On the stability of certain nonparallel flows of a viscous incompressible liquid in a channel

The stability of very simple nonparallel flows of a viscous incompressible liquid in an infinite plane channel described by the exact solutions of the Navier-Stokes equations is studied. Such solutions are realized between two parallel porous plates when the liquid (or gas) is forced in at one wall and drawn out at the same velocity at the other, with a steady flow of liquid along the channel. In this case the transverse velocity component is constant, and the profile of the longitudinal velocity component is independent of the longitudinal con-ordinate x, being an asymmetric function of the transverse coordinate y. A study of the hydrodynamic stability then reduces to the solution of an equation differing from the Orr-Sommerfeld equation by virtue of the presence of additional terms containing the transverse velocity component of the main flow. By numerically solving both this equation and the ordinary Orr-Sommerfeld equation and comparing the corresponding results for various inflowing Reynolds numbers R₀=v₀h/ (v₀ is the inflow velocity, h is the width of the channel), the effect of the nonparallel and asymmetrical nature of such flows on their stability is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 125–129, July–August, 1970.     

Stability of a plane-parallel convective flow of a liquid in a horizontal layer

We consider the stationary plane-parallel convective flow, studied in [1], which appears in a two-dimensional horizontal layer of a liquid in the presence of a longitudinal temperature gradient. In the present paper we examine the stability of this flow relative to small perturbations. To solve the spectral amplitude problem and to determine the stability boundaries we apply a version of the Galerkin method, which was used earlier for studying the stability of convective flows in vertical and inclined layers in the presence of a transverse temperature difference or of internal heat sources (see [2]). A horizontal plane-parallel flow is found to be unstable relative to two critical modes of perturbations. For small Prandtl numbers the instability has a hydrodynamic character and is associated with the development of vortices on the boundary of counterflows. For moderate and for large Prandtl numbers the instability has a Rayleigh character and is due to a thermal stratification arising in the stationary flow.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–100, January–February, 1974.     

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

Theory of slightly perturbed three-dimensional flows in nozzles

The theory of slightly perturbed flows in conical nozzles is used to determine the transverse force and moment generated in the presence of asymmetric perturbations. A system of ordinary differential equations is derived for finding the transverse force and moment. An approximate analytical solution of this system is constructed and its qualitative features are studied. A comparison is made with a numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1977.     

Approximate solution of the problem of the flow of a conducting liquid in a duct with rectangular cross section

An approximate analytic solution is constructed for the problem of the fully developed stationary flow of a viscous incompressible liquid with a finite isotropic conductivity in a duct of rectangular cross section in the presence of an external magnetic field at right angles to nonconducting walls. An investigation is made of the extent to which flows in ducts with two electrodes parallel to the field resemble flows with four nonconducting walls. Theoretical and experimental investigations devoted to this problem are reviewed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 33–40, September–October, 1970.     

Stability of erythrocyte sedimentation in a constant magnetic field

The stability of erythrocyte sedimentation in the presence of a transverse component of the ponderomotive force is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 66–70, November–December, 1989.Tha author is very grateful to S. A. Regirer for his interest in the work and useful discussions.     

Experimental investigation of flow in a trench

An experimental investigation was made of the flow of a viscous incompressible liquid in a trench of square transverse cross section, using a laser Doppler velocimeter. The investigation was made with two values of the Reynolds number Re, corresponding to laminar and turbulent flow conditions in the channel. The experimental data show that a core with a constant vorticity is formed in the trench, that a jet propagates near the walls of the trench, and that there are secondary eddies in the corners of the trench. The motion of a viscous liquid in a trench of rectangular cross section is part of a broad class of breakaway flows. Experimental data on the investigation of flow in trenches are extremely few. A majority of the existing information is limited to visual observations [1–4]. In [2, 5, 6] the question of the unstable character of flow in trenches was discussed. Quantitative measurements of stable eddy flows in trenches were made in [7–9] using a thermoanemometer, and in [7] measurements were made of the pressure at the bottom and walls of trenches; there are data on the distribution of the velocity in the middle sections of trenches. In [8] the mean velocity, the intensity of the turbulence, and the stress of the turbulent flow were obtained in several sections parallel to the side walls of the trench, In [9] a measurement was made of the velocities also in two cross sections of a trench in which one component of the velocity prevails. A brief analysis of the existing experimental results shows that these data are insufficient to form a detailed representation of the character of flow in a trench.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 76–86, March–April, 1976.     

