Development of taylor instability in compressible conducting media in a magnetic field

Article [1] is devoted to the investigation of the interface of viscous incompressible conducting media in the presence of a current and with a magnetic field with a small magnetic Reynolds number. Article [2] discusses the stability of a contact discontinuity in compressible media. In this article, in an analysis of the case of long-wave vibrations in the region z<0, the boundary condition is unsatisfactorily met. Therefore, in this part of it the author actually considered a problem with a mass force f=(0, 0, –g sign z) instead of f=(0, 0, –g). The present, article considers the stability of the interface of compressible conducting media in a magnetic field. It is postulated that the magnetic intensity can undergo a discontinuity at this boundary. The article gives the dependence of the maximal increment of the rise in the instability on the determining parameters. An analysis is made of the stability of a contact discontinuity as a function of the angles formed by the wave vector and the intensity of the magnetic field. The stabilizing effect of the walls on the stability is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, 15–19, September–October, 1975.The author thanks A. G. Kulikovskii for his aid in stating the problem and for his continuing interest in the work.     

Electroconvective jets in liquid dielectrics

The principal feature of electroconvective jets in liquid dielectrics developing under the influence of a high-voltage external field is the large value of the EHD interaction parameter. This leads to the coupling of the hydrodynamic and electric problems. As formulated in [1, 2] the situation is reversed: the EHD interaction parameter is small. In these problems the interest is usually confined to finding the electric characteristics of the jet for a given velocity field. In [3] flows from sharp electrodes in liquid dielectrics were analyzed under two principal assumptions: nonlinear ohmic conductivity and point EHD interaction. This paper deals with the calculation of submerged electroconvective jets with ionic conductivity on the basis of the boundary-value problem formulated in [4]. In this case point EHD interaction is not assumed. It should be noted that in this formulation the problem is of practical as well as theoretical interest, for example, in connection with the problem of designing throttle EHD converters [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 13–19, November–December, 1984.     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Subsonic flow of a compressible gas around a semiinfinite flat plate in a pipe

The subsonic flow of an ideal compressible gas around the rear end of a semiinfinite flat plate in a pipe is considered. The flow pattern is similar to that assumed by Efros [1, 2] with a return stream for the cavitational flows. Fal'kovich's method [3] is used to solve the problem and this makes it possible to obtain the solution to the problems of the gas streams at several typical velocities. The method is a generalization of that of Chaplygin [4] for flow problems at one typical velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 101–108, July–August, 1970.     

Asymptotic analysis of the behavior of an elastic bar under aperiodic intensive loading

It is established that when aperiodic loads of large intensity act on an elastic bar, the higher modes of stability loss have the highest rates of growth of deflections. A method is indicated for determining the numbers of these modes, when the effect of the longitudinal vibrations on the transverse vibrations is taken into account and when it is not taken into account. A comparison of the results obtained with results of other authors [1–7] is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 160–170, May–June, 1972.     

Study of the Resonant Vibrations and Dissipative Heating of Thin-Walled Elements Made of a Physically Nonlinear Material

An approximate formulation is given to a dynamic coupled thermomechanical problem for physically nonlinear inelastic thin-walled structural elements within the framework of a geometrically linear theory and the Kirchhoff–Love hypotheses. A simplified model is used to describe the vibrations and dissipative heating of inhomogeneous physically nonlinear bodies under harmonic loading. Nonstationary vibroheating problem is solved. The dissipative function obtained from the solution for steady-state vibrations is used to simulate internal heat sources. For the partial case of forced vibrations of a beam, the amplitude–frequency characteristics of the field quantities are studied within a wide frequency range. The temperature characteristics for the first and second resonance modes are compared.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Solution of the temperature jump (Knudsen layer flow) and linear heat-transfer problems for two parallel plates in a rarefied gas

The Monte Carlo method [1, 2] is used to solve the linearized Boltzmann equation for the problem of heat transfer between parallel plates with a wall temperature jump (Knudsen layer flow). The linear Couette problem can be separated into two problems: the problem of pure shear and the problem of heat transfer between two parallel plates. The Knudsen layer problem is also linear [3] and, like the Couette problem, can be separated into the velocity slip and temperature jump problems. The problems of pure shear and velocity slip have been examined in [2].The temperature jump problem was examined in [4] for a model Boltzmann equation. For the linearized Boltzmann equation the problems noted above have been solved either by expanding the distribution function in orthogonal polynomials [5–7], which yields satisfactory results for small Knudsen numbers, or by the method of moments, with an approximation for the distribution function selected from physical considerations in the form of polynomials [8–10]. The solution presented below does not require any assumptions on the form of the distribution function.The concrete calculations were made for a molecular model that we call the Maxwell sphere model. It is assumed that the molecules collide like hard elastic spheres whose sections are inversely proportional to the relative velocity of the colliding molecules. A gas of these molecules is close to Maxwellian or to a gas consisting of pseudo-Maxwell molecules [3].     

