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].     

Kinetic theory of recondensation in a binary gas mixture with arbitrary Knudsen numbers

A rigorous solution of the problem of recondensation between two surfaces with arbitrary Knudsen numbers is possible only on the basis of a consecutive kinetic consideration. For the single-component case, this problem was solved in [1] using the BGK model of the collision integral in the kinetic equation. In [2], for the same purpose, the method of moments for Maxwellian molecules was used. The case of a binary mixture, in which one of the components is a noncondensing gas was discussed in [3, 4]. Under these circumstances, in [3], a single-relaxation lumped model was used for each component; the model did not reflect many of the properties of the exact collision integral. A more rigorous model (the collision integral in the Hamel form) was applied in [4]. Here there was written a system of integral equations for the hydrodynamic quantities, and its numerical solution is examined in several specific partial cases. In the present article, the problem of recondensation in a binary mixture is treated by the method of moments for Maxwellian molecules. For the case of small relative difference in the temperature of the surfaces, analytical expressions are obtained for the rates of mass transfer and the heat fluxes, making it possible to shed light on the principal special characteristics of the process of recondensation in a binary mixture.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 150–155, July–August, 1975.The authors thank R. Ya. Kucherov for his useful observations.     

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.     

Vapor Flow with Evaporation–Condensation on Solid Particles

Strongly nonequilibrium vapor (gas) flows in a region filled by solid particles are considered with allowance for particlesize variation due to evaporation–condensation on the particle surface. The study is performed by directly solving the kinetic Boltzmann equation with allowance for the transformation of the distribution function of gas molecules due to their interaction with dust particles.     

Calculation of the transport coefficients and slip velocities for molecules in the form of solid spheres

A solution of the Boltzmann equation is carried out by the Monte Carlo method for problems of rarefied gasdynamics in a linear formulation. The problems are solved by calculating the transport coefficients and slip velocities on a solid wall for molecules in the form of solid spheres. The accuracy of the method due to various parameters of the computational scheme in the solution of the problem is investigated by calculating the transport coefficients for pseudo-Maxwellian molecules.The Boltzmann kinetic equation is a complex integro-differential equation which is very difficult to solve and analyze. Hence, the solution of even one-dimensional problems and for the linearized Boltzmann equation turns out to be quite difficult, and such problems are solved by approximate methods (the expansion in Knudsen numbers, the method of moments, the expansion in series, etc. [1]). A method of solving the linearized Boltzmann equation by the Monte Carlo method is proposed in [2]. An exact solution of a number of problems of rarefied gas dynamics has been obtained by this method [3, 4]. However, the method was applied for pseudo-Maxwellian molecules, for which the collision cross section is inversely proportional to the relative velocity of the colliding particles =₀/g.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 155–158, March–April, 1971.In conclusion, the author is grateful to M. N. Kogan for formulating the problem and for great assistance provided during the research, and also to V. I. Vlasov, S. L. Gorelov and V. A. Perepukhov for assistance in compiling the program.     

Gas flow in cylindrical capillaries under intermediate conditions

At present, there are sufficient solutions of the problem of free-molecular gas flow through a short cylindrical channel, for example, [1–3]. In intermediate flow conditions, for Knudsen number Kn 1, solutions have been obtained for the limiting cases: an infinitely long channel [4] and a channel of zero length (an aperture) [5]. However, no solution is known for short channels for Kn 1. The present work reports a calculation by the Monte Carlo method of the macroscopic characteristics of the gas flow through a short cylindrical channel (for various length—radius ratios), taking into account intermolecular collisions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 187–190, January–February, 1977.     

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.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Spherical expansion of vapor during evaporation of a droplet

The problem of the evaporation of a spherical particle is solved by a numerical finnite-difference method for the stationary and nonstationary cases on the basis of the generalized Krook kinetic equation [1]. Evaporation into a vacuum and into a flooded space are considered taking into account the reduction in size and cooling of the droplet. The minimum mass outflow is determined for stationary evaporation into a vacuum at small Knudsen numbers. The results are compared with those of other authors for both the spherical and plane problems. Most previous studies have used different approximations which reduce either to linearizing the problem [2, 3] or to use of the Hertz-Knudsen equation [4]. The inaccurate procedure of matching free molecular and diffusive flows at some distance from the surface of the droplet [5] is completely unsuitable in the absence of a neutral gas. Equations for the rate of growth of a droplet in a slightly supercooled vapor were obtained in [6] from a solution of the ellipsoidal kinetic model by the method of (expansion of) moments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 97–102, March–April, 1976.     

