Motion of a liquid-gas bubble mixture in the incipient cavitation stage

This article is concerned with the nonsteady motion of a mixture of an ideal incompressible liquid with gas bubbles in the incipient stage of cavitation. Solutions are obtained in the first and second approximations. The solutions are applied to pipe flow situations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 151–153, September–October, 1976.The author is indebted to L. I. Sedov for valuable comments.     

Motion of a liquid containing gas bubbles

In this paper a class of models of continuous media is constructed whose free energy depends on the density, the rate of change of density, and the temperature. Conditions at discontinuities in such models are found. A Kogarko model for a mixture of liquid with gas bubbles is obtained. Moreover, the propagation of small disturbances is investigated. In the third and fourth sections exact solutions are found for the problems of nonsteady and steady motion of a mixture of liquid and bubbles in a tube.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–116, November–December, 1971.The author thanks L. I. Sedov for guidance and valuable advice.     

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.     

New control laws for stabilization of a rigid body motion using rotors system

This paper presents a new class of globally asymptotic stabilizing control laws for dynamics and kinematics attitude motion of a rotating rigid body. The rigid body motion is controlled with the help of a rotor system with internal friction. The Lyapunov technique is used to prove the global asymptotic properties of the stabilizing control laws. The obtained control laws are given as functions of the angular velocity, Cayley–Rodrigues and Modified-Rodrigues parameters. It is shown that linearity and nonlinearity of the control laws depend not only upon the Lyapunov function structure but also the rotors friction. Moreover, some of the results are compared with these obtained in the literature by other methods. Numerical simulation is introduced.     

Effect of nonisothermal conditions of skin friction in subsonic turbulent channel flows

A number of different approaches are available in current literature to the theoretical determination of skin friction in nonisothermal flows in channels and pipes. Some of these (e.g., [1–3]) predict an appreciably stronger dependence of skin friction on temperature ratio =T_w/T (T_w is the wall temperature; T is the core fluid temperature), than those indicated by experimental results [4–7]. Significantly better agreement with existing experimental results is achieved in [8, 9] based on comprehensive numerical analysis of a system of integrodifferential equations. However, the assumptions they make are not always sound nor physically clear. Besides, the use of numerical methods does not allow the authors to relate their analysis to known limiting laws and simultaneously develop reliable numerical expressions to generalize experimental data. Physically quite clear results have been obtained in [10–12] and, in particular, very simple limiting laws for skin friction have been established. At the same time, it appears that based on the same physically clear assumptions, it is possible to obtain even more general results which agree well with experimental data. Simultaneously, these results which coincide with the limiting values at infinite Reynolds number make it possible to indicate the limits of applicability of these laws and extend them to finite Reynolds number range.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 69–76, January–February, 1984.     

Construction of a method for calculating a plane boundary layer in weak solutions of polymers with laminar,transitional, and turbulent flow zones

The investigation of the effect of small polymer additives on the characteristics of the flow of a viscous liquid is, at the present time, one of the most promising approaches to the lowering of the friction resistance. One interesting question in this connection is the study of the effect of small polymer additives on the characteristics of the transitional region of flow in a boundary layer, as well as on the value of the friction resistance with the presence of laminar, transitional, and turbulent sections in the boundary layer. The article sets forth a possible method for calculation of a plane boundary layer and the friction resistance for the case of the motion of a body in weak polymer solutions with a constant concentration, taking account of the change in the flow conditions in the layer and based on the use of integral relationships. Questions connected with the development of a boundary at a body, with the feeding of a polymer in it, as well as with the effect of degradation or destruction of the polymer in the solution, are not discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–48, May–June, 1977.     

Friction and heat transfer with simultaneous convection on a permeable surface

The velocity and heat transfer fields near a vertical permeable surface with simultaneous convection are investigated. A solution is found for the boundary layer equations with known laws of surface temperature and flow velocity change. The transformed boundary layer equations contain the parameter G/R², which determines the effect of free convection on friction and heat transfer for constrained motion. Calculations of friction and heat transfer as functions of draft (suction) with simultaneous convection are presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 96–100, March–April, 1973.     

Slow rotation of a sphere in a confined volume of a dilute gas

The axially symmetric motion of a gas in a volume confined between an external immobile surface of rotation and a coaxial surface of a rotating sphere is considered. A solution is obtained by the moment method based on the Boltzmann equation with a collision integral of Maxwellian molecules. The gas-velocity distribution and an expression for the friction torque exerted on the sphere are obtained for arbitrary Knudsen numbers and for an arbitrary shape of the outer surface. The proportionality of the gas slip velocity over the surface of the sphere to the friction strain is shown. The friction torque is investigated for specific shapes of the outer surface. The motion of a gas filling the space between concentric spheres, each of which rotates about an arbitrary axis, is treated. In the limiting case of small Knudsen numbers the expressions obtained are compared with the corresponding results for a continuous medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–124, July–August, 1978.The authors are grateful to the participants of the seminars guided by G. I. Petrov and A. M. Golovnyi for discussions concerning this work.     

