal cylindrical cavity with a time-varying rotation rate is experimentally investigated. The body is near the cavity boundary under a centrifugal force and undergoes solid-body rotation together with the liquid and the cavity at a fixed rotation rate. The dependence of the body dynamics on the amplitude and frequency of modulation of the rotation rate is investigated. It is found that at a critical amplitude of modulation (at definite frequency), the heavy body repulses from the cavity boundary and comes into a steady state at some distance from the wall. It is found that the average lift force (repulsive one) is generated by the azimuthal oscillation of the body in the rotating frame of reference and manifests itself at a distance comparable to the thickness of the viscous boundary layer. In the experiments, we observed azimuthal drift of the body due to asymmetric azimuthal oscillations of the body. In the limit of high frequency of the rotation rate modulation, the dependence of the lift force coefficient on the gap between the body and the wall is determined.     

Flow reversal of an elastico-viscous liquid between a stationary and a torsionally oscillating disc

The flow of an elastico-viscous liquid contained between two infinite discs, when one is held at rest and the other performs small-amplitude torsional oscillations about their common axis, is considered. The liquid is characterized by equations of state more general than, and containing as special cases, the equations of state used by previous authors who have considered this problem.The phenomenon of flow reversal is examined for large values of the Reynolds number, and the apparently different conclusions of previous authors are explained in terms of their particular choices of material parameters.It is also shown that in general the flow at high frequency is dominated by a particular combination of material parameters.     

Oscillations of a viscous fluid in a thin layer between two coaxial disks rotating together

A study is made of the problem of the motion of an incompressible viscous fluid in the space between two coaxial disks rotating together with constant angular velocity under the assumption that the pressure changes in time in accordance with a harmonic law. The problem is solved using the equations of unsteady motion of an incompressible viscous fluid in a thin layer. It is shown that the velocity field in this case is a superposition on a steady field of damped oscillations with cyclic frequency equal to twice the angular velocity of the disks and forced oscillations with cyclic frequency equal to the cyclic frequency of the oscillations of the pressure field. It is shown that the amplitude of the forced oscillations of the velocity field depends strongly on the ratio of the cyclic frequency of the oscillations of the pressure field to the angular velocity of the disks. It is shown that there is a certain value of the ratio at which the amplitude of the forced oscillations has a maximal value (resonance). It is shown that even for very small amplitudes of the pressure oscillations the amplitude of the oscillations of the relative velocity at resonance may reach values comparable with the mean velocity of the main flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 166–169, January–February, 1984.