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.     

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].     

The regularized Chapman-Enskog expansion for scalar conservation laws

Rosenau [R] has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at low wave numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator.In this paper we study the behavior of the Rosenau regularization of the Chapman-Enskog expansion (R-C-E) in the context of scalar conservation laws. We show that this R-C-E model retains the essential properties of the usual viscosity approximation, e.g., existence of travelling waves, monotonicity, upper-Lipschitz continuity, etc., and at the same time, it sharpens the standard viscous shock layers. We prove that the regularized R-C-E approximation converges to the underlying inviscid entropy solution as its mean-free-path 0, and we estimate the convergence rate.     

Calculation of the Slip Velocity of a Rarefied Gas on a Solid Spherical Surface with Allowance for Accommodation Coefficients

The slip velocity of a rarefied gas with inhomogeneous temperature and mass velocity on a solid spherical surface is calculated with the use of a twomoment boundary condition in the linear approximation in terms of the Knudsen number. The dependence of the slip velocity on accommodation coefficients of the two first moments of the distribution function is studied.     

The behaviour of macromolecules in the flow through a sudden planar contraction

Analytic steady-state results for FENE-P model macromolecules, in the nearly coiled-up and nearly stretched state respectively, in general two-dimensional flow fields are derived. These results are utilized in the flow through a sudden planar contraction. Special emphasis is devoted to the structure tensor R R, which furnishes, among other things, the mean square extension and the average orientation of the macromolecules.     

Multiscale Expansion Method in the Hilbert Problem

The problem of constructing an asymptotic approximation to the solution of the kinetic Boltzmann equation is considered for the hydrodynamic region of low Knudsen numbers. The problem is linearized for one-dimensional perturbations in a gas at rest. The distribution function is sought in the form of a multiscale expansion of the Hilbert asymptotic series type. The construction of a solution uniformly suitable as t is demonstrated with reference to a particular example of sonic wave propagation. It is shown that the multiscale technique makes it possible to extend the domain of applicability of the Hilbert expansion to the entire interval of dissipative relaxation.     

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.     

A method for solving boundary-value problems of transfer at arbitrary knudsen numbers

A new method is proposed, called the method of the smoothed distribution function, which makes possible a considerable simplification in the procedure for calculating moments of the collision integral, and enables us to obtain a solution to the system of Maxvell-Boltzmann moment equations. The approximating distribution function used in the collision integral in the Boltzmann form ensures a limiting process towards continuous expressions for the flux of molecular characteristics. For the example of the solution to the classical problem of heat transfer between two parallel plates with arbitrary Knudsen numbers, a comparison was made of the theoretical results with the results of other analyses, and also with experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 3, pp. 182–185, May–June, 1986.     

Heat transport in a parabolic tube of circular cross-section

The local temperature has been determined for a viscous liquid flowing through a paraboloidal tube. Wall temperature and inlet temperature have been considered constant. The liquid flow was considered as creeping flow and its velocity distribution was determined by solving the biharmonic differential equation of the stream function. The local temperature was evaluated numerically from the analytical results.

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Wärmetransport im Paraboloidrohr

Zusammenfassung Es wird die lokale Temperatur in einer viskosen Strömung durch ein Paraboloidrohr bestimmt. Dabei wird konstante Wand- und Einlauftemperatur angenommen. Die kriechende Strömungsgeschwindigkeit wurde aus der Lösung der biharmonischen Differentialgleichung der Stromfunktion bestimmt. Die lokale Temperatur wurde aus den analytischen Ergebnissen für einige Paraboloidrohre numerisch bestimmt.

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Nomenclature ₁ F ₁ confluent hypergeometric function - diffusivity - T(, , ) temperature - T _w temperature at the paraboloidal wall - T _i temperature at the inlet - u(, ) flow velocity of viscous liquid in -direction - volumetric flow - eigenvalues of confluent hypergeometric function - streamfunction - _o wall of paraboloidal tube - _o inlet of paraboloidal tube - , , paraboloidal coordinates     

Solution of the linearized couette-flow problem in a rarefied gas by the integral-diffusion method

In ordinary diffusion theory the transfer of properties is determined by the local gradients of the corresponding fields. As the mean free path increases, the flux density becomes an integral quantity and is determined by a neighborhood of the point under consideration of the order of a few mean free paths. In a previous article [1], the author proposed a model for a one-dimensional transfer process in linear rarefield-gas problems based on the analogy with radiative transfer. The same approach, though without directional averaging, is used in the present paper to analyze the linearized Couette flow problem. The solution obtained here has the properties of the solution obtained by more exact methods based on the solution of the Boltzmann equation [3-4].Nomenclature p_xy shear stress - c mean thermal velocity of molecules - 2/3 A mean free path - d half-width of channel - ±w₀ plate velocity - c nonequilibriumvalue of momentum flux density - y transverse coordinate - ratio of specific heats - W dimensionless velocity - P_xy shear stress scaled with respect to the shear stress in free-molecule flow - Y dimensionless coordinate - W₁(y) velocity distribution according to Millikan's solution - coefficient of viscosity - R Reynolds number - K Knudsen number     

