Numerical modeling of thermocapillary convection in a liquid layer with a nonlinear dependence of the surface tension on temperature

The principal characteristics of thermocapillary convection in a rectangular channel with one of the boundaries heated to a temperature higher and the other to a temperature lower than T₀ are investigated numerically on the basis of the Navier-Stokes equations. Certain convection characteristics corresponding to normal and anomalous thermocapillary effects are qualitatively compared. The conditions under which self-similar solutions of the type obtained in [10] can be used to describe the flow in a bounded region are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 138–143, January–February, 1991.     

Self-similar solution of the Navier-Stokes equations governing gas flows in rotary logarithmically spiral two-dimensional channels

A self-similar solution of the Navier-Stokes equations governing gas flows with constant transport coefficients in rotary log-spiral two-dimensional channels is obtained and analyzed. The solution and its existence depend on the following dimensionless parameters: the Reynolds number Re; the parameterM _o characterizing the channel rotation; the self-similarity parameters and responsible for the channel shape; the direction of channel rotation; and, finally, the wall temperature ratio. A numerical solution of the system of second-order ordinary differential equations gives the ranges of the governing parameters on which self-similar solutions for the gas flow in a rotary channel can exist.Perm'-Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 44–50, November–December, 1995.     

Analysis of the Applicability of Macroscopic Models of the Shock Wave Structure in a Binary Mixture of Inert Gases

The results of calculating the shock wave structure in Ne–Ar, He–Ar, He–Ne, and He–Xe mixtures by means of the relaxation method on the basis of the system of Navier-Stokes equations and complete and modified systems of Burnett equations are compared with the results of direct statistical simulation (Monte-Carlo method). The domain of applicability of these systems of equations for calculating gas dynamic variable profiles is analyzed as a function of both the molecular mass ratio and the initialconcentrations.     

Derivation of the equations of slow nonisothermal gas flows

In recent years, some new phenomena have been predicted theoretically on the basis of the Burnett approximation. These include thermal-stress and concentration-stress convection [1–3], and also effects due to the influence of a magnetic field in a multiatomic gas (viscomagnetic heat flux, etc., [4]). It has been shown theoretically (see [5]) that under certain conditions various terms of the Burnett approximation must be taken into account in the expression for barodiffusion. The conclusions relating to a viscomagnetic heat flux have recently been confirmed experimentally [4]. The predicted phenomena follow rigorously from the Burnett equations. However, many hydrodynamicists adopt a sceptical attitude to these equations, which is due partly perhaps to attachment to the classical Navier-Stokes equations, which have served theoreticians without fail for a century and a half. In this connection, we discuss the evolution of ideas relating to the validity of the Burnett approximation. We discuss the minimal assumptions which must be made in order to derive the equations of slow [Reynolds number R = 0(1)], essentially nonisothermal [ ln T = 0(1)] flows of a gas as a continuous medium (Knudsen number K O) in the case when the derivatives of the thermal Burnett stresses in the momentum equation have the same order of magnitude as the Euler and Navier-Stokes terms of this equation [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 77–84, November–December, 1979.We thank G. I. Petrov and L. I. Sedov for discussions that stimulated the above analysis.     

Interaction between a distributed source and the free surface of a viscous fluid

A self-similar solution of the Navier-Stokes equations describing steady-state axisymmetric viscous incompressible fluid flow in a half-space is investigated. The motion is induced by sources or sinks distributed over a vertical axis with a constant density. The horizontal plane bounding the fluid is a free surface. It is found that in the presence of sources a solution of the above type exists and is unique for any value of the Reynolds numberR > 0, but in the case of sinks only on the interval –2 R < 0. The results of calculating the self-similar solutions are presented. The asymptotics of the solutions are found asR 0 andR .Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 53–65, March–April, 1996.     

Three-dimensional local separation

A solution to the problem of local separation of a three-dimensional boundary layer from an arbitrary smooth surface is constructed. Separation takes place along the limiting streamline at the points of which the component of the surface friction (calculated from the boundary-layer equations) that is orthogonal to this streamline has a break. An asymptotic expansion of the solution of the Navier-Stokes equations that describes the flow field in the separation region is found. The conclusions for the two-dimensional and self-similar theory of local separation are generalized to the three-dimensional case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 39–47, May–June, 1991.     