Effect of the friction of a thrown piston on the internal ballistics of a one-stage unit

The present article gives a method for the direct measurement of the friction force of cylindrical samples strongly compressed along the axis at the wall of the bore of a gun with a small velocity of the motion, modelling the behavior of a piston in a ballistic unit. As an example, the article gives the results of measurements with a polymer material, i.e., caprolon. On the basis of an experimental investigation, a new friction model is constructed, which is used for the solution of the problem of the internal ballistics of a single stage powder unit with a piston made of caprolon. The laws of similarity of the processes under consideration are investigated. It is established that the value of the maximal pressure of the powder gases and velocity of the piston depend essentially on the value of the friction coefficient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 76–83, May–June, 1978.     

Shock waves in flow of an incompressible liquid in collapsing pipes: Application to large blood vessels

Flow of a liquid in collapsing pipes is of great interest for problems in the mechanics of blood circulation, since collapse can take place in many blood vessels. This effect forms the basis for a large number of diagnostic and therapeutic methods, and also for methods of investigating the system of blood circulation. Consequently the mechanics of collapsing pipes has been studied intensively of late [1], but the available studies are far from exhausting the theoretical or the applied aspects of the problem. This applies also to the study of discontinuous solutions such as shock waves which describe steep fronts of opening or narrowing of a blood vessel. The most studied phenomenon is unsteady flow caused by change in the external pressure [2]. There is an explanation in [3–6] of the effect on the process of formation of discontinuities in collapsing pipes due to such factors as friction on the wall, distributed lateral outflow, the presence of a stagnant zone in the flow, and viscoelasticity of the wall. The origin of some acoustic phenomena in the arteries is connected by some with the propagation of discontinuities; these phenomena include Korotkov sounds, used in the determination of the arterial pressure of blood [1, 7]. The present study considers quasione-dimensional flow of a viscous incompressible liquid in a collapsing pipe of finite length and made of a nonlinear viscoelastic material; there is a study of the conditions in which discontinuities arise in such systems, and an investigation of the structure of shock waves with allowance for the effect of the surrounding tissues.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 44–50, November–December, 1987.     

Experimental investigation of high-speed fluid friction

The experimental investigation of the laws of friction at high relative speeds involves difficulties associated with ensuring that the experimental apparatus provides a sufficiently long slide path. In this paper an automated experimental apparatus is described. This makes it possible to study in a compact fashion the laws of fluid friction at speeds of approximately 150 m/sec under widely variable load and external pressure. The experimental relations for the friction moment coefficient for water in the laminar and turbulent regimes are compared with calculations made by the integral relations method of boundary layer theory. The results of series of experimental and theoretical studies of high-speed friction against ice in the developed fusion layer regime are presented. Questions of the effectiveness of using polymer additives to reduce friction at high speeds are considered. The experimental method described can be used to investigate friction problems involving pastes, emulsions, and other rheologically complex continuous media, in which under conditions of high velocity gradients (strain rates) qualitatively new effects are to be expected.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 74–82, May–June, 1994.     

Turbulent mixing of accompanying and opposing flows

The solution of the problem of the edge of a turbulent jet, obtained in [1] starting from a model of the vortical motion of an ideal liquid, is generalized for the case of the turbulent mixing of two plane semibounded (accompanying or opposing) flows of an incompressible liquid. It is shown that the results of calculation are in qualitative agreement with experimental data and that they are close to them quantitatively. Some of the special characteristics of the method are discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 92–101, November–December, 1973.     

Immersion of a floating deformed body

A study is made of the plane-eddyless motion of an ideal incompressible liquid with the vertical uniformly accelerated immersion of a body, whose bottom is deformed in accordance with a given law. The method of solution of [1] is used. Dependences are given for calculating the pressures, the forces of the resistance to the immersion, and the forms of the free surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 136–137, September–October, 1974.     

