Spin-down of a fluid between infinite cones

This study is concerned with the spin-down of a fluid between stationary cones. It follows on from [7], where solutions were obtained for a fluid spinning down between two infinite disks and where it was shown that under various initial conditions the dependence of the velocity on radius and time tends to a universal Kármán stage. In the case of cones the analogous universal stage is not of the Kármán type, which makes possible an experimental check of the applicability of the self-similar boundary layer equations generalizing the Karman equations previously considered in [11–13]. The experiments confirm the conclusions of the theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 37–44, July–August, 1986.In conclusion, the authors wish to thank A. M. Obukhov and F. V. Dolzhanskii for formulating the problem and constructive discussions.     

An unsteady-state thermal boundary layer on a flat isothermal plate

The Kármán-Polhausen integral method is used to investigate the problem of an unsteady-state thermal boundary layer on an isothermal plate with a stepwise change in the conditions of flow around the plate; analytical expressions are obtained for the thickness of the thermal boundary layer. A dependence is found for the rate of movement of the boundary between the steady-state and unsteady-state regions of the solution on the Prandtl number. A similar problem was solved in [1, 2] for a dynamic layer, Goodman [3] discusses the more partial problem of an unsteady-state thermal boundary layer under steady-state flow conditions. Rozenshtok [4] considers the problem in an adequate statement but, unfortunately, he permitted errors of principle to enter into the writing of the system of characteristic equations; this led to absolutely invalid results. In an evaluation of the advantages and shortcomings of the integral method under consideration, given in [4], it must only be added that the method is applicable to problems in which the initial conditions differ from zero since, in this case, approximation of the velocity and temperature profiles by polynomials is not admissible.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 64–69, July–August, 1970.     

Self-similar reverse flows in the region of separation of a turbulent boundary layer

The Kármán momentum equation and very simple scaling arguments relating to the profiles of the turbulent tangential stresses are used to construct a self-similar reverse flow of an incompressible fluid that, depending on the values of the empirical constants occurring in it, can be realized either behind the separation point of a turbulent boundary layer or in front of its point of reattachment. The empirical constants are determined by means of several independent experimental studies on turbulent separating flows of liquids and gases at subsonic and supersonic velocities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. A, pp. 173–177, July–August, 1988.     

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Effect of surface tension on the free outflow of a wetting fluid from a horizontal tube

The flow structure for which the Kármán hypothesis is valid is explored experimentally and theoretically. It is established that there exists not a point but a region of finite size, adjacent to the upper generator of the tube and including the moving line of contact between the phase interface and the wall, in which the fluid is stationary relative to the line of contact, i.e., in which the no-slip condition is not satisfied. The dimensions of the region depend on the surface tension. The action of this stagnant zone on the flow is fully explained by the effect of the surface tension in experiments [1]. It is established that depending on the ratio of the tube diameter to the dimension of the stagnant zone two flow regimes are possible: in sufficiently wide tubes, an inertial regime for which Kármán's hypothesis holds and the no-slip condition is not satisfied, and, in sufficiently narrow tubes, a creep regime in which the no-slip condition continues to apply. The values of the determining dimensionless parameter corresponding to the change of regime and the cessation of flow are calculated. They are similar to the experimental values.deceasedTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 85–93, March–April, 1988.     

Linear theory of supersonic flow past ruled bodies of three-dimensional configuration

The problem of flow past bodies with a ruled surface connecting an n-ray star in the initial section with a circular midsection is solved in a linear formulation with the use of the equation for resistance obtained earlier by the author [1]. Bodies of the investigated class have a substantial advantage in comparison with an equivalent cone and, for not very slender bodies, also in comparison with Kármán's ogival-shaped body. The resistance of the investigated bodies depends little on n.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–101, July–August, 1978.     

Complex Parametric Vibrations of Flexible Rectangular Plates

In this paper we consider parametric oscillations of flexible plates within the model of von Kármán equations. First we propose the general iterational method to find solutions to even more general problem governed by the von Kármán–Vlasov–Mushtari equations. In the language of physics the found solutions define stress–strain state of flexible shallow shell with a bounded convex space R ² and with sufficiently smooth boundary . The new variational formulation of the problem has been proposed and his validity and application has been discussed using precise mathematical treatment. Then, using the earlier introduced theoretical results, an effective algorithm has been applied to convert problem of finding solutions to hybrid type partial differential equations of von Kármán form to that of the ordinary differential (ODEs) and algebraic (AEs) equations. Mechanisms of transition to chaos of deterministic systems with infinite number of degrees of freedom are presented. Comparison of mechanisms of transition to chaos with known ones is performed. The following cases of longitudinal loads of different sign are investigated: parametric load acting along X direction only, and parametric load acting in both directions X and Y with the same amplitude and frequency.     

