The local transport relations in a rarefied gas

Fluid flows that do not have local equilibrium are characteristic of some of the new frontiers in engineering and technology, for example, high-speed high-altitude aerodynamics and the development of micrometre-sized fluid pumps, turbines and other devices. However, this area of fluid dynamics is poorly understood from both the experimental and simulation perspectives, which hampers the progress of these technologies. This paper reviews some of the recent developments in experimental techniques and modelling methods for non-equilibrium gas flows, examining their advantages and drawbacks. We also present new results from our computational investigations into both hypersonic and microsystem flows using two distinct numerical methodologies: the direct simulation Monte Carlo method and extended hydrodynamics. While the direct simulation approach produces excellent results and is used widely, extended hydrodynamics is not as well developed but is a promising candidate for future more complex simulations. Finally, we discuss some of the other situations where these simulation methods could be usefully applied, and look to the future of numerical tools for non-equilibrium flows.

Basic concepts of rarefied gas dynamics are given in a concise form. Some problems of rarefied gas flows are considered, namely, calculations of velocity slip and temperature jump coefficients, gas flow through a tube due to pressure and temperature gradients, and gas flow through a thin orifice. Results on the two last problems are given over the whole range of gas rarefaction. A methodology for modelling the Holweck pump is described. An extensive list of publications on these topics is given. 1 Brief history of rarefied gas dynamics Rarefied gas dynamics is based on the kinetic approach to gas flows. In 1859 Maxwell [1] abandoned the idea that all gaseous molecules move with the same speed and introduced the statistical approach to gaseous medium, namely, he introduced the velocity distribution function and obtained its expression in the equilibrium state. Thus Maxwell gave the origin to the kinetic theory of gases. Then, in 1872 Boltzmann [2] deduced the kinetic equation which determines the evolution of the distribution function for gaseous systems being out of equilibrium. In 1909 Knudsen [3], measuring a flow rate through a tube, detected a deviation from the Poiseuille formula at a low pressure. Such a deviation was explained by the fact that at a certain pressure the gas is not a continuous medium and the Poiseuille formula is not valid anymore. A description of such a flow required the development of a new approach based on the kinetic theory of gases. This can be considered as the beginning of rarefied gas dynamics. Later, advances were made by Hilbert [4], Enskog [5] and Chapman [6] to solve the Boltzmann equation analytically via an expansion of the distribution function with respect to the Knudsen number. The main result of this solution was a relation of the transport coefficients to the intermolecular interaction potential, but no numerical calculation of rarefied gas flows could be realized at that time. In 1954 the so-called model equations [7,8] were proposed to reduce the computational efforts in calculations of rarefied gas flows. Using these models it was possible to obtain numerical results on rarefied gas flows in the transition regime. Thus in 1960 a numerical investigation of rarefied gas flows began in its systematic form. For a long time, it was possible to solve only the model equations. Practically, all classical problems of gas dynamics (Poiseuille flow, Couette flow, heat transfer between two plates, flow past a sphere, etc.) were solved over the whole range of gas rarefaction by applying the model equations. In 1989 first results based on the exact Boltzmann equation were reported, see, for example, Ref. [9]. However, even using the powerful computers available nowadays, a numerical calculation based on the Boltzmann equation itself is still a very hard task, which requires great computational efforts. Thus, the model equations continue to be a main tool in practical calculations. Below, the main concepts of rarefied gas dynamics and some examples of its application will be given. In the last section, the main results of rarefied gas dynamics that could be applied to vacuum technology are listed.