Die kompressible Grenzschichtströmung am dreidimensionalen Staupunkt bei starkem Absaugen oder Ausblasen

Zusammenfassung Es wird die kompressible, laminare Grenzschichtströmung am dreidimensionalen Staupunkt mit Absaugen oder Ausblasen an der Wand untersucht und daraus Wandschubspannung, Wärmeübergang und Verdrängungsdicke in Abhängigkeit von der Normalgeschwindigkeit an der Wand bestimmt. Besonders ausführlich wkd auf die Grenzfälle sehr starken Absaugens bzw. Ausblasens eingegangen, die auf singuläre Störungsprobleme führen, deren Lösung mit der Methode der angepaßten asymptotischen Entwicklungen erfolgt. – Die Unsymmetrie am Staupunkt wird durch den Parameter c gekennzeichnet mit den Spezialfällen c=0 (ebener Staupunkt) und c=1 (rotationssymmetrischer Staupunkt). Im Grenzfall starken Absaugens sind die beiden Wandschubspannungskomponenten und der Wärmeübergang unabhängig von c, im Grenzfall starken Ausblasens ist nur eine der beiden Wandschubspannungskomponenten von c unbeeinflußt.

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The compressible boundary layer flow at a threedimensional stagnation point with intensive suction or injection

The compressible laminar boundary layer flow at a general three-dimensional stagnation point including large rates of injection or suction on the porous surface is considered. The wall shear stress, heat flux and displacement thickness as function of the mass transfer parameter are determined. The two limiting cases of intensive suction and intensive blowing lead to singular perturbations problems, which are solved by the method of matched asymptotic expansions.—The asymmetry of the stagnation point flow is characterized by the ratio c of the two velocity gradients including the special cases of two-dimensional (c=0) and axisymmetric (c=1) stagnation point flow.-For intensive suction the wall shear stresses and the heat flux become independent of c, whereas for intensive blowing only one of the two wall shear stress components is independent of c.

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Bezeichnungen x, y, z kartesische Koordinaten, siehe Bild 1 - u, v, w Geschwindigkeitskomponenten in x-, y- und z-Richtung innerhalb der Grenzschicht - U, V Geschwindigkeitskomponenten in x- und y-Richtung am Außenrand der Grenzschicht - a =(dU/dx)_x=0 Geschwindigkeitsgradient in x-Richtung der Außenströmung im Staupunkt - b=(dV/dy)_y=0 Geschwindigkeitsgradient in y-Richtung der Außenströmung im Staupunkt - c=b/a Staupunkt-Parameter (c=0: ebene Strömung, c=1: axialsymmetrische Strömung) - Dichte - p Druck - h spezifische Enthalpie - Viskosität - Pr Prandtl-Zahl - _x, _y Wandschubspannungskomponenten in x- bzw. y-Richtung - c_m=–ga Ausblaseparameter, siehe Gl. (21) - t_w= h_w/h_e bezogene Wandenthalpie - Ähnlichkeitsvariable, siehe Gl. (11) - F () G () dimensionslose Funktionen nach den Gln. (12), - (),() (13), (15) und (32) - ¯ Ähnlichkeitsvariable, siehe Gl. (29) - F (¯), G (¯) dimensionlose Funktionen nach Gl. (29) - ₁=1/c_m Störparameter für starkes Absaugen - ₂=1/c _m ² Störparameter für starkes Ausblasen - ^*, ^* Verdrängungsdicken nach Gl. (26) - R, R rechte Seiten der Differentialgln. (40) bzw. (41) - z=/ in Abschnitt 4: bezogener Wandabstand nach Gl. (43) - =t_w·z² Ähnlichkeitsvariable in Abschnitt 4 - ^* (c) Lösung der Gl. (58) - A(c) Definition nach Gl. (63) Indizes w an der Wand - e am Außenrand der Grenzschicht - a äußere Lösung - i innere Lösung     

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.     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Rheology of carbon black suspensions. III. Sol-gel transition system

Linear viscoelastic properties of carbon black (CB) suspensions with various CB volume fractions () in a rosin-modified phenol resin type varnish (Varnish-1) were investigated at various temperatures (T). The CB/Varnish-1 suspensions exhibited a sol-gel transition on an increase in , and the _gel value at the gelation point decreased with increasing T. This T dependence of _gel, being opposite to the dependence seen for usual gelling systems, can be related to a phenol resin type polymeric component included in the Varnish-1. At low T, this polymeric component appeared to be rather well solvated in the Varnish-1 thereby allowing the gelation due to bare attraction between the CB particles at large . In contrast, at high T, the polymeric component appeared to have been less solvated, as evidence from a moderate failure of the time-temperature superposition of pure Varnish-1 and a decrease of its elasticity (in a shifted frequency scale) with increasing T. This less solvated polymeric component would have been adsorbed on the CB particles, thereby allowing the agglomeration of the particles at small _gel at high T.     

