Interaction of two supersonic radial gas flows

The flow resulting from the collision of two spherically symmetric supersonic gas streams generated by three-dimensional sources has been studied within the framework of the inviscid perfect gas model. When the characteristics of both sources are the same, the problem reduces to that of the interaction between a spherically symmetric flow and a plane barrier [1, 2]. By means of a certain passage to the limit, the flow from one of the sources can be reduced to a uniform supersonic gas flow. In this case the problem reduces to the problem of uniform gas flow past a source considered in [1, 3, 4]. The resulting flows are investigated with reference to all the parameters characterizing the two sources. The shock wave structure, both in the neighborhood of and at points remote from the axis of symmetry, is studied, together with the distributions of the gas dynamic quantities in the shock layers; certain similarity laws are established. The astrophysical applications of the problem associated, in particular, with a certain x-ray radiation mechanism in binary stellar systems [5] are of particular interest.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 159–165, July–August, 1990.The authors are grateful to V. B. Baranov for his constant interest and to S. A. Zhekov and V. V. Usov for useful discussions.     

Turbulent jet mixing of supersonic flows in profiled flow-equalization channels

A numerical model for the calculation of gas dynamic systems with turbulent mixing of supersonic jets is proposed. The problem of designing a transitional flow-equalization channel of minimum length is solved for the viscous turbulent mixing of two parallel or mutually inclined supersonic flows. The problem is solved in two stages. In the first stage the flow-equalization channel is designed by solving the inverse problem in the ideal gas approximation. In the second stage the basic problem is solved for the channel thus obtained on the basis of the parabolized Navier-Stokes equations. Investigations have demonstrated the validity of this approach to the equalization of nonuniform flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 175–178, July–August, 1987.The authors are grateful to V. I. Kopchenov for supplying the program for solving the basic problem by a first-order Godunov method and to A. I. Kraiko and Yu. V. Kurochkin for their interest and advice.     

Boundary layer development in supersonic MHD generators

A numerical model of the turbulent boundary layers in the gas dynamic channel of a supersonic MHD generator is constructed. This model describes the development and structure of the layers in the nozzle, on the electrode and insulating walls of the duct, in the two-dimensional approximation. The characteristics of the boundary layers in various generator operating regimes are investigated numerically. The integral boundary layer thicknesses characterizing the nonuniformity of the gas dynamic and electrodynamic quantities are calculated. The limits of applicability of the integral calculation method are determined for typical MHD generator operating conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 35–41, May–June, 1990.The authors wish to thank A. B. Vatazhin, V. A. Bityurin, and V. A. Zhelnin for discussing the formulation of the problem, A. A. Yakushev for participating in the discussion of the results, and Yu. V. Rakseeva and L. V. Yashina for preparing the article.     

Spin-down of a fluid in a low cylinder at large Reynolds numbers

The problem of the axisymmetric motion of a fluid between infinite disks is solved by the method of matched asymptotic expansions without introducing model assumptions. For the strongly nonlinear stage of spin-down solutions are found that correspond to initial states different from rigid-body rotation, when the boundary layer is not a Kármán layer. The experimental results obtained are in qualitative and quantitative agreement with the theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–46, May–June, 1986.The authors wish to thank A. M. Obukhov and F. V. Dolzhanskii for formulating the problem and for constructive discussion.     

Bending of a packet of transversely isotropic circular plates

A mechanical-mathematical model of the bending of transversely isotropic plates is developed on the basis of the model constructed by é. I. Grigolyuk and V. M. Tolkachev. The original model is generalized with allowance for anisotropy. The new model is used to solve the problem of the axisymmetric bending of a packet of circular plates. The existence of regions of adhesion, slip, and separation is taken into account. The problem of determining the contact stresses in the model is reduced to the integration of a system of differential equations. Ivano-Frankovsk Division, Institute of Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 71–77, August, 1999.     

Optimum supersonic profile of given thickness

The problem of determining the profile possessing minimum wave drag in a uniform supersonic free gas stream is considered. The length, thickness, and angle of attack of the body are considered given. The analysis is limited to the class of bodies for which attached shocks originate during the flow. It is assumed that the flow is supersonic in the domains of influence of the components of the desired contour and there are no internal shocks. The problem is studied within the framework, of model and exact formulations. Results of numerical computations are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 89–96, January–February, 1976.The author is grateful to A. V. Shipilin for constant attention to the research.     

Asymptotic analysis of the unsteady problem of waves in a two-layer flow

The problem investigated is the unsteady problem of the internal waves generated in a two-layer flow by a certain periodic perturbation which leads to small deviations from the basic flow. A method of constructing an approximate solution uniformly valid throughout the region of variation of the variables and the parameters of the problem is indicated. It is confirmed that for large times and near-resonance parameters the motion of the fluid is described by the mixed problem for a cubic Schrödinger equation. Certain qualitative properties of the solution of this nonlinear problem are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 82–90, November–December, 1987.The author is grateful to V. I. Bukreev and to I. V. Sturova for their interest in his work.     

