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.     

Control of the motion of a vessel with a heavy inhomogeneous liquid

The problem of controlling the internal waves inside a closed rectangular cavity filled with a heavy two-layer fluid is considered in the linear approximation. The fluid is assumed to be stably stratified, ideal, and incompressible. The controlling horizontal force is applied to the body containing the cavity. It is assumed that at the initial moment there are no oscillations of the fluid and the interface is horizontal. The problem is to bring the vessel as a whole into a prescribed state of linear motion without relative wave motion of the fluid. The Cauchy-Poisson problem and the self-consistent integro-differential equation of the vessel motion are solved using the Fourier method and taking into account the reaction of the internal waves. On the basis of an analysis of the corresponding generalized momentum problem, approaches are proposed for solving the problem of control. It is shown that a control action with a sufficiently high order of Steklov smoothness ensures the approximate solution of the control problem with the required accuracy for all the characteristics of the motion of the hybrid oscillating system under consideration.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–22, May–June, 1995.The study was performed with financial support from the Russian Foundation for Fundamental Research (project No. 94-01-01368).     

Dynamics of a flat cavity in a low-viscosity liquid

The problem of the motion of a cavity in a plane-parallel flow of an ideal liquid, taking account of surface tension, was first discussed in [1], in which an exact equation was obtained describing the equilibrium form of the cavity. In [2] an analysis was made of this equation, and, in a particular case, the existence of an analytical solution was demonstrated. Articles [3, 4] give the results of numerical solutions. In the present article, the cavity is defined by an infinite set of generalized coordinates, and Lagrange equations determining the dynamics of the cavity are given in explicit form. The problem discussed in [1–4] is reduced to the problem of seeking a minimum of a function of an infinite number of variables. The explicit form of this function is found. In distinction from [1–4], on the basis of the Lagrauge equations, a study is also made of the unsteady-state motion of the cavity. The dynamic equations are generalized for the case of a cavity moving in a heavy viscous liquid with surface tension at large Reynolds numbers. Under these circumstances, the steady-state motion of the cavity is determined from an infinite system of algebraic equations written in explicit form. An exact solution of the dynamic equations is obtained for an elliptical cavity in the case of an ideal liquid. An approximation of the cavity by an ellipse is used to find the approximate analytical dependence of the Weber number on the deformation, and a comparison is made with numerical calculations [3, 4]. The problem of the motion of an elliptical cavity is considered in a manner analogous to the problem of an ellipsoidal cavity for an axisymmetric flow [5, 6]. In distinction from [6], the equilibrium form of a flat cavity in a heavy viscous liquid becomes unstable if the ratio of the axes of the cavity is greater than 2.06.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 15–23, September–October, 1973.The author thanks G. Yu. Stepanov for his useful observations.     

Asymptotic of the flow upon shock incidence on a wedge cavity

The asymptotic of the motion originating because of shock incidence on a wedge cavity in a metal is investigated as the wave amplitude tends to zero. It has been shown in [1] that the flow is hence divided into two domains. The principal term governing the flow in the first domain agrees with the acoustic approximation. The flow in the second domain is described by incompressible fluid equations in the principal term. Determination of the flow in the second domain is reduced herein to the solution of a singular nonlinear integral equation. A numerical solution is found for a series of values of the cavity aperture.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 129–138, May–June, 1972.     

Creeping flow of viscoplastic materials through a planar expansion followed by a contraction

In this work, the creeping flow of a viscoplastic fluid through a planar channel with an expansion followed by a contraction is analyzed numerically. The solution of the conservation equations of mass and momentum is obtained via the finite volume method. In order to model the non-Newtonian behavior of the fluid, it was used the generalized Newtonian fluid constitutive equation. The viscosity function was the one proposed by Souza Mendes and Dutra [Souza Mendes, P.R., Dutra, E.S.S., 2004. Viscosity function for yield-stress liquids. Appl. Rheol. 14, 296–302]. The yielded and unyielded regions are obtained for several combinations of rheological parameters. The influence of these parameters on pressure drop through the cavity is also obtained and analyzed.     

