Construction of optimal laws of variation of the angular momentum vector of a rigid body

We consider the problem of constructing optimal preset laws of variation of the angular momentum vector of a rigid body taking the body from an arbitrary initial angular position to the required terminal angular position in a given time. We minimize an integral quadratic performance functional whose integrand is a weighted sum of squared projections of the angular momentum vector of the rigid body. We use the Pontryagin maximum principle to derive necessary optimality conditions. In the case of a spherically symmetric rigid body, the problem has a well-known analytic solution. In the case where the body has a dynamic symmetry axis, the obtained boundary value optimization problem is reduced to a system of two nonlinear algebraic equations. For a rigid body with an arbitrarymass distribution, optimal control laws are obtained in the form of elliptic functions. We discuss the laws of controlled motion and applications of the constructed preset laws in systems of attitude control by external control torques or rotating flywheels.     

Effect of a spectrum of relaxation times on the capillary thinning of a filament of elastic liquid

The capillary thinning of a filament of viscoelastic liquid, which is the basis of a microrheometer, is analyzed in terms of a multi-mode FENE fluid. After a short time of viscous adjustment, the stress becomes dominated by the elastic contribution and the strain-rate takes on a value equal to two-thirds the rate at which the stress would relax at fixed strain. This strain-rate decreases as the dominant mode changes. At late times, modes reach their finite extension limit. The fluid then behaves like a suspension of rigid rods with a large extensional viscosity, and the liquid filament breaks. Predictions are compared with the experiments of Liang and Mackley (1994).     

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.     

Draft of a nozzle with the circulating outflow of a vortical flow of gas

The presence of circulation in an outflowing gas leads to a change in the working parameters of a nozzle. The question of the mass flow rate and the draft of a nozzle without a diffusor (a point) for twisted flows has been studied theoretically and experimentally [1–6]. The use of nozzles with a supersonic part introduces a considerable degree of complication into the method for the analytical calculation of the draft characteristics and the program for their experimental investigation. In [2, 7], a theory of a nozzle is formulated for a model of a potential circulating flow of gas; in [5, 8], an electronic computer was used to solve the complete system of the equations of gasdynamics for the motion of a rotating flow along a nozzle; in [7, 9], an investigation was made of a variational problem of the shaping of a diffusor for a circulation flow. The calculation of the draft, carried out in the above-mentioned communications (with the exception of [2], in which a study was made of a partial model of an eddyless rotational motion), is bound up with labor-consuming computer calculations. In the present article, in a development of [3, 6], a quasi-one-dimensional theory of a supersonic nozzle for a vortical flow of gas is formulated and verified experimentally.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–149, September–October, 1975.     

Topology of Foliations and Decomposition of Stochastic Flows of Diffeomorphisms

Let M be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno et al. (Stoch Dyn 13(4):1350009, 2013) it is shown that, up to a stopping time \(\tau \), a stochastic flow of local diffeomorphisms \(\varphi _t\) in M can be written as a Markovian process in the subgroup of diffeomorphisms which preserve the horizontal foliation composed with a process in the subgroup of diffeomorphisms which preserve the vertical foliation. Here, we discuss topological aspects of this decomposition. The main result guarantees the global decomposition of a flow if it preserves the orientation of a transversely orientable foliation. In the last section, we present an Itô-Liouville formula for subdeterminants of linearised flows. We use this formula to obtain sufficient conditions for the existence of the decomposition for all \(t\ge 0\).     

Effect of thickness of film of fuel on the rate of heterogeneous (gas-film) detonation

The article discusses the process of detonation in a tube filled with a gaseous oxidizer (oxygen), over whose whole perimeter there is deposited a film of fuel (a hydrocarbon with a low vapor pressure). It is assumed that the rate of heat evolution is determined by the diffusional (noninstantaneous) mixing process of the vaporizing fuel and the oxidizer. The dependence of the detonation rate on the thickness of the film of deposited fuel is obtained.     

