Nonlinear wave motions on liquid surfaces

A study is made of the formation of a shock wave (bore), produced by the movement of an initially weak discontinuity in the spatial derivatives of velocity and liquid depth in an area of stationary current in a channel of constant inclination. The formation of shock waves from compression waves was first studied by Riman [1]. Frictional resistance was considered in the Chezy form. The equations obtained therein for determination of the moment in time and spatial coordinates of the point at which the shock wave is formed, as well as the laws for propagation of shock waves are applicable to the problem of one-dimensional transient motion in a gas, the pressure of which is dependent on density. Instantaneous collapse of waves, as well as formation and movement of bores in rivers for an idealized flow model in a channel with horizontal bottom, neglecting friction, were described by Khristianovich, Mikhlin, and Devison [2], and Stoker [3]. Recently in the work of Sachdev and Bhatnagar [4], using numerical integration of the equation for bore intensity, the problem of shock wave propagation in a channel of constant inclination with consideration of fluid resistance in the Chezy form was studied. Gradual wave collapse and the bore formation mechanism were studied by Stoker [3] on the basis of the shallow-water theory. Neglecting friction on the horizontal channel bottom, he calculated the moment of time and coordinates of the point at which the shock wave is formed in the case where the initial disturbance is sinusoidal. The dependence of these values on wave amplitude for a channel of constant inclination was obtained by Jeffrey [5], who also neglected friction on the channel bottom and considered the initial disturbance to be sinusoidal. Lighthill and Whitham [6] discovered that for Froude numbers greater than two, the linear theory led to unlimited growth in the intensity of the flood wave. We note that the studies of flood-wave motion in the region of the first characteristic, performed in [3, 6], differ only in the forms of the resistance laws and dependences of the unknown functions on the variables. Physical peculiarities of various liquid wave motions were also examined by Lighthill in [7].Saratov. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 62–66, March–April, 1972.     

On the shape of a slender axisymmetric cavity in a ponderable liquid

A solution is obtained to the equation for the shape of a slender axisymmetric cavity in a heavy liquid. The minimal cavitation numbers are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 133–136, November–December, 1979.I thank E. N. Kapankin, V. P. Karlikov, and Yu. L. Yakimov for interesting and helpful discussions of the work.     

Numerical simulation of the dynamics of a liquid in a moving axisymmetric cavity

The problem of the motion of an ideal liquid with a free surface in a cavity within a rigid body has been most fully studied in the linear formulation [1, 2]. In the nonlinear formulation, the problem has been solved by the small-parameter method [3] and numerically [4–7]. However, the limitations inherent in these methods make it impossible to take into account simultaneously the large magnitude and the threedimensional nature of the displacements of the liquid in the moving cavity. In the present paper, a numerical method is proposed for calculating such liquid motions. The results of numerical calculations for spherical and cylindrical cavities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–177, March–April, 1984.     

Stability of axisymmetric motions in a rotating inviscid atmosphere

Summary We study the set of ordinary differential equations representing the truncation of the spectral vorticity equation in spherical geometry for two-dimensional, non divergent motions in a homogeneous, inviscid rotating atmosphere. The set of included spectral components represent an arbitrary axisymmetric field of motion (zonal flow) interacting with a couple of non zonal, wave-like perturbations. We analize the stability of the equilibrium configuration described by purely zonal flow, by an esplicit representation of the linear and non-linear parameters which characterize the unstable oscillations induced by perturbations of the equilibrium.

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Sommario Si studiano le soluzioni di un sistema di equazioni differenziali ordinarie che rappresentano il troncamento a poche componenti di Fourier dell'equazione di conservazione della vorticità per moti bidimensionali, non divergenti in una atmosfera rotante, inviscida e omogenea. Tale troncamento include tutte le componenti spettrali che rappresentano flussi a simmetria assiale (moti zonali) in interazione non lineare con una coppia di componenti asimmetriche che rappresentano flussi ondulatori. Si studia la stabilità della configurazione di equilibrio caratterizzata da moto zonale stazionario, analizzando i parametri lineari e non lineari che descrivono le oscillazioni instabili di ampiezza finita del campo di flusso che nascono da condizioni iniziali nell'intorno dell'equilibrio.

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This work was supported by the C.N.R. through the Gruppo Nazionale per la Fisica Matematica.     

Developed steady wave motions of a liquid film falling along a vertical plane

The falling of a thin viscous fluid layer (film) along a vertical plane under the effect of gravity is accompanied by wave motions in which capillary forces play an essential part. An equation for the film thickness h(x, t) is used extensively in analyses of these motions. This equation, obtained from the Navier—Stokes equations and the boundary conditions under different assumptions, reduces to an ordinary third-order nonlinear differential equation [1–7] for steady plane motions. Periodic solutions of this equation were sought by the methods of asymptotic expansions in the amplitude or by Fourier series expansions [1–7], which assumes a sequential accounting of the nonlinearity as a small perturbation. This limits the validity of the results obtained to the domain of small amplitudes. The case of arbitrary amplitudes is considered in this paper. A solution of the problem, based on an asymptotic expansion in the parameter ε is constructed. In this expansion the equation for the first approximation remains nonlinear but admits of integration, which discloses the class of bounded periodic solutions. Moreover, strict integral relations (for any ε) are obtained, and a variational problem about seeking the lower bound of values of the mean film thickness and other characteristics of the ultimately developed optimal motions is formulated and solved on their basis. The results obtained agree with experiments.     