Evaporation of spherical drops in a binary gas mixture at arbitrary Knudsen numbers

The quasisteady evaporation of drops in a binary gas mixture is investigated at arbitrary Knudsen numbers. The analysis is based on the solution of kinetic equations with collision integrals in the Boltzmann form by Lees's method [9]. The obtained solution makes it possible to consider an arbitrary model of intermolecular interactions. Formulas for the evaporation time of the drops are analyzed, the model of rigid elastic spheres being used for the interaction of the molecules.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–118, January–February, 1982.     

Group-invariant solutions of kinetic equations

In the development of analytic methods of solution of kinetic equations, it is expedient to use group raetliods. The establishment of a symmetry group makes it possible to justify the choice of a definite model of kinetic equation corresponding to the physical formulation of the problem, to solve the Cauchy problem in a number of cases, and to obtain classes of new exact solutions that can be used as standards in the construction of numerical algorithms for solving kinetic equations. Bobylev [1–4] and Krook and Wu [5, 6] used group methods to analyze the spatially homogeneous Boltzmann equation in the case of isotropy with respect to the velocities and Maxwellian molecules. They obtained exact solutions and investigated the asymptotic behavior of the main equation. In the present paper, group methods are used to find and analyze exact solutions of the Bhatnagar-Gross-Krook kinetic equation, which successfully simulates the basic properties of the Boltzmann equation. Conclusions are drawn about the symmetries of the Boltzmann equation. To simplify the calculations, the exposition is presented for the case of the one-dimensional Bhatnagar-Gross-Krook equation with constant effective collision frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 135–140, July–August, 1982.     

Boundary conditions for flows with chemical reactions occurring on a surface

With the use of a solution of a model Boltzmann equation for a binary mixture in the Knudsen layer, we obtain the boundary conditions for the equations of gas dynamics when the reactionl _iA_il _jA_j (l _i molecules of A_i change intol _j molecules of A_j, and vice versa) is occurring on a surface. The boundary condition that we obtain differs from those that are usually applicable by the presence of terms of the same order. This confirms the conclusion arrived at by the authors in [1], where it was shown that if the Knudsen layer is left out of account, which is precisely what is usually done, it is impossible to obtain correct boundary conditions.Moscow. Translated from Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 129–138, January–February, 1972.     

气体运动论数值算法在微槽道流中的应用研究

简要介绍基于Boltzmann模型方程的气体运动论数值算法基本思想及其对二维微槽道流动问题数值计算的推广，并阐述适用于微尺度流动问题的气体运动论边界条件数值处理方法。通过对压力驱动的二维微槽道流动问题进行数值模拟，将不同Knudsen数下的微槽道流计算结果分别与有关DSMC模拟值和经滑移流理论修正的N—S方程解进行比较分析，表明基于Boltzmann模型方程的气体运动论数值算法对微槽道气体流动问题具有很好的模拟能力。     

Flow of a viscous fluid in a closed cavity in the presence of slip effects

Slip at the wall is observed in the flow of non-Newtonian fluids [1–4] and rarefied gases [5]. The most complete information on the phenomenon is obtained in capillary viscosimetry. For small radii of the capillaries and in porous media the slip effect is manifested even for Newtonian fluids (water, kerosene, for example) [6]. Experiments [2, 4] show that the influence of the entrance section can be ignored if the length of the capillary exceeds its radius by about 100 times. For the measurement of the rheological characteristics of high-viscosity fluids the use of long capillaries is difficult, and it is necessary to calculate the two-dimensional flow at the entrance section with allowance for slip. The need for such calculations also arises, for example, when one is choosing the optimal parameters of the screw devices employed in the processing of polymers [7]. Two-dimensional flows of a viscous incompressible fluid are frequently calculated with the flow function and vorticity =– used as variables [8–14]. The expressions for the vorticity on the boundary are usually obtained from the viscous no-slip condition [8, 9]. In the present paper, expressions are obtained for the vorticity on a wall in the presence of slip. The obtained expressions are used to solve a test problem on the flow of a viscous incompressible fluid in a cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–16, January–February, 1980.     

