Interface Dynamics of Immiscible Fluids under Horizontal Vibration

The average dynamics of two immiscible fluids of different densities in a rectangular cavity oscillating horizontally with a high frequency are investigated experimentally. The fluids are characterized by a small surface tension coefficient. The regularities of the quasi-steady spatial relief formation on the fluid interface are studied. It is shown that, in addition to the capillary and vibrational parameters, the pattern excitation threshold is determined by the dimensionless vibration frequency. In the limit of high dimensionless frequency, good agreement with the well-known theoretical results is obtained.     

Vibrational dynamics of a light body in a liquid-filled rotating cylinder

The behavior of a light free cylindrical body in a rapidly rotating horizontal cylinder containing a liquid under vibrational action (the vibration direction is perpendicular to the rotation axis) is investigated. An intense rotation of the body relative to the cavity is detected. Depending on the vibration frequency, the body rotation velocity in the laboratory reference system may be higher or lower than the cavity rotation velocity and in the resonance region they may differ by several times. The mechanism of motion generation is theoretically described. It is shown that the motion is related with the excitation of inertial oscillations of the body: the cause of the motion is an average vibrational force generated due to nonlinear effects in the Stokes boundary layer near the oscillating body. The formation of large-scale axisymmetric vortex structures periodic along the rotation axis, which appear under conditions of inertial oscillation of the body during its motion, both leading and lagging, is detected.     

Some properties of multi-fluid stokes flows

The solution of Stokes' equations for a rotating axisymmetric body which possesses reflection symmetry about a planar interface between two infinite immiscible quiescent viscous fluids is shown to be independent of the viscosities of the fluids and identical with the solution when the fluids have the same viscosity. The result is generalized to a rotating axisymmetric system of bodies which possesses reflection symmetry about each interface of a plane stratified system of fluids. An analogous result for two-fluid systems with a nonplanar static interface is also derived. The effect on torque reduction produced by the presence of a second fluid layer adjacent to a rotating axisymmetric body is considered and explicit calculations are given for the case of a sphere. A proof of uniqueness for unbounded multi-fluid Stokes' flow is given and the asymptotic far field structure of the velocity field is determined for axisymmetric flow caused by the rotation of axisymmetric bodies.     

Interface Dynamics of Immiscible Fluids under Circularly Polarized Vibration (Experiment)

This paper is one of a series of experimental studies of immiscible fluid interfaces under vibration of different types. A comparison between the average interface dynamics for high-frequency rotational vibration [1] and translational unidirectional vibration [2, 3] shows that a complication of the oscillation pattern leads to qualitatively novel effects. In this study, the cavity oscillates circularly and translationally in a horizontal plane. Two average vibration effects are discovered: a uniform azimuthal rotation of the fluid relative to the cavity and the excitation of a periodic hexagonal relief on the initially flat fluid interface.     

Evolution Equations for the Surface Concentration of an Insoluble Surfactant; Applications to the Stability of an Elongating Thread and a Stretched Interface

The general form of the convection–diffusion equation governing the evolution of the surface concentration of an insoluble surfactant over an evolving interface is reviewed and discussed for three-dimensional, axisymmetric, and two-dimensional configurations. The linearized form of the evolution equation is then derived around cylindrical and planar shapes in a framework that is suitable for carrying out a linear stability analysis for axisymmetric or two-dimensional perturbations. Particular attention is paid to the cases of quiescent unperturbed fluids, unidirectional shear flow, and elongational flow. By way of application, the linearized transport equations are combined with Stokes-flow hydrodynamics to investigate the stability of an elongating cylindrical viscous thread suspended in an ambient viscous fluid or in a vacuum, and the stability of a two-dimensional interface separating two semi-infinite fluids and stretched under the action of an orthogonal stagnation-point flow. The results illustrate the possibly important role of the surfactant on the linear growth of periodic waves on the cylindrical interface, and reveal that the surfactant has no effect on the stability of the planar interface.     

Sand-Fluid Interface under Vibration

The average dynamics of the interface between a pure fluid and a granular medium with fluid-filled pores in a closed vibrating cavity are investigated experimentally. Three types of vibration, namely, linear and circular translational in a horizontal plane and rotational about a vertical axis, are considered. In all cases, the excitation of a dynamic relief on the surface of the granular medium, preceded by fluidization of the sand, is observed. For more complicated vibration types, additional average effects are manifested, such as the generation of an average granular-medium motion relative to the cavity under circular vibration and the displacement of the fluidized granular medium toward the rotation axis under rotational vibration. In the cases considered, the regularities of the average dynamics of the fluidized granular medium are found. It is shown that the phenomena in a granular-medium-fluid system can be analyzed using the two-fluid theoretical model.     

