Thermal slip of a nonuniformly heated gas along a solid planar surface

Two methods are used to study the solution of a linearized model of the Boltzmann equation in the problem of thermal slip of a nonuniformly heated gas along a solid flat wall.The first method involves analytic solution of the integral equation for the average gas velocity. In the case of purely diffuse or purely specular reflection of the molecules from the wall surface the first method makes it possible to obtain analytically two important results; namely, the average gas velocity at the surface and at a large distance from the wall. The average gas velocity profile cannot be constructed analytically with this method. The second approximate method involves expanding the distribution function into a series in Sonine polynomials in velocity space and formulation of half-space moment equations from which the correction to the distribution function is determined. This method is used to obtain a simple analytic expression for the distribution function, from which we can find the average velocity profile for the gas for any arbitrary tangential momentum accommodation coefficient. In particular cases in which analytic solution of the problem by the first method is possible, good agreement is obtained between the two computational methods.It is known that a gas in a temperature gradient field tangent to the wall must begin to move in the direction of the temperature gradient (thermal slip). The first attempt to solve the thermal slip problem was made by Maxwell [1]. In his analysis Maxwell assumed that the distribution function of the molecules incident on the wall near the surface does not differ from the bulk distribution at a large distance from the wall. As a result Maxwell obtained the following expression for the thermal slip velocity for any tangential momentum accommodation coefficient u ^*=3/4 grad lnT.Here is the kinematic viscosity.However, in the case of molecular reflection from the wall which is not purely specular, the distribution of the incident molecules in the Knudsen layer differs from the bulk distribution because of collisions with the molecules reflected from the wall. Thus, Maxwell's assumption is not valid in the general case.For the exact solution of the problem it is necessary to find the distribution function in the Knudsen layer by solving the Boltzmann equation. Several investigators have used the Grad method [2] to find the distribution function in the Knudsen layer. However, the use of Grad's method in the thermal-slip problem leads to Maxwell's result [3].The solution of the thermal-slip problem obtained by Sone [4] is more exact than the analyses noted above. A comparison of the results obtained by Sone with those of this investigation is given at the end of our paper.     

Effect of Temperature on Slip Coefficients of Molecular Gases

Results obtained by accurate analytical methods applied to the problem of molecular-gas slip over a rigid spherical surface are reported. The Boltzmann equation is modified to take into account rotational degrees of freedom in the BGK model is used as a master kinetic equation. The calculated slip coefficients are shown to depend on the Prandtl number and on the gas temperature. Slip coefficients for several molecular gases are plotted as functions of temperature. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 58–65, January–February, 2006.     

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.     

On the Solution of a Boltzmann System for Gas Mixtures

We study the Boltzmann equation for a mixture of two gases in one space dimension with initial condition of one gas near vacuum and the other near a Maxwellian equilibrium state. A qualitative-quantitative mathematical analysis is developed to study this mass diffusion problem based on the Green’s function of the Boltzmann equation for the single species hard sphere collision model in Liu andYu (Commun Pure Appl Math 57:1543–1608, 2004). The cross-species resonance of the mass diffusion and the diffusion-sound wave is investigated. An exponentially sharp global solution is obtained.     

Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches

Numerical simulations of shock wave propagation in microchannels and microtubes (viscous shock tube problem) have been performed using three different approaches: the Navier–Stokes equations with the velocity slip and temperature jump boundary conditions, the statistical Direct Simulation Monte Carlo method for the Boltzmann equation, and the model kinetic Bhatnagar–Gross–Krook equation with the Shakhov equilibrium distribution function. Effects of flow rarefaction and dissipation are investigated and the results obtained with different approaches are compared. A parametric study of the problem for different Knudsen numbers and initial shock strengths is carried out using the Navier–Stokes computations.      

Case Method Employed for Solving the Problem of Thermal Creep of a Rarefied Gas along a Solid Cylindrical Surface

An analytical solution of a halfspace boundaryvalue problem is constructed for an inhomogeneous kinetic Boltzmann equation with the collision operator in the form of an operator of an ellipsoidal statistical model in the problem on thermal creep of a rarefied gas along a solid cylindrical surface. Corrections to the thermal creep coefficient are obtained for the cases of longitudinal and transverse flow past a straight circular cylinder in the approximation linear with respect to the Knudsen number, allowing for the interfacial curvature. The results are compared with available data.     

Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian

We consider the initial value problem for the Fokker–Planck–Boltzmann equation namely, viewed as the Boltzmann equation with an additional diffusion term in velocity space to describe, for instance, the transport in thermal baths of binary elastic collisional particles. The strong solution for initial data near an absolute Maxwellian is proved to exist globally in time and tends asymptotically in the -norm to another time dependent self-similar Maxwellian in large time. The effect of the diffusion in phase space is investigated. It produces a diffusion process in velocity space and results in a heating process on the macroscopic fluid-dynamic observable, accelerating the convergence of solutions to the equilibrium of a self-similar Maxwellian at a faster time-decay rate than the Boltzmann equation. This phenomena is also observed for homogeneous Fokker–Planck–Boltzmann equations, where the time-decay rate in the -norm to the self-similar Maxwellian is proved to be faster than exponential. Moreover, the Fokker–Planck–Boltzmann equation is shown to converge (under an appropriate scaling) strongly to the Boltzmann equation in the process of the zero diffusion limit.     

