Jet Outflow from a Small Hole in a Plane

On the basis of the method of matched asymptotic expansions, the problem of the outflow of a nonswirling axisymmetric laminar jet from a hole in a plane is solved for large Reynolds numbers. Since directly matching the leading terms of the asymptotic expansions for the axial boundary layer and the main flow region is impossible, the problem is solved by introducing an intermediate region. In the axial region the solution is the Schlichting solution [1] for an axisymmetric jet in the boundary-layer approximation, in the intermediate region the solution is found analytically, and in the main flow region the problem is reduced to that of viscous flow induced by a sink line in the presence of a transverse wall [2].     

One-dimensional nonstationary motions of a liquid

An effective method is developed for solving the problem of the nonstationary motion of a liquid with plane, cylindrical, and spherical symmetry [1]. It is based on the idea of dividing the region of disturbed motion into two parts and using matched asymptotic expansions. Solutions are presented to typical problems associated with the motion of a piston, and these make it possible to obtain the solution to problems of an explosion in a liquid, oscillations of a bubble, and so forth. It is also shown that the well-known solutions for such problems given, for example, in the book of Naugol'nykh and Roi on the basis of the acoustic approximation with allowance for nonlinear terms are incorrect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1980.     

Nonlinear analysis of the flow initiated by the sudden motion of a wedge

Certain self-similar problems involving the sudden motion of a wedge which were treated in the linear approximation in [1–3] are studied by the method of matched asymptotic expansions. The nature of the wave boundary of the perturbed region is determined. Second-approximation solutions are constructed which describe flows behind weak shock fronts propagating in a stationary gas and behind fronts of weak discontinuity lines propagating by known uniform flows. A boundary-value problem is formulated whose solution describes, in first approximation, flows in the neighborhoods of points of interaction of the fronts. The existence of similarity rules of flows in these nieghborhoods is estimated. An approximate solution of the problems is given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–47, May–June, 1976.     

Friction and heat transfer in an incompressible fluid near a plane stagnation point

In the neighborhood of a plane stagnation point, the flow and heat transfer of an incompressible fluid are studied. In the inner flow region, the velocity and pressure fields are described by the complete Navier-Stokes equations, and the temperature field is described by the complete energy equation. In the outer flow region, a two-term asymptotic solution of the corresponding equations is obtained. The problem is reduced to the numerical solution of ordinary differential equations. Numerical results are discussed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–65, July–August, 1996.     

Optimization of suction-blowing in a viscous fluid

The problem of drag minimization in a viscous fluid by means of controlled suction (blowing) is considered. In the low Reynolds number approximation matched asymptotic expansions are used to construct the second approximation and analytic solutions of the optimization problem are found for a sphere and a circular cylinder. Transition from unseparated to separated flow is accompanied by a qualitative restructuring of the optimal solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–32, May–June, 1989.     

Flow past a plate under the free surface of a weightless liquid

A study is made of the nonlinear problem of the flow without separation of a perfect weightless liquid past a plate near the free surface. This problem was first posed by Gurevich [1]. At present, there are only a general solution to the problem [2–4] and some numerical calculations [5], which have been made under definite restrictions and are inadequate for detailed information about the interaction between the free surface and the plate. In the present paper, a complete investigation of the problem is given. Convenient computational formulas are obtained together with asymptotic expansions of them, and detailed calculations are made for all depths of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–162, January–February, 1980.     

Asymptotic nonlinear solution of the problem of the penetration of a compressible fluid by a slender body

A combined solution of the problem of the penetration of a compressible fluid by a slender wedge and a cone is found by the method of matched asymptotic expansions. The new solution is based on taking into account the nonlinear terms in the Cauchy-Lagrange integral and is uniformly applicable in the neighborhood of the nose.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 49–57, May–June, 1993.     

Asymptotic theory of flow attachment for a viscous incompressible fluid

A study is made of the two-dimensional laminar flow of a viscous incompressible fluid in the neighborhood of the point of attachment of the flow to a solid surface. The case of large Reynolds numbers is considered. It is assumed that the dimensions of the separation region are of the same order of magnitude as the characteristic dimension of the body around which the flow takes place. The asymptotic theory of such flow is constructed by applying the method of matched asymptotic expansions to the analysis of the Navier-Stokes equations. It is shown that in the neighborhood of the attachment point the flow is locally inviscid and can be described by the complete system of Euler equations. A solution to the corresponding boundary-value problem is constructed numerically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 63–71, November–December, 1980.     

Influence of transverse slits on the hydrodynamic coefficients of a wing of finite span in stationary and nonstationary motion near a wall

The method of matched asymptotic expansions is used to construct an approximate solution to the problem of the influence of narrow transverse slits on the hydrodynamic coefficients of a thin rectangular wing moving near a wall. The flow in the neighborhood of a slit is described by a local asymptotic solution satisfying the condition of continuity of the pressure on the leading edge of the slit and matched to the main solution. Results of the calculations illustrate the influence of the slits on the hydrodynamic characteristics of the wing at different Strouhal numbers and aspect ratios.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–128, November–December, 1980.     

Transonic gas flow over a flat plate

The solution of the two-sided Tricomi problem in the hodograph plane is constructed with satisfaction of the entire set of boundary conditions, which ensures its correct asymptotic behavior with respect to vanishing angle of attack. As a result, it is found that the deviation from the Guderley solution begins with the singular terms.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–137, January–February, 1987.     

