Rarefied systems

In some cases, in particular for rarefied systems and/or low Reynolds number flows a velocity discontinuity at wall can be observed. This is the shp flow regime. The Reynolds number can be low enough for the classical boundary layer theory not to be completely valid but not low enough for the inertia terms to be neglected. The fluid velocity at the wall is not zero and... 

In rarefied systems of particles, drops, or bubbles, the particle-particle interaction can be neglected in the first approximation then one deals with the behavior of a single particle moving in fluid. In this case, the streamline pattern depends on the particle shape, the flow type (translational or shear), and a number of other geometric factors. 

In Section 2.9, various aspects were considered of the hydrodynamics of a constrained flow past a system of particles based on the cell model. Here we briefly describe mass and heat transfer in such systems at high Peclet numbers. We investigate either sufficiently rarefied systems of particles or systems with an irregular structure, where the diffusion interaction of isolated particles can be neglected. (Regular disperse systems, where the interaction between diffusion wakes and boundary layers must be taken into account, were investigated in [172, 365].)... 

Strictly speaking, dispersion interaction is valid only for two highly rarefied systems, i.e., gages. Extension of the principal of additivity of forces to condensed systems that do not represent a simple sum of free molecules has not yet been justified by theory. The experimental value found by Bradley [25] for the force on interaction between two quartz and borate spheres, however, was close to the value calculated on the basis of his assumption of additivity of molecular interaction. Hence, we may a priori accept the additivity of London interaction and extend this principal to condensed systems since at the present time there are no other methods for evaluating molecular interaction of such bodies when they are separated by a small gap. 

Strictly speaking, London interaction is valid for two very rarefied systems, i.e., gases. The extension of the additivity of the forces to condensed systems not constituting a simple sum of free molecules has not yet been given a firm theoretical basis. 

One possibility for this was demonstrated in Chapter 3. If impact theory is still valid in a moderately dense fluid where non-model stochastic perturbation theory has been already found applicable, then evidently the continuation of the theory to liquid densities is justified. This simplest opportunity of unified description of nitrogen isotropic Q-branch from rarefied gas to liquid is validated due to the small enough frequency scale of rotation-vibration interaction. The frequency scales corresponding to IR and anisotropic Raman spectra are much larger. So the common applicability region for perturbation and impact theories hardly exists. The analysis of numerous experimental data proves that in simple (non-associated) systems there are three different scenarios of linear rotator spectral transformation. The IR spectrum in rarefied gas is a P-R doublet with either resolved or unresolved rotational structure. In the process of condensation the following may happen. 

When Bernie Shizgal arrived at UBC in 1970, his research interests were in applications of kinetic theory to nonequilibrium effects in reactive systems. He subsequently applied kinetic theory methods to the study of electron relaxation in atomic and molecular moderators,46 hot atom chemistry, nucleation,47 rarefied gas dynamics,48 gaseous electronics, and other physical systems. An important area of research has been the kinetic theory description of the high altitude portion of planetary atmospheres, and the escape of atmospheric species.49 An outgrowth of these kinetic theory applications was the development of a spectral method for the solution of differential and integral equations referred to as the quadrature discretization method (QDM), which has been used with considerable success in statistical, quantum, and fluid dynamics.50... 

Analogy in molecular system Rarefied gas flow Molecular theory of liquids or kinetic theory of gas... 

Multilayer adsorption models have been used by Asada [147,148] to account for the zero-order desorption kinetics. The two layers are equilibrated. Desorption goes from the rarefied phase only. This model has been generalized [148] for an arbitrary number of layers. The filling of the upper layer was studied with allowance for the three neighboring molecules being located in the lower one. The desorption frequency factor (CM) was regarded as being independent of the layer number. The theory has been correlated with experiment for the Xe/CO/W system [149]. Analysis of the two-layer model has been continued in Ref. [150], to see how the ratios of the adspecies flows from the rarefied phases of the first and the second layers vary if the frequency factors for the adspecies of the individual layers differ from one another. In the thermodynamic equilibrium conditions these flows were found to be the same at different ratios of the above factors. 

No exact general criterion is available when it is necessary to include the relaxation terms in the equations of change however, relaxation terms are necessary for viscoelastic fluids, dispersed systems, rarefied gases, capillary porous mediums, and helium, in which the frequency of the fast variable transients may be comparable to the reciprocal of the longest relaxation time. 

For gas-filled granular beds, the thermal conductivity of the gas may be very low. Since gas phase heat conduction mainly occurs near the points of contact between adjacent solid particles, the distance for heat conduction over the gas phase may approach the mean free path of the gas molecules. This reduces the thermal conductivity of the gas further, since the whole system may become rarefied for evacuated beds of fine powders. 