Diffraction of a plane acoustic wave by a rigid sphere in a cylindrical cavity: An axisymmetric problem

The paper studies the interaction of a rigid spherical body and a cylindrical cavity filled with an ideal compressible fluid in which a plane acoustic wave of unit amplitude propagates. The solution is based on the possibility of transforming partial solutions of the Helmholtz equation between cylindrical and spherical coordinates. Satisfying the interface conditions between the cavity and the acoustic medium and the boundary conditions on the spherical surface yields an infinite system of algebraic equations with indefinite integrals of cylindrical functions as coefficients. This system of equations is solved by reduction. The behavior of the system is studied depending on the frequency of the plane wave     

Three-dimensional problem of unsteady wave motions of a liquid in a region of variable depth

The three-dimensional problem of unsteady wave motions of a liquid above a plane inclined floor in the framework of a linear dispersion model was considered for the first time in [1] for the particular case =/4, where is the angle of inclination of the floor plane to the free surface of the liquid. The class of exact self-similar solutions of the problem for =/2(2m + 1), m=0, 1, 2,..., for the case of an initial perturbation of a free surface of a special type which is constant in the direction of the normal to the shoreline was found in paper [2]. The present paper is devoted to the investigation of wave motions of a liquid due to an initial perturbation of arbitrary form for the angles of inclination of the floor assumed in [2]. A complete system of eigenfunctions corresponding to the continuous and discrete spectra is found. The theorem of the expansion of an arbitrary absolutely integrable function with respect to the boundary values of the eigenfunctions is proved. An exact solution of the problem is obtained and its asymptotic analysis is carried out.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 104–112, March–April, 1986.     

Supercritical convective motions in a cubic cavity

The article reports the results of an experimental investigation of the convective instability of air in a cubic cavity with a height of 4 cm, at whose horizontal boundaries there was set up a homogeneous distribution of the temperature, while, at the vertical boundaries, the temperature varied linearly from a maximal value at the lower horizontal boundary to a minimal value at the upper boundary. Differential thermocouples were installed inside the cavity, whose readings were used to record the development of convection and to determine the form of the convective motion and its intensity. A study is made of the effect of small angular deviations of the model from the position with which mechanical equilibrium is possible in the model.     

Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel

In this paper, a power series and Fourier series approach is used to solve the governing equations of motion in an elastic axisymmetric vessel with the assumption that the fluid is incompressible and Newtonian in a laminar flow. We obtain solutions for the wave speed and attenuation coefficient, analytically where possible, and show how these differ under a number of different conditions. Viscosity is found to reduce the wave speed from that predicted by linear wave theory and the nonlinear terms to increase the wave speed in comparison to the linear solution. For vessels with a wall stiffness in the arterial range, the reduction in the wave speed due to the viscous terms is approximately 10% and the increase due to the nonlinear terms is approximately 5%. This difference between the linear and nonlinear wave speeds was found to be largely constant irrespective of the number of terms considered in the power series for the velocity profile. The linear wave speed was found to vary weakly with stiffness, whilst the nonlinear wave speed was found to vary significantly with the stiffness, especially at low values of stiffness. The 10% variation in the wave speed due to the viscous terms was found to be constant with wall stiffness whilst the 5% variation due to the nonlinear terms was found to vary with wall stiffness. The importance of the number of terms considered in the power series is discussed showing that only a relatively small number is required in the viscous case to obtain accurate results.     

Slow motions of a liquid drop in a viscous liquid

The problem of the fully established slow translational motion of a round drop (bubble) in a viscous liquid was solved by Adamar and Rybchinskii [1, 2]. The results of experimental measurements are rarely in agreement with the Adamar-Rybchinskii formula. This is connected with braking of the flow due to surface-active impurities, which are usually rather numerous in liquids. Nevertheless, we shall consider the problem of the not fully established motion of a drop in the simplest case, assuming that there are no surface-active substances. The article discusses problems of the vibrations and motions of a spherical drop in a viscous liquid, with arbitrary accelerations. An analysis is made of a formula for the force of resistance of a drop of liquid with a high viscosity, an elastoviscous drop, and a particle with slipping-through.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 32–37, November–December, 1975.     

Motion of a liquid droplet in a vibrating fluid

The motion of a liquid droplet in a liquid with density different from that of the liquid composing the droplet and subjected to harmonic excitations is investigated. Nonlinear equations are obtained that describe the translational motion of the droplet and its oscillations. It is shown by numerical means that, under the essential resonance conditions, the droplet ascends by a cascade method if the value of the load coefficient is small.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 78–83, July, 1995.     

Influence of the boundary of a column of incompressible liquid in investigating axisymmetric oscillations of a solid sphere in a cavity

Axisymmetric oscillations of a rigid spherical body in a column of ideal incompressible liquid with a plane boundary in the form of a free liquid surface or a rigid wall within a round cylindrical cavity are considered. The potential and pressure fields are plotted; expressions are obtained for the kinetic energy of the system and the hydrodynamic forces acting on the body. The resistance of the liquid to accelerated movement of the body is determined as a function of the distance to the boundary, for various parameter values. For specified oscillations of the body, the results obtained for axisymmetric conditions in a halfspace are compared with those obtained in an infinite cylindrical cavity. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 11–18, December, 1999.     

Impact of a spherical rigid body on the surface of a cavity in a compressible liquid: An axisymmetric problem

The shock interaction of a spherical rigid body with a spherical cavity is studied. This nonstationary mixed boundary-value problem with an unknown boundary is reduced to an infinite system of linear Volterra equations of the second kind and the differential equation of motion of the body. The hydrodynamic and kinematic characteristics of the process are obtained __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 11–19, January 2008.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].