Investigation of thermocapillary and thermogravitational convection in a fluid with local heating

The results of a numerical solution of the problem of the unsteady convective motion generated in a fluid layer by the formation at the initial instant of a heated zone in the form of a thin cylindrical column, extending from the surface into the interior of the fluid, are presented. The problem is formulated with allowance for both thermocapillary and thermogravitational convection. The influence of the thermocapillary and thermogravitational effects on the fluid motion for various layer thicknesses is subjected to parametric analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 25–29, November–December, 1989.     

Mass transfer between bubbles and the continuous phase in a pseudofluidized layer

The problem of mass transfer between an isolated bubble and the continuous phase in a pseudofluidized layer is considered, when the rising velocity of the bubble exceeds the pseudofluidization rate. In this case the bubble with the surrounding region, a so-called two-phase system, is surrounded by a surface current impermeable to the liquid [1–3], and the problem reduces to determining the concentration field and the total flow on the material surface. The problem is solved for large and small Peclet numbers by a boundary layer diffusion method and by asymptotic expansion matching.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–49, July–August, 1973.     

Group Stratification and Exact Solutions of the Equation of Transonic Gas Motions

Exact solutions of the Kármán–Guderley equation that describes spatial gas flows in the transonic approximation are considered. A group stratification of the equation with respect to the infinite-dimensional part of the admissible group is constructed. New invariant and partly invariant solutions are obtained. The possibility of existence of solutions continuous in the entire space is analyzed for invariant submodels with one independent variable. A solution of the Kármán–Guderley equation of the double-wave type is constructed.     

First-order approximation in a steady fluid flow problem with surface forces taken into account

The plane problem of finding the shape of the free surface of an ideal incompressible fluid is considered for a fully developed discharge from an infinite layer through a point sink. The gravitational and surface stress forces are taken into account.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 166–169, November–December, 1974.The author thanks G. V. Shcherbina for formulation of the problem and valuable advice.     

Design of a wing airfoil in a flow with a separated turbulent boundary layer

The problem of designing the contour of an airfoil in a viscous (incompressible and compressible) flow with a separated turbulent boundary layer from a pressure distribution given on the separationless part of the contour is solved using the boundary layer theory together with the separated flow model proposed in [1]. Numerical calculations are carried out to demonstrate the possibilities of the method.Kazan'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 83–91, May–June, 1994.     

Radiative-convective heat transfer in a plane layer of a selectively absorbing mist

This paper deals with the thermal field in a plane layer of selectively absorbing gas which has been injected into a steady turbulent stream of high-temperature gas flowing around a porous plate. The boundary-value problem in terms of the energy equation reduces to a nonlinear integral equation in terms of a dimensionless temperature, and this equation is solved numerically by the Newton-Kantorovich method. The results are presented on graphs of temperature and thermal flux in the absorbing gas layer as functions of the space coordinate. Such a problem has been analyzed in [1] for the case of an injected gray gas.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 3, pp. 179–182, May–June, 1972.     

Magnetohydrodynamic lubrication theory for a cylindrical bearing

Liquid metal, which is a conductor of electric current, may be used as a lubricant at high temperatures. In recent years considerable attention has been devoted to various problems on the motion of an electrically conducting liquid lubricant in magnetic and electric fields (magnetohydrodynamic theory of lubrication), Thus, for example, references [1–3] study the flow of a conducting lubricating fluid between two plane walls located in a magnetic field. An electrically conducting lubricating layer in a magnetohydrodynamic bearing with cylindrical surfaces is considered in [4–8] and elsewhere.The present work is concerned with the solution of the plane magnetohydrodynamic problem on the pressure distribution of a viscous eletrically conducting liquid in the lubricating layer of a cylindrical bearing along whose axis there is directed a constant magnetic field, while a potential difference from an external source is applied between the journal and the bearing. The radial gap in the bearing is not assumed small, and the problem reduces to two-dimensional system of magnetohydrodynamic equations.An expression is obtained for the additional pressure in the lubricating layer resulting from the electromagnetic forces. In the particular case of a very thin layer the result reported in [4–8] is obtained. SI units are used.     

