Accuracy of model kinetic equations

The accuracy of model kinetic equations is analyzed using the exact moment solutions of the Boltzmann–Maxwell equation for homoenergetic affine flows of a monatomic gas of Maxwellian molecules in the absence of external forces. Solutions of the third-order kinetic-moment equations for homogeneous shear flow and one-dimensional homogeneous expansion-collapse flow are considered. The principal advantages of the domestic Shakhov and, especially, Larina–Rykov models are demonstrated.     

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.     

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].     

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.     

Method of outer and inner asymptotic expansions in the theory of Brownian motion of aerosol particles

We examine the Brownian motion of particles in a gaseous medium, complicated by the influence of inertial forces. The equation for the distribution function in phase space describing motion of this type was obtained in [1]. Also presented in [1] are the solutions of this equation for certain simple particular cases. The approximate equations of motion of aerosol particles in coordinate space were first obtained in [2] and solved for certain concrete problems in [3,4]. More exact equations of motion in coordinate space, and also the limits of applicability of the equations of [2], are presented in [5].     

Asymptotic analysis of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder

The method of asymptotic integration of equations of elasticity [1] is used to study the behavior of the solution of a 3D elasticity problem for a radially inhomogeneous transversally isotropic hollow cylinder of small thickness. Under the assumption that the load is sufficiently smooth, the asymptotic method [1] is used to construct inhomogeneous solutions. An algorithm for constructing exact particular solutions of the equilibrium equations is given for loads of specific types in the case where the cylinder lateral surface is loaded by forces polynomially depending on the axial coordinate. Then the homogeneous solutions are constructed. The asymptotic expansions of homogeneous solutions are obtained, and the above analysis is used to explain the character of the stress-strain state.     

Approximate kinetic equations in rarefied gas theory

The fundamental kinetic equation of gas theory, the Boltzniann equation, is a complex integrodiffcrential equation. The difficulties associated with its solution are the result not only of the large number of independent variables, seven in the general case, but also of the very complicated structure of the collision integral. However, for the mechanics of rarefied gases the primary interest lies not in the distribution function itself, which satisfies the Boltzmann equation, but rather in its first few moments, i.e., the averaged characteristics. This circumstance suggests the possibility of obtaining the averaged quantities by a simpler way than the direct method of direct solution of the Boltzmann equation with subsequent calculation of the integrals.It is well known that if a distribution function satisfies the Boltzmann equation, then its moments satisfy an infinite system of moment equations. Consequently, if we wish to obtain with satisfactory accuracy some number of first moments, then we must require that these moments satisfy the exact system of moment equations. However, this does not mean that to determine the moments of interest to us we must solve this system, particularly since the system of moment equations is not closed. The closure of the system by specifying the form of the distribution function (method of moments) can be considered only as a rough approximate method of solving problems. First, in this case it is not possible to satisfy all the equations and we must limit ourselves to certain of the equations; second, generally speaking, we do not know which equation the selected distribution function satisfies, and, consequently, we do not know to what degree it has the properties of the distribution function which satisfies the Boltzmann equation.A more reliable technique for solving the problems of rarefied gasdynamics is that based on the approximation of the Boltzmann equation, more precisely, the approximation of the collision integral. The idea of replacing the collision integral by a simpler expression is not new [1–4]. The kinetic equations obtained as a result of this replacement are usually termed model equations, since their derivation is usually based on physical arguments and not on the direct use of the properties of the Boltzmann collision integral. In this connection we do not know to what degree the solutions of the Boltzmann equation and the model equations are close, particularly since the latter do not yield the possibility of refining the solution. Exceptions are the kinetic model for the linearized Boltzmann equation [5] and the sequence of model equations of [6], constructed by a method which is to some degree analogous with that of [5].In the present paper we suggest for the simplification of the solution of rarefied gas mechanics problems a technique for constructing a sequence of approximate kinetic equations which is based on an approximation of the collision integral. For each approximate equation (i.e., equation with an approximate collision operator) the first few moment equations coincide with the exact moment equations. It is assumed that the accuracy of the approximate equation increases with increase of the number of exact moment equations. Concretely, the approximation for the collision integral consists of a suitable approximation of the reverse collision integral and the collision frequency. The reverse collision integral is represented in the form of the product of the collision frequency and a function which characterizes the molecular velocity distribution resulting from the collisions, where the latter is selected in the form of a locally Maxwellian function multiplied by a polynomial in terms of the components of the molecular proper velocities. The collision frequency is approximated by a suitable expression which depends on the problem conditions. For the majority of problems it may obviously be taken equal to the collision frequency calculated from the locally Maxwellian distribution function; if necessary the error resulting from the inexact calculation of the collision frequency may be reduced by iterations.To illustrate the method, we solve the simplest problem of rarefied gas theory-the problem on the relaxation of an initially homogeneous and isotropic distribution in an unbounded space to an equilibrium distribution.The author wishes to thank A. A. Nikol'skii for discussions of the study and V. A. Rykov for the numerical results presented for the exact solution.     

Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Exact solution of the kinetic equation in the problem of nonisothermal flow in the neighborhood of a weakly curved surface

An analytic method for solving the half-space boundary value problem for the inhomogeneous Boltzmann equation with the collision operator in the form of an elliptico-statistical model (the ES-model of the Boltzmann equation) is proposed for the problem of nonisothermal rarefied gas flow in the neighborhood of a curved surface. An exact analytic expression is derived for the thermal slip of a monatomic gas along the surface of a rigid spherical aerosol particle. A numerical value of the gas-kinetic coefficient which takes into account the effect of the curvature of the surface on the thermal slip coefficient is obtained. A comparison with published data is carried out. Moscow, Arkhangelsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 165–173, March–April, 1998.     

