Spherical flow of a diatomic gas from an evaporating drop

The spherical expansion of gas from an evaporating drop is investigated on the basis of the numerical solution to a model kinetic equation for a gas with rotational degrees of freedom. Examples considered are the stationary evaporation of a drop with given temperature into the vacuum and evaporation of a drop into a gas-filled space under the condition of an energy balance on the drop surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 184–187, July–August, 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.     

Numerical simulation of the dynamics of a liquid in a moving axisymmetric cavity

The problem of the motion of an ideal liquid with a free surface in a cavity within a rigid body has been most fully studied in the linear formulation [1, 2]. In the nonlinear formulation, the problem has been solved by the small-parameter method [3] and numerically [4–7]. However, the limitations inherent in these methods make it impossible to take into account simultaneously the large magnitude and the threedimensional nature of the displacements of the liquid in the moving cavity. In the present paper, a numerical method is proposed for calculating such liquid motions. The results of numerical calculations for spherical and cylindrical cavities are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–177, March–April, 1984.     

Role of the Knudsen layer in the problem of droplet evaporation

A study is made of the part played by the Knudsen layer in the problem of weak unsteady evaporation of a spherical droplet in its own vapor. It is shown that use of the classical Hertz—Knudsen formula may lead to appreciable errors, in particular, in the determination of the time required by the droplet temperature to relax to the state corresponding to steady evaporation.Translated from Izvestiya Akademii Nauk SSSR; Mekhanika Zhidkosti i Gaza, No. 1, pp. 127–131, January–February, 1984.     

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].     

Stability of heat transfer regimes at the front stagnation point of a body in a stream of dissociated gas

The formulation and solution of the stationary problem of heat transfer in the neighborhood of the front point of a body at constant temperature in a stream of dissociated air are given in [1]. In [2], the results are given of numerical solution of this problem in the nonstationary formulation; the establishment of a stationary heat transfer regime was established for all the calculated variants. In the present paper, we investigate the stability of stationary heat transfer regimes at the front stagnation point of a body in a stream of dissociated air using the Lyapunov functional method [3, 4] and the method of [2, 5], which is based on the use of Meksyn's method in boundary-layer theory [6, 7]. It is established that an arbitrarily strong growth of the Damköhler number does not lead to instability and multiplicity of the stationary regimes, in contrast to the case when a hot mixture of gases flows over the front point of a thermostat [2, 5, 8]. Numerical solution of the boundary-layer equations for a wide range of Damköhler numbers confirms the results of the approximate qualitative analysis and shows that in a number of cases the time of establishment of the stationary state is a nonmonotonic function of the Damköhler number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 97–106, September–October, 1979.     

A spherical wave in a viscoelastic half-space

The propagation of spherical waves in an isotropie elastic medium has been studied sufficiently completely (see, e.g., [1–4]). it is proved [5, 6] that in imperfect solid media, the formation and propagation of waves similar to waves in elastic media are possible. With the use of asymptotic transform inversion methods in [7] a problem of an internal point source in a viscoelastic medium was investigated. The problem of an explosion in rocks in a half-space was considered in [8]. A numerical Laplace transform inversion, proposed by Bellman, is presented in [9] for the study of the action of an explosive pulse on the surface of a spherical cavity in a viscoelastic medium of Voigt type. In the present study we investigate the propagation of a spherical wave formed from the action of a pulsed load on the internal surface of a spherical cavity in a viscoelastic half-space. The potentials of the waves propagating in the medium are constructed in the form of series in special functions. In order to realize viscoelasticity we use a correspondence method [10]. The transform inversion is carried out by means of a representation of the potentials in integral form and subsequent use of asymptotic methods for their calculation. Thus, it becomes possible to investigate the behavior of a medium near the wave fronts. The radial stress is calculated on the surface of the cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 139–146, March–April, 1976.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

Hydrodynamic drag of an ellipsoidal droplet at small Reynolds numbers

At present, there is an absence of the accurate data on the influence of the shape of a droplet on its hydrodynamic drag and mass transfer without which the design of mass transfer apparatus is impossible [1–3]. Most often it is assumed that the drag of an ellipsoidal liquid droplet as it moves along the axis of symmetry is determined by the product of the drag of a spherical liquid droplet and a coefficient which takes into account the shape and is determined from the drag of a solid ellipsoid for which the exact solutions are known. It is shown below that this assumption is not always valid.Translated from Tzvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–8, May–June, 1987.     

Diffraction of a spherical acoustic wave on a cone

An exact analytic solution of the problem of diffraction of a plane acoustic wave on a cone of arbitrary aperture angle was obtained and studied in [1]. For the case of spherical wave diffraction on a cone a formula is known [2] which relates the solutions of the spherical and plane wave diffraction problems. This study will employ the results of [1, 2] to derive and investigate an exact analytical solution of the problem of diffraction of a spherical acoustic wave on a cone of arbitrary aperture angle. Results of numerical calculations will be presented and compared with analogous results for a plane wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 200–204, March–April, 1976.The author is indebted to S. V. Kochura for her valuable advice.     

