Stationary travelling waves on a film flowing down an inclined plane

At small flow rates, the study of long-wavelength perturbations reduces to the solution of an approximate nonlinear equation that describes the change in the film thickness [1–3]. Steady waves can be obtained analytically only for values of the wave numbers close to the wave number _n that is neutral in accordance with the linear theory [1, 2]. Periodic solutions were constructed numerically for the finite interval of wave numbers 0.5_{n n} in [4]. In the present paper, these solutions are found in almost the complete range of wave numbers 0 _n that are unstable in the linear theory. In particular, soliton solutions of this equation are obtained. The results were partly published in [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–146, July–August, 1980.     

Strong convergence to global solutions for multidimensional flows of compressible,viscous fluids with polytropic equations of state and discontinuous initial data

We extend to general polytropic pressures P() = K, > 1, the existence theory of [8] for isothermal (= 1) flows of Navier-Stokes fluids in two and three space dimensions, with fairly general initial data. Specifically, we require that the initial density be close to a constant in L ² and L , and that the initial velocity be small in L ² and bounded in L ^{2 n} (in two dimensions the L ² norms must be weighted slightly). Solutions are obtained as limits of approximate solutions corresponding to mollified initial data. The key point is that the approximate densities are shown to converge strongly, so that nonlinear pressures can be accommodated, even in the absence of any uniform regularity information for the approximate densities.     

On elastic-plastic structures with associated stress-strain relations allowing for work softening

Summary For the elementary (finite or infinitesimal) constituents of the structure are assumed generalized stress-strain relations which satisfy the condition of normality but which may exhibit work-softening, concavity of yield surfaces, variation of elastic coefficients with stress and/or plastic strain.The usual phenomena of geometric instability are excluded. Sufficient conditions are formulated for overall stability in spite of the presence of unstable elements, and for uniqueness of the incremental boundary-values problem. Conditions are discussed with a view to applications and expressed in terms of positive definiteness of appropriate quadratic forms.Finally, yield surfaces and flow laws for the structure are examined, and among other things their necessary association is shown.

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Sommario Per i costituenti elementari (finiti o infinitesimi) della struttura si assumono legami incrementali tra sforzi e deformazioni generalizzati che soddisfano alla condizione di normalità ma che per il resto sono generici, cioè tali da presentare eventualmente incrudimento negativo, concavità del campo elastico, variazione dei coefficienti elastici con gli sforzi e/o con le deformazioni plastiche.Esclusi per il sistema i fenomeni usuali di instabilità geometrica si formulano condizioni sufficienti per la stabilità del complesso nonostante la presenza di parti a funzionamento instabile, e per l'unicità del problema incrementale al contorno. Le condizioni sono discusse in vista delle applicazioni ed espresse in termini di definizione positiva di forme quadratiche opportune.Si esaminano infine le superfici di snervamento e le leggi di scorrimento per l'intera struttura e, tra l'altro, se ne dimostra la necessaria associazione.

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First published in Italian in Rendiconti dell'Istituto Lombardo Classe Scienze e Lettere, A 100, 1966.The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza delle Costruzioni of the Facoltà di Architettura, Politecnico of Milano. Part of it was carried out at Brown University (Providence R. I. - U.S.A.). The author acknowledges with tanks the NATO Fellowship assigned by the C.N.R. in 1964, the encouragement and kind hospitality he received from Professor D.C. Drucker, and the interesting discussions he had with Professor Drucker and Dr. A. C. Palmer.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

Stability and Transition on a Swept Cylinder in a Supersonic Flow

Results of experimental investigations of the evolution of natural disturbances and laminar–turbulent transition in a supersonic boundary layer on the attachment line of a circular cylinder with a sweep angle of 68° and a freestream Mach number M = 2 are presented. The experimental studies are supplemented by calculations of the mean flow and stability characteristics. Flow regimes in the boundary layer on the attachment line are determined by a hotwire technique as functions of the Reynolds number and height of twodimensional roughness elements. The results are compared with NASA (Ames) experiments.     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Investigation of the flow past wings of infinite span with a blunt leading edge in the vorticity interaction regime

An asymptotic analysis of the Navier-Stokes equations is carried out for the case of hypersonic flow past wings of infinite span with a blunt leading edge when 0, Re , and M . Analytic solutions are obtained for an inviscid shock layer and inviscid boundary layer. The results of a numerical solution of the problems of vorticity interaction at the blunt edge and on the lateral surface of the wing are presented. These solutions are compared with the solution of the equations of a thin viscous shock layer and on the basis of this comparison the boundaries of the asymptotic regions are estimated.deceasedTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–127, November–December, 1987.     

Numerical solution of the Navier-Stokes equations for flow of gas around a rectangle

A difference scheme of the Lax-Wendroff type was used to solve the Navier-Stokes equations for the problem of flow of a perfect gas around a rectangle (cylinder of rectangular cross section) with and without a front plate for subsonic, transonic, and small supersonic flow velocities (Mach numbers 0.3M2 and Reynolds numbers 1R250). The occurrence and development of front, lateral, and rear separation of the flow are discussed, and the aerodynamic characteristics of the considered bodies are determined.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–17, July–August, 1972.     