Slip coefficients and macroparameter jumps of a diatomic gas with rotational degrees of freedom on a weakly curved spherical surface

Using the half-space moment method, the problem of the slip of a diatomic gas along a rigid spherical surface is solved within the framework of a model kinetic equation previously proposed which takes into account the rotational degrees of freedom of the gas. Second-order slip coefficients (correctionsC _m ^′ , β _R ^′ , and β _R to the isothermal and thermal slip which are linear with respect to the Knudsen number Kn) are obtained. The gas macroparameter jump coefficientsC _v andC _q, which are of the second order in the Knudsen number and characterize the discontinuity of the normal mass and heat fluxes on the gas-rigid phase interface, are calculated. These coefficients are given as functions of the tangential momentum accommodation coefficient, the translational and rotational energy accommodation coefficients, and the Prandtl number. The coefficients are calculated for certain diatomic gases. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–173, January–February, 2000.     

Rotation of a sphere in an unbounded gas

The problem of the slow rotation of a sphere in an unbounded gas is solved at arbitrary Knudsen numbers. The kinetic equation with collision integral in the form of the Bhatnagar-Gross-Krook model (BGK-model) describing the state of the surrounding medium is solved by the Lees method, all the moments of the distribution function which ensure the asymptotic passage of the distribution function into the Chapman-Enskog distribution function at large distances from the sphere being taken into account. In the particular case of moderately large spherical particles a value of the isothermal slip coefficient similar to the exact value is obtained. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 165–171, January–February, 1997.     

Properties of the nonequilibrium high-velocity “Tails” of the molecular distribution function in plane couette flow of a compressible gas

The results of an analytic and numerical investigation of the properties of the high-velocity “tails” of the distribution function are given for the solution of the BGK model of the kinetic Boltzmann equation for plane Couette flow of a compressible gas. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–190, July–August, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00573; grant in support of leading science schools No. 96-15-9603).     

Analytic solution of the ellipsoidal-statistical model Boltzmann equation

An exact solution of the ellipsoidal-statistical model Boltzmann equation is constructed. The problem of the temperature jump in a rarefied gas is considered by way of illustration. By expanding the distribution function in two orthogonal directions the problem is reduced to the solution of a vector transport equation with polynomial boundary conditions. The Case approach reduces the equation to a characteristic equation for which generalized eigenvectors and eigenvalues are found. A theorem of existence and uniqueness of the solution, represented in the form of an expansion in eigenvectors, is proved. The proof reduces to solving a Riemann-Hilbert vector boundary-value problem with a matrix coefficient whose diagonalizing matrix has branch points in the complex plane. The value of the temperature jump is found from the conditions of solvability of the boundary-value problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 151–164, March–April, 1992.     

Focusing of a shock wave in a rarefied gas A numerical study

The process of focusing of a shock wave in a rarefied noble gas is investigated by a numerical solution of the corresponding two dimensional initial–boundary value problem for the Boltzmann equation. The numerical method is based on the splitting algorithm in which the collision integral is computed by a Monte Carlo quadrature, and the free flow equation is solved by a finite volume method. We analyse the development of the shock wave which reflects from a suitably shaped reflector, and we study influence of various factors, involved in the mathematical model of the problem, on the process of focusing. In particular, we investigate the pressure amplification factor and its dependence on the strength of the shock and on the accommodation coefficient appearing in the Maxwell boundary condition modelling the gas-surface interaction. Moreover, we study the dependence of the shock focusing phenomenon on the shape of the reflector, and on the Mach number of the incoming shock. Received 25 May 1998 / Accepted 4 January 2000     

Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects

The constitutive relations of gases in a thermal nonequilibrium (rarefied and microscale) can be derived by applying the moment method to the Boltzmann equation. In this work, a model constitutive relation determined on the basis of the moment method is developed and applied to some challenging problems in which classical hydrodynamic theories including the Navier–Stokes–Fourier theory are shown to predict qualitatively wrong results. Analysis of coupled nonlinear constitutive models enables the fundamentals of gas flows in thermal nonequilibrium to be identified: namely, nonlinear, asymmetric, and coupled relations between stresses and the shear rate; and effect of the bulk viscosity. In addition, the new theory explains the central minimum of the temperature profile in a force-driven Poiseuille gas flow, which is a well-known problem that renders the classical hydrodynamic theory a global failure.     