The steady-state flow of a rarefied gas from a spherical source or sink

The method of characteristics is used to solve problems in the steady-state flows of a rarefied gas on the basis of approximating the kinetic equations. Numerical results are given for the solution of the problem of the flow from a spherical source or sink using the generalized Kruk equation [1] and approximating the Boltzmann equation by the method proposed by the author [2, 3], Various flow conditions are discussed: flow into a vacuum, flow into a flooded volume, flow from a sink.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–66, March–April, 1971.     

Asymptotic solution to the problem of hypersonic flow over bodies in the neighborhood of the separation point of a thin shock layer

A study is made of the problem of hypersonic flow of an inviscid perfect gas over a convex body with continuously varying curvature. The solution is sought in the framework of the asymptotic theory of a strongly compressed gas [1–4] in the limit M when the specific heat ratio tends to 1. Under these assumptions, the disturbed flow is situated in a thin shock layer between the body and the shock wave. At the point where the pressure found by the Newton-Buseman formula vanishes there is separation of the flow and formation of a free layer next to the shock wave [1–4]. The singularity of the asymptotic expansions with respect to the parameter ₁ = ( –1)/( + 1) associated with separation of the strongly compressed layer has been investigated previously by various methods [3–9]. Local solutions to the problem valid in the neighborhood of the singularity have been obtained for some simple bodies [3–7]. Other solutions [7, 9] eliminate the singularity but do not give the transition solution entirely. In the present paper, an asymptotic solution describing the transition from the attached to the free layer is constructed for a fairly large class of flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–105, January–February, 1982.     

Unsteady flows of an erosion plasma in irradiation of solid targets by annular beams

In recent times high-pressure physics has made ever wider application of constructions which use convergent shock waves [1–8]. The study of gas dynamic flows with convergent shock waves imposes the need for more careful calculation of the motions of a gas in regions whose dimensions are much less than the characteristic dimensions of the flow. In the present study the numerical method is used to study the gas dynamic phenomena accompanying the irradiation of solid obstacles by annular beams of monochromatic radiation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 179–182, November–December, 1988.In conclusion we note that at very short durations t t_k the solution to the problem is similar to the flow during separation of a gaseous toroid [19].     

High-altitude aerothermodynamics

Hypersonic high-altitute flight can be conventionally divided into three regimes: the continuum regime, when the Knudsen number Kne1, the free-molecule regime (Kn1), and the transitional regime (K1). In general, each of these regimes differs with respect to both the structure of the flow and the method of determining the aerodynamic and thermal characteristics. For Knudsen numbers Kne1 the Navier-Stokes equations or models with slip and temperature jump boundary conditions are widely used. When Kn1 the methods employed are mainly directed towards determining the distribution function of the molecules reflected from the surface of the body. On the transition interval between these two limiting regimes numerical methods of solving the Boltzmann equation and its model equations are being used with success. Together with the experimental techniques, these various methods, which complement each other, make it possible to investigate gas flows fairly effectively from the continuum to the free-molecule regime (see, for example, [1]).Based on a paper presented to the Fluid Mechanics Section of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 142–152, March–April, 1993.     

A model and kinetic equations for a charged mixture of gases in the presence of phase transitions

A model of a gas mixture is studied in which one of the components can carry electric charge and undergo phase transitions. Under a number of assumptions, Boltzmann kinetic equations are written down and the form of the collision integral determined. Conservation equations for the components of the mixture are found. The conservation equations for a charged mixture of gases in the absence of phase transitions have been discussed earlier [1]. Collision integrals for a reacting gas mixture in the case of chemical reactions of bimolecular type and when the mixture is described by Boltzmann kinetic equations are derived in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 3, pp. 118–127, May–June, 1980.