At constant temperature of the system, when r -> 0 (because of the van der Waals repulsion between two molecules at small intermolecular distances, Un -> oo), the ratUal rUstribution function g(ri tends to zero for impermeable atoms and simple molecules. In cases when r 2 -> at T = const., or when T- oo, the molecules are completely independent (highly rarefied gases and vapours, U12 0), and 12) 1- The above... 

Since 1980, a large body of research has been performed using the DSMC technique. Applications include hypersonic flows, spacecraft propulsion systems, materials processing, astrophysics, and flows through micromachines. Recent reviews of the method and applications are provided in Refs. 27-29. It is the purpose of this article to review the status of the DSMC technique specifically in relation to its ability to accurately model the nonequilibrium, chemically reacting flows that are characteristic of rarefied hypersonic conditions. 

Actually, computational convenience has almost always suggested using pairwise additive potentials for simulations of condensed phases also, though strictly two-body potentials are only acceptable for rarefied gases. The computational convenience of two-body potentials is maintained, however, if non-additive effects are included implicitly, i. e. with the so called two-body effective potentials. All empirical or semi empirical functions whose parameters have been optimized with respect to properties of the system in condensed phase belong to this class. As already observed, this makes these potentials state-dependent, with unpredictable performance under different thermodynamic conditions. 

Some other theoretical methods for investigating rarefied and concentrated disperse systems, based on equations of mechanics of multiphase systems, are described in the books [86,183,205, 312, 313],... 

For the sedimentation of rarefied monodisperse systems of spherical particles, drops, or bubbles, the mean Sherwood number can be calculated by using formulas (4.6.8) and (4.6.17), where the Peclet number must be determined on the basis of the constrained flow velocity. 

For we can now write down, without neglecting anything, the vapour pressure of any system which gives off a monatomic gas. Let A0 be the amount of heat which is developed when the infinitely rarefied gas condenses at the absolute zero if we consider its condensation from the finite volume V, we then have... 

One of the earliest particle-based schemes is the Direct Simulation Monte Carlo (DSMC) method of Bird [126]. In DSMC simulations, particle positions and velocities are continuous variables. The system is divided into cells and pairs of particles in a cell are chosen for collision at times that are determined from a suitable distribution. This method has seen wide use, especially in the rarefied gas dynamics community where complex fluid flows can be simulated. 

To begin with, let us consider a rarefied gas its condensation into a liquid, or into a solid crystal leads to a decrease in the system energy due to the saturation of interaction forces between molecules in the condensed phase. Such a decrease taken per mole of a substance, is equivalent to the heat of evaporation (or heat of sublimation, taken with the opposite sign) and can be expressed as... 

Stechelmacher, W., Knudsen flow 75 years on The current state of the art for flow of rarefied gases in tubes and systems. Reports on Progress in Physics, 1986. 49 1083. 

A pressure-driven gas flow is generated by applying a pressure difference between the inlet and the outlet of a fluidic system, for example, a channel. Due to very small hydraulic diameters, pressure-driven gas flows in microchannels are laminar. Gas microflows are distinct from gas macroflows by rarefaction effects which appear as soon as the mean free path of the molecules is no longer negligible compared with the hydraulic diameter of the microchannel. For such rarefied flows, the classic Poiseuille model is no longer valid, and other models should be used, according to the rarefaction level, which is quantified by the Knudsen number. 

From statistical mechanics, it follows that temperature is weU defined when the velocity distribution is Maxwellian. Systems for which this condition is fulfilled are complex reactions where the rate of elastic collisions is larger than the rate of reactive collisions. This is generally true for reactions in not too rarefied media and for many biological and transport processes. It may be noted that molecular collisions are responsible for attainment of Maxwellian distribution. Normally, significant deviations from the Maxwellian distribution are observed only under extreme conditions. Distribution is perturbed when physical processes are very rapid. Thus for a gas, local equilibrium assumption would not be valid when the relative variation of temperature is no longer small within a length equal to mean free path. 

The above conditions, however, are not fulfilled by other processes such as shock waves, highly rarefied gases. Many biological systems have gradients of density, energy density etc. over a distance of 10 A (such as material near or in a membrane or a cell nucleus) which would not have true local equilibrium. 

In the presence of only one phase, rarefied DPD gas with attractive tail in interparticle interaction forces, we can simulate condensation phenomenon. As shown in Figure 26.27, the microstructures appearing are different than those for binary fluids. The average cluster size S(f) R t) increases much slower than in binary systems. Condensation patterns are more regular and resemble separate droplets rather than shapeless cluster structures. Therefore one can suppose that the mechanisms of growth in condensing gas must also be different than in separation of binary mixture. 

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