Onset of convection regimes with double period in a periodic field of external forces

We investigate the onset of secondary convective flows in the Rayleigh problem (in a horizontal layer of a viscous incompressible liquid with free boundaries) in the presence of a parameter that varies with time and has a period T, namely the temperature gradient or the intensity of the gravitational field.Translated from Zhurnal Prikladnoi Mekhanlki i Tekhnicheskoi Fizlki, No. 6, pp. 65–70, November–December, 1972.     

The effect of transverse pressure on the stability of a plate

The stability problem of a centrally compressed infinite plate is solved with allowance for the transverse normal deformation caused by uniform load for various boundary conditions at the edges. The linearized nonlinear equations of elastic deformation of thin plates taking into account transverse shear and transverse normal deformation are used. The obtained critical loads are compared with existing solutions.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 170–178, March–April, 2005.     

Some exact solutions of the ferrohydrodynamics equations

The theoretical study of nonisothermal flows of magnetizable liquids presents serious matheical difficulties, which are intensified as compared to to the study of normal liquids by the necessity of simultaneous solution of both the hydrodynamics and Maxwell's equations, with corresponding boundary conditions for the magnetic field. Thus, in most cases problems of this type are solved by neglecting the effect of the liquid's nonisothermal state on the field distribution within the liquid, and also, as a rule, with use of an approximate solution for Maxwell's equations and fulfillment of the boundary conditions for the field [1–3]. The present study will present easily realizable practical formulations of the problem which permit exact one-dimensional solutions of the complete system of the equations of thermomechanic s of electrically nonconductive incompressible Newtonian magnetizable liquids with constant transfer coefficients. A common feature of the formulations is the presence of a longitudinal temperature gradient along the boundaries along which liquid motion is accomplished. Plane-parallel convective flows of this type in a conventional liquid and their stability were considered in [4–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–133, May–June, 1979.     

Asymptotic solution of the Orr-Sommerfeld equation in the outer region of the boundary layer with allowance for nonparallelism

There have been many publications devoted to the investigation of the hydrodynamic stability of nonparallel flows on the basis of the modified Orr-Sommerfeld equation [1–4]. Taking into account the additional terms associated with the presence in the flow of a transverse component of velocity and acceleration can lead not only to a significant quantitative discrepancy as compared with calculations based on the usual Orr-Sommer-feld equation but also to qualitatively new results (nonclosure of the neutral curves for flow on a permeable surface in the presence of strong injection [4]). In this paper an asymptotic solution of the Orr-Sommer-feld equation, valid in the outer region of boundary layer flow, is constructed for self-similar gradient flow over a surface (Falkner-Skan flow). The continuity of the eigenvalue spectrum for an unbounded increase in the perturbation propagation velocity is demonstrated on the basis of the solution obtained. For the ordinary Orr-Sommerfeld equation a continuous transition of the spectrum through the value of the perturbation propagation velocity C_r=1 (which coincides with the velocity of the external flow) is impossible [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–173, January–February, 1987.     

Electrosolenoidal flows in a cylindrical cavity

The main difficulties in investigating three-dimensional magnetohydrodynamic (MHD) flows with vorticity arise, first, because it is necessary to solve an independent boundary-value problem in order to find the field of the electromagnetic forces and, second, because the regimes of these flows are strongly nonlinear for the majority of high-power technological MHD processes and a number of natural phenomena. Particular importance attaches to MHD flows generated by the interaction of an electric current applied to the fluid with the magnetic self-field. This class of MHD flows has become known as electrosolenoidal flows [1]. The presence of a definite symmetry in the distribution of the electromagnetic forces and the geometry of the region of the liquid conductor makes it possible to find a solution in self-similar form. The present paper is devoted to exact solutions of the nonlinear equations for axisymmetric electrosolenoidal flows of a conducting incompressible fluid in infinite cylindrical cavities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 48–53, May–June, 1991.     

Convection in an oscillating force field and microgravity

Convective flows in a plane layer of viscous fluid in the presence of an oscillating external force are investigated numerically [1 – 8]. The layer is assumed to be placed in a gravitational field. The cases in which the external field oscillations are generated by rotation about the horizontal axis or by vibration in the longitudinal direction are considered. The Navier-Stokes equations and the Boussinesq approximation are used for describing the fluid motion. The flows developing in the layer in the presence of a transverse temperature gradient are determined, the stability boundaries of these flows are found, and the supercritical motion regimes are studied. These investigations are carried out using the averaging method (in order to find the stability limits for high rotation velocities and vibration frequencies) and the Galerkin method.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–106, September–October, 1994.     