Stability of convective motion in a two-dimensional vertical fluid layer with permeable boundaries

The stability of the convective motion of a viscous incompressible fluid in a channel between permeable vertical planes heated to different temperatures is considered under the assumption of homogeneous transverse air blasting. Stability boundaries for different values of the Prandtl number Pr and Peclet number Pe that characterize the intensity of transverse motion are numerically determined. The results demonstrate that transverse blasting substantially influences both the hydrodynamic instability mechanism and instability due to the growth of thermal waves in the flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 94–101, January–February, 1976.In conclusion, I wish to express my appreciation to E. M. Zhukhovitskii for supervising the study. and G. Z. Gershuni for useful discussion of the results.     

Collapse of a cylindrical cavity in a fluid layer under impact

The case of impact on a thin annular fluid layer with a gas-filled cavity is considered. The solution of the problem reduces to integrating a system of two first-order ordinary differential equations. The equations are analyzed qualitatively, and some exact solutions are found. Cases are noted of pulsation of the cavity, and the influence of counter-pressure and viscosity is investigated. The experimental results obtained are in agreement with the numerical computations carried out herein.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 98–106, November–December, 1970.The authors are grateful to A. M. Kogan and L. V. Mostovaya for performing the computations.     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Linearized equations of motion of a viscous fluid in a porous medium of periodic structure

The investigation of flow in essentially inhomogeneous porous systems through the analysis of model periodic structures [1] is considered. In the acoustic approximation, an integrodifferential equation is obtained that describes the motion of a viscous fluid in a rigid porous medium of periodic structure. The velocity vector and pressure are represented in the form of asymptotic series with respect to a small parameter that characterizes the size of the periodicity cell, and the well-known procedure for averaging linearized hydrodynamic equations with small coefficients of viscosity [2, 3] is also used. A solution is presented to the local problem in the periodicity cell for a structure consisting of a doubly periodic system of infinitely long rods of circular section and a compressible viscous fluid that fills the space between them, and also for a structure formed by a system of orthogonal rectilinear channels, filled with viscous fluid, in a solid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 123–130, March–April, 1988.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

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.     

Instability analysis of nonlinear surface waves in a circular cylindrical container subjected to a vertical excitation

Singular perturbation theory of two-time scale expansions was developed both in inviscid and weak viscous fluids to investigate the motion of single surface standing wave in a liquid-filled circular cylindrical vessel, which is subject to a vertical periodical oscillation. Firstly, it is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear evolution equation of slowly varying complex amplitude, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from solvability condition of high-order approximation. It shows that when forced frequency is low, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is high, the influence of surface tension is significant, and can not be neglected. This proved that the surface tension has the function, which causes free surface returning to equilibrium location. Theoretical results much close to experimental results when the surface tension is considered. In fact, the damping will appear in actual physical system due to dissipation of viscosity of fluid. Based upon weakly viscous fluids assumption, the fluid field was divided into an outer potential flow region and an inner boundary layer region. A linear amplitude equation of slowly varying complex amplitude, which incorporates damping term and external excitation, was derived from linearized Navier–Stokes equation. The analytical expression of damping coefficient was determined and the relation between damping and other related parameters (such as viscosity, forced amplitude and depth of fluid) was presented. The nonlinear amplitude equation and a dispersion, which had been derived from the inviscid fluid approximation, were modified by adding linear damping. It was found that the modified results much reasonably close to experimental results. Moreover, the influence both of the surface tension and the weak viscosity on the mode formation was described by comparing theoretical and experimental results. The results show that when the forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when the forcing frequency is high, the surface tension of the fluid is prominent. Finally, instability of the surface wave is analyzed and properties of the solutions of the modified amplitude equation are determined together with phase-plane trajectories. A necessary condition of forming stable surface wave is obtained and unstable regions are illustrated.     

Motion of a variable-volume sphere in an ideal fluid near a plane surface

The motion of a spherical cavity in a fluid is investigated. The radius of the sphere varies under the action of a constant pressure at infinity. The problems of the collapse of a cavity moving in an unbounded fluid and of the collapse of a cavity near a plane are solved in the exact formulation. The occurrence of an initial translational velocity or the presence of a solid surface, by contrast with the collapse of a sphere at rest in an unbounded fluid [1], yields a limiting radius at which the process of collapse ceases. A sphere initially at rest near a plane always comes into contact with the plane as a result of collapse. The radius and velocities at which the sphere arrives the plane are calculated for various initial distances from the latter. The possible mechanism of the action of a cavitation bubble on a solid surface is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1971.     