Determining the form of a body resulting in minimum heat flux,with laminar flow in the boundary layer

The exact solution of the problem of determining the optimal body shape for which the total thermal flux will be minimal for high supersonic flow about the body involves both computational and theoretical difficulties. Therefore, at the present time wide use is made of the inverse method, based on comparing the thermal fluxes for bodies of various specified form [1, 2]. The results of such calculations cannot always replace the solution of the direct variational problem. Therefore it is advisable to consider the direct variational problem of determining the form of a body with minimal thermal flux by using the approximate Newton formula for finding the gasdynamic parameters at the edge of the boundary layer. This approach has been used in finding the form of the body of minimal drag in an ideal fluid [3–5] arid with account for friction [6], and also for determining the form of a thin two-dimensional profile with minimal thermal flux for given aerodynamic characteristics [7].     

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.     

Advance of the interface of liquids with pressure filtration

The article is a continuation of the investigations of a number of authors [1–3] on the advance of the interface between different liquids: it gives a solution of the problem in the case of one-dimensional filtration of liquids with different viscosities in distorted layers of variable permeability; it indicates a method for determining the interface between multicolored liquids, and the method is generalized for the case of a series of two-dimensional flows, connected by a conformai transformation. The article discusses problems that reduce to formation of the functions determining the flow, and to calculation of the integrals.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 185–191, September–October, 1971.     

Solute dispersion in a pulsating flow

The one-dimensional model proposed by Taylor [1] of the dispersion of soluble matter describes approximately the distribution of the solute concentration averaged over the tube section in Poiseuille flow. Aris [2] obtained more accurately the effective diffusion coefficient in Taylor's model and solved the problem for the general case of steady flow in a channel of arbitrary section. Many papers have been published in the meanwhile devoted to particular applications of this theory (for example, [3–5]). Various dispersion models have been constructed [6–8] that make the Taylor—Aris model more accurate at small times and agree with it at large times. The acceleration of the mixing of the solute considered in these models in the presence of the simultaneous influence of molecular diffusion and convective transport also operates in unsteady flows. In particular, the presence of velocity pulsations influences the growth of the dispersion even if the mean flow velocity is equal to zero at every point of the flow. In the present paper, the Taylor—Aris theory is extended to the case of laminar flows with periodically varying flow velocity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 24–30, September–October, 1982.     

Solution of the problem of a flow of rarefied gas of constant density around a semiinfinite plate by the integral diffusion method

Several papers [1–4] have proposed approximate diffusion models which can be used to examine the transport process in a rarefied gas where the mean free path is large and transport is not determined by the local gradient of the particular quantity.In this paper the integral diffusion model [2] is used to solve the problem of determination of the friction stress and velocity of a flow of an incompressible gas around a plane semi-infinite plate in the whole range of Knudsen numbers. The obtained solution is compared with published solutions and experimental data [9].     

Simulation of ordered structures in open turbulent flows

Extensive experimental material [1–4] indicates that ordered (coherent) structures play an important part in determining the nature of the flow, the generation of Reynolds stresses and turbulence energy, and the transport of heat, momentum, and passive admixtures in a turbulent flow. In the present paper, a model is constructed for describing coherent structures in which, given the profile of the mean velocity, one can determine the characteristic sizes, the propagation velocities, and also the frequency and amplitude characteristics of these ordered motions. The model is based on the analogy between the ordered formations and secondary flows in a subsidiary laminar flow whose velocity profile is the same as the turbulent profile of the mean velocity. The influence of small-scale pulsations is described by the introduction of the coefficient of turbulent viscosity. In the framework of the model, numerical calculations are made for two-dimensional turbulent flows in a mixing layer, a jet, and a wake behind a cylinder. The results of the calculations are compared with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 45–52, July–August, 1981.     