Impact of a strip of compressible liquid on a barrier

The exact solution of the plane problem of the impact of a finite liquid strip on a rigid barrier is obtained in the linearized formulation. The velocity components, the pressure and other elements of the flow are determined by means of a velocity potential that satisfies a two-dimensional wave equation. The final expressions for them are given in terms of elementary functions that clearly reflect the wave nature of the motion. The exact solution has been thoroughly analyzed in numerous particular cases. It is shown directly that in the limit the solution of the wave problem tends to the solution of the analogous problem of the impact of an incompressible strip obtained in [1]. A logarithmic singularity of the velocity parallel to the barrier in the corner of the strip is identified. A one-dimensional model of the motion, which describes the behavior of the compressible liquid in a thin layer on impact and makes it possible to obtain a simple solution averaging the exact wave solution, is proposed. Inefficient series solutions are refined and certain numerical data on the impact characteristics for a semi-infinite compressible liquid strip, previously considered in [2–4] in connection with the study of the earthquake resistance of a dam retaining water in a semi-infinite basin, are improved. The solution obtained can be used to estimate the forces involved in the collision of solids and liquids. It would appear to be useful for developing correct and reliable numerical methods of solving the nonlinear problems of fluid impact on solids often examined in the literature [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 138–145, November–December, 1990.The results were obtained by the author under the scientific supervision of B. M. Malyshev (deceased).     

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.     

Nonsteady supersonic flow over spiked bodies

In longitudinal supersonic flow over spiked cylinders nonsteady regimes can occur in which a separation zone is periodically generated at the spike, grows vigorously in size, and then vanishes. Several authors [1–6] have investigated the physical pattern of flow with separation zone fluctuations (using shadowgraphs) and have determined the boundaries of existence of the nonsteady regime as a function of the ratio between the spike length and diameter of the cylinder. The authors, however, did not systematically study the dependence of the pulsation frequency on the freestream Mach and Reynolds numbers or on the relative diameter and tip angle of the spike. We have undertaken such an investigation. We are concerned primarily with the influence of the dimensionless parameters on the Strouhal number Sh of the separation zone pulsations at a spike attached to the front of a flat-ended cylinder. Earlier investigations [4–6] have been carried out using motion pictures with film speeds up to 32·10³ frames/sec. In the present study we used high-speed motion pictures with a speed of 6.25· 10⁵ frames/sec. This speed allowed us to determine the precise sequence of phases of the pulsations and their relative durations, as well as the speed at which the boundaries of the separation zone move.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 118–124, September–October, 1976.     

Gas flow from a finite volume through a laval nozzle

Self-similar nonsteady flow in a Laval nozzle is considered; the flow is established when an ideal gas issues from a volume into a space at a sufficiently small pressure. The flow in the nozzle is assumed to be one-dimensional. Qualitative conclusions are formed on the effect of the nonsteady flow conditions on the distribution of M. For the case when the nonsteady properties have little effect, an asymptotic solution is obtained in quadratures and an example of a calculation is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 71–76, March–April, 1978.     

Comparison of the drag coefficients of bodies moving in liquids with various stratification profiles

For bodies moving in liquids with various stratification profiles, the relation between the drag coefficients considered as functions of the Froude number is investigated. The problems of stratified liquid dynamics have not previously been studied from this viewpoint, either experimentally or theoretically. On the range of the Froude numbers F₁-1, the drag force coefficients obtained from bench measurements of the towing resistance to the uniform horizontal motion of models in two-layer and continuously stratified liquids are compared. The experimental data obtained in a thermocline are then compared with the results of [1,2] for linearly stratified and two-layer liquids.Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 4–11, July–August, 1996.     

The effect of polymer additives on the components of the energy balance of a turbulent flow

On the basis of the experimental data obtained, an analysis is made of the effect of polymer additives directly on the generation of turbulent energy, on the dissipation of the energy of the averaged motion, and on the density of the flux of the kinetic energy of the turbulence. The presence of polymer additives in the turbulent flow significantly changes the relationship between the generation of the turbulent energy and the dissipation of the energy of the averaged motion. Under the action of polymer additives, the density of the flux of kinetic energy decreases over the depth of the channel, which, in turn, brings about a decrease in the influx of energy from the averaged motion to the pulsed motion. The following definitions are adopted below: the x₁ axis of a Cartesian system of coordinates coincides with the horizontal axis of symmetry of the channel and with the direction of the averaged motion of the liquid; the x₂ axis is directed upward; the x₃ axis is perpendicular to the lateral wall of the channel; the origin of coordinates, O, coincides with the lower plane (bottom) of the channel. Further, U₀ is the mean velocity of the flow of liquid in the channel; U_i is the local component of the averaged velocity (i=1, 2, 3); H is the height of the channel; z=2x₂/H; Re is the Reynolds number of the averaged flow;v is the coefficient of kinematic viscosity; u_i is the pulsation component of the velocity (i=1, 2, 3); u_* is rate of dynamic friction; A=(₀–_p0 ^–1 is the coefficient of the lowering of the friction resistance with the flow of polymer solutions; ₀ and _p are the coefficients of the friction resistance with the motion of water and polymer solutions in a channel, respectively; c is the weight concentration of the polymer solution (%); ₀ is the friction stress at the wall; U₊ is the velocity of the flow at the axis of the channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 53–58, January–February, 1973.