Measuring the momentum flux of molecules reflected from a surface of given roughness in free molecule flow

The magnitude of the force transferred to a solid surface by reflected molecules in free molecule flow depends firstly on the mechanism of the interaction of the molecules and the element of the surface, or, more precisely, is determined by the velocity distribution function of the reflected molecules, and secondly on the collisions of the molecules with the irregularities of the surface, the number of which depends on the angles of inclination of the sides of these irregularities [1]. Thus, if the microrelief of a plate placed in a molecular flow is known, then certain information about the character of the interaction of the molecules and the surface element can be extracted from measurements of the force acting on it. The aim of this paper is the experimental study of the transfer of momentum by molecules reflected from a plate, depending on the degree of roughness of the surface. Most experimental studies of the transfer of momentum and energy between a flow of rarefied gas and a surface have been made with the aim of revealing the qualitative features of the influence of roughness on the interaction [2–5]. Quantitative estimates of the angles of inclination of the irregularities on test models have been given in very few papers [6–8]. An integral measure is proposed in [6] for estimating the roughness. This is the measure used in the present study.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 158–162, July–August, 1985.     

Design of molecular pumps

Molecular pumps are used to obtain high and superhigh vacuum and are widely applied because of several advantages [1–3]. Several specific schemes are described for calculating molecular pumps on digital computers, based on the technique suggested in [4]. Examples are presented of calculations of specific designs of molecular pumps of the cylindrical and disk types, and also turbomolecular pump types. An example is also given of optimization of the geometry of a disk molecular pump for given pump dimensions, rotor speed, and magnitude of the gas flow rate through the pump. The results of the calculations agree with experiment within the accuracy of the latter.     

Use of the Monte Carlo method to calculate the flow of a highly rarefied gas in systems with arbitrary wall configuration

A method is described for modeling collisions of gas molecules with the walls of a system whose geometry is given numerically in the computer memory from a graphical representation of the walls (for example, from their working drawings). When changing from the calculation of one system to another only the information on the wall changes; the computational program remains the same. The method is applicable to problems of rarefied gasdynamics which are solvable by the Monte Carlo method; in the following it is used to calculate the conductance of elements of high vacuum lines and the compression ratio created by molecular vacuum pumps, and also to calculate the forces acting on a body which rotates in a cavity filled with a highly rarefied gas.In conclusion the authors wish to thank Yu. I. Neimark for formulating the problem and discussions of the results obtained.     

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.     

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1].The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients C_m and K_TS, respectively.In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10].In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients C_m and K_TS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 176–182. Original Russian Text Copyright © 2004 by Poddoskin.     

Possibility of forming an inverted distribution over the rotational levels of nitrogen in an expanding gas stream

A number of theoretical papers have been devoted to an investigation of the relaxation kinetics of the population of a system of rotational levels of molecules in a stream of gas freely expanding from a sonic nozzle [1–3]. The complexity of the task of constructing models of relaxation and of collisions consistent in accuracy, however, as well as the difficulties in solving the resulting system of kinetic and gas-dynamic equations, lead to the necessity of using substantial approximations. Some disagreement between the experimental data and calculated results [1, 2] requires an evaluation of the accuracy of the various approximations used and further refinement of the theoretical models. In contrast to [1], in order to bring out the possible mutual influence of nonequilibrium energy exchange between the degrees of freedom of nitrogen molecules and the variation of the gas-dynamic parameters, the calculation presented below is based on a numerical solution of a self-consistent system of kinetic and gas-dynamic equations for the populations of rotational states and the temperature, density, and velocity of gas in the stream. Collisional probabilities of rotational transitions, calculated with allowance for the long-range part of the potential of the interaction between molecules [4], are used for this.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 9–16, May–June, 1986.     

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].     

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.     