Development of a steady relief at the interface of fluids in a vibrational field

The vibrations of a vessel strongly influence the behavior of the interface of the fluids in it. Thus, vertical vibrations can lead both to the parametric excitation of waves (Faraday ripples) and to the suppression of the Rayleigh-Taylor instability [1–2]. At the present time, the influence of vertical vibrations on the behavior of a fluid surface have been studied in sufficient detail (see, for example, review [3]). The behavior of an interface of fluids in the case of horizontal vibrations has been studied less. An interesting phenomenon has been revealed in the experimental papers [4, 5]: in the case of fairly strong horizontal vibrations of a vessel containing a fluid with a free surface, the fluid collects near one of the vertical vessel walls, the free surface being practically plane and stationary with respect to the vessel, while its angle of inclination to the horizon depends on the vibration rate. But if there is a system of immiscible fluids with comparable but different densities in the vessel, horizontal vibrations lead to the formation of a steady wave relief at the interface. An explanation of the behavior of a fluid with a free boundary was given in [6] on the basis of averaged equations of fluid motion in a vibrational field. The present paper is devoted to an analysis of the behavior of the interface of fluids with comparable densities in a high-frequency vibrational field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 8–13, November–December, 1986.     

Nonlinear analysis of governing laws of realization of the wave motion on the surface of a charged jet moving with respect to a material medium

On the base of analytic asymptotic calculations which are quadratic with respect to the ratio of the wave amplitude and the jet radius it is shown that the presence of a tangential jump in the velocity field on the jet surface leads to generation of a periodic wave motion on the interface between the media and has the destabilizing effect for both axisymmetric and bending and bending-deformation waves. It is found that there is a degenerate internal nonlinear resonance interaction between waves on the jet surface. This interaction may be of six different types in which the energy can be transferred between the interacting waves including waves of different symmetry. In the last case the energy is transferred from waves determining the initial deformation to axisymmetric waves.     

Effect of imperfect bonding on axisymmetric elastodynamic response of a lined circular tunnel in poroelastic soil due to a moving ring load

A two-dimensional elasticity analysis for steady-state axisymmetric dynamic response of an arbitrarily thick elastic homogeneous hollow cylinder of infinite length, which is imperfectly bonded to the surrounding fluid-saturated permeable formation, subject to an axially moving ring load, is presented. The problem solution is derived by using Biot’s dynamic theory of poroelasticity in conjunction with double Fourier transformation with respect to time (frequency) and axial coordinate (axial wave number). The analytical results are illustrated with numerical examples in which a concrete tunnel lining of uniform wall thickness is imperfectly bonded to a surrounding water-saturated poroelastic formation of soft/stiff frame characteristic. Numerical solutions for the radial shell mid-plane and formation displacements are calculated by analytical (numerical) inversion of the Fourier transformation with respect to the frequency (axial wave number). Primary attention is focused on the influence of bonding condition at the liner/soil interface, formation material type, and load velocity on the system’s dynamic response. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

Effects of rotary inertia on sub- and super-critical free vibration of an axially moving beam

The most important issue in the vibration study of an engineering system is dynamics modeling. Axially moving continua is often discussed without the inertia produced by the rotation of the continua section. The main goal of this paper is to discover the effects of rotary inertia on the free vibration characteristics of an axially moving beam in the sub-critical and super-critical regime. Specifically, an integro-partial-differential nonlinear equation is modeled for the transverse vibration of the moving beam based on the generalized Hamilton principle. Then the effects of rotary inertia on the natural frequencies, the critical speed, post-buckling vibration frequencies are presented. Two kinds of boundary conditions are also compared. In super-critical speed range, the straight configuration of the axially moving beam loses its stability. The buckling configurations are derived from the corresponding nonlinear static equilibrium equation. Then the natural frequencies of the post-buckling vibration of the super-critical moving beam are calculated by using local linearization theory. By comparing the critical speed and the vibration frequencies in the sub-critical and super-critical regime, the effects of the inertia moment due to beam section rotation are investigated. Several interesting phenomena are disclosed. For examples, without rotary inertia, the study overestimates the stability of the axially moving beam. Moreover, the relative differences between the super-critical fundamental frequencies of the two theories may increase with an increasing beam length.     

Mode identification of a rotating disk

A modal testing method permitting identification of the natural frequencies, the number of nodal diameters and wave motions in a rotating disk is presented in this paper. This method is applicable at arbitrary rotation speed without requiring a priori information about the vibration modes of the stationary disk. The influence of disk rotation speed on the prediction of mode shapes with this method is shown, and experimental predictions of modal parameters are presented for both axisymmetric and asymmetric disks.     

Dynamical analysis of axially moving plate by finite difference method

The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.     

Structure of Averaged Flow Driven by a Vibrating Body with a Large-Curvature Edge

The averaged viscous incompressible fluid flow driven by a vibrating body with a large-curvature edge is investigated experimentally and numerically. The case of an axisymmetric body immersed in fluid and performing translational vibrations along its axis is considered. Experiments carried out on fluids of various viscosity over a wide vibration frequency and amplitude range and direct numerical calculations based on the complete time-dependent equations of viscous fluid dynamics show that the global structure of the averaged flow significantly depends on the relation between the curvature radius of the body edge and the viscous skin-layer thickness. Different averaged flow regimes are detected and the flow restructuring process is investigated as a function of the vibration amplitude and frequency.     