Convective motion of a conducting liquid carrying a current in a transverse magnetic field

The convective instability of a layer of conducting liquid carrying a current and lying in a magnetic field perpendicular to the current is considered. The problem of the nonconductive approximation in a linear setting is solved. The relationships between the Rayleigh number and the Hartmann number (determining the neutral stability) are derived.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza. No. 2, pp. 20–26, March–April, 1971.     

Convection currents in a vertical layer with a space-modulated temperature distribution on the boundaries

Steady convective motions in a plane vertical fluid layer are investigated. The temperature along the boundaries of the layer varies harmonically and has different average values on each of the boundaries. Thus space-period modulation of the temperature of the walls is assigned along with average lateral heating of the layer. The form of the plane steady motions and regions of existence of through currents and currents of cellular structure are found for various values of the parameters of the problem by the finite difference grid-point method. The dependence of the main characteristics of fluid motion on the Grashof number is determined. The results presented in the article pertain to the case when the period of modulation of the temperature of the boundaries coincides with the wavelength of the critical mode of a plane-parallel current. A numerical investigation of supercritical motions in a vertical layer with plane isothermal boundaries heated to a different temperature was carried out in [1–3]. The effect of a space-periodic inhomogeneity due to curvature of walls on the form and stability of convective motions in a vertical layer with lateral heating was examined in [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–25, September–October, 1978.The author thanks E. M. Zhukhovitskii for formulating the problem and supervising the work and G. Z. Gershuni for discussions and useful comments.     

Intense evaporation of a gas from a two-dimensional periodic surface

Evaporation (or condensation) of a gas is said to be intense when the normal component of the velocity of the gas in the Knudsen layer has a value of the order of the thermal velocity of a molecule, c_T=(2kT/m)^1/2. In this case the distribution function of the molecules with respect to their velocities in the Knudsen layer differs from the equilibrium (Maxwellian) value by its own magnitude. As a result of this, over the thickness of the Knudsen layer the macroparameters also vary by their own magnitudes. So in order to obtain the correct boundary conditions for the Euler gas dynamic equations, it is necessary to solve the nonlinear Boltzmann equation in the Knudsen layer. The problem of obtaining such boundary conditions for the case of a plane surface was considered in [1–11]. In the present study this problem is solved for a two-dimensional periodic surface in the case when the dimensions of the inhomogeneities are of the order of the mean free path of the molecules and the inhomogeneities have a rectangular shape. The flow in the Knudsen layer becomes two-dimensional, and this leads to a considerable complication of the solution of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 132–139, March–April, 1985.In conclusion the author would like to express his gratitude to V. A. Zharov for his valuable advice, and also V. S. Galkin, M. N. Kogan, and N. K. Makashev for discussion of the results obtained.     

Supersonic condensation of a monatomic gas

The problem of steady supersonic condensation of a monatomic gas on a plane evaporating surface is solved in the Knudsen layer by the direct statistical modeling method. The domain of existence of the solution of the problem is determined. The results of calculating the structure of the Knudsen layer near the surface are presented. A topological picture of the solutions of the strong evaporation and subsonic and supersonic strong condensation problems is given as a function of the Mach number, determined from the normal velocity component, and the other governing parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 171–175, May–June, 1990.     

Universal equations of the laminar boundary layer on a rotating blade

The article discusses the problem of the motion of an incompressible liquid in a boundary layer on a blade rotating uniformly around an axis perpendicular to the swing of the blade. A parametric method is used to solve the problem, and three series of parameters are introduced, on which depend the characteristics of the boundary layer. A corresponding system of universal equations is set up, which is integrated over a broad range of change in the parameters. The results obtained permit investigating the principal laws governing flow in a boundary layer on a rotating blade. The effect of rotation on breakaway and other characteristics of the boundary layer is clarified.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 84–93, March–April, 1971.     

Dynamic problem for a two-layer medium with a vibrational source at the boundary interface

The present study is concerned with an analysis of gravitational and acoustic waves which are excited by a vibrational source deeply placed in a liquid covered by ice. An analysis of the rigidity characteristics of ice modeled by an elastic layer or by a Kirchhoff plate is done by factorization of the solution to the integral equation equivalent to an initially combined boundary value problem. The uncombined boundary condition is used to solve problems for unrestricted ice fields in [1–3], whereas combined conditions with vibrational sources positioned at the boundary of the medium are used in [4].Translated from Zhurnal Prikladnoi Mekhaniki, No. 3, pp. 125–129, May–June, 1986.