Method of representation of solution of the Blasius equation and its applications

In the articles published on the Blasius problem [1–3] the solutions are investigated either by some numerical method or by joining the series solution for small values of the argument with the asymptotic expansion for large values. In the present article we give a new method of representing the solutions for different boundary-value problems associated with the Blasius equation. The properties of the obtained solution are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–146, May–June, 1975.The author expresses deep gratitude to Yu. A. Dem'yanov for discussion of the work and for valuable comments.     

Application of the method of matched asymptotic expansions to the calculation of the stationary thermal propagation of the front of an exothermic reaction in a condensed medium

In this paper we use the method of matched asymptotic expansions to establish a two-term formula for the speed of propagation of the front of an exothermic reaction in a condensed medium whose thermophysical characteristics depend on the concentration of the reacting matter and the temperature. As the parameter of the expansion we use the ratio of the activation temperature to the adiabatic combustion temperature. The results are applied to the case of the combustion of nonvolatile condensed systems. We compare the approximate formula obtained with the results of a numerical integration.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 106–112, September–October, 1972.     

Asymptotic law of expansion of a cavity in the presence of hydrodynamic singularities in the flow

The asymptotic law of expansion of an axisymmetric half-body of finite resistance exposed to a flow of an ideal incompressible fluid and its relation with the force acting on the half-body were obtained in [1]. It is obvious that the deformation of the nose of the body will not change the form of the expressions found. It will be shown below how the asymptotic law and magnitude of hydrodynamic reactions change for half-bodies of finite resistance in the presence of sources (sinks) in the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 153–155, May–June, 1977.The author thanks G. Yu. Stepanov for attention to the work.     

Analysis of the transition of one-dimensional flows of a viscous heat conducting gas to asymptotic regimes

Asymptotic solutions are presented which describe the formation of one-dimensional flows of a viscous heat conducting gas at short and long times. Comparison of them with numerical calculations made in the study for the system of Navier-Stokes equations in conjunction with an estimate of the ratio of each term in the asymptotic expansion to the preceding term leads to dimensionless-number dependences that determine the limits of applicability of the obtained solutions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 110–118, September–October, 1981.     

Supersonic gas flow over a semi-infinite plate with small harmonic spanwise disturbances

The method of matched asymptotic expansions is used to investigate the problem of supersonic perfect-gas flow over a semi-infinite surface with longitudinal ribbing formed by imposing small transverse harmonic perturbations on a flat plate. The ratio of the maximum amplitude of the surface perturbations to the thickness of the boundary layer is of the order of Re^–1/4. The problem is solved with allowance for four terms of the expansion.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 146–156, January–February, 1993.     

Stability of the integral manifold of an oscillation system with slowly varying frequencies and pulse action

Using uniform estimates for oscillation integrals and sums, we prove theorems on asymptotic and conditional asymptotic stability of the integral manifold of a multifrequency system of ordinary differential equations with pulse action at fixed times.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 365–394, July–September, 2004.     

Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation

This paper deals with the most important characteristics of a generalized Van der Pol–Duffing oscillator in resonance with a periodic excitation. We use an asymptotic perturbation method based on Fourier expansion and time rescaling and demonstrate through a second order perturbation analysis the existence of one or two limit cycles. Moreover, we identify a sufficient condition to obtain a doubly periodic motion, when a second low frequency appears, in addition to the forcing frequency. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.     

Estimates of the normal velocities of propagation of laminar and very small-scaled turbulent flames

On the most general assumptions (taking account of the Lewis-Semenov number, thermal expansion, variability of thermophysical parameters, etc.), analytical estimates are obtained for the normal velocities of combustion of laminar and turbulent flames. In the case of an Arrhenius dependence of the reaction velocity on the temperature, the combustion velocity is represented by an asymptotic series with respect to the Frank-Kamenetskii dimensionless temperature; for turbulent flames, with respect to a parameter of the relative scale of turbulence. The final results over a wide range of change of parameters are compared with a numerical calculation on a computer of the exact equations and with the relations obtained by the method of combined asymptotic expansions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 28–37, May–June, 1976.     