A comprehensive review on micro- and nano-scale gas flow effects: Slip-jump phenomena, Knudsen paradox, thermally-driven flows, and Knudsen pumps

The investigation of fluid flow in a tube or channel commenced from studying experimentally the liquid flow along a circular tube under the pressure difference imposed at both the ends, the results were verified by the exact solution of the Navier-Stokes Eq. under constant pressure gradient assumption (Poiseuille flow). The experimental work has been playing a pioneering role in the investigation of channel flows. In the case of gas for the tube (or channel) flow, according to the global mass conservation law, the pressure gradient is never a constant. The theoretical studies on rarefied gas Poiseuille flow based on constant pressure gradient assumption still compared well with the early experimental work because when pressure difference between the inlet and the outlet is small in comparison with the average pressure, the pressure distribution is approximately a linear one. Recently non-linear pressure distribution was experimentally found in the integrated micro- channel/pressure sensor systems with gas flow in the transitional regime. The mass flow rate of micro channel was exactly measured, challenging the rarefied gas dynamics community to put forward computation or simulation means for the calculation of micro-channel flow, and the flow characteristics in other micro devices. This paper reviews some of these computational efforts. It also presents a strict kinetic solution of the finite length micro-channel flow problem based on the global mass conservation and the exact kinetic theoretical solution of the Poiseuille flow at each section. Calculations and simulations are compared with reviewed experiments to check the agreement between the computational or theoretical results with experimental data. Keywords: Microchannel flow, Poiseuille flow, finite length channel flow, gas flow in channels, lattice Boltzmann method (LBM), information preservation (IP) method, kinetic theoretical solution of microchannel flow, global mass conservation for channel flow

An object-oriented serial implementation of a DSMC simulation package

Flow of a fluid through a parallel channel is one of the simplest types of flow. However, the equations of flow and energy are far from simple and can only be solved in a closed form in the simplest cases when nonlinear effects such as the inertia and convective terms can be neglected (i.e., for zero Reynolds number) and when the energy and momentum equations are uncoupled. A numerical iterative method is described in which the coupled momentum and energy equations are solved when the viscosity, thermal conductivity and specific heat are functions of temperature, and the density a function of temperature and pressure; inertia terms are retained in the momentum equation and the convective terms, compression work term and the predominant dissipation term retained in the energy equation. Results are obtained for a variety of boundary temperatures up to about 2400 deg F and the effect of variable fluid properties and various terms in the energy and momentum equation are shown.

Comparison of measured and predicted thermal mixing tests using improved finite difference technique

ONLINEAR problems of thermal conduction in monatomic rarefied gas confined within two concentric cylinders and spheres are studied by the four-moment method coupled with the bimodal two-stream distribution function of Beck. Finite temperature difference between the two boundary surfaces is included in the present theory. Therefore, present results are presumed to be complete solutions obtainable by the fourmoment technique with the assumed velocity distribution of the gas. Content Recently, Lees' four-moment approach1 for heat conduction problem in rarefied gases was further extended by Lou and Shih2 to investigate nonlinear problems. Instead of linearizing the problem, they employed the technique of series expansion. It was shown that the nonlinear solutions could be obtained in analytic forms. In principle, their method is suitable for any temperature difference. However, as the temperature difference is getting larger, the solutions will have to include more higherorder terms in the series; hence, they become cumbersome. In order to overcome this inconvenience, another modified fourmoment approach is proposed. Let us consider the steady heat conduction problem in a rarefied Maxwellian gas confined between two concentric cylinders, or spheres. The surface of the inner cylinder (or sphere) with radius Rf is maintained at a uniform constant temperature 7}; and the inner surface of the outer cylinder (or sphere) with radius Rn is maintained at Tu. The velocity distribution function of gas molecules f(V9 R) is assumed to be the two-steam bimodal distribution function.3 In region 1

