Dynamic motion of a conical frustum over a rough horizontal plane

An analytical and numerical study of the dynamic motion of a conical frustum over a planar surface is presented resulting to a non-linear system of ordinary differential equations. Wobbling and rocking components of motion are discussed in detail concluding that, in general, the former component dominates the latter. For small inclination angles an asymptotic approximation of the angular velocities is possible, revealing the main characteristics of wobbling motion and its differences from rocking. Connection is made of the analysis with the behavior of the ancient classical columns, whose three dimensional dynamic response challenges the accuracy of the two dimensional models, usually applied in practice. The consideration of such discrete-blocky systems can benefit from the present study, through qualitative results and benchmarks for more complicated numerical methods, like the Distinct Element Method.     

Stability of motion of the plane boundary separating two fluids

The effect of viscosity on the disintegration of liquid jets can be considered in two ways. First, viscous forces alter the basic flow: they form a boundary layer whose presence necessarily alters wave formation. Second, viscous forces can have a direct effect on the development of perturbations for a given velocity profile of the basic flow. In this case the study of stability must be based on the Navier-Stokes equations instead of the equations of an ideal fluid. This complicates analysis considerably. Available data [1] indicate that this influence is very minor in the case of moderately viscous fluids. It appears, therefore, that the principal role is played by changes in the velocity profile alone, and that the behavior of the perturbations is described by the equations of an ideal fluid.In the present study we investigated the stability of motion and wave formation at the boundary between two fluids in order to determine the effect of viscosity on the drop formation mechanism. The simplest case of oscillation of the boundary is chosen in order to keep the analysis as simple as possible.     

Spreading of a paraboloid elevation of liquid over a horizontal base

The literature contains studies [1–4] of the problem of the spreading of an axisymmetric elevation of ground water with conservation of the initial mass of the liquid and under the condition that some of the liquid remains in the previously occupied volume. The investigations used the Boussinesq equation with constant and discontinuous (at the point where ?h/?t = O, where h is the height of the elevation) permeability of the medium. For the first problem, there is an exact analytic solution of the type of an instantaneous source; the solution to the second problem was sought in the form of a self-similar solution of the second kind as an asymptotic solution for the corresponding Cauchy problem. In the present paper, the solution to the problem of the spreading of an axisymmetric elevation over a horizontal base is generalized to the case of an elevation having the shape of an elliptic paraboloid.     

Stability and stabilization of motion in the case of a single-wheel robot

A model of a single-wheel robot is considered. Several particular cases of its steady motions are analyzed. The stability and controllability of the robot is studied in a neighborhood of the steady motion.     

Stability of unsteady rotational motion of a plane layer of ideal fluid with free boundaries

The exact solution of the equations of an ideal incompressible fluid describing the unsteady rotational motion of a plane layer with free boundaries is obtained. For constant vorticity the stability problem is studied in the linear approximation. The asymptotic behavior of the free boundaries of the layer as t is calculated. It is shown that the vorticity of the basic motion stabilizes the boundaries of the layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–21, March–April, 1986.     

Stability analysis of viscoelastic thin shallow hyperbolic paraboloid shells

The stability of linearly viscoelastic flexible shallow hyperbolic paraboloid shell is analysed under transverse load. Allowances are made for geometrical nonlinearity and initial imperfections of the surface shape. By application of the method of finite differences with respect to geometrical variables and the method of differentiation with respect to a parameter (time) the solution for the system of equilibrium non-linear integro-differential equations is reduced to Cauchys problem which can be solved numerically. The critical time was shown to depend on the load, curvature, initial imperfections and edge elements compressibility. Critical loads for an outlying time moment are determined.     

Rotation of a paraboloid in a conducting fluid

An exact solution is found for a rotating paraboloid of revolution in a viscous conducting fluid, and in the presence of a uniform aligned magnetic field.     

Stability of thermocapillary motion in a cylindrical layer

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 50–52, July–August, 1989.     

Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1–4].     

Stability of a plane ionizing shock in a longitudinal magnetic field

The influence of a permanent transverse magnetic field on the stability of a plane shock relative to small displacements of its front from the equilibrium position was examined in [1, 2]. Under the same simplifying assumptions, the stability of a shock in a longitudinal magnetic field (the induction vector is directed along the normal to the discontinuity) is investigated in this paper. The boundaries of the stability domain are determined. It is shown that the whole domain of neutral oscillations which exist in the gasdynamic case makes the transition into the stability domain in the presence of a longitudinal field. The boundaries of the stability domain are independent of the interaction parameter in contrast to the case of shock motion in a transverse field [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 195–198, March–April, 1976.     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

The energy balance in turbulent motion in a plane curvilinear channel

We determine the component energy equations for mean and pulsating motion in a plane curvilinear channel. We discuss the characteristics of the turbulence in curvilinear channels by comparison with those of rectilinear flows.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5,pp. 150–153, September–October, 1970.     

Thermocapillary motion of a two-bubble cluster near a plane solid wall

The thermocapillary motion of two bubbles near a plane solid wall at uniform temperature is investigated by solving five boundary integral-equations. Preliminary computations show that wall–bubble interactions dictate the migration of equal bubbles with line of centers parallel to the wall. To cite this article: A. Sellier, C. R. Mecanique 333 (2005).