Wall pressure fluctuations due to mean-shear-turbulence interaction in a turbulent boundary layer

The cross-correlated power spectrum of wall pressure fluctuations and the derivative of the turbulent velocity component normal to the wall is measured at a longitudinal coordinate; this spectrum, along with the mean shear, determines the contribution of the mean-shear-turbulence interaction to pressure fluctuations at a wall under a turbulent boundary layer. The spectrum is used to calculate the power spectrum and the transverse cross-correlated power spectrum of pressure fluctuations at the wall. Comparison of calculated and directly measured pressure spectra indicates that wall pressure fluctuations are almost completely determined by the mean-shear-turbulence interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 28–34, July–August, 1976.The author thanks G. P. Morozov-Rostovskii and Yu. A. Konokhov for fabricating a miniature hot-wire anemometer and help in carrying out the experiment.     

Construction of approximate solutions to two-dimensional potential problems of electrohydrodynamics

After transformation to new variables the system of equations describing planar potential electrohydrodynamic flows with a small interaction parameter is converted to a single equation. The particular solution of this equation, which is the electrohydrodynamic analog of Hamel's solution in the dynamics of a viscous liquid, is found. Two types of flows, described by simplified equations, can be distinguished when certain constraints are imposed on the manner in which the electrical parameters vary along the coordinate lines and the terms of the equation correspondingly estimated. These flows are the jet and quasione-dimensional flows of the charged component in a curvilinear electrostatic field and a supper-posed two-dimensional potential flow of the carrier medium. Solutions to the approximate equations are obtained for certain particular cases.Kiev. Transklated from Izvestiya Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 140–147, July–August, 1972.     

Interaction of a turbulent stream of gas with a liquid film

We consider the turbulent motion of a gas in contact with a liquid film next to a wall. We assume that the stream of gas excites in the liquid a complex system of motions which are analogous in principle to the motions in the near-wall zone of a homogeneous turbulent stream with transverse shear. As a result of these motions, the gas stream has considerable turbulence even at the gas-film boundary. On this assumption, we calculate the relation between the pressure drop and the average gas velocity and find that it is in satisfactory qualitative and quantitative agreement with experimental results. As our scale of turbulence at the boundary, we took a linear variation as a function of the film thickness, which enabled us to describe the available experimental results satisfactorily, making use of two empirical constants.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 67–74, March–April, 1976.In conclusion, the authors take this opportunity to express their grateful recollection of conversations with the late Professor P. A. Semenov, who drew their attention to this problem, and also to thank G. G. Chernyi and G. A. Lyubimov for their comments and their interest in the work.     

Stress concentration around a circular hole in a transversely isotropic spherical shell

Analytical expressions for the stresses near a circular hole in a transversely isotropic shallow spherical shell under uniform pressure are derived. The form of the solution depends on the range of change in the compliance to transverse shear. The influence of the relative radius of the hole and the compliance to transverse shear on the stress concentration is analyzed.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 99–106, December 2004.     

Influence of nonparallel flow on the development of disturbances in a supersonic boundary layer

The linear theory is used to solve the problem of the development of two-dimensional disturbances in the boundary layer of compressible fluid. In contrast to the stability theory of plane-parallel flows, the present paper takes into account the presence in the boundary layer of transverse (at right angles to the flow direction) motions, the dependence of the averaged flow parameters on the longitudinal coordinate, and also the deformation of the amplitude distribution profile of the disturbances as a function of the longitudinal coordinate. The calculations are made for Mach number M = 4.5.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 26–31, March–April, 1980.     

Steady-state wave flows of a weakly conducting gas in transverse electromagnetic fields

Gas flow in a MHD channel in transverse magnetic and electric fields is considered. The steady-state flows associated with the establishment of equilibrium between the hydraulic resistence and ponderomotive forces are investigated. The conditions of existence and the properties of such a steady-state flow regime are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 123–129, July–August, 1990.     

Plasma flow accompanied by blowing and pumping of plasma across electrodes

The article discusses plane stationary slowly varying flows of a nonviscous plasma with good conductivity in a channel in a transverse magnetic field; the flows are accompanied by blowing in and pumping plasma across solid metallic electrodes. The Hall effect is taken into consideration. It is shown that the potential jump near the anode, which appears in an accelerated plasma flow in an ordinary channel with solid electrodes, can be eliminated in flows accompanied by blowing in (pumping) of plasma. It is also shown that flows are possible in which the velocity, density, and the transverse electric field increase in the direction of the accelerator cathode.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 26–34, November–December, 1970.