In this paper the equations of motion for a steady, one-dimensional, viscous and compressible gas are simplified according to two procedures and are then integrated. In one procedure the viscosity term is retained in the momentum equation and omitted in the energy equation. In the second procedure the viscosity term is omitted in the momentum equation and retained in the energy equation. The equations used in the first case are the momentum equation, into which is introduced the frictional force [= (4/3)judfc-rdx]; the continuity equation; and a polytropic relationship between pressure and density, the last expression replacing the energy equation in its usual form. The value of the exponent n, in the polytropic relationship for which the Rankine-Hugoniot points are satisfied, is shown to be a function of the initial Mach Number, Mo, only. I t is strongly indicated that retaining the viscosity terms in the momentum equation and neglecting them in the energy equation would offer a significant and considerably simplified mathematical representation of a steady compressible viscous flow in two and three dimensions. In the second case, the Eulerian equation, together with the continuity and energy equations for a one-dimensional motion, describes approximately the flow of a viscous, heat-conducting gas. This is analogous to the treatment of Boley and Lieber for two-dimensional flow. An exact solution is obtained for the shock-wave structure in both cases, and, although the velocity distribution in the latter case is continuous everywhere, the singularity in the velocity gradient is not removable. However, while the present report shows that retaining the viscous term in the energy equation alone is not sufficient for the removal of this singularity, it is shown that the frictional force, when considered in the momentum equation, does remove the singularity. Since the shock-wave thickness, according to the Prandtl definition, is zero in the second treatment (a physical impossibility), it was not found necessary to calculate the variation of the remaining flow variables (p, p, S, T) with Mach Number. Good agreement with the exact one-dimensional solutions of Morduchow and Libby and H. Reissner and MeyerhofF with regard to the structure of the shock wave and its thickness is obtained when the effect of viscosity is considered only in the momentum equation. Moreover, indications are such as to justify the conclusion that the presence of the viscous stress term in the momentum equation is a necessary condition for the removal of the singular velocity gradient in one-dimensional flow.

Multiscale simulation of molecular gas flows by the general synthetic iterative scheme

The effect of a stationary solid body on the flow of rarefied gases is considered. We also consider the effect of an oscillating solid body on a rarefied gas. For a gas of particles of arbitrary statistics we derive coupled integral equations for the chemical potential, velocity field, and temperature. We show that the stationary value of a functional is closely related to the drag exerted on the body. When the mean free path is much larger than a typical dimension of the body, we show that the drag is given by a simple surface integral.

<p><span>Natural gas hydrates, which are ice like crystalline solids, contain tremendous amount of potential hydrocarbon gas. Gas recovery through hydrate dissociation can be achieved through depressurization, inhibitor injection and thermal stimulation. The hydrate dissociation by depressurization involves significant pressure and temperature gradients as the dissociation process is highly endothermic. The destabilization of solid hydrate into fluid constituents causes loss of cementation which can alter the stress field which in turn changes the porosity and permeability of the hydrate bearing medium causing subsidence. In the present study, a thermo-hydro-mechanical-chemical (THMC) coupled numerical simulator is developed accounting for the hydrate phase change kinetics, non-isothermal multiphase flow and geomechanics. The point centered or node centered finite volume method is used for space discretization of flow and energy equations while the finite element method is used for stress equilibrium equation. This procedure requires the flow and mechanics variables to be co-located. The finite volumes are constructed around the flow variables defined at nodes while the finite element is defined by the corner nodes. The volumetric strain rate term in the flow equations, which couples the flow and geomechanics equations, is evaluated by interpolating the volumetric strains calculated over the finite elements to the finite volumes. Our simulations show that this procedure results in a stable convergence of the solution without the need for any stabilizing terms due to co-located variable arrangement. Our simulations also show that the iterative coupled approach, where the flow and geomechanics equations are solved separately and sequentially, gives stable convergence without any additional split terms due to sequential but iterative solving of the coupled equations.</span></p>