Waves from an underground explosion

The problem of the propagation of a spherical detonation wave in water-saturated soil was solved in [1, 2] by using a model of a liquid porous multicomponent medium with bulk viscosity. Experiments show that soils which are not water saturated are solid porous multicomponent media having a viscosity, nonlinear bulk compression limit diagrams, and irreversible deformations. Taking account of these properties, and using the model in [2], we have solved the problem of the propagation of a spherical detonation wave from an underground explosion. The solution was obtained by computer, using the finite difference method [3]. The basic wave parameters were determined at various distances from the site of the explosion. The values obtained are in good agreement with experiment. Models of soils as viscous media which take account of the dependence of deformations on the rate of loading were proposed in [4–7] also. In [8] a model was proposed corresponding to a liquid multicomponent medium with a variable viscosity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 34–41, May–June, 1984.     

Influence of excitation of internal degrees of freedom of molecules on the process of weak evaporation or condensation

A study is made of the process of weak evaporation (or condensation) with allowance for excitation of vibrational and rotational degrees of freedom of diatomic molecules. The solution to the corresponding Knudsen layer problem is obtained on the basis of a model kinetic equation of the type of the Morse equation [1]. A relation is obtained that establishes the connection between the rate of evaporation (or condensation) and the parameters of the surface and the gas above it. The boundary conditions of slip for the equations of gas dynamics are analyzed. The results are compared with the evaporation or condensation in the case of a monatomic gas. The introduction of accommodation coefficients for an evaporating surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 98–110, November–December, 1979.     

Convective diffusion to a drop under arbitrary conditions of absorption. Diffusion boundary layer approximation

The diffusion boundary layer approximation is used to investigate the stationary convective diffusion of substance dissolved in a flow to a spherical drop under arbitrary conditions of absorption on its surface, in particular when a chemical reaction of arbitrary order takes place on the surface. An integral equation is obtained for the local diffusion flux to the surface of the drop. It is shown that 1) the total Sherwood number increases with increasing rate of the reaction and decreases with increasing exponent of the reaction rate; 2) with increasing Péclet number, saturation occurs (i.e., the total diffusion flux to the surface of the drop tends to a limiting value, which depends only on the reaction kinetics). The case of total absorption of diffusing, substance on the surface of reacting solid and liquid particles in a homogeneous Stokes flow at large Péclet numbers was investigated in [1]. The problem of convective diffusion to the surface of a solid spherical particle in the case of mixed kinetics was considered in [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 64–69, November–December, 1979.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

Asymptotic solution to the problem of drag of a fluid by a moving plate

An analogy is established between the formulations of the problem of the drag of a fluid by a moving plate [1–3] and the problem of propagation of a stationary flame [4, 5]. The theory of singular perturbations is used to a find a two-term asymptotic expression for the film thickness h₀. The expansion parameter is the Bond number Bo 1. The limited applicability of the well-known formula of [1, 2] is estimated quantitatively. Such an estimate has been obtained earlier experimentally [3]. The approach used in the present paper should also be fruitful for the solution of other problems in capillary hydrodynamics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 52–56, November–December, 1980.     

Computation of the pressure on the surface of a fuselage with a wing in an ideal fluid

The flows around complex three-dimensional bodies by an ideal fluid are computed by methods [1–3] using approximation of the surface by a set of plane elements. A layer of surface singularities, whose intensity is found by solving a system of linear algebraic equations of very high order, is distributed continuously over each element. Evaluation of the system coefficients and its solution require significant machine time expenditures on powerful electronic computers. If in the method of [2] the total system of equations is separated successfully into several subsystems by simplifications and an approximate solution of the problem is obtained more rapidly than by the method in [1], then the same author practically used the method in [1] to design specific fuselages in [3]. A method [4] developed for a fuselage is expanded in this paper to design a wing-fuselage combination. This method turns out to be less tedious, without being inferior in accuracy, by being different from the method in [1] in the means of solving the fundamental integral equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 110–115, May–June, 1977.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Numerical study of unsteady evaporation and heat transfer from a spherical surface

An unsteady radial problem of evaporation and heat transfer from a spherical surface is considered for a model kinetic equation. The problem is solved numerically using a second-order implicit conservative method.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 181–192. Original Russian Text Copyright © 2005 by Titarev and Shakhov.     

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.     

Hypersonic similitude in the case of flow past a combination of a circular cone and a delta-shaped wing

The validity of the well-known law of hypersonic similitude [1, 2] for a combination of a circular cone and a delta-shaped wing has hitherto been verified only for the integral characteristics [3]. The law is verified in this paper for both the integral and local parameters of the flow. The posed problem has been solved numerically using the stationary analog of Godunov's method [4]. The shock waves and characteristic surfaces bounding the region of the properly conical flow were separated. As in the paper of Ivanov and Kraiko [5], the required distributions of the parameters were found by stabilization with respect to the coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–190, March–April, 1984.I thank A. N. Kraiko for his interest in the work and for discussing the results.