Methods of calculation and numerical analysis of conducting-gas flow in high-current electric arcs

The mechanism of conducting-gas acceleration in an electric arc by intrinsic magnetic field was first investigated in [1]. Further theoretical study of this question was associated with the numerical calculation of arcs [2–7]. A more general approach to the solution of the problem was realized in [4], where the finite-difference method was used. Integral calculational models were developed in [5–7]. The present work proposes a modified version of the difference method [4] and a series of integral methods for the calculation of the conducting-gas flow in a high-current electric arc. The development of integral methods is of interest in that they are usually associated with adequate accuracy in determining integral values and values averaged over the cross section by a relatively simple calculation, and also allow the solution of the problem to be obtained in a number of situations when the realization of a difference method is complicated. The results of different calculation methods are compared. The effect of conditions in the initial cross section of the calculation region of the arc on its characteristics is investigated and a numerical analysis of the heating and acceleration of conducting gas is carried out.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 103–110, September–October, 1978.     

Calculation of unsteady supersonic flow past a two-dimensional cascade of plates ween vorticity inhomogeneities of the flow act on it

In the framework of the linear theory of small perturbations the problem of unsteady subsonic flow past a two-dimensional cascade of plates has been considered in a number of papers. Thus, the unsteady aerodynamic characteristics of a cascade of vibrating plates were calculated in [1] by the method of integral equations, while the same method was used in [2, 3] to calculate the sound fields that are excited when sound waves Coming from outside or vorticity inhomogeneities of the oncoming flow act on the cascade. The problem of a two-dimensional cascade of vibrating plates in a supersonic flow was solved in [4, 5]. In [4] the solution was constructed on the basis of the well-known solution of the problem of vibrations of a single plate, while in [5] a variant of the method of integral equations was used which differed slightly from the usual formulation of this method [1–3]. The approach proposed in [5] is used below to calculate the unsteady flow past a two-dimensional cascade of plates in the case when vorticity inhomogeneities of a supersonic oncoming flow act on it. Equations are obtained for the strength of the unsteady pressure jumps arising in such a flow and the vortex wakes shed from the trailing edges of the plates. Examples of the calculations illustrating the accuracy of the method and its possibilities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp, 152–160, May–June, 1986.     

Recombination of electrons in a plasma expanding into a vacuum

As noted in a paper by one of the authors [1], when a hot ionized gas expands into a vacuum, at a certain moment ionization equilibrium must necessarily break down. Shortly after this point, which may be found by the method indicated in [1], ionizing events become very rare and only recombination occurs in the gas. In [1] photorecombination and triple collisions with the capture of an electro to the ground level of the atom were considered. Here the recombination did not proceed to the end: on expanding to infinity and cooling to zero the gas remained partially ionized.Papers have recently appeared [2–7] in which the significant role of triple collisions with the capture of electrons to upper atomic levels is noted. The recombination process has a cascade character at low temperatures and densities which are not excessively small. At first, the electron is captured by one of the upper atomic levels in a triple collision with an ion and another electron. Subsequently, as a result of electron collisions of the second kind, and later also as a result of radiative transitions, the bound electron descends through the energy levels to the atomic ground state. The recombination coefficient for such a process depends much more strongly on the electron temperature T than for a triple collision with capture directly by the ground level (as T^–9/2 as opposed to T^–1), and at low temperatures cascade recombination proceeds much more quickly than capture to the ground level. Since this casts doubt upon the conclusions of [1] regarding the residual ionization when a plasma expands into a vacuum, we were led to re-examine the question, which, as will be clear from what follows, is not considerably more complicated.     

Flow processes with change of regime in fractured reservoirs

Experimental and industrial observations indicate a strong nonlinear dependence of the parameters of the flow processes in a fractured reservoir on its state of stress. Two problems with change of boundary condition at the well — pressure recovery and transition from constant flow to fixed bottom pressure — are analyzed for such a reservoir. The latter problem may be formulated, for example, so as not to permit closure of the fractures in the bottom zone. For comparison, the cases of linear [1] and nonlinear [2] fractured porous media and a fractured medium [3] are considered, and solutions are obtained in a unified manner using the integral method described in [1]. Nonlinear elastic flow regimes were previously considered in [3–6], where the pressure recovery process was investigated in the linearized formulation. Problems involving a change of well operating regime were examined for a porous reservoir in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–73, May–June, 1991.     