Thermoelastic problem of two collinear cracks

A two-dimensional problem of two collinear cracks influenced by concentrated heat sources is studied. The considerations are limited to a problem of normal opening of the crack (Mode I) assuming that cracks of different lengths are thermally insulated and free of loads. The formulas describing displacements and stresses at any point of the body are given. The stress intensity factors at the crack tips and the openings of the cracks are obtained. To avoid possible overlapping of the cracks boundaries, the position of the heat sources and the lengths of the cracks are analyzed to ensure non-negative stress intensities factors (SIF's) at all the crack tips. The solution to the above problem has been derived using the method given by Panasuyk et al. [V.V. Panasuyk, M.P. Savruk, A.P. Datshshyn, Stress Distribution around Cracks in Plates and Shells, Naukova Dumka, Kiev, 1976 (in Russian); V.V. Panasuyk, M.P. Savruk, A.P. Datsyshyn, A general method of solution of two-dimensional problems in the theory of cracks, Eng. Fracture Mech. 9 (1977) 481–497.     

Motion of shock waves in a channel of variable cross section in the presence of a steady flow regime

The motion of shock waves against a steady flow in a channel of variable cross section is considered. Situations for which Chisnell's hypothesis [1] or a quasistationary flow model are valid are considered. The problem is of interest, in particular, in connection with the investigation of the starting of shock wind tunnels.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 103–110, July–August, 1981.I am very grateful to A. N. Kraiko and V. T. Grin' for valuable advice and support during the work.     

Poisson–Nernst–Planck Systems for Ion Flow with Density Functional Theory for Hard-Sphere Potential: I–V Relations and Critical Potentials. Part II: Numerics

We consider a one-dimensional steady-state Poisson–Nernst–Planck type model for ionic flow through membrane channels. Improving the classical Poisson–Nernst–Planck models where ion species are treated as point charges, this model includes ionic interaction due to finite sizes of ion species modeled by hard sphere potential from the Density Functional Theory. The resulting problem is a singularly perturbed boundary value problem of an integro-differential system. We examine the problem and investigate the ion size effect on the current–voltage (I–V) relations numerically, focusing on the case where two oppositely charged ion species are involved and only the hard sphere components of the excess chemical potentials are included. Two numerical tasks are conducted. The first one is a numerical approach of solving the boundary value problem and obtaining I–V curves. This is accomplished through a numerical implementation of the analytical strategy introduced by Ji and Liu in [Poisson–Nernst–Planck systems for ion flow with density functional theory for hard-sphere potential: I–V relations and critical potentials. Part I: Analysis, J. Dyn. Differ. Equ. (to appear)]. The second task is to numerically detect two critical potential values V _c and V ^c .The existence of these two critical values is first realized for a relatively simple setting and analytical approximations of V _c and V ^c are obtained in the above mentioned reference. We propose an algorithm for numerical detection of V _c and V ^c without using any analytical formulas but based on the defining properties and numerical I–V curves directly. For the setting in the above mentioned reference, our numerical values for V _c and V ^c agree well with the analytical predictions. For a setting including a nonzero permanent charge in which case no analytic formula for the I–V relation is available now, our algorithms can still be applied to find V _c and V ^c numerically.     

Quasi-three-dimensional calculation of flow in blade passages of turbines

The flow in turbomachines is currently calculated either on the basis of a single successive solution of an axisymmetric problem (see, for example, [1-A]) and the problem of flow past cascades of blades in a layer of variable thickness [1, 5], or by solution of a quasi-three-dimensional problem [6–8], or on the basis of three-dimensional models of the motion [9–11]. In this paper, we derive equations of a three-dimensional model of the flow of an ideal incompressible fluid for an arbitrary curvilinear system of coordinates based on averaging the equations of motion in the Gromek–Lamb form in the azimuthal direction; the pulsation terms are taken into account in the equations of the quasi-three-dimensional motion. An algorithm for numerical solution of the problem is described. The results of calculations are given and compared with experimental data for flows in the blade passages of an axial pump and a rotating-blade turbine. The obtained results are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 69–76, March–April, 1991.I thank A. I. Kuzin and A. V. Gol'din for supplying the results of the experimental investigations.     