Operating regimes of a chemical reactor with longitudinal and transverse mixing in the case of large Péclet numbers

A two-dimensional model of a chemical reactor with longitudinal and transverse mixing is investigated in the case of large Péclet numbers calculated from the effective thermal conductivity in the transverse direction. For this model the existence of at least one steady-state regime has been demonstrated [1], sufficient criteria of its uniqueness have been determined, an asymptotic expansion of the solution has been constructed in the case of small Péclet numbers, and the critical ignition and quenching parameters have been found. In this paper the other limiting case of the model, in which heat is propagated in the transverse direction much more slowly than it is transported by the flow along the reactor (large Péclet numbers), is analyzed in detail. An asymptotic expansion of the solution which closely coincides with the data of numerical calculations is constructed. The critical quenching and ignition conditions of the process are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 120–127, January–February, 1987.     

Certain Aspects of an “Incompressible Fluid“ Model in Gas Dynamics

An “incompressible fluid” model in gas dynamics is developed in the linear approximation. Using the dissipative relaxation time as a characteristic scale, we arrive at another form of the dimensionless Boltzmann equation. In the limiting case of small Knudsen numbers an approximate solution is obtained in the form of a Hilbert multiple-scale asymptotic expansion. It is revealed that for slow, weakly nonisothermal processes the asymptotic expansion for the linearized Boltzmann equation leads in a first stage to equations for the velocity, pressure and temperature that do not contain the density (quasi-incompressible approximation). The density depends on the temperature and can, if necessary, be found from the equation of state. The next-approximation equations contain the Burnett effects, the velocity calculation being reduced to the general problem of finding a vector field from a given divergence and rotation. With reference to a simple case of the heating of a stationary gas in a half-space it is shown that the temperature establishment process is accompanied by gas flow from the wall.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 170–178.Original Russian Text Copyright © 2005 by Chekmarev.     

Supersonic flow past a blunt wedge

A study is made of the asymptotic solution of the problem of flow past a blunt wedge by a uniform supersonic stream of perfect gas. By separation of variables it is shown that at large distances the disturbance of the flow is damped exponentially. In the case of subsonic flow behind the shock wave the exponent of the leading correction term in the expansion of the shock front is calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1984.     

Consideration of longitudinal diffusion in flow within a channel

The stationary problem of convective diffusion in a channel with absorbent walls is considered. It is assumed that a Poiseuille flow exists. Two methods are employed in the solution, the method of separation of variables, and the method of expansion in eigenfunctions of the corresponding problem with piston profile (expansion method). It is established by comparison with independently obtained solutions for high Peclet number that for the first eigenfunctions and eigenvalues the expansion method gives satisfactory results over the entire Peclet-number range. For approximate calculation of subsequent eigenfunctions and eigenvalues a modification of the smooth asymptotic expansion method is used. The results are used to calculate matter flow density on the wall, to evaluate the length of the entrance region, and to obtain an analytical expression for the limiting Nusselt number in terms of the Peclet number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 63–73, November–December, 1973.     

Development of a two-dimensional turbulent jet in an infinite space

The second and third terms in the asymptotic expansion of the stream function in the nonsimilar problem of the development of a two-dimensional turbulent jet in an unbounded space are found in final form. Results of experimental investigations of free turbulent jets are cited, and the effect of the initial velocity profile on the aerodynamic characteristics of the jet is considered. The problem of the development of a two-dimensional turbulent jet in an unbounded space has been considered in [1–3]. The existing solution is similar, and is valid only at a sufficiently large distance from the slit. Allowance for the finite dimensions of the slit leads to a nonsimilar problem. The papers [4–6] are devoted to the experimental investigation of the free two-dimensional turbulent jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1971.     