The results are presented of a study of the application of analytical methods to the solution of two phase flow into single wells. Approximate analytical expressions for the pressure distribution in two-phase flow are found for a number of conditions. The results obtained from the analytical solutions are found to be in good agreement with results obtained by finite difference techniques using a high speed digital computer. Mathematical solutions for four sets of boundary conditions are presented. All of these solutions are composed of a short-term transient plus a steady or quasi-steady state. The rates of decay of the short-lived transients are analyzed. It is found that the durations of the short-term transients may be characterized by a parameter defined as the time constant which can be determined from simple relations. It is shown also that if the outer radius is much greater than the radius of the well, the short term transients decay at rates which are proportional to the square of the exterior radius, and the rates of decay are only slightly dependent upon the radius ratio. Numerical solutions based on finite difference techniques are presented for a number of conditions. The numerical solutions are in good agreement with the predictions based on the theoretical analysis for small and moderate drawdowns. Examples involving large drawdowns indicate that the nonlinearities in the equations of flow do not appreciably alter the longevity of the short-term transients. In all cases the time required for the short-term transients to disappear is predicted satisfactorily. Introduction The mechanism by which oil and gas flow into a single well is of vital interest to the petroleum industry. The fundamental equations of two-phase flow which describe this mechanism are nonlinear partial differential equations. Numerical solutions of these equations describing pressure transients have been obtained with the aid of electronic computers. Although solutions obtained in this manner take into account a large number of effects, the reduction of this information to useful generalities is difficult. One method of obtaining generalities is the use of linearized approximations of the nonlinear equations. Since it is possible to obtain explicit solutions of the linearized equation, general properties of the role of pressure in the flow mechanism may be ascertained. Results obtained from this approach are limited to some extent by the linearizing assumptions. The severity of these limitations may be evaluated by comparing solutions of the linear equation with numerical solutions of the more exact nonlinear equations of two-phase flow. In the past considerable amount of work has been devoted to studying pressure build-up using the single-phase flow theory. Unfortunately, most pressure build-up tests involve multiphase flow. A small amount of work has been done studying pressure build-up where the flow is two-phase. The encouraging results of these studies suggest that useful results may be found from additional studies of not only pressure build-up, but also the rapid transients associated with placing a well on production. This paper presents the basic theory of the pressure transients associated with placing a well on production and with closing it in. The paper is concerned chiefly with two-phase compressible flow; however, the results also apply to single-phase flow, The results are based on analytical solutions of the flow equations, and they are verified by numerical solutions using finite difference techniques. Much of the previous work on compressible flow into wells has been confined to single-phase flow. Some work has been done on compressible two-phase steady state flow, and solutions of the equations of flow have been found by finite difference techniques using high-speed computers. Muskat presents some rather general solutions to the equations of single-phase compressible flow into wells. Much work has been done on pressure build-up in wells (see, for example, Refs. 2–6). Almost all of the work on pressure build-up concerns single-phase flow with the exception of Ref. 5 and part of Ref. 2. Some work has been done on pressure fall-off in injection wells. Muskat presents the solution of the equations for radial steady state two-phase compressible flow. SPEJ P. 116^

A particle, shaped as an ellipsoid of revolution and rotating about an arbitrary axis, experiences a torque in a rarefied monatomic gas because of the impact with the gas atoms. This torque is calculated for the case where the linear dimensions of the ellipsoid are small compared with the mean free path of the gas. The interaction of the gas atoms with the ellipsoidal surface is taken into account by means of the boundary condition for the distribution function in terms of an accommodation coefficient (diffuse and specular reflection). The torque is considered for prolate and oblate ellipsoids. The formulae obtained are of interest in the theory of the Brownian motion of a rotating particle in a rarefied gas.

The energy equation for constant density fluid flow with the viscous dissipation term is often used for the governing equations of gas flow with low velocity in microchannels. If the gas is an ideal gas with low velocity, the average temperatures at the inlet and the outlet of an adiabatic channel are the same based on the first law of the thermodynamics. If the gas is a real gas with low velocity, the average temperature at the outlet is higher or lower than the average temperature at the inlet. However, the outlet temperature which is obtained by solving the energy equation for constant density fluid flow with the viscous dissipation term is higher than the inlet gas temperature, since the viscous dissipation term is always positive. This inconsistency arose from choice of the relationship between the enthalpy and temperature that resulted in neglecting the substantial derivative of pressure term in the energy equation. In this paper, the energy equation which includes the substantial derivative of pressure term is proposed to be used for the governing equation of gas flow with low velocity in microchannels. The proposed energy equation is verified by solving it numerically for flow in a circular microtube. Some physically consistent results are demonstrated.