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances.The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M=1.50– and dimensionless oscillation frequency l/V=0–1.0. A comparison is made with the results obtained from the Newton theory.     

Some properties of homogeneous flows of rarefied gases

A generalization of the existence conditions for homogeneous flows of a rarefied monatomic gas mixture [2, 3] to the case where external forces are present is presented in [1]. Below we obtain for this case the solution of the Cauchy problem for the Boltzmann equation under free molecular (collisionless) conditions, when the collision integrals may be neglected (Knudsen number K 1). On the basis of this solution we construct a general solution for the equations of the kinetic moments of a Maxwellian monatomic gas mixture in the form of a series in inverse powers of K. Some additional remarks are made concerning the properties of the solutions of the second-order kinetic moment equations, and on the applicability of the Grad 13-moment equations and the Chapman-Enskog method [in particular, for the calculation of slow (Stokesian) motions of a gas mixture].The authors wish to thank M. N. Kogan and A. A. Nikol'skii for their comments.     

Self‐Sustained Oscillations in Supersonic Overexpanded Impact Jets

The influence of geometric and gasdynamic parameters on the flow structure and parameters of selfsustained oscillations in supersonic overexpanded jets interacting with a normally located plane finite obstacle is studied experimentally and theoretically. It is found that the geometric Mach number and the halfangle of nozzle expansion exert a significant effect on the interaction process. A comparison of experimental and numerical data allows one to find a possible reason for the emergence of selfsustained oscillations in overexpanded impact jets.     

On Regularity of Stationary Stokes and Navier-Stokes Equations near Boundary

We obtain local estimates of the steady-state Stokes system without pressure near boundary. As an application of the local estimates, we prove the partial regularity up to the boundary for the stationary Navier-Stokes equations in a smooth domain in five dimension.     

Effect of heat flow on heat‐transfer characteristics for a supersonic spatial flow around a spherically blunted cone

Heattransfer processes for a supersonic spatial flow around a spherically blunted cone were studied by solving direct and inverse threedimensional problems taking into account heat flow along the longitudinal and circumferential coordinates. It is shown that highly heatconducting materials can be used to advantage to decrease the maximum temperatures on the windward side of streamline bodies.     

A particle transport problem with a quadratic nonlinearity and general boundary conditions: The stationary solution

Summary We study a stationary, nonlinear, particle transport problem in slab geometry with general boundary conditions. The existence and uniqueness of the solution is proved by means of fixed point techniques, provided that the source term is sufficiently small.

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Sommario Si studia un problema stazionario nonlineare di particelle in geometria piana con condizioni al contorno generali. L'esistenza e unicitá della soluzione è dimostrata con tecniche di punto fisso purchè il termine di sorgente sia sufficientemente piccolo.

     

Laminar boundary layer in axisymmetric swirling ducted gas flow

Several studies have been published [1–3] in which the authors solve the problem of the laminar boundary layer in an incompressible fluid on the walls of an axisymmetric duct in the presence of swirl in the outer flow. In [3], Loitsyanskii's parametric method of [4, 5], generalized to the case of three-dimensional flow, is used to solve this problem.In this article the parametric method for integrating the universal equations is extended to the solution of the problem of the laminar boundary layer on the wall of an axisymmetric channel in the case of swirling gas flow.     

Asymptotic theory of wave-propagation

A general method is presented for finding asymptotic solutions of problems in wave-propagation. The method is applicable to linear symmetric-hyperbolic partial differential equations and to the integro-differential equations for the electromagnetic field in a dispersive medium. These equations may involve a large parameter . In the electromagnetic case is a characteristic frequency of the medium. The parameter may also appear in initial data or in the source terms of the equations, in a variety of different ways. This gives rise to a variety of different types of asymptotic solutions. The expansion procedure is a ray method, i.e., all the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. In general, these rays do not lie on characteristic surfaces, but may, for example, fill out the interior of a characteristic hypercone. They are associated with an appropriately defined group velocity. In subsequent papers the ray method developed here will be applied to the analysis of transients, Cerenkov radiation, transition radiation, and other phenomena of wave-propagation.An interesting by-product of the ray method is the conclusion, derived in section 6.3, that the theory of relativity imposes no restriction on the speed of energy transport in anisotropic media.This research was supported by the Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract No. AF 19(628)4065     

Some self-similar solutions of the Leibenzon equation

Self-similar one-dimensional solutions of the Leibenzon equation c²_t= _zz ^k (z 0, k 2) are considered. Approximate solutions are constructed for the two cases in which the initial value = ₁ = const > 0 and on the boundary either a constant value = ₂ < ₁ is maintained or the flow (directed outwards) is given. In the first problem the dependence of the boundary flow on the governing parameters is determined. A characteristic property of the types of motion in question is the existence near the boundary of a region, expanding with time, in which the flow is almost independent of the coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–150, September–October, 1991.