Some methods of calculating the steady motion of a rarefied gas

Vacuum molecular pumps have been long known and have several advantages [1–3].Several studies have been devoted to the design of vacuum molecular pumps [7–10]. The methods developed in these studies have been based either on the formulas for gas diffusion in long pipes or on the integral equations of material balance. However, these theories do not permit obtaining design data for real designs of molecular pumps which are close to the experimental data, and, moreover, do not permit solving the practically important problem of optimizing the parameters and geometry of the molecular and turbomolecular pumps with respect to output and compression ratio. The calculations made in [8–10] are valid only for rotor speeds which are much less than the average velocities of the gas molecules. However, the studies in the second direction cannot be continued to a final result in view of the extreme complexity of the solution of the resulting system of integral equations.In the following we describe the calculation of vacuum molecular pumps, based on the Monte-Carlo method (the Monte-Carlo method has been used to calculate the conductance of the elements of vacuum lines in the free molecular regime in [4, 5, 6] and to calculate using the method of sequential approximations the flow of a rarefied gas with account for the collisions between molecules in [11]).We shall apply this method not only to systems with a high vacuum, when the collisions between molecules may be neglected, but also to systems in which in addition to the molecule collisions with the wall it is necessary to consider the possibility of a small number of mutual collisions.     

Plane Problem of Three-Dimensional Stability for a Hinged Plate with Two Symmetric End Cracks

The plane stability problem for a rectangular plate with two symmetric end cracks is solved in three-dimensional formulation. The three-dimensional linearized theory of stability and the finite-difference method are used. The effect of the crack parameter on the critical load is examined__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 47–52, April 2005.     

Stability of a gas boundary layer on a heated surface with a weak negative pressure gradient

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 72–76, March–April, 1992.     

Electromagnetic disturbance created by an expanding conducting cylinder in a magnetic field

We examine the electromagnetic disturbance created by a linearly expanding, ideally conducting cylinder in an external uniform magnetic field with account for effects of order v/c (v is the boundary velocity).The problem of uniform magnetic field compression in a contracting cylindrical cavity with ideally conducting walls with radius decreasing linearly with time was solved in [1]. As far as we know the corresponding external problem has not been investigated.     

Effect of a longitudinal magnetic field on the characteristics of a stabilized channel arc

The results of calculations of the temperature profiles and volt-ampere characteristics of a long cylindrical argon arc in a longitudinal uniform magnetic field are presented. The calculation was made for the following parameters: pressure p =0.1–10.0 atm; temperatures T = 1000-20,000°K; magnetic field induction B =0-10 T; diameter of cylindrical channel d = 1.0 cm. It is shown that for strongly radiating arcs (p1.0 atm) the temperature profiles become more inflated with an increase in the magnetic field, while for weakly radiating arcs (p 0.1 atm) the appearance of loops in the volt-ampere characteristics is typical for certain conditions (14,000T20,000°K, B1.0 T), indicating the impossibility of arcing under these conditions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 147–153, March–April, 1975.     

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.     

MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, B, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the layer type [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations.The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number ==/ have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations).The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities ( and ) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions.The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and numbers (Alfvén and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration.A critical value of the Alfvén number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.     

Principle of correspondence of static boundary-value problems of nonlinear viscoelasticity with aging to boundary-value problems of the theory of elasticity

The principle of correspondence of boundary-value problems of the nonlinear nonuniform anisotropic theory of viscoelasticity to boundary-value problems of the theory of elasticity is formulated. The correspondence is established by means of integral transforms with previously unknown kernels. A class of viscoelastic materials for which these transforms can be reduced to boundary-value problems of fictitious elasticity is determined. Samara State Aerospace University, Samara 443086. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 155–161, July–August, 1998.     

Investigation of pressure-discontinuity system with one and two triplets,taking into account equilibrium physicochemical transformations of air

It is known that the interaction of pressure discontinuities preceding the projecting elements of a supersonic object considerably increase the pressure in the interaction region [1–3]. Existing methods of estimating this excess pressure at the leading edge of the projecting element are based on the calculation of the configuration of pressure-discontinuity intersections with two or one triple points for a perfect gas with a constant adiabatic modulus . The calculation reduces to the successive solution of two transcendental equations for the determination of the angles of slope of the discontinuities at the node points [2, 4]. The present paper states the formulation of the problem and results of flow calculations in pressure-discontinuity configurations with triple points, taking into account the equilibrium dissociation of air. The Predvoditelev approximation is used to calculate the thermodynamic function of the pressure p, as proposed in [5]. The formulation of the problem is considered for the calculation of the flow taking into account the equilibrium dissociation of air in the interference region of pressure discontinuities with two and one triple points — interactions of types I and II, according to the classification of [4]. Some results of the computer solution of the resulting system of equations are given both for a flow of cold unperturbed air (the interaction region w of the leading shock wave of an object with its projecting elements) and for a flow of hot dissociating air (the interaction region O with the boundary-layer breakaway region at the surface of the supersonic object). It is shown that, both in region w and in region O, the relative pressure is considerably affected not only by the velocity and the angle of the incident pressure discontinuity but also by the density of the incoming flow (the flight altitude of the object). Depending on this parameter, the relative pressure in the interaction region may be less or more than the pressure calculation for a perfect gas with = 1.4 to analogous flow conditions. The results obtained indicate the need to take account of the real properties of air in determining the mechanical and thermal loads in the interaction region of the pressure discontinuities at the surface of projecting elements of a hypersonic object.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 111–116, September–October, 1978.     

Laminar flow between the surfaces of two eccentric circular cylinders

The problem of laminar fluid flow in an asymmetric annular gap has been solved in earlier work in an approximation with error difficult to gauge. In the present paper, the problem is solved exactly. The velocity distribution of the flow is obtained, the mean velocity is determined, and an exact expression is found for the coefficient which determines the flow rate. An approximate expression proposed for this coefficient holds asymptotically at small eccentricities. The results are compared with experimental data.Translated from Izvestiya Akademil Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–20, January–February, 1984.