Construction of asymmetric nozzles of maximum moment given additional conditions on the geometrical and force characteristics

A solution is given to the variation problem of constructing asymmetric plane nossles which realize the maximum moment relative to some point. The contours of the nozzle are assumed to be noninteracting. The method of the undetermined control contour is used [1]. The solution of this problem contains as a special case the solution to the problem of constructing a nozzle of maximum thrust, including also the case of a given lifting force [1–3]. It is shown that the construction of a nozzle of maximum moment under additional conditions on the thrust and the lifting force, or on the moment relative to another point, reduces to the construction of a nozzle of maximun moment relative to some auxiliary point.Translated from Izvestiya Akademi Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 148–152, September–October, 1984.The author thanks A. N. Kraiko for useful discussions and for his appraisal of the study.     

Motion of a suspension in the laminar boundary layer on a flat plate

The boundary layer motion of a weak suspension is investigated with allowance for the effect on the particles not only of the Stokes force but also of the additional transverse force resulting from the transverse nonuniformity of the flow over the individual particle. As distinct from studies [1–3], in which the limiting values of the transverse force (Saffman force) were used [4], the velocity and density of the dispersed phase have been determined with allowance for the dependence of the Saffman force on the ratio of the Reynolds numbers calculated from the velocity of the flow over the individual particle and the transverse velocity gradient of the undisturbed flow, respectively [5, 6].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–73, January–February, 1992.In conclusion the authors wishes to thank M. N. Kogan, N. K. Makashev, and A. Yu. Boris for useful discussions of the results.     

Interaction of a One-Dimensional Continuum with an Inertial Object Moving over the Continuum

Free transverse oscillations in a system consisting of an infinite moment continuum, such as the Euler-Bernoulli beam lying on the Winkler foundation, and a rigid body moving along the beam with a constant velocity and having a point contact with the guide are studied. The range of the considered velocities of the concentrated inertial object along the continuum is limited by the requirement of a finite energy of elastic deformation of the infinite continuum, corresponding to cojoint free osillations of an unbounded system. An analytical solution of the corresponding spectral problem in a system with a mixed spectrum is constructed. Limiting situations are analyzed, where the inertial rigid object moving along the beam is devoid of one “oscillatory” degree of freedom for some reasons. In particular, an inertial object devoid of mass but having a nonzero tensor of inertia is considered. Dependences of all characteristics of the discrete spectrum of oscillations and their shapes on the magnitude of object velocity along the moment elastoinertial guide are given.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 88–97, July– August, 2005.     

Supersonic flow past a star-shaped body without symmetry planes at angle of attack

The author's solution [2] for the flow past a star-shaped body without symmetry planes is generalized. It is shown that in this case also the vector of the resultant force is situated in the plane of the angle of attack (there is no lateral force). The twisting moment acted on the star-shaped body is calculated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 164–173, May–June, 1993.     

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the transmitted Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.     

Three-Dimensional Stress State of an Orthotropic Rectangular Prism under a Transverse Force Applied at its End

The exact three-dimensional elastic solution is found for an orthotropic composite rectangular prism (plate) bent by a transverse force applied to its end. The three-dimensional distribution of stresses and displacements is obtained. The distribution of tangential stresses in the prism and plate is analyzed numerically. New qualitative features of the distribution are established. The strength of the plate is estimated, and the transverse force and torque are determined. The Kirchhoff-Love relations of plate bending are partially justified__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 30–37, April 2005.     

Dispersed phase motion in laminar boundary layer flow over a wedge

The motion of a dispersed phase in the laminar boundary layer on a wedge is considered with allowance for the effect of not only the Stokes force, which coincides in direction with the flow velocity, but also the transverse force (Saffman force) resulting from the transverse nonuniforrnity of the flow over the individual particle [1–3].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 34–42, November–December, 1993.In conclusion, the authors wishes to thank S. V. Manuilovich for assisting with the numerical calculations.     

Self-oscillational motion of a cylinder towed in weak aqueous polymer solution

A study is made of the effect of polymer additions on the nonstationary vibrational motion of a cylinder towed in a weak aqueous solution of the polymer and capable of making transverse displacements under the action of the force arising from asymmetry of the periodic detachment of the boundary layer. The Reynolds number was 3·10³–10⁴.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, 1973.The authors wish to thank V. P. Borovikov for his assistance in organizing the experiments.     

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.     

Numerical Analysis of the Branched Forms of Bending for a Rod

Nonlinear boundary–value problems of plane bending of elastic arches subjected to uniformly distributed loading are solved numerically by the shooting method. The problems are formulated for a system of sixth–order ordinary differential equations that are more general than the Euler equations. Four variants of rod loading by transverse and longitudinal forces are considered. Branching of the solutions of boundary–value problems and the existence of intersected and isolated branches are shown. In the case of a translational longitudinal force, the classical Euler elasticas are obtained. The existence of a unique (rectilinear) form of equilibrium upon compression of a rod by a following longitudinal force is shown.     

Discontinuous solutions of an interacting three-dimensional hypersonic boundary layer

The paper is a mathematical study of the three-dimensional flow of viscous gas in a hypersonic boundary layer that develops along a flat wing whose leading edge has a step shape. The flow interacts with a flap on the wing set at a small angle. A linear solution to the problem is constructed under the assumption that the deflection angle of the flap is small and the difference between the length of the plates is of order unity. It is shown that an important part in the formation of the flow near and behind the flap may be played by the change in the pressure along the span of the wing due to the step shape of the leading edge. It is significant that although the pressure and displacement thickness are continuous functions of the transverse coordinate, the longitudinal and transverse components of the friction force have discontinuities.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 19–26, March–April, 1991.I thank V. V. Sychev and A. I. Ruban for suggesting the problem, for valuable advice, and assistance.     