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique (GITT) is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations (ODEs) in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.     

Hydrodynamic action of the fluid slowly flowing around a spheroidal particle covered by a viscous film

A steady problem of a slow axisymmetric flow of a viscous incompressible fluid around an oblate spheroid covered by a viscous film is solved analytically with the use of the Stokes approximation. Surface tension on the interface between the fluids is taken into account. Expressions for velocity components and stream functions are presented. A formula for determining the force action of the incoming flow onto the oblate spheroid is derived.     

Modeling and analysis of an axially acceleration beam based on a higher order beam theory

In this paper, a higher order model equation is presented for an axially accelerating beam. Based on a new kinematic frame of the beam and continuum mechanics theory, the coupled governing equations of nonlinear vibration for axially accelerating beam are obtained with the aid of the generalized Hamilton principle. The governing equations take into account the characteristic of the material, the shear strain, the rotation strain and the effect of longitudinally varying tension due to the axial acceleration. The equations are decoupled into a nonlinear partial-integro-differential equations when the transverse nonlinear vibration is small. For the principal parametric resonances, the steady-state frequency responses are obtained by the multiple scales method. The stable and unstable interval are analyzed for the trivial and nontrivial steady-state response. Effects of the system parameters on the amplitude have been investigated. The results show that the material parameter (i.e, in-plane Poisson ratio) has a significant effect on the amplitude and the nonlinear vibration behavior type. The amplitude decrease with the growth of the in-plane Poisson ratio. The total potential energy has play a very important role in determining the amplitude of frequency response according to model analysis. Lastly, comparisons among the analytical solutions and numerical solutions are made and good agreements for the amplitude are found.     

Creep buckling of axially compressed circular cylindrical shells of a zirconium alloy: Experiment and computer simulation

Experiments were performed to study the deformation and buckling of axially compressed circular cylindrical shells of Zr2.5Nb zirconium alloy under creep conditions. Computer simulation using the MSC.Marc 2012 software was conducted by step-by-step integration of the equations of quasistatic deformation of thin shells using Norton’s law of steady creep. The results of the experiment and computer simulation show that the buckling modes are a combination of axisymmetric bulges located near one end or both ends of the shell and axisymmetric buckling modes with the formation of three or four waves in the circumferential direction. A comparison is made of the time dependences of the axial strain of the shells obtained in the experiment and by computer simulation. It is shown that for large axial compressive stresses, these dependences are in satisfactory agreement. For lower values of these stresses, the difference between the theoretical and experimental dependences is greater.     

Stability of the Interface of a Liquid-Suspension System under High-Frequency Nonlinearly Polarized Vibration

The stability of the state of a two-layer system consisting of a homogeneous liquid and a solid particle suspension in the same liquid with a plane interface between them is investigated. The system performs both horizontal and vertical high-frequency vibration with an arbitrary phase shift. It is shown that a mean flow develops under the simultaneous action of both horizontal and vertical vibration; quantitative flow stability characteristics are determined numerically using the differential sweep method. It is shown that a traveling wave relief exists on the liquid-suspension interface. Transverse oscillations in phase with longitudinal oscillations destabilize the entire system. The presence of an oscillation phase shift can lead to an increase in the stability limit; the direction of motion of the wave relief can differ depending on the value of the phase shift. Instability of the system in the presence of strictly vertical vibration is observed, the crisis being associated with longwave monotonic perturbations.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 3–13.Original Russian Text Copyright © 2005 by Lobov, Lyubimov, and Lyubimova.     

Formation and Amplification of Shock Waves in a Bubble “Cord”

The structure and dynamics of the wave field generated by a bubble system in the form of an axial bubble cylinder (cord) excited by a plane shock wave propagating along the axis in an axisymmetric shock tube are numerically examined. It is shown that consecutive excitation of oscillations of the bubble zone results in formation of a quasi-steady shock wave in the cord and in the ambient liquid. Results of the numerical analysis of the maximum amplitude of the resulting wave as a function of problems parameters are described.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 46–52, September–October, 2005.     

Unsteady nonaxisymmetric flows in the hydrodynamic Czochralski model at high Prandtl numbers

Thermal-gravitational and thermocapillary convection is numerically modeled in the axisymmetric and three-dimensional approximations for the hydrodynamic model configuration corresponding to technological regimes of oxide crystal growth and taken as the basis in an international test. The salient features of the interaction between the convection and the flow driven by crystal rotation are studied at high Prandtl numbers. The flow and temperature fields occurring upon the generation of axisymmetric oscillations and the loss of axial symmetry are studied, analyzed, and compared with the results of other authors.