Asymptotic analysis of a pulsating flow of viscous fluid in a symmetric deformed channel

The time-periodic flow of a viscous incompressible fluid in a two-dimensional symmetric channel with slightly deformed walls is considered. The solution of the Navier-Stokes equations is constructed by means of the method of matched asymptotic expansions [1] at large characteristic Reynolds numbers. It is shown that in an unsteady flow a region of nonlinear perturbations surrounds the line of zero velocity inside the fluid. The formation and development of such nonlinear zones with respect to time is considered. An alternation of the topological features of the streamline pattern in the nonlinear perturbation zone is discovered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 17–23, July–August, 1987.The author is deeply grateful to V. V. Sychev for his formulation of the problem and his attentive attitude to my work.     

Influence of moisture transfer in the aeration region on the thermal regime of soil

One-dimensional heat and moisture transfer in the aeration region of reclaimed land is considered for two limiting cases. In the first, the heat transfer is mainly due to the thermal conductivity of the soil; in the second, to the motion of moisture. The influence of the water-table depth on the thermal regime is investigated. A solution to the problem of unsteady heat transfer in soil is found analytically by the method of matched asymptotic expansions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 63–71, September–October, 1981.We thank V. S. Berman for discussions and valuable comments.     

Turbulent flow of a fluid in the neighborhood of the trailing edge of a plate at zero angle of attack

The laminar flow regime of an incompressible fluid at the trailing edge of a plate was studied by Stewartson and Messiter [1, 2] by means of the method of matched asymptotic expansions. In. the present paper, this method is used to analyze the same problem, but in the case of turbulent flow in the boundary layer and the wake. A system of linear equations of elliptic type with variable coefficients is obtained for the averaged values of the flow parameters in the main part of the boundary layer and the wake that is responsible for the change in the displacement thickness. A solution of this system is constructed by the Fourier method in the case of a power law of the velocity in front of the interaction region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 17–23, November–December, 1983.     

Asymptotic theory of short separation regions on the leading edge of a slender airfoil

The two-dimensional flow of a viscous incompressible fluid near the leading edge of a slender airfoil is considered. An asymptotic theory of this flow is constructed on the basis of an analysis of the Navier—Stokes equations at large Reynolds numbers by means of matched asymptotic expansions. A central feature of the theory is the region of interaction of the boundary layer and the exterior inviscid flow; such a region appears on the surface of the airfoil in a definite range of angles of attack. The boundary-value problem for this region is reduced to an integrodifferential equation for the distribution of the friction. This equation has been solved numerically. As a result, closed separation regions are constructed, and the angle of attack at which separation occurs is found.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 42–51, January–February, 1981.I thank V. V. Sychev and Vik. V, Sychev for assistance.     

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.     

Indentation of a Rigid Body into an Elastic Plate

The problem of a punch shaped like an elliptic paraboloid pressed into an elastic plate is studied under the assumption that the contact region is small. The action of the punch on the plate is modeled by point forces and moments. The method of joined asymptotic expansions is used to formulate the problem of one–sided contact for the internal asymptotic representation; the problem is solved with the use of the results obtained by L. A. Galin. The coordinates of the center of the elliptic contact region, its dimensions, and the angle of rotation are determined. The moments which ensure translational indentation of the punch are calculated and an equation that relates displacements of the punch to the force acting on it is given.     

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.     

Wave propagation in the combustion of a low-concentration aerial suspension

In the general case the convective combustion of aerial suspensions is described by the equations of mechanics of multiphase media [1]. If the volume particle content is neglected and it is assumed that in the initial stage of convective front propagation the particles are stationary, and that during combustion their temperature is constant, then the equations for describing the combustion process reduce to the equations of gas dynamics for a distributed supply of heat and mass [2, 3]. The equations and model constant mass burning rate kinetics are used to solve the plane one-dimensional problem of the combustion of an aerial suspension in part of a region bounded on one side by a fixed wall. A small parameter proportional to the mass concentration and the heat value of the fuel is introduced. The method of matched asymptotic expansions [4] is used to construct a uniformly applicable first approximation. The solution obtained describes the wave propagation in aerial suspension combustion processes. The resulting pattern includes an inclined compression wave propagated with the speed of sound followed by a convective hot reaction product front whose propagation velocity is much less (in conformity with the small parameter introduced) than the speed of sound.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 63–73, March–April, 1986.     

On the asymptotic behavior of localized perturbations in the presence of Kelvin-Helmholtz instability

The asymptotic behavior of localized two-dimensional perturbations of the surface of a shear discontinuity separating two homogeneous steady flows of ideal incompressible fluid is studied in the linear approximation. The effect of surface tension and gravity forces is taken into account. Mathematically the problem reduces to the investigation by the method of steepest descent of the asymptotic behavior of a double integral for various values of parameters which are the components of the group velocity vector. In this problem the principal difficulty is to find the two-dimensional steepest descent contour in the space of two complex variables that determines which of the various saddle points gives the asymptotic form. First, for the Fourier component with respect to one of the variables with allowance for all the saddle points we find an asymptotic form which parametrically depends on the second variable. The choice of the second variable makes it possible to prove analytically that in the absence of gravity the asymptotic behavior of the growing perturbations is determined by a single saddle point in the plane of that variable. In this way it is possible to justify the authors' previous conclusions [1] concerning the shape of the boundary L of the region D in the group velocity plane occupied by growing perturbations. In the presence of gravity the growth rates of perturbations corresponding to different group velocities are found numerically and the region D occupied by the growing perturbations is indicated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 23–30, March–April, 1985.     

Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.