Transonic flows around a carrying airfoil profile

Unsymmetrical gas flows around a carrying airfoil, with velocities close to sonic at infinity, are discussed. In a region situated a certain distance from the profile, an asymptotic solution of the flow problem is constructed. A detailed analysis is made of the dependence of the terms of the asymptotic sequences of the parameters characterizing the transverse and longitudinal dimensions of the body. The law of change in the lifting force as a function of the difference 1 – M, which is assumed to be small, is established. The connection between the theoretical results and experimental data is discussed in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 104–112, January–February, 1978.     

Dynamics of a spherical particle in a laminar boundary layer

The problem of the motion of an individual spherical particle in a laminar boundary layer is considered for small Reynolds numbers determined from the relative velocity and the transverse velocity gradient of the flow undisturbed by the particle. The dependence of the transverse force acting on the particle, which results from the nonuniformity of the free stream, on the distance of the particle from the surface of a flat plate is calculated. It is shown that the direction of the transverse force changes with the distance of the particle from the plate: near the surface the force is positive, i.e., directed away from the plate, and at greater distances negative.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 91–96, November–December, 1990.The author wishes to thank M. N. Kogan and N. K. Makashev for useful discussions.     

Effect of radial motion of erythrocytes on their sedimentation in a tube in an external magnetic field

The effect of the transverse component of the ponderomotive force on the aggregation and sedimentation of erythrocytes is investigated. It is assumed that the only mechanism of action of a constant magnetic field on the blood is magnetization of the erythrocytes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 120–129, September–October, 1991.The author is most grateful to S. A. Regirer for his interest in the work and useful observations.     

Isothermal viscous incompressible flow in a hydrodynamic model of an electrophoresis chamber

A numerical solution and approximate analysis of the system of Navier-Stokes equations averaged over the transverse coordinate has made it possible to obtain the dependence of the length of the hydrodynamic flow stabilization interval in a thin cell of rectangular cross section on the Reynolds number, the relative thickness of the cell, and the relative size of the inlet opening. The principal and secondary flow regimes are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 14–20, March–April, 1990.     

Conservation of the resultant force vector in the plane of the angle of attack for slender star-shaped bodies in supersonic flows

At high supersonic flight speeds bodies with a star-shaped transverse and power-law longitudinal contour are optimal from the standpoint of wave drag [1–3]. In most of the subsequent experimental [4–6] and theoretical [6–9] studies only conical star-shaped bodies have been considered. For these bodies in certain flow regimes ascent of the Ferri point has been noted [10]. In [11] the boundary-value problem for elongated star-shaped bodies with a power-law longitudinal contour was solved for the case of supersonic flow. The present paper deals with the flow past these bodies at an angle of attack. It is found that for arbitrary star-shaped bodies with any longitudinal (in particular, conical) profile the aerodynamic forces can be reduced to a wave drag and a lift force, the lateral force on these bodies being equal to zero for any position of the transverse contour.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–141, November–December, 1989.     

A realization of the moment method

The difficulties involved in solving a system of moment equations using two-sided distributions are analyzed. The properties of these distributions do not permit the realization of the moment method (in the case of a collision integral in Boltzmann form) in specific transport boundary-value problems. A method of obtaining analytic solutions of the system of moment equations for linearized transport problems is proposed. The accuracy of the method is analyzed with reference to a classical transport problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 185–188, May–June, 1991.     

Unsteady flow over a rough bottom

The unsteady motion of an ideal incompressible fluid with a free surface, developing from a state of rest, is considered. The flow is assumed to be irrotational, continuous and two-dimensional; it may be the result either of an initial disturbance of the free boundary or of a given boundary pressure distribution. The rigid boundaries of the flow region are fixed, and the free surface does not cross them at any time during the motion. The fluid is located in a uniform gravity force field and there is no surface tension. A method which in the case of localized roughness of the bottom makes it possible to find the shape of the free surface at any moment of time with predetermined accuracy is proposed. The method involves reducing the initial linear problem to a Volterra integral equation of the second kind, the kernel of this equation being a nonlocal operator. This operator has a smoothing effect, which makes it possible to reduce the solution of the initial problem to the solution of an infinite, perfect lyregular system of Volterra integral equations for a denumerable set of auxiliary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–119, November–December, 1989.The author is grateful to I. V. Sturova and B. E. Protopopov for useful discussions and criticism.     

Further developments in determining the dynamic contact law

A force/strain relation (incorporating shear effects) for the transverse impact of beams is established in the frequency domain. Inversion by use of a fast-Fourier-transform (FFT) algorithm is shown to allow the force history to be determined. The effects of sampling rate and size are considered when analyzing experimental data.Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Canada on June 10–15, 1984.     

Slow motions of a rigid sphere in incompressible viscoelastic fluids

Problems concerned with the force of resistance and the moment of forces acting from the side of a viscoelastic fluid on a sphere moving with acceleration are considered in a linear formulation. Fairly simple relations are obtained for a fluid with a single relaxation time or a single after-action time. A discussion of the asymptotic expressions is given for a fluid with a large number of times.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–16, March–April, 1976.