Effect of accuracy of solving integral equations for the thermal and viscous Chapman functions on the value of gas-kinetic coefficients

The dependence of the gas-kinetic coefficients on the accuracy of calculating the thermal and viscous Chapman functions for the case of a simple gas in the neighborhood of a plane rigid surface is studied. Expressions for the gas-kinetic coefficients are obtained by solving the Boltzmann equation using the Loyalka method. In order to find the temperature jump we use boundary conditions which take into account the accommodation both on the energy and the momentum. The effect of the accuracy of solving the integral equations for the thermal and viscous functions on the value of the temperature jump and the thermal and isothermal slip coefficients was studied by taking into account one, two or three terms in expansions of these functions in Sonine polynomials. The dependence of the results on the choice of the molecule interaction potential model is analyzed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–198, March–April, 1995.     

Exact solutions of model BGK Boltzmann equation in temperature jump and weak evaporation problems

An exact solution to the model Boltzmann equation with Bhatnagar-Gross-Krook (BGK) collision operator is obtained in the problems of weak evaporation and temperature and density jumps of a rarefied gas in a half-space. Case's method is used to find generalized eigenvectors of the corresponding characteristic equation. An existence and uniqueness theorem for the solution of the posed problems with boundary conditions on a flat surface and far from it is proved. For this, we develop a formalism of diagonalization and factorization of the vector Riemann-Hilbert boundary-value problem with matrix coefficient whose diagonalizing matrix has branch points in the complex plane.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 163–171, January–February, 1992.     

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].     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: II Computational Validation

In Part I Moyne and Murad [Transport in Porous Media 62, (2006), 333–380] a two-scale model of coupled electro-chemo-mechanical phenomena in swelling porous media was derived by a formal asymptotic homogenization analysis. The microscopic portrait of the model consists of a two-phase system composed of an electrolyte solution and colloidal clay particles. The movement of the liquid at the microscale is ruled by the modified Stokes problem; the advection, diffusion and electro-migration of monovalent ions Na⁺ and Cl⁻ are governed by the Nernst–Planck equations and the local electric potential distribution is dictated by the Poisson problem. The microscopic governing equations in the fluid domain are coupled with the elasticity problem for the clay particles through boundary conditions on the solid–fluid interface. The up-scaling procedure led to a macroscopic model based on Onsager’s reciprocity relations coupled with a modified form of Terzaghi’s effective stress principle including an additional swelling stress component. A notable consequence of the two-scale framework are the new closure problems derived for the macroscopic electro-chemo-mechanical parameters. Such local representation bridge the gap between the macroscopic Thermodynamics of Irreversible Processes and microscopic Electro-Hydrodynamics by establishing a direct correlation between the magnitude of the effective properties and the electrical double layer potential, whose local distribution is governed by a microscale Poisson–Boltzmann equation. The purpose of this paper is to validate computationally the two-scale model and to introduce new concepts inherent to the problem considering a particular form of microstructure wherein the clay fabric is composed of parallel particles of face-to-face contact. By discretizing the local Poisson–Boltzmann equation and solving numerically the closure problems, the constitutive behavior of the diffusion coefficients of cations and anions, chemico-osmotic and electro-osmotic conductivities in Darcy’s law, Onsager’s parameters, swelling pressure, electro-chemical compressibility, surface tension, primary/secondary electroviscous effects and the reflection coefficient are computed for a range particle distances and sat concentrations.     

Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 110–123, March–April, 2000.     

Analytic Solution of an Inhomogeneous Kinetic Equation in the Problem of Second-Order Thermal Creep

An analytic solution of the problem of second-order thermal creep is obtained. A method for solving the half-space boundary value problem for an inhomogeneous linearized kinetic BGK equation forms the basis of the solution. The general solution of the input equation is constructed in the form of an expansion of the corresponding characteristic equation in terms of the eigenfunctions. Substitution of the solution in the boundary conditions leads to a Riemann boundary value problem. The unknown thermal creep velocity is found from the condition of solvability of the boundary value problem. The numerical analysis performed confirms the existence of negative thermophoresis (in the direction of the temperature gradient) for high-conductivity aerosol particles at low Knudsen numbers.     

Gas-kinetic theory of lubrication

An equation of the gas-kinetic theory of lubrication is obtained under the assumption of incompressibility of the gas on the basis of solution of the Boltzmann equation by the moment method with a special approximating function. In the limit of a small Knudsen number calculated using the minimal gap, the equation goes over into Reynolds's wellknown equation. Reynolds's problem of a lubricating layer of gas between two closely spaced planes is considered. In the limit of a small Knudsen number, agreement with the well-known solution of the hydrodynamic theory is obtained. A comparison is made with the solution obtained by the hydrodynamic method with slip boundary conditions under neglect of the compressibility of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 161–166, January–February, 1984.I thank L. P. Smirnov for constant interest in the work, and also the participants of G. I. Petrov's seminar for helpful discussions.