Spatial numerical simulations of linear and weakly nonlinear wave instabilities in supersonic boundary layers

The spatial development of disturbances with small and moderate amplitudes in a two-dimensional (2-D) supersonic flat-plate boundary layer at Mach 4.8 is investigated using direct numerical simulations based on the compressible 3-D Navier-Stokes equations. Disturbances are introduced into the boundary layer by blowing and suction within a narrow disturbance strip at the wall. In response to the timewise periodic forcing, two types of disturbance waves are generated, a first-mode wave and a multiple-viscous-solution. The multiple-viscous-solution was described by Mack (1969, 1984) but was not seen before in a direct numerical simulation. The results of the simulations are compared with results of linear stability theory, and the agreement is very good. In simulations for larger amplitudes, fundamental resonance is observed, where both types of 3-D waves are nonlinearly amplified and synchronize their phase velocities with the 2-D disturbance waves. Subharmonic resonance is found for 3-D waves with large wave numbers, where the phase velocities of the linear 2-D and 3-D waves are nearly the same.This work was supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn-Bad Godesberg, as part of SFB 259.     

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.     

New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

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Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

Gaussian Closure of Transient Unsaturated Flow in Random Soils

The Gaussian closure approximation, previously used by the authors to solve steady state stochastic unsaturated flow problems in randomly heterogeneous soils, is extended here to transient flow. The method avoids linearizing the governing flow equations or the soil constitutive relations. It places no theoretical limit on the variance of constitutive parameters and applies to a broad class of soils with flow properties that scale according to a linearly separable model. Closure is obtained by treating the dimensionless pressure head as a multivariate Gaussian function. It yields a system of coupled nonlinear differential equations for the first and second moments of . We apply the Gaussian closure technique to the problem of transient infiltration into a randomly stratified soil. In each layer, hydraulic conductivity and water content vary exponentially with . Elsewhere we show that application of the technique to other constitutive relations is straightforward. Our solution for the mean and variance of in a one-dimensional layer with random conductivity compares well with Monte Carlo results over a wide range of parameters, provided that the spatial variability of the constitutive exponent is small. The solution provides considerable insight into the behavior of the transient unsaturated stochastic flow problem.     

Numerical Simulation of Ferrofluid Flow for Subsurface Environmental Engineering Applications

Ferrofluids are suspensions of magnetic particles of diameter approximately 10nm stabilized by surfactants in carrier liquids. The large magnetic susceptibility of ferrofluids allows the mobilization of ferrofluid through permeable rock and soil by the application of strong external magnetic fields. We have developed simulation capabilities for both miscible and immiscible conceptualizations of ferrofluid flow through porous media in response to magnetic forces arising from the magnetic field of a rectangular permanent magnet. The flow of ferrofluid is caused by the magnetization of the particles and their attraction toward a magnet, regardless of the orientation of the magnet. The steps involved in calculating the flow of ferrofluid are (1) calculation of the external magnetic field, (2) calculation of the gradient of the external magnetic field, (3) calculation of the magnetization of the ferrofluid, and (4) assembly of the magnetic body force term and addition of this term to the standard pressure gradient and gravity force terms. We compare numerical simulations to laboratory measurements of the magnetic field, fluid pressures, and the twodimensional flow of ferrofluid to demonstrate the applicability of the methods coded in the numerical simulators. We present an example of the use of the simulator for a fieldscale application of ferrofluids for barrier verification.     

High-altitude aerothermodynamics

Hypersonic high-altitute flight can be conventionally divided into three regimes: the continuum regime, when the Knudsen number Kne1, the free-molecule regime (Kn1), and the transitional regime (K1). In general, each of these regimes differs with respect to both the structure of the flow and the method of determining the aerodynamic and thermal characteristics. For Knudsen numbers Kne1 the Navier-Stokes equations or models with slip and temperature jump boundary conditions are widely used. When Kn1 the methods employed are mainly directed towards determining the distribution function of the molecules reflected from the surface of the body. On the transition interval between these two limiting regimes numerical methods of solving the Boltzmann equation and its model equations are being used with success. Together with the experimental techniques, these various methods, which complement each other, make it possible to investigate gas flows fairly effectively from the continuum to the free-molecule regime (see, for example, [1]).Based on a paper presented to the Fluid Mechanics Section of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 142–152, March–April, 1993.