Similitude of hypersonic flows of low-density gas over blunt bodies

Laws of similitude of hypersonic flows of monatomic gases have been obtained earlier from asymptotic analysis of the equations as S and confirmed by experimental data and numerical results [1], For diatomic gases, dimensionless numbers have not been deduced by analyzing the equations but by general arguments based on analogy with monatomic gases; they were used to compare experimental and calculated results in [1–3]. In the present paper, dimensionless numbers are derived on the basis of model kinetic equations for a diatomic gas, and limits of their applicability are established. Numerical calculations confirm the exact and approximate laws of similitude and permit a comparison with experimental results. The influence of the laws of viscosity on the drag for a sphere as a function of the Reynolds number Re₀ determined using the viscosity at the stagnation point is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 130–135, March–April, 1981.     

Two-dimensional flows of a viscous binary fluid between moving solid boundaries

A class of exact solutions of hydrodynamic equations with additional Korteweg stresses is obtained which is characterized by a linear dependence of part of the velocity components on the space variable. In this class, exact solutions of two problems of binary fluid flow between moving flat solid boundaries was found. A family of particular exact solutions is obtained for the problem of viscous fluid flow between planes which approach or move away from each other according to a special law.     

Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics

Four basic flow configurations are employed to investigate steady and unsteady rarefaction effects in monatomic ideal gas flows. Internal and external flows in planar geometry, namely, viscous slip (Kramer’s problem), thermal creep, oscillatory Couette, and pulsating Poiseuille flows are considered. A characteristic feature of the selected problems is the formation of the Knudsen boundary layers, where non-Newtonian stress and non-Fourier heat conduction exist. The linearized Navier–Stokes–Fourier and regularized 13-moment equations are utilized to analytically represent the rarefaction effects in these boundary-value problems. It is shown that the regularized 13-moment system correctly estimates the structure of Knudsen layers, compared to the linearized Boltzmann equation data.     

Non-stationary two-dimensional potential flows by the Broadwell model equations

The two-dimensional Broadwell model of discrete kinetic theory is studied in order to clarify the physical relevance of its solutions in comparison to the solutions of the continuous Boltzmann equation. This is achieved by determining completely, in closed form, all non-stationary potential flows with steady limiting conditions and isotropic pressure tensor at infinity. Several classes of exact solutions are also constructed when some of the above hypotheses are dropped. Most results are made possible by suitable transformations, which reduce essentially a complicated overdetermined system of partial differential equations to solving explicitly a Liouville equation. The structure of the obtained solutions, and especially the unphysical features that they exhibit, are finally commented on. It is remarkable that, for the problem considered here, there is no solution showing the typical qualitative features which characterize the continuous Boltzmann equation.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

Uniform rarefied gas flows at low Knudsen numbers

In [1,2] the exact solutions of the Boltzmann equation for stresses in shear [1] and uniform unsteady flows of a monatomic Maxwellian gas [2] were used to analyze, the region of applicability of the Chapman-Enskog method (some conclusions of these studies are summariized in [3]). In this paper we examine the second of these flows. In contrast with the cited studies, we consider the region of applicability of the Hilbert method and solve the problem of the initial Knudsen layer (where the time t^* is of the order of the mean time between collisions=/p). The results of the Chapman-Enskog and Hilbert methods are compared, and certain conclusions of [1, 2] are refined. The conclusions obtained are also basically valid for shear flow.The author thanks M. N. Kogan for discussions of this study.     

Exact solutions for buckling of non-uniform columns under axial concentrated and distributed loading

In this paper, the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading is studied. The expression for describing the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a non-uniform column with arbitrary distribution of flexural stiffness or axial forces is reduced to a second-order differential equation without the first-order derivative by means of functional transformations. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 12 cases, several of which are important in engineering practice. The exact solutions that represent a class of exact functional solutions for the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading are obtained. In order to illustrate the proposed method, a numerical example is given in the last part of this paper.     

Water Redistribution in Irrigated Soil Profiles: Invariant Solutions of the Governing Equation

Exact solutions for a class of nonlinear partial differential equations modelling soil water infiltration and redistribution in irrigation systems are studied. These solutions are invariant under two-parameter symmetry groups obtained by the group classification of the governing equation. A general procedure for constructing invariant solutions is presented in a way convenient for investigating numerous new exact solutions.     

Exact solutions in 1+1 dimensions of the general two-velocity discrete illner model

The Illner model is the most general two-velocity model of the discrete Boltzmann equation. It includes, as particular cases, both the Carleman and the McKean model. Exact solutions in 1+1 dimensions of the general two-velocity discrete Illner model can be studied in a concise way. The conclusions of the precursors need ameliorating. A new type of exact solutions in 1+1 dimensions is obtained. This gives a general method for studying non-trivial exact solutions for the similar discrete Boltzmann equation. Project supported by the National Natural Science Foundation of China (19631060) and the China Post-Doctoral Science Foundation     

Exact solution of the Navier-Stokes equations in a fluid layer between the moving parallel plates

This paper studies exact solutions of the Navier-Stokes equations for a layer between parallel plates the distance between which increases proportionally to the square root of time. A countable set of exact solutions and their derived countable set of continuous families of exact solutions are obtained. It is shown that certain intervals of the Reynolds parameter have two solutions and some of them one solution.