Dynamics of strings made of phase-transforming materials

This paper presents a theory to describe the dynamical behavior of a string made of a phase-transforming material like a shape-memory alloy. The study of phase boundaries, the driving force acting on them and the kinetic relation governing their propagation is of central concern. The paper proposes a qualitative experimental test of the notion of a kinetic relation, as well as a simple experimental method for measuring it quantitatively. It presents a numerical method for studying general initial and boundary value problems in strings, and concludes by exploring the use of phase transforming strings to generate motion at very small scales.     

Physics of the formation of fiber lightguides

A fiber lightguide is a fine continuous or tubular transparent filament. Fiber lightguides are formed from the liquid mass exuded through a dye or drawn from a suitable blank. Both of these processes can be considered using the equations of the hydrodynamics of an incompressible Newtonian liquid. (Polymers, which are not Newtonian liquids, are not considered here.) The drawing of a continuous glass fiber from a dye is considered in [1]. The drawing of a microcapillary from a dye is considered in [2], where a qualitative consideration is given which is insufficient for an understanding of the effect of different parameters of the process on the dimensions of the drawn microcapillary. In this paper we consider the formation of a microcapillary from a tubular blank using the approximation of an incompressible Newtonian liquid with a variable viscosity determined by the given temperature distribution. The effect of surface tension and of the excess pressure produced in the channel to counteract the surface tension are taken into account. It is assumed that the drawing process is steady, the blank has thin walls and is axisymmetrical, and the transition to a microcapillary occurs smoothly. With these assumptions the problem of obtaining the shape of the transition and the dimensions of the microcapillary obtained is reduced to a system of ordinary differential equations. The dependence of the dimensions of the microcapillary on the dimensions of the blank and the parameters of the process is established, thereby enabling the process to be optimized.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 167–174, March–April, 1976.The authors thank B. Z. Katsenelenbaum and A. D. Shatrov for useful discussions, A. I. Leonov for a number of observations, and I. V. Aleksandrov, T. V. Bukhtiarov, and A. A. Dyachenko for discussing the results at various stages.     

Behavior of perturbations of solitary and periodic waves on the surface of a heavy liquid

In [1] a system of equations was obtained for the case of a potential motion of an ideal incompressible homogeneous fluid; the system described the propagation of a train of waves in a medium with slowly varying properties, the motion in the train being characterized by a wave vector and a frequency. A solitary wave is a particular case of a wave train in which the length of the waves in the train is large. In [2, 3] a quasilinear system of partial differential equations was obtained which described two-dimensional and three-dimensional motion of a solitary wave in a layer of liquid of variable depth. It follows from this system that if the unperturbed state of the liquid is the quiescent state, then some integral quantity (the average wave energy [2–4]), referred to an element of the front, is preserved during the course of the motion. This fact is also valid for a train of waves, and can be demonstrated to be so upon applying the formalism of [1] to a Lagrangian similar to that used in [2]. In the present paper we obtain, for the case of a layer of liquid of constant depth, a solution in the form of simple waves for a system, equivalent to the system obtained in [3], describing the motion of a solitary wave and also the motion of a train of waves. We show that it is possible to have tilting of simple waves, leading in the case considered here to the formation of corner points on the wave front. We consider several examples of initial perturbations, and we obtain their asymptotics as t→∞. We make our presentation for the solitary wave case; however, in view of our statement above, the results automatically carry over to the case of a train of waves.     

On the comparison of two different solutions in the form of series of the governing equation of an unsteady flow of a second grade fluid

In this paper, two different solutions in the form of series of the governing equation of unsteady flow of a second grade fluid are considered. These are series expansions with respect to inverse power of time and a perturbation expansion. Two illustrative examples are given. One of them is the unsteady flow of a second grade fluid over a plane wall suddenly set in motion and the other is the diffusion of a line vortex in a fluid of second grade. It is a remarkable fact that the expression of the series expansion with respect to inverse power of time is exactly in the same form as that of the perturbation expansion. Thus, it is possible to replace a series expansion with respect to inverse power of time with a perturbation expansion.     