Nonisothermal displacement of high-paraffin oils by water with allowance for solid-phase precipitation

A mathematical model of the nonisothermal displacement of oil by water with allowance for solid-phase precipitation is proposed. Self-similar solutions of the problem of nonisothermal displacement of oil from a homogeneous, thermally insulated formation are obtained. The inverse problems of determining the relative phase permeabilities and the temperature dependence of the paraffin saturation concentration from laboratory displacement data are solved. Exact solutions of the non-self-similar problems of the displacement of high-paraffin oil by a slug of hot water and of the thermal delay problem are obtained. The nonisothermal displacement of high-paraffin oils by water with allowance for heat transfer to the surrounding strata is subjected to qualitative analysis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–137, May–June, 1989.The authors are grateful to A. K. Kurbanov and Yu. V. Kapyrin for useful discussions and their interest in the work.     

A plane steady-state free-boundary problem for the navier-stokes equations

The model problem of the plane slow steady-state motion of a viscous incompressible fluid with a free boundary is investigated. It is assumed that the free boundary does not have any points in common with the solid surfaces confining the fluid. By the solution of the auxiliary fixed-boundary problem for the Navier-Stokes equations the problem is reduced to an operator equation describing the form of the free surface. The existence and uniqueness problems for the solution and its qualitative behavior are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 91–102, May–June, 1972.In conclusion, the authors would like to thank R. M. Garipov and V. Kh. Izakson for affording an opportunity to become acquainted with the results of their unpublished work.     

Fresh water lenses formed by seepage from irrigation channels

A two-dimensional model of a steady-state fresh water lens, formed above saline water as a result of mole irrigation, is investigated in the hydrodynamic formulation [1]. The corresponding boundaryvalue problem is solved by means of Kochina's method [1] based on the use of the analytical theory of linear differential equations. The results of numerical calculations are presented and the effect of the model parameters on the shape and size of the lens is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–106, July–August, 1990.The authors are grateful to V. M. Entov and A. R. Tsitskishvili for advice and comments.     

Stability of a nonstationary round jet of an ideal incompressible fluid

The stability of transient flow in a cylinder of an ideal incompressible fluid with a free boundary is studied. There are 20 different cases of the behavior of small disturbances as a function of the parameters of the problem. In particular, if surface tension is not taken into account a round jet is stable with respect to axially symmetrical disturbances, but the introduction of capillary forces leads to a strong instability.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 80–84, July–August, 1972.In conclusion the author thanks V. V. Pukhnachev for formulation of the problem and valuable advice.     

On the Lagrange problem about the optimal form for circular hollow columnsSur le problème de Lagrange de la forme optimale d'une colonne circulaire aux parois minces

A new approach to the problem of finding the form of the strongest clamped circular column with thin walls of fixed volume and height is proposed. The same model describes also the form of the horizontal beam with rectangular vertical sections of a fixed height and variable widths. The existence and the uniqueness of the optimal column are proved here for the first time and the optimal form is found. To cite this article: Yu.V. Egorov, C. R. Mecanique 331 (2003).     

Use of Galerkin's method to investigate transitions in a problem of Rayleigh convection

Galerkin's method is used to study transitions in the problem of convection of a fluid in a horizontal layer with free isothermal boundaries (Rayleigh's model).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 22–27, March–April. 1984.I thank V. I. Yudovich for constant interest in the work.     

Calculation of stresses and deformations of bellows by initial parameter method of numerical integration

By reducing the boundary value problem in stress analysis of bellows into initial value problem, this paper presents a numerical solution of stress distribution in semi-circular arc type bellows based upon the toroidal shell equation of V. V. Novozelov^[8]. Throughout the computation, S. Gill’s method^[1O] of extrapolation is used. The stresses and deformations of bellows under axial load and internal pressure are c-alculated, the results of which agree completely with those derived from the general solution of Prof. Chien Wei-zang^[1-4]. The extrapolation formula presented in this paper greatly promotes the accuracy of discrete calculation.The computer program in BASIC language of Wang 2200 VS computer is included in the appendix.     

Unsteady motion of rotating ring of viscous incompressible liquid with free boundary

We examine the plane problem of rotationally symmetric motion of a rotating ring of viscous incompressible liquid with free boundary. The theorem for the existence and uniqueness of the problem solution is obtained. The qualitative properties of the solution and its asymptotic behavior as t are studied.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 82–88, May–June, 1970.This study was made under the guidance of V. V. Pukhnachev.     

Dispersion of a plasma cloud in a uniform magnetic field

The numerical solution of the two-dimensional gasdynamical problem of the dispersion of a plasma cloud in a magnetic field which is uniform to infinity is described. The disturbance of the field and the deformation of the cloud are taken into account self-consistently.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 146–148, July–August, 1974.In conclusion the authors thank A. I. Barchenkov, G. V0 2harova, S. A. Kuchai, V. G. Rogachev, and V. P. Statsenko for valuable advice and assistance in the work.