Burnett simulations of gas flow in microchannels

The Burnett equations with slip boundary conditions are used to model the gas flow in microchannels in transition flow regime. As the Navier-Stokes equations are not appropriate to model the gas flow in this regime, the higher-order Burnett equations are adopted in the present study. In earlier studies, convergent solutions of the Burnett equations of microPoiseuille flow could only be obtained when Knudsen number is less than 0.2. By using a relaxation method on the boundary values, convergent solutions of the Burnett equations can be obtained even when Knudsen number reaches 0.4. The solutions of Burnett equations agree very well with experimental data and direct simulation Monte Carlo (DSMC) results. The pressure distributions and velocity profiles are then discussed in detail.     

Viscous incompressible flow over the surface of a rotary disk

Viscous incompressible film flow over the surface of an impermeable rotary disk is studied. An exact self-similar solution of the complete Navier-Stokes system of equations is obtained and the velocity and pressure fields together with the radial profiles of the fluid film are determined. A physical interpretation of the results obtained is given.Volgograd. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 39–43, November–December, 1995.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

Memory of body shape in a far wake

The possibility of memory of the body shape in a far wake is investigated within the framework of the complete Navier-Stokes equations. It is shown that for a far wake there can be no memory of the shape as a result of the laminarization of the flow. On the interval of intermediate asymptotics the presence of memory leads to the problem of nonequivalence of the momentum sources and sinks for a self-similar turbulent wake, since there can be no memory of the shape for a source of momentum. The asymptotics for the average velocity defect in the wake are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 43–46, July–August, 1990.     

Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.     

Singularities in the boundary layer on a cone at incidence

The singularities in the three-dimensional laminar boundary layer on a cone at incidence are studied. It is shown that these singularities are formed in the outer part of the boundary layer and described by linear equations whose solutions are obtained in analytic form. The known results for the plane of symmetry are classified on this basis. Two solutions of the non-self-similar problem are found, one of which has a singularity at zero incidence and in the sink plane. The second branch goes over continuously into the solution for axisymmetric flow. However, as the angle of attack increases, in the sink plane a singularity is formed and all the self-similar solutions existing here lose their meaning. Starting from the critical angle of attack, the flow in the vicinity of the sink plane is no longer described by the boundary layer equations, so that the results can be used to construct an adequate physical model.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 25–33, November–December, 1993.     

Screw flow of a liquid in a cylindrical tube

A study is made of the flow of a viscous incompressible liquid with helical streamlines in an infinite cylindrical tube within which a screw rotates (auger). Generalized linearized Oseen equations are derived, and one class of the exact solutions of these is identical with the corresponding class of exact solutions of the complete Navier-Stokes equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 6, pp. 3–7, November–December, 1979.     

Asymptotic analysis of the equations of a three-dimensional boundary layer

The asymptotic solutions of the self-similar equations of two- and three-dimensional boundary layers have been investigated by many authors (see, for example, [1–3]). In [4, 5], asymptotic solutions were found for non-self-similar equations for two-dimensional flow, and the propagation of perturbations near the external edge of the boundary layer was analyzed. In the present paper, asymptotic solutions are obtained for the non-self-similar equations of a three-dimensional laminar boundary layer of an incompressible fluid. It is shown that the conclusion drawn in [5] — that the boundary conditions can be transferred from infinity to a finite distance from the wall — is also true for three-dimensional flow. The obtained solutions explain the experimentally well-known phenomenon of the conservativeness of the secondary currents. The essence of this phenomenon is that a change in the sign of the transverse (along the normal to a streamline of the external flow) pressure gradient is accompanied by a very rapid change in the direction of the secondary flow near the wall, whereas in the upper layers of the boundary layer the direction remains unchanged for a substantial time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 155–157, September–October, 1979.     