Regimes of sedimentation of a consolidated system of solid spherical particles

The results of the experimental investigation of gravitational sedimentation of a consolidated system of solid monodisperse spherical particles in a viscous liquid are represented over wide ranges of the particle number density and the Reynolds and Stokes numbers. Empirical dependences of the velocity of sedimentation of a particle aggregate and the drag coefficient of a system of particles as functions of the initial volume number density are obtained. The boundary values of the particle number density separating the sedimentation regimes are determined.     

Estimate of the applicability limits of a linear theory of internal waves

Using a perturbation method, the applicability limits of a linear theory of internal gravity waves are estimated. It is shown that over a wide range of wavelengths, typical of a real ocean, in studying the dynamics of internal gravity waves it is possible to use a linear approximation, which confirms the validity and adequacy of this approximation for the corresponding spatial and temporal scales of the linear model of wave dynamics.     

Saturation of absorption of the powerful radiation of a system of anharmonic oscillators

Recently, the theory of nonequilibrium systems simulated by a set of anharmonic oscillators has received significant development. The investigation of such kinds of systems is especially important in the study of problems associated with the stimulation of chemical reactions and the development of effective molecular lasers. The systematic analysis of the kinetics of anharmonic oscillators assumes the simultaneous solution of a large number of nonlinear equations describing the population balance of the vibrational levels. Realization of this approach is associated with cumbersome numerical calculations and does not permit obtaining a qualitative picture of the behavior of the system as a function of the different parameters (pressure, temperature, etc.). An approximate analytical theory has been formulated in [1, 2] which permits finding the distribution function over the vibrational states with the effects of anharmonicity taken into account. We will employ the approach developed in these papers to describe a system of anharmonic oscillators under conditions of powerful optical pumping. This problem was discussed in [3], where it was found that such a system changes into a saturation mode in the case of high pumping levels. The existence of this mode is explained by the fact that the maximum rate of energy input into a vibrational degree of freedom is determined by the rate of distribution of this energy over all the vibrational levels, i.e., by the constant of V—V-exchange. For sufficiently large pumpings the approximation of the Boltzmann distribution function adopted in [3] in connection with the calculation of the saturation parameters is too crude. The goal of this paper is to derive in explicit form expressions for the vibrational energy supply, the absorbed power, and so on, under saturation conditions without the use of the approximation indicated above [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 10–15, September–October, 1978.     

Applicability of a model of the formation of a molecular beam under conditions of translational disequilibrium

In the model of the formation of a molecular beam proposed in [1], Kantrowitz and Grey assumed ideal conditions, namely, the gas stream expands in accordance with an isentropic law, the introduction of the skimmer into the gas stream does not disturb the flow upstream, and the entrance section of the skimmer is the surface of the last collisions. In reality, these assumptions are not always satisfied. In the case of rapid expansion of a supersonic stream of rarefied gas there is a departure from isentropic behavior because of the freezing of the relaxation processes, and the formation of a molecular beam by means of a skimmer changes the gas-dynamic parameters [2, 3]. The aim of the present work was to make a direct experimental verification of the applicability of the model of molecular beam formation from a supersonic stream of rarefied gas under conditions of translational disequilibrium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 170–173, May–June, 1980.     

Experiments of free vibration of shells of revolution

This paper presents experimental techniques used in and results obtained during a series of experiments performed to investigate the free-vibration behavior of 15-in. base-diameter spherical and paraboloidal shell models of various geometric parameters and boundary conditions. The models were manufactured from polyvinyl-chloride (P.V.C.) sheets by a thermo-vacuum process using special molds and templates. To provide the necessary excitation power, a variable control amplitude/frequency audio oscillator was used to drive a loudspeaker which was mounted underneath the models. A recently developed noncontact fiber-optics instrument, referred to as the “Fotonic Sensor”, was employed in determining the motion of the models. Thirty-two models of different configurations and boundary conditions were tested in the series. The natural frequencies were observed for all models, while the mode shapes associated with the first four-to-six resonant frequencies were mapped only for some of the models, using special scanning devices.