Experimental investigation of the interaction of a supersonic annular jet and an obstacle

There has been much interest in recent years in gas-dynamic problems involving the interaction of gas jets with obstacles, and there have been studies of combinations of individual jets, systems of jets, and also annular jets. Various papers have been published with the results of theoretical and experimental investigations of the interaction of axisymmetric continuous jets with obstacles [1–3]. However, there have been only a few experiments on the fluctuations of an annular system of jets that encounter an obstacle [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 109–117, May–June, 1983.     

Solution of the problem of optimum boundary layer control on the porous surface of a turbine blade

Problems of the optimization of coolant injection into a laminar compressible gas boundary layer are considered within the framework of the integral method of boundary layer calculation [1].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 17–24, November–December, 1993.     

Applicability of a model of the formation of a molecular beam under conditions of translational disequilibrium

In the model of the formation of a molecular beam proposed in [1], Kantrowitz and Grey assumed ideal conditions, namely, the gas stream expands in accordance with an isentropic law, the introduction of the skimmer into the gas stream does not disturb the flow upstream, and the entrance section of the skimmer is the surface of the last collisions. In reality, these assumptions are not always satisfied. In the case of rapid expansion of a supersonic stream of rarefied gas there is a departure from isentropic behavior because of the freezing of the relaxation processes, and the formation of a molecular beam by means of a skimmer changes the gas-dynamic parameters [2, 3]. The aim of the present work was to make a direct experimental verification of the applicability of the model of molecular beam formation from a supersonic stream of rarefied gas under conditions of translational disequilibrium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 170–173, May–June, 1980.     

Two-phase three-component flow through a porous medium with variable total flow

In analyzing the processes of the displacement of oil, in which intensive interphase mass transfer takes place, it is normally assumed that the partial volumes of the components as they mix are additive (Amagat's Law) [1, 2]. Then the equations of motion have an integral, which is the total volume flow rate through the porous medium, and the basic problems of frontal displacement, if there are not too many components in the system, permit an exact analytical study to be made [3–5]. If this assumption is rejected, the total flow becomes variable [3, 6, 7]. It appears that the consequences of this as applied to the processes of the displacement of oil by high pressure gases have not previously been considered. The results of such a study, developing the approach outlined in [4], are given below. The initial multicomponent system is simulated by a three-component one which contains oil (the component being displaced), gas (the neutral or main displacing component), and intermediate hydrocarbon fractions or solvent (the active component). It is shown that instead of the triangular phase diagram (TPD) normally used where the partial volumes of the components are additive, in this case it is convenient to use a special spatial phase diagram (SPD) of the apparent volume concentrations of the components to construct the solutions and to interpret them graphically. The method of constructing the SPD and its main properties are explained. A corresponding graphoanalytical technique is developed for constructing the solutions of the basic problems of frontal displacement which correspond to motions with variable total flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1985.     