Motion of particles in a nonstationary layered flow of an incompressible fluid

The motion of spherical particles in a nonstationary layered flow are considered. It is assumed that the fluid is incompressible and that the particles do not interact with one another or influence the parameters of the fluid. Allowance is made for the influence of the pressure gradient, the apparent mass, the Magnus force, and the viscosity of the fluid on the motion of the particles. The formulation of the problem corresponds to the conditions of motion of the two-phase mixture in the channels of the rotatory-pulsatory apparatus [1] used in technology to realize various processes such as solution, emulsification, dispersing, etc. The processes in such an apparatus are strongly nonsteady and have hitherto been hardly investigated at all.Translated from 'Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 53–58, July–August, 1981.We thank A. R. Gurvich for making the calculations.     

Simulation of the motion of thin rotating jets of a viscous incompressible fluid

A regular procedure is proposed for deriving approximate equations of motion of straight thin rotating jets of a viscous incompressible fluid, and similarity parameters of such flows are established. The problem of the stability of free steady motion of a finite jet is considered in the framework of a model that takes into account in the zeroth approximation the effects of viscosity, the rotation of the jet, and capillarity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 51–59, March–April, 1984.     

Capillary waves in a relaxing conducting viscous fluid with a surface charge

The influence of weak viscosity and dynamical surface tension relaxation effect on the structure of the wave motion spectrum in a conducting viscous fluid with a surface charge is investigated. Taking these phenomena into account leads to the appearance of additional branches of both wave and aperiodic motions associated with fluid elasticity effects and the finite time taken by the fluid surface layer to respond to fast external excitation.Yaroslavl. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 98–105, January–February, 1996.     

Application of a differential equation for turbulent viscosity to the analysis of plane non-self-similar flows

A differential equation of the kinetic-energy balance of turbulence is used in a number of papers to close the equations describing average motion in turbulent flows. On the basis of this relation, a differential equation for turbulent viscosity is obtained herein. Numerical computations are carried out for incompressible non-self-similar turbulent and transition flows in awake, a jet, and a boundary layer; universal constants in the equation for the viscosity are refined. The flow in a wake and boundary layer with high longitudinal pressure gradients is investigated by analytical and numerical methods. Dimensionless criteria determining the nature of the effect of the pressure gradient on the average flow and turbulent viscosity are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–127, September–October, 1971.The author is greateful to I. P. Smirnov, S. Yu. Krasheninnikov, and V. B. Kuz'mich for aid in compiling the program for the numerical computations and to L. L. Bychkov for processing the computational results and plotting the graphs.     

Plane-parallel flow around a gas cavity

The stationary motion of a gas cavity in an ideal incompressible fluid is studied taking account of surface tension by using a variational equation. Approximate analytical dependences of the dimensionless parameters on the degree of cavity deformation are obtained. It is shown that the variational equation admits of an exact analytical solution. The stability of motion corresponding to the exact solution is proved relative to arbitrary perturbations in the cavity shape. A solution is given for the problem of stationary motion of an elliptical cavity in a gravity viscous fluid and the stability problem is investigated. Dependences are found for the velocity of cavity rise, the Reynolds number, and the Froude number as a function of the cavity size.     

Particle deposition on a channel wall in a turbulent disperse flow with gradient

Kinetic ideas about the motion of a set of particles (droplets) in a turbulent gas flow with gradient are used to derive a Fokker-Planck equation for the case of sufficiently large particles (more than few microns). This equation describes the process in which they are deposited on the wall of a channel. Satisfactory agreement has been obtained between the numerical solution to this equation for the deposition rate and the experimental data published in the literature. Under the assumption that the parameters of the carrier gaseous flow vary fairly slowly, a generalized equation is derived for particle diffusion in turbulent flow. This takes into account the intensity gradient of transverse pulsations in the velocity of the carrier gaseous flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 57–63, July–August, 1985.     

Slender axisymmetric cavities in a supersonic flow

Slender axisymmetric cavities in a subsonic flow of compressible fluid were investigated in [1–4]. In [5] a finite-difference method was used to calculate the drag coefficient of a circular cone, near which the shape of the cavity was determined for subsonic, transonic, and supersonic water flows; however, in the supersonic case the entire shape of the cavity was not determined. Here, on the basis of slender body theory an integrodifferential equation is obtained for the profile of the cavity in a supersonic flow. The dependence of the cavity elongation on the cavitation number and the Mach number is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–181, January–February, 1989.