A class of solutions of the boundary layer equations

Exact solutions of the boundary layer equations can be obtained in closed form only in rare cases. These generally involve self-similar solutions for which the corresponding ordinary differential equation can be integrated exactly. In this paper solutions of more general form, containing additive functions of the longitudinal x coordinate in the expression's for the stream function and the self-similar variable, are considered in relation to two-dimensional steady boundary layers. This makes it possible to enlarge the class of problems whose solutions are analytic expressions and in a number of cases can be obtained in the form of expressions containing arbitrary functions of x, which makes possible various interpretations of the solution. In order to introduce arbitrary functions into the solutions of the equations of the axisymmetric boundary layer the problem is reduced to an overdetermined system of ordinary differential equations. This method is also capable of being applied more widely.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 45–51, March–April, 1990.     

Choice of a self-similar solution in boundary-layer theory

If the speed of the outer flow at the edge of the boundary layer does not depend on the time and is specified in the form of a power-law function of the longitudinal coordinate, then a self-similar solution of the boundary-layer equations can be found by integrating a third-order ordinary differential equation (see [1–3]). When the exponent of the power in the outerflow velocity distribution is negative, a self-similar solution satisfying the equations and the usually posed boundary conditions is not uniquely determinable [4], A similar result was obtained in [5] for flows of a conducting fluid in a magnetic field. In the present paper we study the behavior of non-self-similar perturbations of a self-similar solution, enabling us to provide a basis for the choice of a self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–46, July–August, 1974.     

Analysis of flow in an annular duct with large injection at the walls

Viscous heat-conducting compressible fluid flow in an annular duct formed by two coaxial cylinders with large injection at the walls is investigated. An asymptotic solution exhibiting the influence of the axial symmetry of the duct is obtained in the vicinity of the y axis and is compared with the results of exact numerical calculations. Asymptotic solutions of the Navier-Stokes equations have been obtained earlier for flows in a plane channel with various rates of wall injection in the case of an incompressible gas [1, 2] and a compressible gas [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 135–139, May–June, 1976.     

Boundary-layer separation on a cooled body and interaction with a hypersonic flow

In previous papers, e.g., [1, 2], boundary-layer separation was investigated by analyzing solutions of the boundary-layer equations with a given external pressure distribution. In general, this kind of solution cannot be continued after the separation point. Study of the asymptotic behavior of solutions of the Navier-Stokes equations [3–5] shows that, in boundarylayer separation in supersonic flow over a smooth surface, the main effect on the flow in the immediate vicinity of the separation point is a large local pressure gradient induced by interaction with the external flow. The solution can be continued beyond the separation point and linked to the solutions in the other regions, located downstream [5]. Similar results for incompressible flow were recently obtained in [6]. We can assume that in general there is always a small region near the separation point in which separation is self-induced, and where the limiting solution of the Navier-Stokes equations does not contain unattainable singular points. However, this limiting slope picture can be more complex and can contain more regions where the behavior of the functions differed from that found in [3–6]. The present paper investigates separation on a body moving at hypersonic speed, where the ratio of the stagnation temperature to the body temperature is large. It is shown that both. for hypersonic and supersonic speeds the flow near the separation point is appreciably affected by the distribution of parameters over the entire unperturbed boundary layer, and not only in a narrow layer near the body, as was true in the flows studied earlier [3–6]. Regions may appear with appreciable transverse pressure drops within the zone occupied by layers of the unperturbed boundary layer. Similarity parameters are given, the boundary problems are formulated, and the results of computer calculation are presented. The concept of subcritical and supercritical boundary layers is refined, and the dependence of pressure coefficients responsible for separation on the temperature factor is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 99–109, November–December 1973.     

Nonequilibrium hypersonic flow of carbon dioxide gas over blunt bodies

The Navier-Stokes equations are used to investigate hypersonic flow of carbon dioxide gas over blunt bodies with allowance for nonequilibrium development of chemical reactions and vibrational relaxation of the CO₂ molecules. The problem is solved by the method of stabilization by means of an implicit difference scheme that includes the use of Newton's iterative process. The results are given of calculations of the flow field, the convective heat flux, and the frictional stresses on the surface of blunt cones with spherical noses. The influence of admixtures on the flow field and the heat fluxes is investigated. The results of the calculations are compared with the locally self-similar solution for the neighborhood of the front stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1979.