Generalization of the Krook kinetic relaxation equation

One of the most significant achievements in rarefied gas theory in the last 20 years is the Krook model for the Boltzmann equation [1]. The Krook model relaxation equation retains all the features of the Boltzmann equation which are associated with free molecular motion and describes approximately, in a mean-statistical fashion, the molecular collisions. The structure of the collisional term in the Krook formula is the simplest of all possible structures which reflect the nature of the phenomenon. Careful and thorough study of the model relaxation equation [2–4], and also solution of several problems for this equation, have aided in providing a deeper understanding of the processes in a rarefied gas. However, the quantitative results obtained from the Krook model equation, with the exception of certain rare cases, differ from the corresponding results based on the exact solution of the Boltzmann equation. At least one of the sources of error is obvious. It is that, in going over to a continuum, the relaxation equation yields a Prandtl number equal to unity, while the exact value for a monatomic gas is 2/3.In a comparatively recent study [5] Holway proposed the use of the maximal probability principle to obtain a model kinetic equation which would yield in going over to a continuum the expressions for the stress tensor and the thermal flux vector with the proper viscosity and thermal conductivity.In the following we propose a technique for constructing a sequence of model equations which provide the correct Prandtl number. The technique is based on an approximation of the Boltzmann equation for pseudo-Maxwellian molecules using the method suggested by the author previously in [6], For arbitrary molecules each approximating equation may be considered a model equation. A comparison is made of our results with those of [5].     

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.     

Flow around a sphere by a two-phase supersonic jet

Results are presented of a calculation of the flow around a sphere of a two-phase supersonic jet, discharging into a vacuum. Calculations were performed by the determination method with use of a difference grid constructed on the basis of characteristic ratios [1], The parameters of the unperturbed jet were determined with the two-velocity and two-temperature model of mutually penetrating flows of continuous media (gas and particles) [2, 3] by the network method [4]. In calculating the flow around the sphere, as in [5–7], it was assumed that the particles do not affect the gas flow in the shock layer. An analysis of the effect of particles on gasdynamic parameters in a shock layer was performed in [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 171–176, November–December, 1978.The authors are grateful to A. N. Nikulin for providing the program for calculation of flow about a blunt body by a uniform supersonic flow.     

Radiating hydrogen flow in axisymmetric nozzles

A considerable number of studies published in recent years have been devoted to the study of gas in channels and pipes. In view of the complexity of the question and the lack of analytic techniques, individual aspects of the problem are generally considered. The determination of the radiant field characteristics in regions of simple geometric form filled with a stationary radiating-absorbing medium has been carried out in several studies. The articles [1–3] are devoted to the calculation of the radiant field and the temperature field for a given flow of a perfect inviscid nonheat-conducting radiating gas with constant absorption coefficient. The flow is assumed to be irrotational [1, 2] or nearly potential [3]. The authors investigated the accuracy of the solution obtained with the aid of various approximate methods and found that the diffusion approximation yields a small error in calculating the radiation density field and the values of the radiant thermal fluxes for a quite broad class of wall reflecting properties. We may note also [4, 5], in which a calculation is made of one-dimensional steady flow of a viscous heat-conducting radiating perfect gas with constant transport coefficients.In [1–5] the absorption coefficient is considered constant. This assumption simplifies the solution process considerably, since as the independent variables we can take the corresponding optical thicknesses. The study [3] contains a remark that the calculation method proposed there may be used with a variable absorption coefficient. However, this possibility was not used in the calculations presented.For a constant absorption coefficient these studies yield a rather complete analysis of the methods for solving two-dimensional problems in geometrically simple regions in the absence of mechanical motion and one-dimensional problems with motion. They contain results obtained for the exact integral or integrodlfferential equations and present an analysis of the approximate methods. The study [3] considers broader possibilities of solving two-dimensional problems (using the Monte-Carlo method), but the flow is assumed known ahead of time.In the following we present a method for calculating the two-dimensional equilibrium flow of an inviscid non-heat-conducting radiating gas with variable absorption coefficient. As an example, we consider the flow of radiating-absorbing hydrogen in axisymmetric nozzles. It is assumed that the radiation is gray and is in local thermodynamic equilibrium. The transport equation is considered in the diffusion approximation. The nozzles examined have a semi-infinite cylindrical inlet section. The initial gas flow in the cylindrical section is supersonic. In the solution process we determine the radiation density field and all the flow parameters within the nozzle.The author wishes to thank Yu. D. Shmyglevskii for his continued interest in this study.