Mass transfer between bubbles and the continuous phase in a pseudofluidized layer

The problem of mass transfer between an isolated bubble and the continuous phase in a pseudofluidized layer is considered, when the rising velocity of the bubble exceeds the pseudofluidization rate. In this case the bubble with the surrounding region, a so-called two-phase system, is surrounded by a surface current impermeable to the liquid [1–3], and the problem reduces to determining the concentration field and the total flow on the material surface. The problem is solved for large and small Peclet numbers by a boundary layer diffusion method and by asymptotic expansion matching.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–49, July–August, 1973.     

Asymptotic Solutions of a First-Order Differential Equation with Deviating Argument and Slowly Varying Coefficients

The object of this paper is to study the problem of constructing an approximate solution of a first-order weakly nonlinear ordinary differential equation with deviating argument and slowly varying coefficients. On the basis of asymptotic techniques in nonlinear mechanics, we construct an algorithm for the asymptotic integration of the differential equation under consideration.__________Published in Neliniini Kolyvannya, Vol. 7, No. 4, pp. 475–486, October–December, 2004.     

Asymptotic of the electromagnetic field above a layerlike anisotropic medium of high conductivity

The model of an anisotropic, layerlike medium is often employed in problems of electromagnetic probing, and many papers have been written on the propagation of an electromagnetic field in such media. A systematic exposition of such problems was set out by Tikhonov, Sku-garevskaya, and Dmitriev [1–6]. In this paper we shall construct an asymptotic for the electromagnetic field of a point source lying above a layerlike medium of finite anisotropy having a fairly high longitudinal and transverse conductivity, or in which the source lies at a considerable height above the medium. The principles here laid down for the construction of the asymptotic indicate quite clearly under what circumstances the asymptotic is feasible, and if necessary allow the next approximations to be taken into account.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 175–188, March–April, 1966.     

Heat and mass transfer to a reacting particle in a gas flow for arbitrary temperature dependence of the transport coefficients

A study is made of simultaneous heat and mass transfer to a reacting particle of any shape in a translational (and shear) flow of a viscous heat conducting compressible gas for which the thermal conductivity and diffusion coefficient, and also the specific heat depend on the temperature. The first two terms of the asymptotic expansion with respect to the small Reynolds number are obtained for the mean Sherwood and Nusselt numbers. The case of a power-law temperature dependence of the gas viscosity is considered in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–119, January–February, 1984.     

Contact Problem for a Narrow Annular Punch. Unknown Region of Contact

The nonlinear problem of determining the contact stresses and the contact zone under the base of a narrow annular punch is studied. An asymptotic model of one–sided contact along the line is constructed by the method of matched asymptotic expansions. Explicit asymptotic formulas for the line–pressure density are obtained. The asymptotic representation of the contact arc is given.     

Nonaxisymmetric elastoplastic impact of a parabolic body on a spherical shell

A method is proposed to calculate a spherical shell under nonaxisymmetric impact of a massive body. The motion of the shell is described by momentless equations, which are solved using the Laplace transformation and an asymptotic expansion of the required quantities in a small parameter. The contact interaction force P(t) was determined for the elastoplastic model of local bearing deformation for a parabolic impactor. Plots of the solution are given. The validity of the results is confirmed by good agreement between the solution and the limiting cases — an axisymmetric impact and an impact on a half-space.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 181–186, January–February, 2005.     

Asymptotic analysis of stationary distribution of the front of a two-stage consecutive exothermic reaction in a condensed medium

An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in the article. The method of joined asymptotic expansions is used in constructing the solutions. The ratio of the sum of the activation energies of the reactions to the final adiabatic combustion temperature is a parameter of the expansion. The characteristic limiting states of the stationary distribution of the wave corresponding to different values of the parameters figuring in the problem are shown. Approximate analytical expressions for the wave velocity and distribution of concentrations are obtained for each of the states.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 75–87, January–February, 1973.