Resonance Vibrations and Dissipative Heating of Thin-Walled Laminated Elements Made of Physically Nonlinear Materials

The dynamic thermomechanical problem for thin-walled laminated elements is formulated based on the geometrically linear theory and Kirchhoff–Love hypotheses. A simplified model of vibrations and dissipative heating of structurally inhomogeneous inelastic bodies under harmonic loading is used. The mechanical properties of materials are described using strain-dependent complex moduli. A nonstationary vibration-heating problem is solved. The dissipative function, derived from the stationary solution, is used to specify internal heat sources. The amplitude–frequency characteristics and spatial distributions of the main field variables are studied for a sandwich beam subjected to forced vibrations     

Multizone combustion of condensed systems

In the general case the combustion of condensed systems is of a stage-wise character and the combustion front is multizone [1, 2]. Following the investigation of two-zone models [3–5] it became clear that, during multizone combustion, one of the zones of heat evolution is the controlling zone. The velocity of the front is equal to the velocity of the controlling zone; however, with a change in the parameters of the system, there is the possibility of a transition of the controlling role from one zone to another, as well as of the coalescence and splitting of zones. This paper discusses a generalization of the two-zone problem which makes it possible to go over to the analysis of a complex, multizone front and shows that, for a front with two reactions (in the condensed phase and in the gas) and with dispersion, there are in all three possible arrangements of the zones of heat evolution (two three-zone variants and one two-zone variant). All possible types of dependence of the combustion rate on the depth of the dispersion are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 99–105, November–December, 1972.     

Small vibrations of a sphere in a spherical volume of viscous fluid

Problems of the vibration of bodies in confined viscous fluids have been solved to determine the added masses and damping coefficients of rods [1–3] and floats [4–5]. The solutions of these problems, based on the use of simplifications of the boundary-layer method [4–6], are obtained analytically in general form and are in good agreement with the experimental data. However, in each specific case the possibility of using such solutions for given values of the fluid viscosity and vibration frequency must be justified either experimentally [2, 4, 5] or theoretically as, for example, in [1], where an analytic solution was obtained for concentric cylinders. The present paper offers a general solution of the problem of the small vibrations of a sphere in a spherical volume of fluid valid over a broad range of variation of the dimensionless kinematic viscosity. The limiting cases of this solution for both high and low viscosity are considered. The asymptotic expressions obtained are compared with calculations based on the analytic solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 29–34, March–April, 1986.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Two methods for calculating the load on the surface of a slender body executing axisymmetric vibrations in a sonic gas flow

Low-frequency axisymmetric vibrations of the surface of a slender body in a sonic flow are considered. The distribution of the stationary longitudinal velocity on the body is assumed to be linear. The linear equation with variable coefficients for the nonstationary part of the velocity potential is solved by two methods: by separation of the variables, as was done in [1] for a two-dimensional flow, and by the method of superposition of sources. Particular solutions with the required singularity are obtained.Translated from Izvestiya Akaderaii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 151–154, March–April, 1980.     

Asymptotic theory of the flow around an obstacle by a sonic flow

The first investigation of the problem of the flow around an obstacle by a gas flow whose velocity is equal to the speed of sound at infinity was carried out in [1, 2], where it is shown in particular that the principal term of the appropriate asymptotic expansion is a self-similar solution of Tricomi's equation, to which the problem reduces in the first approximation upon a hodographic investigation. The requirement that the stream function be analytic as a function of the hodographic variables on the limiting characteristic was an important condition determining the selection of the self-similarity exponent n (xy^–n is an invariant of the self-similar solution). The analytic nature of the velocity field everywhere in the flow above the shock waves, which arise from necessity upon flow around an obstacle, follows from this condition. The latter was found in [3], where one of the branches of the solution obtained in [1] was used in the region behind the shock waves. The principal and subsequent terms of the asymptotic expansion describing a sonic flow far from an obstacle were discussed in [4], where the author restricted himself to Tricomi's equation. Each term of the series constructed in [4] contains an arbitrary coefficient (we will call it a shape parameter) which is not determined within the framework of a local investigation, and consideration of the problem of flow around a given obstacle as a whole is necessary in order to determine these shape parameters. It follows from the results of [4] that the problem of higher approximations to the solution of [1] coincides with the problem, of constructing a flow in the neighborhood of the center of a Laval nozzle with an analytic velocity distribution along the longitudinal axis (a Meyer-type flow). Along with the Meyer-type flow in the vicinity of the nozzle center, which corresponds to a self-similarity exponent n=2, two other types of flow are asymptotically possible with n=3 and 11, given in [5]. The appropriate solutions are written out in algebraic functions in [6]. The results of [5] show that the condition that the velocity vector be analytic on the limiting characteristic in the flow plane is broader than the condition that the stream function be analytic as a function of the hodographic variables, which is employed in [1, 2, 4]. Therefore, the necessity has arisen of reconsidering the problem of higher approximations for the obstacle solution of F. I. Frankl'. It has proved possible for the region in front of the shock waves to use a series which is more general than in [4], which implies the inclusion of an additional set of shape parameters. The solution is given in the hodograph plane in the form of the sum of two terms; the series discussed in [4] corresponds to the first one, and the series generated by the self-similar solution with n=3 or with n=11 corresponds to the second one.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 99–107, May–June, 1979.The authors thank S. V. Fal'kovich for a useful discussion.